ee351: spectrum analysis and discrete time systems...
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EE351: Spectrum Analysis and Discrete Time Systems
(Signals, Systems and Transforms)
Dr. Ha H. Nguyen
Associate Professor
Department of Electrical Engineering
University of Saskatchewan
August 2005
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Introduction
• The concepts of signals and systems arise in a wide variety of areas, such as
communications, circuit design, biomedical engineering, power systems, speech
processing, etc.
• The ideas and techniques associated with these concepts play an important role
in such diverse areas.
• Although the physical nature of the signals and systems that arise in these
various disciplines may be drastically different, two basic features in common are:
– The signals, which are functions of one or more independent variables,
contain information about the behavior or nature of some phenomenon.
– The systems respond to particular signals by producing other signals or some
desired behavior.
• Examples of signals and systems:
– Voltages and currents as functions of time in an electrical circuit are
examples of signals. A circuit is itself an example of a system, which
responds to applied voltages and currents.
– A camera is a system that receives light from different sources and produces
a photograph.
Dr. H. Nguyen Page 1
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• Example problems of signal and system analysis:
– Analyzing existing systems: We are presented with a specific system and are
interested in characterizing it in detail to understand how it will respond to
various inputs (analysis of a circuit).
– Designing systems to process signals in particular ways. For example, to
design systems to enhance or restore signals that have been degraded in
some way (image restoration, image enhancement).
– Designing systems to extract specific pieces of information from signals.
Examples include the estimation of heart rate from an electrocardiogram,
weather forecast.
– Designing of signals with particular properties. For example, the design of
communication signals must take into account the need for reliable reception
in the presence of distortion (due to transmission media) and interference
(such as noise).
Dr. H. Nguyen Page 2
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Chapter I: Signals and Systems
There is an analytical framework–that is, a language for describing signals and
systems and an extremely powerful set of tools for analyzing them–that applies
equally well to problems in many fields.
This chapter begins the development of such an analytical framework for signals and
systems by introducing their mathematical description and representations.
Signals are represented mathematically as functions of one or more
independent variables.
Examples:
• A speech signal can be represented mathematically by acoustic pressure as a
function of time.
• A picture can be represented by brightness as a function of two spatial variables.
This course focuses only on signals involving a single independent variable.
For convenience, the independent variable will generally referred to as time, although
it may not in fact represent time in specific applications.
Dr. H. Nguyen Page 3
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Examples of SignalsArterial Blood Pressure
0 0.5 1 1.5 2 2.5
60
70
80
90
100
110
120
Time (sec)
AB
P (m
mH
g)
Portland State University ECE 222 Signal Fundamentals Ver. 1.06 7
Microelectrode Recording
2 2.01 2.02 2.03 2.04 2.05 2.06
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Time (sec)
Portland State University ECE 222 Signal Fundamentals Ver. 1.06 5
Speech
1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Time (sec)
Linus: Philosophy of Wet Suckers
Portland State University ECE 222 Signal Fundamentals Ver. 1.06 8
Electrocardiogram
0 0.5 1 1.5 2 2.5
6.5
7
7.5
8
8.5
Time (sec)
Portland State University ECE 222 Signal Fundamentals Ver. 1.06 6
31
1.4
.6Sta
bility
Consid
erbounded–input
bounded–output
(BIB
O)
stability
Stable
systemif
foran
ybounded
input
signal
|x(t)|
≤B
x<
∞,
∀t
the
outp
ut
signal
isbounded
|y(t)|
≤B
y<
∞,
∀t
Exam
ple
:
–Stab
lesystem
Averag
er:y[n
]=
1
2N
+1
N∑k=−
N
x[n
−k]
Bounded
input|x
[n]|
<B
x⇒
bounded
outp
ut|y
[n]|
<B
y=
Bx
–In
stable
system
Integ
rator:y(t)
=
t∫−∞
x(τ
)dτ
E.g
.bounded
inputx(t)
=u(t)
⇒unbounded
outp
uty(t)
=t
System
stability
isim
portan
tin
engin
eering
applicatio
ns,
unstab
le
systems
need
tobe
stabilized
.
Lam
pe,Schober:
Sig
nals
and
Com
munic
atio
ns
32
Exam
ple
:T
he
first
Taco
ma
Narrow
ssu
spen
sion
bridge
collap
seddue
tow
ind-in
duced
vibrations,
Novem
ber
1940.
(Photos from http :/ / w w w .e n m .b ris.a c .u k / rese a rch/ n on lin e a r/ ta com a / ta com a .htm l)
Lam
pe,Schober:
Sig
nals
and
Com
munic
atio
ns
Dr. H. Nguyen Page 4
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Discrete-time & Continuous-time Signals
• The course will cover concepts and techniques associated both with
continuous-time and discrete-time signals (and systems).
• Continuous-time signals are defined for a continuum of values of the
independent variable (the independent variable is continuous).
– Will always be treated as a function of t.
– Parentheses are used to denote continuous-time functions, for example x(t).
– The independent variable t is a real-valued and continuous.
• Discrete-time signals are only defined at discrete times (the independent variable
takes on only a discrete set of values).
– Will always be treated as a function of n.
– Square brackets are used to denote discrete-time functions, for example x[n].
– The independent variable n is an integer.
• Examples:
– The speech signal as a function of time and atmospheric pressure as a
function of altitude are examples of continuous time signals.
– The daily closing stock market index is an example of discrete-time signal.
Dr. H. Nguyen Page 5
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• It is often useful to represent the signals graphically:
Arterial Blood Pressure
0 0.5 1 1.5 2 2.5
60
70
80
90
100
110
120
Time (sec)
AB
P (m
mH
g)
Portland State University ECE 222 Signal Fundamentals Ver. 1.06 7
Microelectrode Recording
2 2.01 2.02 2.03 2.04 2.05 2.06
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Time (sec)
Portland State University ECE 222 Signal Fundamentals Ver. 1.06 5
Speech
1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Time (sec)
Linus: Philosophy of Wet Suckers
Portland State University ECE 222 Signal Fundamentals Ver. 1.06 8
Electrocardiogram
0 0.5 1 1.5 2 2.5
6.5
7
7.5
8
8.5
Time (sec)
Portland State University ECE 222 Signal Fundamentals Ver. 1.06 6Figure 1: A continuous-time signal (electrocardiogram).
3
Discrete– time sig nals
– S ymbol n for independent variable
– U se brackets [·]
Discrete– time sig nal: x[n]
– Graph ical representation
x[0]
x[1]x[−1]
0 321
x[2]x[−2]
n−3 −2 −1
54
x[n]
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
4
1.1.2 S ign a l E n e rgy a n d Powe r
O ften classifi cation of sig nals according to en erg y and power
– Terminolog y en erg y and power u sed for any sig nal x(t), x[n]
– N eed not necessarily h av e a ph y sical meaning
S ig nal energ y
– E nerg y of a possibly complex continu ou s– time sig nal x(t) in
interv al t1 ≤ t ≤ t2
E(t1, t2) =
t2∫
t1
|x(t)|2 d t
– E nerg y of a possibly complex discrete– time sig nalx[n] in interv al
n1 ≤ n ≤ n2
E(n1, n2) =
n2∑n=n1
|x[n]|2
– Total energ y
E∞ = E( − ∞,∞) =
∞∫
− ∞
|x(t)|2 d t
E∞ = E( − ∞,∞) =
∞∑n= − ∞
|x[n]|2
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
Figure 2: An example of discrete-time signal.
Dr. H. Nguyen Page 6
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• Continuous-time signals (and systems) have very strong roots in problems
associated with physics, and, more recently, with electrical circuits and
communications.
• The techniques of discrete-time signals (and systems) have strong roots in
numerical analysis, statistics and time series analysis (associated with such
applications as the analysis of economic and demographic data).
• In the past decades, there has been a growing interrelationship between
continuous-time signals and systems and discrete-time signals and systems. This
has come from the dramatic advances in technology for the implementation of
systems and for the generation of signals. For example, it is increasingly
advantageous to consider processing continuous-time signals by representing
them with time samples.
• This course develops the concepts of continuous-time and discrete-time signals
and systems in parallel. Since many of the concepts are similar, by treating them
in parallel, insight and intuition can be shared and both the similarities and
differences between them become better focused.
Dr. H. Nguyen Page 7
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Signal Energy and Power
• In many (but not all) applications, the signals are directly related to physical
quantities that capturing power and energy in a physical system.
• Example: Let v(t) and i(t) be the voltage and current across a resistor with
resistance R. Then the instantaneous power is
p(t) = v(t)i(t) =1
Rv2(t)
The total energy expended over the time interval t1 ≤ t ≤ t2 is∫ t2
t1
p(t)dt =
∫ t2
t1
1
Rv2(t)dt
The average power over this time interval is
1
t2 − t1
∫ t2
t1
p(t)dt =1
t2 − t1
∫ t2
t1
1
Rv2(t)dt
• For most of this course we will use a broad definition of power and energy that
applies to any signal x(t) or x[n]. Such definitions need not necessarily have a
physical meaning.
Dr. H. Nguyen Page 8
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• Signal energy
– The total energy over the time interval t1 ≤ t ≤ t2 of a possibly complex
continuous-time signal x(t) is:
E(t1, t2) =
∫ t2
t1
|x(t)|2dt
– Similarly, the total energy of a possibly complex discrete-time signal x[n] over
the time interval n1 ≤ n ≤ n2 is:
E(n1, n2) =
n2∑
n=n1
|x[n]|2
– The total energy is the energy in a signal over an infinite time interval:
E∞ = E(−∞,∞) =
∫ ∞
−∞
|x(t)|2dt
E∞ = E(−∞,∞) =
∞∑
n=−∞
|x[n]|2
Dr. H. Nguyen Page 9
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Example: Find the total energy of the following discrete-time signal:
x[n] =
an, n ≥ 0
0, n < 0
where |a| < 1. The answer is:
E∞ =∞∑
n=−∞
|x[n]|2 =∞∑
n=0
(|a|2)n =1
1− |a|2
• Signal power
– Consider the time-averaged signal power.
– The average powers of x(t) and x[n] over the intervals t1 ≤ t ≤ t2 and
n1 ≤ n ≤ n2 are:
P (t1, t2) =1
t2 − t1
∫ t2
t1
|x(t)|2dt and P (n1, n2) =1
n2 − n1 + 1
n2∑
n=n1
|x[n]|2
respectively.
Dr. H. Nguyen Page 10
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
– Analogously, when the limits are taken over an infinite time interval, then:
P∞ = limT→∞
1
2T
∫ T
−T
|x(t)|2dt, P∞ = limN→∞
1
2N + 1
N∑
n=−N
|x[n]|2
• Classification of signals based on their energy and power
– Signals with finite total energy, E∞ <∞, are known as energy signals.
∗ The energy signals have zero average power: P∞ = 0.
∗ Examples of energy signals: All signals seen previously, any signal with
finite amplitude and finite duration (x(t) = 0 for |t| > τ and
max(|x(t)|) <∞).
– Signals with finite average power, P∞ > 0, are known as power signals.
∗ The power signals have infinite total energy: E∞ =∞ if P∞ > 0.
∗ Examples of power signals: periodic signals such as x(t) = cos(t) and
x[n] = sin(5n).
– Signals with infinite power (P∞ =∞) and infinite energy (E∞ =∞).
∗ These signals are not desirable in engineering applications.
∗ Examples: x(t) = et and x[n] = n5.
Dr. H. Nguyen Page 11
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Signal Transformations
• Transformation of a signal is a central concept in signal and system analysis. For
example, an audio system takes an input signal representing music recorded on a
compact disc, and modifies it to enhance desirable characteristics.
• This section focuses on a very limited but important class of elementary signal
transformations that involve simple modification of the independent variable
(i.e., the time axis).
• Elementary signal transformations:
– Time shift: x(t)→ x(t− t0) and x[n]→ x[n− n0]
∗ If t0 > 0 or n0 > 0, signal is shifted to the right (i.e., delayed)
∗ If t0 < 0 or n0 < 0, signal is shifted to the left (i.e., advanced)
– Time reversal : x(t)→ x(−t) and x[n]→ x[−n]
– Time scaling : x(t)→ x(αt) and x[n]→ x[αn]
∗ If α > 1, signal appears compressed
∗ If α < 1, signal appears stretched
Dr. H. Nguyen Page 12
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Example: The following figure plots x(t) and its transformations x(−t),
x(t− 1), x(t + 2) and x(t/2).
−5 −4 −3 −2 −1 0 1 2 3 4 5−1
0
1
t
x(t)
−5 −4 −3 −2 −1 0 1 2 3 4 5−1
0
1
t
x(−t)
−5 −4 −3 −2 −1 0 1 2 3 4 5−1
0
1
t
x(t−1)
−5 −4 −3 −2 −1 0 1 2 3 4 5−1
0
1
t
x(t)
−5 −4 −3 −2 −1 0 1 2 3 4 5−1
0
1
t
x(t+2)
−5 −4 −3 −2 −1 0 1 2 3 4 5−1
0
1
t
x(t/2)
−5 −4 −3 −2 −1 0 1 2 3 4 5−1
0
1
t
x(t)
−5 −4 −3 −2 −1 0 1 2 3 4 5−1
0
1
t
x(2t)
−5 −4 −3 −2 −1 0 1 2 3 4 5−1
0
1
t
x(2(t−1))
Dr. H. Nguyen Page 13
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
More Example:7
1.1.3 Tran sform ation s of th e In d e p e n d e n t Variab le
Time sh ift
– Replace t → t − t0 x(t) → x(t − t0)
n → n − n0⇒
x[n] → x[n − n0]
– D elay : t0, n0 > 0, A dvan ce: t0, n0 < 0
Time rev ersal
– Replace t → −t x(t) → x(−t)
n → −n⇒
x[n] → x[−n]
Time scalin g
– Replace t → αt , α ∈ IR x(t) → x(αt)
n → αn , α ∈ ZZ⇒
x[n] → x[αn]
– C o n tin u o u s– time case: |α| < 1 : sig n al is lin early stretch ed
|α| > 1 : sig n al is lin early compressed
Time sh ift, time rev ersal, an d time scalin g o peratio n s arise n atu rally
in th e pro cessin g o f sig n als
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
8
E x am p le :
n
nt
n
n
t
t
Time-scaled sig n als
Time-rev ersed sig n als
Time-sh ifted sig n als
S ig n als
t
x[2 n]
x[−n]
x[n − 4 ]
x(2 /3t)
x(−t)
x(t − t0)
x(t) x[n]
t0
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
Dr. H. Nguyen Page 14
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• A more general signal transformation is x(t)→ y(t) = x(αt + β) . For this
transformation, a systematic approach to obtain the plot of y(t) is as follows:
– First shift x(t) in accordance with the value of β: The signal x(t) is shifted to
the right if β < 0, shifted to the left if β > 0.
– Then perform time scaling and/or time reversal on the resulting signal in
accordance with the value of α: The resulting signal is linearly stretched if
|α| < 1, linearly compressed if |α| > 1 and reversed in time if α < 0.
Example: The following figures draw x(t), x(2t) and x(2(t− 1)).
−5 −4 −3 −2 −1 0 1 2 3 4 5−1
0
1
t
x(t)
−5 −4 −3 −2 −1 0 1 2 3 4 5−1
0
1
t
x(2t)
−5 −4 −3 −2 −1 0 1 2 3 4 5−1
0
1
t
x(2(t−1))
Dr. H. Nguyen Page 15
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Even & Odd Symmetry
There is a set of useful properties of signals that relate to their symmetry under time
reversal.
• Even signal: x(−t) = x(t) or x[−n] = x[n].
• Examples: The signal plotted below, cos[kω0n].
9
1.1.4 Pe riod ic S ign a ls
Periodic continu ou s– time sig na l
x(t) = x(t + T ) , ∀t
– T > 0: Perio d
– x(t) periodic w ith T ⇒ x(t) a lso periodic w ith mT , m ∈ IN
– S ma llest period of x(t): Fu ndamenta l perio d T0.
– E x ample (T0 = T ):
0
x(t)
t4T3T2T−3T −2T −T T
Periodic discrete– time sig na l
x[n] = x[n + N ] , ∀n
– Integ er N > 0: Perio d
– x[n] periodic w ith N ⇒ x[n] a lso periodic w ith mN , m ∈ IN
– S ma llest period of x[n]: Fu ndamenta l perio d N0.
– E x ample (N0 = 4 ):
n
3 6
x[n]
0 1
2
54
A sig na l th a t is not periodic is referred to a s a perio dic.
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
1 0
1.1.5 E v e n a n d O d d S ign a ls
E v en sig na l
x(−t) = x(t) o r x[−n] = x[n]
– E x ample:x(t)
t
O dd sig na l
x(−t) = −x(t) o r x[−n] = −x[n]
– E x ample:
n
x[n]
– Necessa rily : x(0) = 0 or x[0] = 0
D ecomposition of a ny sig na l into a n ev en a nd odd pa rt:
x(t) = E vx(t) + O dx(t) or x[n] = E vx[n] + O dx[n]
w ith
E vx(t) =1
2(x(t) + x(−t)) or E vx[n] =
1
2(x[n] + x[−n])
a nd
O dx(t) =1
2(x(t) − x(−t)) or O dx[n] =
1
2(x[n] − x[−n])
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
• Odd signal: x(−t) = −x(t) or x[−n] = −x[n]. Note that an odd signal must
be zero at t = 0 or n = 0.
• Examples: The signal plotted below, sin(kω0t).
9
1.1.4 Pe riod ic S ign a ls
Periodic continu ou s– time sig na l
x(t) = x(t + T ) , ∀t
– T > 0: Perio d
– x(t) periodic w ith T ⇒ x(t) a lso periodic w ith mT , m ∈ IN
– S ma llest period of x(t): Fu ndamenta l perio d T0.
– E x ample (T0 = T ):
0
x(t)
t4T3T2T−3T −2T −T T
Periodic discrete– time sig na l
x[n] = x[n + N ] , ∀n
– Integ er N > 0: Perio d
– x[n] periodic w ith N ⇒ x[n] a lso periodic w ith mN , m ∈ IN
– S ma llest period of x[n]: Fu ndamenta l perio d N0.
– E x ample (N0 = 4 ):
n
3 6
x[n]
0 1
2
54
A sig na l th a t is not periodic is referred to a s a perio dic.
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
1 0
1.1.5 E v e n a n d O d d S ign a ls
E v en sig na l
x(−t) = x(t) o r x[−n] = x[n]
– E x ample:x(t)
t
O dd sig na l
x(−t) = −x(t) o r x[−n] = −x[n]
– E x ample:
n
x[n]
– Necessa rily : x(0) = 0 or x[0] = 0
D ecomposition of a ny sig na l into a n ev en a nd odd pa rt:
x(t) = E vx(t) + O dx(t) or x[n] = E vx[n] + O dx[n]
w ith
E vx(t) =1
2(x(t) + x(−t)) or E vx[n] =
1
2(x[n] + x[−n])
a nd
O dx(t) =1
2(x(t) − x(−t)) or O dx[n] =
1
2(x[n] − x[−n])
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
Dr. H. Nguyen Page 16
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• Any signal can be written as a sum of an odd signal and an even signal:
x(t) = xe(t) + xo(t), where
xe(t) =1
2[x(t) + x(−t)]
xo(t) =1
2[x(t)− x(−t)]
−5 −4 −3 −2 −1 0 1 2 3 4 5−1
0
1
t
x(t)
−5 −4 −3 −2 −1 0 1 2 3 4 5−1
0
1
t
x(−t)
−5 −4 −3 −2 −1 0 1 2 3 4 5−1
0
1
t
xe(t)
−5 −4 −3 −2 −1 0 1 2 3 4 5−1
0
1
t
x(t)
−5 −4 −3 −2 −1 0 1 2 3 4 5−1
0
1
t
−x(−t)
−5 −4 −3 −2 −1 0 1 2 3 4 5−1
0
1
t
xo(t)
Dr. H. Nguyen Page 17
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Periodic Signals
An important class of signals that are encountered frequently in this course is the
class of periodic signals.
A signal is periodic if there is a positive value of T or N such that
x(t) = x(t + T ) or x[n] = x[n + N ]
T and N are called the periods of x(t) and x[n], respectively.
• For any integer m, mT and mN are also the periods.
• For continuous-time signals, the fundamental period T0 is the smallest positive
value of T such that x(t) = x(t + T ). For a special case where x(t) is a
constant, the fundamental period is undefined.
• For discrete-time signals, the fundamental period N0 is the smallest positive
integer of N such that x[n] = x[n + N ].
• Signals that are not periodic are said to be aperiodic.
Dr. H. Nguyen Page 18
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Examples of periodic signals are shown below:
9
1.1.4 Pe riod ic S ign a ls
Periodic continu ou s– time sig na l
x(t) = x(t + T ) , ∀t
– T > 0: Perio d
– x(t) periodic w ith T ⇒ x(t) a lso periodic w ith mT , m ∈ IN
– S ma llest period of x(t): Fu ndamenta l perio d T0.
– E x ample (T0 = T ):
0
x(t)
t4T3T2T−3T −2T −T T
Periodic discrete– time sig na l
x[n] = x[n + N ] , ∀n
– Integ er N > 0: Perio d
– x[n] periodic w ith N ⇒ x[n] a lso periodic w ith mN , m ∈ IN
– S ma llest period of x[n]: Fu ndamenta l perio d N0.
– E x ample (N0 = 4 ):
n
3 6
x[n]
0 1
2
54
A sig na l th a t is not periodic is referred to a s a perio dic.
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
1 0
1.1.5 E v e n a n d O d d S ign a ls
E v en sig na l
x(−t) = x(t) o r x[−n] = x[n]
– E x ample:x(t)
t
O dd sig na l
x(−t) = −x(t) o r x[−n] = −x[n]
– E x ample:
n
x[n]
– Necessa rily : x(0) = 0 or x[0] = 0
D ecomposition of a ny sig na l into a n ev en a nd odd pa rt:
x(t) = E vx(t) + O dx(t) or x[n] = E vx[n] + O dx[n]
w ith
E vx(t) =1
2(x(t) + x(−t)) or E vx[n] =
1
2(x[n] + x[−n])
a nd
O dx(t) =1
2(x(t) − x(−t)) or O dx[n] =
1
2(x[n] − x[−n])
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
9
1.1.4 Pe riod ic S ign a ls
Periodic continu ou s– time sig na l
x(t) = x(t + T ) , ∀t
– T > 0: Perio d
– x(t) periodic w ith T ⇒ x(t) a lso periodic w ith mT , m ∈ IN
– S ma llest period of x(t): Fu ndamenta l perio d T0.
– E x ample (T0 = T ):
0
x(t)
t4T3T2T−3T −2T −T T
Periodic discrete– time sig na l
x[n] = x[n + N ] , ∀n
– Integ er N > 0: Perio d
– x[n] periodic w ith N ⇒ x[n] a lso periodic w ith mN , m ∈ IN
– S ma llest period of x[n]: Fu ndamenta l perio d N0.
– E x ample (N0 = 4 ):
n
3 6
x[n]
0 1
2
54
A sig na l th a t is not periodic is referred to a s a perio dic.
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
1 0
1.1.5 E v e n a n d O d d S ign a ls
E v en sig na l
x(−t) = x(t) o r x[−n] = x[n]
– E x ample:x(t)
t
O dd sig na l
x(−t) = −x(t) o r x[−n] = −x[n]
– E x ample:
n
x[n]
– Necessa rily : x(0) = 0 or x[0] = 0
D ecomposition of a ny sig na l into a n ev en a nd odd pa rt:
x(t) = E vx(t) + O dx(t) or x[n] = E vx[n] + O dx[n]
w ith
E vx(t) =1
2(x(t) + x(−t)) or E vx[n] =
1
2(x[n] + x[−n])
a nd
O dx(t) =1
2(x(t) − x(−t)) or O dx[n] =
1
2(x[n] − x[−n])
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
Dr. H. Nguyen Page 19
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Elementary Signals: Complex Exponential and Sinusoidal
Several classes of signals play prominent role:
• They model many physical signals.
• They serve as building blocks for many other signals.
• They serve for system analysis.
1. Continuous-Time Complex Exponential Signal: x(t) = Ceat where, in
general, both C and a are complex numbers.
• If both C and a are real ⇒ Real exponential signal.
Example: The following figures plot Ceat with C = 1 and a = ± 110 .
−10 −5 0 5 10 15 20 25 300
5
10
15
20
25
t−10 −5 0 5 10 15 20 25 300
0.5
1
1.5
2
2.5
3
t
Dr. H. Nguyen Page 20
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• If a is imaginary (i.e., a = jω0) ⇒ Periodic complex exponential.
To see that the signal is indeed periodic, let C = Aejφ. Then
x(t) = Aej(ω0t+φ) ?= Aej(ω0(t+T )+φ) = Aej(ω0t+φ)ejω0T
where T is chosen such that ejω0T = 1. Excluding the trivial solution of
ω0 = 0, the fundamental period is T0 =2π
|ω0|.
Example: The following figures plot Ceat with C = 1 and a = j.
−100
1020
30 −1
0
1−1
−0.5
0
0.5
1
Imaginary PartTime (s)
Rea
l Par
t
Dr. H. Nguyen Page 21
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• A harmonically related set of complex periodic exponentials is a set of
exponentials with fundamental frequencies that are all multiples of a single
positive frequency ω0:
φk(t) = ejkω0t where k = 0,±1,±2, . . .
– For k = 0, φ0(t) is a constant
– For all other values of k, φk(t) is periodic with fundamental frequency
|k|ω0 and fundamental period
2π
|k|ω0=
T0
|k|
– This is consistent with how the term harmonic is used in music
– Sets of harmonically related complex exponentials are used to
represent many other periodic signals (Fourier series)
Dr. H. Nguyen Page 22
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Example: The figure below plots the real parts of several harmonically
related complex exponentials.
−20 −15 −10 −5 0 5 10 15 20−1
0
1φ 0(t
)
−20 −15 −10 −5 0 5 10 15 20−1
0
1
φ 1(t)
−20 −15 −10 −5 0 5 10 15 20−1
0
1
φ 2(t)
−20 −15 −10 −5 0 5 10 15 20−1
0
1
φ 3(t)
−20 −15 −10 −5 0 5 10 15 20−1
0
1
φ 4(t)
Time (s)
Dr. H. Nguyen Page 23
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• General complex exponential signal: For the most general case, C = Aejφ
and a = r + jω0, then
x(t) = Ceat = Aertej(ω0t+φ) = Aert cos(ω0t + φ) + jAert sin(ω0t + φ)
– If r > 0, then x(t) is exponentially growing signal
– If r < 0, then x(t) is exponentially decaying signal
Example: The following figures plot the real parts of Ceat with C = 1 and
a = ±0.1 + j0.5.
−20 −15 −10 −5 0 5 10 15 20−8
−6
−4
−2
0
2
4
6
8
t
Rea
l Par
t of
x(t)
−20 −15 −10 −5 0 5 10 15 20−8
−6
−4
−2
0
2
4
6
8
t
Rea
l Par
t of
x(t)
Dr. H. Nguyen Page 24
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
2. Continous-Time Sinusoidal Signals
xc(t) = A cos(ω0t + φ) = ReAej(w0t+φ)
xs(t) = A sin(ω0t + φ) = ImAej(w0t+φ)Of course, both xc(t) and xs(t) also have fundamental period T0 = 2π
|ω0| .Periodic signals have infinite total energy, but finite average power. This
can be seen for the exponential x(t) = Aejω0t (assuming A is real) as follows:
• The energy over one period T0 is
E(0, T0) =
∫ T0
0
A2|ejω0t|2dt = A2T0
Thus, the total energy is E∞ =∞.
• The average power over one period is
P (0, T0) =E(0, T0)
T0= A2
• The average power is
P∞ = limT→∞
1
2T
∫ T
−T
A2|ejω0t|2dt = A2 2T
2T= A2.
Dr. H. Nguyen Page 25
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
3. Discrete-Time Complex Exponential and Sinusoidal Signals
• Complex exponential signal:
x[n] = Cαn = Ceβn (where α = eβ)
– Real complex signal if both C and α are real.
– General complex exponential signal: With C = Aejφ and α = |α|ejω0 , then
x[n] = A|α|nej(ω0n+φ) = A|α|n cos(ω0n + φ) + jA|α|n sin(ω0n + φ)
∗ If |α| > 1, then x[n] is exponentially growing signal
∗ If |α| < 1, then x[n] is exponentially decaying signal
∗ If |α| = 1, then
x[n] = Aej(ω0n+φ) = A cos(ω0n + φ) + jA sin(ω0n + φ)
• Sinusoidal signal
xc[n] = A cos(ω0n + φ) = ReAej(ω0n+φ)
xs[n] = A sin(ω0n + φ) = ImAej(ω0n+φ)The functions Aej(ω0n+φ), A cos(ω0n + φ) and A sin(ω0n + φ) are
discrete-time signals with finite average power but infinite total energy.
Dr. H. Nguyen Page 26
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Example: The figures below plot the real parts of Ceβn for C = 1 and
β = ±0.1 + j0.5.
−20 −15 −10 −5 0 5 10 15 20−10
−5
0
5
10
Time Index (n)
Rea
l Par
t
−20 −15 −10 −5 0 5 10 15 20−10
−5
0
5
10
Time Index (n)
Rea
l Par
t
Dr. H. Nguyen Page 27
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• DT complex exponential vs. CT complex exponential: There are
important differences between the properties of continuous-time and
discrete-time exponential signals ejω0t and ejω0n.
– The DT exponential signals are not distinct for distinct values of ω0:
x[n] = ejω0n = ej(ω0+k2π)n, k = 0,±1,±2, . . .
∗ Need only consider a frequency interval of 2π for ω0, typically
0 ≤ ω0 < 2π.
∗ As ω0 increases from 0, the signals oscillate more and more rapidly until
ω0 = π. As we continue to increase ω0, we decrease the rate of
oscillation until ω0 = 2π.
∗ Low-frequency exponentials have ω0 near 0, 2π and other even multiples
of π.
∗ High-frequencies are near ±π and other odd multiples of π.
– The exponential ejω0n is periodic if ω0/2π is a rational number:
x[n] = ejω0n = ejω0(n+N) ⇒ ω0
2π=
m
N, for some integer m
– If ω0 6= 0 and if N and m have no factors in common, then N is the
fundamental period and the fundamental frequency is 2πN = ω0
m .
Dr. H. Nguyen Page 28
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Examples:
(a) Let x(t) = cos(8πt/35).
Then ω0 =
The fundamental period T0 = 2π/ω0 =
(b) Let x[n] = cos(8πn/35).
Then ω0 =
If the signal is periodic?
The fundamental period N0 = m(2π/ω0) =
for m =
(c) x[n] = cos(n/6).
Then ω0 =
If the signal is periodic?
The fundamental period N0 = m(2π/ω0) =
for m =
Dr. H. Nguyen Page 29
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• Discerete-Time Complex Exponential Harmonics: A harmonically related set
of discrete-time complex exponentials is a set of exponentials with a common
period N :
φk[n] = ejk(2π/N)n where k = 0,±1,±2, . . .
– All the harmonics are not all distinct for all the values of k:
φk+N [n] = ej(k+N)(2π/N)n = ejk(2π/N)nej2πn = ejk(2π/N)n = φk[n]
– There are only N distinct periodic exponentials:
φ0[n] = 1
φ1[n] = ej2πn/N
φ2[n] = ej4πn/N
· · ·φN−1[n] = ej2π(N−1)n/N
Dr. H. Nguyen Page 30
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Discrete-Time Unit Impulse and Unit Step
• The discrete-time unit impulse is defined as δ[n] =
0, n 6= 0
1, n = 0
-
6r
r r r r r r r r
1
δ[n]
n
– It is also known as the unit sample or Kronecker delta
– It is an even function: δ[n] = δ[−n]
• The discrete-time unit step is defined as u[n] =
0, n < 0
1, n ≥ 0
-
6r r r r r
r r r r
1
u[n]
n
Dr. H. Nguyen Page 31
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• There is a close relationship between δ[n] and u[n]
– First order difference:
δ[n] = u[n]− u[n− 1]
– Running sum:
u[n] =
n∑
k=−∞
δ[k]
• The unit impulse can be used to sample the discrete time signal x[n] (sampling
property):
x[n]δ[n− n0] = x[n0]δ[n− n0]
This ability to use the unit impulse to extract a single value of x[n] through
multiplication will play an important role later.
Dr. H. Nguyen Page 32
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Continuous-Time Unit Step
-
61
u(t)
t
The continuous-time unit step is defined as
u(t) =
0, t < 0
1, t > 0
• Discontinuous at t = 0.
• u(0) is not defined.
• Not of consequence because it is undefined for an infinitesimal period of time.
Dr. H. Nguyen Page 33
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Continuous-Time Unit ImpulseUnit Step for Switches
vs
LinearCircuit
t=0
vsu(t)LinearCircuit
LinearCircuit
t=0
is
isu(t)LinearCircuit
• u(t) useful for representing the opening or closing of switches
• We will often solve for or be given initial conditions at t = 0
• We can then represent independent sources as though they wereimmediately applied at t = 0. More later.
Portland State University ECE 222 Signal Fundamentals Ver. 1.06 47
Discrete-Time Basis Functions
• There is a close relationship between δ[n] and u[n]
δ[n] = u[n] − u[n − 1]
u[n] =n
∑
k=−∞
δ[k]
u[n] =∞∑
k=0
δ[n − k]
• The unit impulse can be used to sample a discrete-time signalx[n]:
x[0] =∞∑
k=−∞
x[k]δ[k] x[n] =∞∑
k=−∞
x[k]δ[n − k]
• This ability to use the unit impulse to extract a single value of x[n]through multiplication will play an important role later in the term
Portland State University ECE 222 Signal Fundamentals Ver. 1.06 45
Continuous-Time Unit Impulse
t
ue(t)
t-e e -e e
t
u(t)
t
1 1
δe(t)
δ(t)
• δe(t) ≡due(t)
dt
• As e → 0 ,
– ue(t) → u(t)
– δe(t) for t = 0 becomes very large
– δe(t) for t = 0 becomes zero
• δ(t) ≡ lime→0 δe(t)
Portland State University ECE 222 Signal Fundamentals Ver. 1.06 48
Continuous-Time Unit Step
t
u(t)
1
u(t) ≡
0 t < 0
1 t > 0
• Sometimes known as the Heaviside function
• Discontinuous at t = 0
• u(0) is not defined
• Not of consequence because it is undefined for an infinitesimalperiod of time
Portland State University ECE 222 Signal Fundamentals Ver. 1.06 46
• Define δe(t) ≡ due(t)dt
• As e→ 0:
ue(t)→ u(t)
δe(t) for t = 0 becomes very large
δe(t) for t 6= 0 becomes zero
• The continuous-time unit impulse function (also known as Dirac delta impulse)
is defined as δ(t) = lime→0 δe(t).
Dr. H. Nguyen Page 34
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Continuous-Time Unit Impulse (Continued)
-
66
1
δ(t)
t
• An equivalent definition of the unit-impulse function:
δ(t) ≡
0, t 6= 0
∞, t = 0and
∫ e
−e
δ(t)dt = 1 for any e > 0
• The function is zero everywhere, except zero.
• The most important property of an impulse is its area. The impulse area serves
as a measure of the impulse amplitude.
Dr. H. Nguyen Page 35
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• Graphical representations:
19
Remark :
We u se th e sh ort-h and notation:
dx(t)
dt= x(t)
R elation between δ(t) and u(t)
– F irst ord er d eriv ativ e
δ(t) = u(t)
– R u nning integ ral
u(t) =
t∫
−∞
δ(τ ) dτ
Forma l d iffi cu lty: u(t) is not d iff erentiable in th e conventional sense
becau se of its d iscontinu ity at t = 0.
S ome more th ou g h ts on δ(t)
– C onsid er fu nctions u∆(t) and δ∆(t) instead of u(t) and δ(t):
u∆(t)
∆ t
δ∆(t)
1
∆
t∆
1
wh ereδ∆(t) = u∆(t)
u∆(t) =
t∫
−∞
δ∆(τ ) dτ
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
2 0
– L imit ∆ → 0
∗ u(t) = lim∆→0
u∆(t)
∗ δ(t) :
t
δ∆1(t)
δ∆3(t)
δ∆2(t)
∆3 ∆2 ∆1
1
∆1
1
∆3
1
∆2
O bserv e: A rea u nd er δ∆(t) always 1
⇒ δ(t) is an infi nitesimally narrow impu lse with area 1.
δ(t) = lim∆→0
δ∆(t)∞∫
−∞
δ(τ ) dτ = 1
– R epresentation
a
t
aδ(t)
t0
1
t
δ(t − t0)
1
t
δ(t)
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s• Real systems do not respond instantaneously. Most systems will respond nearly
the same to sharp pulses regardless of their shape–if
– They have the same amplitude.
– Their duration is much briefer than the system’s response.
• The unit impulse is an idealization of such pulses, which is short enough for any
system.
Dr. H. Nguyen Page 36
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Properties of Unit Impulse
• The relationship between δ(t) and u(t):
– First order derivative: δ(t) =du(t)
dt
– Running integral: u(t) =
∫ t
−∞
δ(τ)dτ
• Sampling properties:
x(t)δ(t− t0) = x(t0)δ(t− t0)∫ ∞
−∞
x(t)δ(t− t0)dt =
∫ ∞
−∞
x(t0)δ(t− t0)dt = x(t0)
∫ ∞
−∞
δ(t− t0)dt = x(t0)
• Time scaling:
δ(at) =1
|a|δ(t), (a 6= 0)
This is because
∫ ∞
−∞
δ(at)dt =
∫ ∞
−∞
1
|a|δ(ν)dν =1
|a| .
Dr. H. Nguyen Page 37
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Continuous-Time and Discrete-Time Systems
x(t) - CTsystem
- y(t) x[n] - DTsystem
- y[n]
System: A process in which input signals are transformed by the system or cause the
system to respond in some way, resulting in other signals as outputs.
• All of the systems that we will consider have a single input and a single output
• Continuous-time system transforms continuous-time signals.
• Discrete-time system transforms discrete-time signals.
• We will use the notation x(t) −→ y(t) to mean the input signal x(t) causes the
output signal y(t).
• Similar meaning is used for the notation x[n] −→ y[n].
Dr. H. Nguyen Page 38
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• Simple examples of systems are given below.
– Quadratic system:
x(t) −→ y(t) = (x(t))2
– Delay system:
x[n] −→ y[n] = x[n− 1]
– System represented by a first order differential equation:
dy(t)
dt+ ay(t) = bx(t)
where a and b are constants.
– System described by a first order difference equation:
y[n] = ay[n− 1] + bx[n]
where a and b are constants.
Dr. H. Nguyen Page 39
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Interconnections of Systems
• Many real systems are built as interconnections of several subsystems.
• It is useful to use the understanding of the component systems and of how they
are interconnected to analyze the operation and behavior of the overall system.
• Basic system interconnections:
23
1.3 C on tin u ou s–T im e a n d D isc re te –T im e S y ste m s
Unified representation of ph ysical processes by systems
S y ste m : E ntity th at transforms inpu t sig nals into new ou tpu t sig nals
– O n e or more inpu t and ou tpu t sig nals
– C o n tin u o u s– time system transforms continu ou s– time sig nals
– D iscrete– time system transforms discrete– time sig nals
Formal representation of inpu t– ou tpu t relation
– C ontinu ou s– time system
x(t) −→ y(t)
– D iscrete– time system
x[n] −→ y[n]
R emark : A noth er popu lar notation th at you may find in book s is
y(t) = Sx(t), wh ere S· represents th e system operator.
P ictorial representation of systems
Continuous−timesystemx(t) y(t)
systemDiscrete−time
x[n] y[n]
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
24
1.3.1 S im p le E x a m p le s of S y ste m s
Q u adratic system
y(t) = (x(t))2
S ystem represented by a first order diff erential eq u ation
y(t) + ay(t) = bx(t)
with constants a and b
D elay system
y[n] = x[n − 1 ]
S ystem described by a first order diff erence eq u ation
y[n] = ay[n − 1 ] + bx[n]
with constants a and b
1.3.2 In te rcon n e c tion s of S y ste m s
O ften convenient: break down a complex system into smaller su bsystems
S eries (cascade) interconnection
System 1 System 2Input Output
E xamples: C ommu nication ch annel and receiv er, detector and de-
coder in commu nications
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
25
Parallel interconnection
System 1
System 2
OutputInput
E x ample: Diversity tran smissio n : transmission of th e same sig nal
ov er two antennas and receiv ing it with one antenna
Feedback interconnection
OutputInput
System 2
System 1
E x amples: Closed-loop freq u ency/ ph ase/ timing synch ronization in
commu nications, h u man motion control
1.4 B a sic S yste m P rope rtie s
S imple math ematical formu lation of basic (ph ysical) system proper-
ties
Classifi cation of systems
For conciseness: only defi nitions for continu ou s-time systems
R eplacing “(t)” by “[n]” ⇒ defi nitions for discrete-time systems
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
26
1.4.1 L in e a rity
L et x1(t) −→ y1(t) and x2(t) −→ y2(t)
L in ear system if
1 . A dditiv ity
x1(t) + x2(t) −→ y1(t) + y2(t)
2 . H omog eneity
ax1(t) −→ ay1(t) , ∀a ∈ C
L inear systems possess property of su perposition
L et xk(t) −→ yk(t), th en
K∑
k= 1
akxk(t) = x(t) −→ y(t) =K∑
k= 1
akyk(t)
“N ot linear” systems are referred to as n o n lin ear.
E x a m ple :
1 . S ystem y(t) = tx(t) is linear.
To see th is let
x1(t) −→ y1(t) = tx1(t)
x2(t) −→ y2(t) = tx2(t)
and
x3(t) = ax1(t) + bx2(t) ,
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
25
Parallel interconnection
System 1
System 2
OutputInput
E x ample: Diversity tran smissio n : transmission of th e same sig nal
ov er two antennas and receiv ing it with one antenna
Feedback interconnection
OutputInput
System 2
System 1
E x amples: Closed-loop freq u ency/ ph ase/ timing synch ronization in
commu nications, h u man motion control
1.4 B a sic S yste m P rope rtie s
S imple math ematical formu lation of basic (ph ysical) system proper-
ties
Classifi cation of systems
For conciseness: only defi nitions for continu ou s-time systems
R eplacing “(t)” by “[n]” ⇒ defi nitions for discrete-time systems
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
26
1.4.1 L in e a rity
L et x1(t) −→ y1(t) and x2(t) −→ y2(t)
L in ear system if
1 . A dditiv ity
x1(t) + x2(t) −→ y1(t) + y2(t)
2 . H omog eneity
ax1(t) −→ ay1(t) , ∀a ∈ C
L inear systems possess property of su perposition
L et xk(t) −→ yk(t), th en
K∑
k= 1
akxk(t) = x(t) −→ y(t) =K∑
k= 1
akyk(t)
“N ot linear” systems are referred to as n o n lin ear.
E x a m ple :
1 . S ystem y(t) = tx(t) is linear.
To see th is let
x1(t) −→ y1(t) = tx1(t)
x2(t) −→ y2(t) = tx2(t)
and
x3(t) = ax1(t) + bx2(t) ,
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
Dr. H. Nguyen Page 40
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Basic System Properties
• The basic system properties of continuous-time and discrete-time systems have
important physical interpretations.
• They also have relatively simple mathematical descriptions using the signals and
systems language.
1. Systems with and without memory: A system is said to be memoryless if and
only if the output y(t) at any time t0 depends only on the input x(t) at the
same time, i.e., x(t0).
• Memory indicates the system has the capability to store (remember)
information about input values at times other than the current time.
• In many physical systems, memory is directly associated with the storage of
energy.
• As examples, capacitors and inductors store energy and therefore create
systems with memory. In contrast, resistors have no such mechanism and
therefore create memoryless systems.
Dr. H. Nguyen Page 41
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• Other examples of memoryless systems:
(a) Limiter: y[n] =
x[n], −A ≤ x[n] ≤ A
−A, x[n] < −A
A, x[n] > A
(b) Amplifier: y(t) = Ax(t)
• Other examples of systems with memory:
(a) Accumulator: y[n] =∑n
k=−∞x[k] = x[n] + y[n− 1]
(b) Delay: y(t) = x(t− t0)
• While the concept of memory in a system typically suggest storing past input
and output values, our formal definition of systems with memory also
includes the ones whose current output is dependent on the future values of
the input and output.
• Such systems can be found in applications in which the independent variable
is not time, such as in image processing, in processing signals that have been
recorded previously (speech, geophysical, meteorological signals).
Dr. H. Nguyen Page 42
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
2. Invertibility: A system is invertible if distinct inputs lead to distinct outputs.
• If the system is invertible, then an inverse system exists.
• When the inverse system is cascaded with the original system, the output is the
same as the input:
x[n] - System -y[n] Inverse
system- x[n]
• Examples of invertible systems:
(a) Amplifier: y(t) = Ax(t), A 6= 0. The inverse system is w(t) = 1Ay(t).
(b) Accumulator: y[n] = y[n− 1] + x[n]. The inverse system is
w[n] = y[n]− y[n− 1] (which is the diffirentiator)
• Examples of non-invertible systems:
(a) Limiter: y[n] =
x[n], −A ≤ x[n] ≤ A
−A, x[n] < −A
A, x[n] > A
(b) Slicer: y[n] =
1, x[n] ≥ 0
−1, else
Dr. H. Nguyen Page 43
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
3. Causal: A system is causal if the output y(t) at any time t0 depends on values
of the input x(t) at only the present and past times, −∞ < t ≤ t0.
• If two inputs to a causal system are identical up to some point in time, the
corresponding outputs must also be equal up to this same time:
If x1(t) = x2(t) for t ≤ t0 then y1(t) = y2(t) for t ≤ t0, ∀t0
• All analog circuits are causal.
• All memoryless systems are causal, since the systems only respond to the
current value of the input.
• Not all causal systems are memoryless (very few are).
• Note that causality is not often an essential constraint in applications in which
the independent variable is not time, such as in image processing, in processing
data that have been recorded previously.
• Examples:
(a) Accumulator is a causal system: y[n] =∑n
k=−∞x[k] = x[n] + y[n− 1]
(b) A smoothing averager is a noncausal system: y[n] =1
2M + 1
M∑
k=−M
x[n− k]
Dr. H. Nguyen Page 44
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
4. Stability: A system is bounded-input bounded-output (BIBO) stable if all
bounded inputs (|x(t)| ≤ Bx <∞, ∀t) result in bounded outputs
(|y(t)| ≤ By <∞, ∀t).• Informally, stable systems are those in which small inputs do not lead to
outputs that diverge (grow without bound).
• All physical circuits are technically stable.
• Ideal op amp without negative feedback are usually unstable.
• Examples:
(a) The smoothing averager is a stable system: y[n] =1
2M + 1
M∑
k=−M
x[n− k].
For bounded input |x[n]| ≤ Bx, the output |y[n]| ≤ By = Bx, which is also
bounded.
(b) The integrator is unstable system: y(t) =
∫ t
−∞
x(τ)dτ .
Let the bounded input be x(t) = u(t), then the output y(t) = t is
unbounded.
• System stability is important in engineering applications. Unstable systems need
to be stabilized.
Dr. H. Nguyen Page 45
EE351–Spec
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31
1.4.6 S ta b ility
Consider bo u n d ed – in p u t bo u n d ed – o u tp u t (B IB O ) stability
S ta ble system if for any bou nded inpu t sig nal
|x(t)| ≤ Bx < ∞ , ∀t
th e ou tpu t sig nal is bou nded
|y(t)| ≤ By < ∞ , ∀t
E x a m p le :
– S table system
A v erag er: y[n] =1
2N + 1
N∑k=−N
x[n − k]
B ou nded inpu t |x[n]| < Bx ⇒ bou nded ou tpu t |y[n]| < By =
Bx
– Instable system
Integ rator: y(t) =
t∫
−∞
x(τ ) d τ
E .g . bou nded inpu t x(t) = u(t) ⇒ u nbou nded ou tpu t y(t) = t
S ystem stability is important in eng ineering applications, u nstable
systems need to be stabilized.
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
32
E x a m p le : Th e fi rst Tacoma N arrow s su spension bridg e collapsed du e
to w ind-indu ced v ibrations, N ov ember 1 9 4 0 .
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Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
Dr.
H.N
guye
nPage
46
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
5. Time Invariance: A system is time invariant if it produces identical response to
the same input signal no matter when input signal is applied. Mathematically:
x(t)→ y(t) implies x(t− t0)→ y(t− t0).
x[n]→ y[n] implies x[n− n0]→ y[n− n0].
• In other words, a system is time invariant if a time shift in the input signal
results in a corresponding time shift in the output signal.
• Circuits that have non-zero energy stored on capacitors or in inductors at time
t = 0 are generally not time-invariant (i.e., they are time-variant).
• Memoryless does not imply time-invariant. For example, y(t) = x(t)× f(t).
• Examples:
(a) The system y(t) = (x(t))2 is?
(b) The system y[n] = nx[n] is?
Dr. H. Nguyen Page 47
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
6. Linearity: Consider a system with x1(t)→ y1(t) and x2(t)→ y2(t). The
system is said to be linear if:
a1x1(t) + a2x2(t)→ a1y1(t) + a2y2(t)
for any constant complex coefficients a1 and a2.
x(t) - CTsystem
- y(t) x[n] - DTsystem
- y[n]
a1x1(t) + a2x2(t)→ a1y1(t) + a2y2(t)
a1x1[n] + a2x2[n]→ a1y1[n] + a2y2[n]
• There are two related properties:
– Additive: x1(t) + x2(t)→ y1(t) + y2(t)
– Scaling : ax1(t)→ ay1(t)
• Linear systems enable the application of superposition: If the input consists of a
linear combination of different inputs, the output is the same linear combination
of the corresponding outputs.
• “Not linear” systems are referred to as nonlinear.
Dr. H. Nguyen Page 48
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Linear Time-Invariant (LTI) Systems
A system is said to be linear time invariant (LTI) if it is both linear and time
invariant.
• The linearity and time-invariance properties play a fundamental role in signal
and system analysis because of the following two main reasons:
– Many physical processes posses these properties ⇒ can be modeled as LTI
systems.
– LTI systems can be analyzed in considerable detail, providing both insight
into their properties and a set of powerful tools for signal and system analysis.
• Key idea: If one can represent the input to an LTI system in terms of a linear
combination of a set of basic signals, one can apply the superposition principle to
compute the output of the system in terms of its responses to these basic signals.
• As will be seen shortly, the basic signals can be chosen to be the delayed
impulses ⇒ an LTI system is completely characterized by its response to a unit
impulse, i.e., the impulse response.
Dr. H. Nguyen Page 49
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Impulse Response of an LTI System
• Impulse response is the system’s response to a unit impulse.
• The impulse response is denoted by h(t) or h[n]. Thus:
δ(t) −→ h(t)
δ[n] −→ h[n]
• For any input x(t) (or x[n]), it is possible to use the impulse response h(t) (or
h[n]) to find the output y(t) (or y[n]):
x(t) - h(t) - y(t) x[n] - h[n] - y[n]
• This method is called convolution sum in the discrete-time case and convolution
integral in the continuous-time case.
• Impulse response is an important concept (for example, it is used to implement
digital filters).
Dr. H. Nguyen Page 50
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Discrete-Time Convolution Sum
• Any discrete-time input signal x[n] can be expressed as a sum of scaled and
delayed unit impulses (sampling property of the unit-impulse):
x[n] =
∞∑
k=−∞
x[k]δ[n− k]
• By linearity and time-invariance properties, the output of an LTI system is the
corresponding scaled sum of the outputs due to the delayed impulses:
x[k]δ[n− k]→ x[k]h[n− k]
x[n] =
∞∑
k=−∞
x[k]δ[n− k]→ y[n] =
∞∑
k=−∞
x[k]h[n− k] = x[n] ∗ h[n]
• The above operation is called the discrete-time convolution sum.
• Observe that the impulse response h[n] completely characterizes a discrete-time
LTI system: If we know h[n] then we can calculate the output y[n] for any
input x[n].
Dr. H. Nguyen Page 51
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Example: The impulse response and the input of a discrete-time LTI system are
given below. Find the output.
h[n] = anu[n]
an, n ≥ 0
0, n < 0
(where a = 0.5)x[n] =
1, n = 0
−1, n = 2
2, n = 5
0, otherwise
0 5 10−1
0
1
2
x[n]
0 5 10−1
0
1
2
h[n]
• In terms of unit-impulses, the input signal can be expressed as
x[n] = 1 · δ[n]− 1 · δ[n− 2] + 2 · δ[n− 5]
Dr. H. Nguyen Page 52
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• Since x1[n] = δ[n]→ y1[n] = h[n], x2[n] = −δ[n− 2]→ y2[n] = −h[n− 2] and
x3[n] = 2 · δ[n− 5]→ y3[n] = 2 · h[n− 5]. Therefore,
y[n] = y1[n] + y2[n] + y3[n] = h[n]− h[n− 2] + 2 · h[n− 5]
0 5 10−1
0
1
2x 1[n
]Input
0 5 10−1
0
1
2
y 1[n]
Output
0 5 10−1
0
1
2
x 2[n]
0 5 10−1
0
1
2
y 2[n]
0 5 10−1
0
1
2
x 3[n]
0 5 10−1
0
1
2
y 3[n]
0 5 10−1
0
1
2
x[n]
Time (n)0 5 10
−1
0
1
2
y[n]
Time (n)
Dr. H. Nguyen Page 53
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Technique to Find Discrete-Time Convolution Sum
x[n] - h[n] - y[n] y[n] =∞∑
k=−∞
x[k]h[n− k]
Consider the evaluation of the output value at some specific time n, say n = n0:
y[n0] =
∞∑
k=−∞
x[k]h[n0 − k]
• Plot the two signals (or two sequences) x[k] and h[n0 − k] as functions of k.
• Multiplying these two functions to obtain a sequence g[k] = x[k]h[n0 − k].
• Summing all the samples in the sequence g[k] yields the output value at the
selected time n0.
To plot h[n0−k] as a function of k, it is convenient to follow the following two steps:
• Plot the signal h[−k] first.
• Obtain h[n0 − k] simply by shifting h[−k] to the right (by n0) if n0 is positive,
or to the left (by |n0|) if n0 is negative.
Dr. H. Nguyen Page 54
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Example: Compute y[6] for the previous example.
−5 0 5 10−2
0
2
x[k]
−5 0 5 100
0.5
1
h[k]
−5 0 5 100
0.5
1
h[−
k]
−5 0 5 100
0.5
1
h[6−
k]
Time (k)
y[6] = h[6]− h[4] + 2h[1] = 0.56 − 0.54 + 2 · 0.51 = 0.9531 ≈ 1.
Dr. H. Nguyen Page 55
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Convolution Sum Derivation: Summary
- h[n] -x[n] =
∞∑
k=−∞
x[k]δ[n− k] y[n] =
∞∑
k=−∞
x[k]h[n− k]
Linearity
- h[n] -x[k]δ[n− k] x[k]h[n− k]
Linearity
- h[n] -δ[n− k] h[n− k]
Time Invariance
- h[n] -δ[n] h[n]
Definition of h[n]
- h[n] -x[n] y[n]
LTI SystemInput Output
Dr. H. Nguyen Page 56
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Continuous-Time Convolution Integral
x(t) - h(t) - y(t)
• Recall that if the input x(t) = δ(t), the output of the system is called the
impulse response, denoted by h(t).
• The goal is to obtain a complete characterization of a continuous-time LTI
system in terms of its impulse response.
• This means that, for any input x(t), we must be able to use the impulse
response h(t) to find the output y(t). This method is called convolution integral.
• To derive the convolution integral, we shall decompose and approximate the
input signal by rectangular pulses (or rectangles).
Dr. H. Nguyen Page 57
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Derivation of the Convolution Integral
x(t)
0
( )rx t
( )x kw
kw( )1
2k w+( )12k w−
w
( )( ) ( )( )1 12 2( )x kw u t k w u t k w − − − − +
t
Approximate the input signal x(t) by a sum of weighted delayed rectangular pulses:
x(t) ≈ xr(t) =
∞∑
k=−∞
w · x(kw)
[u(t−
(k − 1
2
)w)− u
(t−
(k + 1
2
)w)]
w
Since δ(t− kw) =du(t− kw)
dt= lim
w→0
[u(t−
(k − 1
2
)w)− u
(t−
(k + 1
2
)w)]
w
Dr. H. Nguyen Page 58
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
So for very small w, x(t) can also be approximated as a sum of impulses:
x(t) ≈ xw(t) =
∞∑
k=−∞
w · x(kw) · δ(t− kw)
By linearity and time-invariance properties, we have
y(t) ≈ yw(t) =
∞∑
k=−∞
w · x(kw) · h(t− kw)
Finally, in the limit w → 0 the above approximations become exact representations
and the summations become the integrals:
x(t) = limw→0
xw(t) =
∫ ∞
−∞
x(τ)δ(t− τ)dτ
y(t) = limw→0
yw(t) =
∫ ∞
−∞
x(τ)h(t− τ)dτ
The last equation above is the continuous-time convolution integral.
Dr. H. Nguyen Page 59
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Example: Approximations of a triangle with rectangular pulses and impulses:
0 1 20
0.5
1
w=0.50
Approximation with Rectangles
0 1 20
0.5Approximation with Impulses
0 1 20
0.5
1
w=0.25
0 1 20
0.1
0.2
0 1 20
0.5
1
w=0.10
0 1 20
0.05
0.1
0 1 20
0.5
1
w=0.05
Time (s)0 1 2
0
0.05
Time (s)
Dr. H. Nguyen Page 60
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Example:R
x(t) m
r
r+
−+− y(t)C
The impulse response of the above RC circuit can be shown to be:
h(t) = RC · e− t
RC u(t)
= e−tu(t) =
e−t, t ≥ 0
0, t < 0(for RC = 1 s)
Let the input signal to the circuit be the triangle considered in the previous page :
x(t) =
t, 0 ≤ t ≤ 1
−(t− 2), 1 ≤ t ≤ 2
0, otherwise
Approximations of this signal by impulses were also shown in the previous page.
Dr. H. Nguyen Page 61
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
The following figures show the responses of the circuit to different approximations of
the input signal by impulses.
−5 0 5 100
0.2
0.4
0.6
0.8Input
−5 0 5 100
0.2
0.4
0.6
0.8Output
True Approximation
−5 0 5 100
0.1
0.2
0.3
0.4
−5 0 5 100
0.2
0.4
0.6
0.8True Approximation
−5 0 5 100
0.05
0.1
Time (s)−5 0 5 100
0.2
0.4
0.6
0.8
Time (s)
True Approximation
Dr. H. Nguyen Page 62
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Convolution Integral: An Alternative Form
- h(t) -x(t) y(t) =
∫ ∞
−∞
x(τ)h(t− τ)dτ
Let u ≡ t− τ , then τ = t− u and du = −dτ .
y(t) =
∫ −∞
∞
x(t− u)h(u)(−du)
=
∫ ∞
−∞
x(t− u)h(u)du
=
∫ ∞
−∞
x(t− τ)h(τ)dτ
Both forms are called the convolution integral. It is often written as:
y(t) = x(t) ∗ h(t) =
∫ ∞
−∞
x(τ)h(t− τ)dτ =
∫ ∞
−∞
x(t− τ)h(τ)dτ
Dr. H. Nguyen Page 63
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Convolution Integral: Solving Graphically
- h(t) -x(t) y(t) = x(t) ∗ h(t) =
∫ ∞
−∞
x(τ)h(t− τ)dτ
• To calculate this integral graphically, perform the following steps for any specific
value of t:
1. First obtain the signal h(t− τ) (regarded as a function of τ with t fixed)
from h(τ) by reflection about the origin and a shift to the right by t if t > 0
or a shift to the left by |t| for t < 0.
2. Next, multiply together the signals x(τ) and h(t− τ) to obtain the function
g(τ) = x(τ)h(t− τ).
3. Finally, y(t) is obtained by integrating the function g(τ) from τ = −∞ to
τ =∞.
• Generally, it is sufficient to plot both x(τ) and h(t− τ) on the same axis.
Dr. H. Nguyen Page 64
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Example: Let x(t) =
1, 0 ≤ t ≤ 2T
0, otherwiseand h(t) =
2, 0 ≤ t ≤ T
3, T ≤ t ≤ 2T
0, otherwise
.
Find y(t) = x(t) ∗ h(t) =
∫ ∞
−∞
x(τ)h(t− τ)dτ .
41
Gra ph ica l in terpreta tio n
– T ime-rev erse h(τ ) ⇒ h(−τ )
– S h ift h(−τ ) by t ⇒ h(t − τ )
– In teg ra te pro d u cts o f o v erla ppin g compo n en ts
Example:
∗ In pu t: x(t) =
1, 0 ≤ t ≤ 2T
0, o th e rw ise
Impu lse respo n se: h(t) =
2, 0 ≤ t < T
3, T ≤ t ≤ 2T
0 o th e rw ise
∗ t < 0
t − 2T 2TTt
h(t − τ)
x(τ)
τ
N o o v erla p: y(t) = 0
∗ 0 ≤ t < T
t T 2T τ
y(t) =
t∫
0
1 · 2 d t = 2t
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
42
∗ T ≤ t < 2T
tT 2T τ
y(t) =
t−T∫
0
3 d t +
t∫
t−T
2 d t = 3(t − T ) + 2T
∗ 2T ≤ t < 3T
tT 2T τ
y(t) =
2T∫
t−T
2 d t +
t−T∫
t−2T
3 d t = 2(3T − t) + 3T
∗ 3T ≤ t < 4T
tT 2T τ
y(t) =
2T∫
t−2T
3 d t = 3(4T − t)
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
41
Gra ph ica l in terpreta tio n
– T ime-rev erse h(τ ) ⇒ h(−τ )
– S h ift h(−τ ) by t ⇒ h(t − τ )
– In teg ra te pro d u cts o f o v erla ppin g compo n en ts
Example:
∗ In pu t: x(t) =
1, 0 ≤ t ≤ 2T
0, o th e rw ise
Impu lse respo n se: h(t) =
2, 0 ≤ t < T
3, T ≤ t ≤ 2T
0 o th e rw ise
∗ t < 0
t − 2T 2TTt
h(t − τ)
x(τ)
τ
N o o v erla p: y(t) = 0
∗ 0 ≤ t < T
t T 2T τ
y(t) =
t∫
0
1 · 2 d t = 2t
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
42
∗ T ≤ t < 2T
tT 2T τ
y(t) =
t−T∫
0
3 d t +
t∫
t−T
2 d t = 3(t − T ) + 2T
∗ 2T ≤ t < 3T
tT 2T τ
y(t) =
2T∫
t−T
2 d t +
t−T∫
t−2T
3 d t = 2(3T − t) + 3T
∗ 3T ≤ t < 4T
tT 2T τ
y(t) =
2T∫
t−2T
3 d t = 3(4T − t)
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
41
Gra ph ica l in terpreta tio n
– T ime-rev erse h(τ ) ⇒ h(−τ )
– S h ift h(−τ ) by t ⇒ h(t − τ )
– In teg ra te pro d u cts o f o v erla ppin g compo n en ts
Example:
∗ In pu t: x(t) =
1, 0 ≤ t ≤ 2T
0, o th e rw ise
Impu lse respo n se: h(t) =
2, 0 ≤ t < T
3, T ≤ t ≤ 2T
0 o th e rw ise
∗ t < 0
t − 2T 2TTt
h(t − τ)
x(τ)
τ
N o o v erla p: y(t) = 0
∗ 0 ≤ t < T
t T 2T τ
y(t) =
t∫
0
1 · 2 d t = 2t
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
42
∗ T ≤ t < 2T
tT 2T τ
y(t) =
t−T∫
0
3 d t +
t∫
t−T
2 d t = 3(t − T ) + 2T
∗ 2T ≤ t < 3T
tT 2T τ
y(t) =
2T∫
t−T
2 d t +
t−T∫
t−2T
3 d t = 2(3T − t) + 3T
∗ 3T ≤ t < 4T
tT 2T τ
y(t) =
2T∫
t−2T
3 d t = 3(4T − t)
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
41
Gra ph ica l in terpreta tio n
– T ime-rev erse h(τ ) ⇒ h(−τ )
– S h ift h(−τ ) by t ⇒ h(t − τ )
– In teg ra te pro d u cts o f o v erla ppin g compo n en ts
Example:
∗ In pu t: x(t) =
1, 0 ≤ t ≤ 2T
0, o th e rw ise
Impu lse respo n se: h(t) =
2, 0 ≤ t < T
3, T ≤ t ≤ 2T
0 o th e rw ise
∗ t < 0
t − 2T 2TTt
h(t − τ)
x(τ)
τ
N o o v erla p: y(t) = 0
∗ 0 ≤ t < T
t T 2T τ
y(t) =
t∫
0
1 · 2 d t = 2t
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
42
∗ T ≤ t < 2T
tT 2T τ
y(t) =
t−T∫
0
3 d t +
t∫
t−T
2 d t = 3(t − T ) + 2T
∗ 2T ≤ t < 3T
tT 2T τ
y(t) =
2T∫
t−T
2 d t +
t−T∫
t−2T
3 d t = 2(3T − t) + 3T
∗ 3T ≤ t < 4T
tT 2T τ
y(t) =
2T∫
t−2T
3 d t = 3(4T − t)
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
Dr. H. Nguyen Page 65
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
41
Gra ph ica l in terpreta tio n
– T ime-rev erse h(τ ) ⇒ h(−τ )
– S h ift h(−τ ) by t ⇒ h(t − τ )
– In teg ra te pro d u cts o f o v erla ppin g compo n en ts
Example:
∗ In pu t: x(t) =
1, 0 ≤ t ≤ 2T
0, o th e rw ise
Impu lse respo n se: h(t) =
2, 0 ≤ t < T
3, T ≤ t ≤ 2T
0 o th e rw ise
∗ t < 0
t − 2T 2TTt
h(t − τ)
x(τ)
τ
N o o v erla p: y(t) = 0
∗ 0 ≤ t < T
t T 2T τ
y(t) =
t∫
0
1 · 2 d t = 2t
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
42
∗ T ≤ t < 2T
tT 2T τ
y(t) =
t−T∫
0
3 d t +
t∫
t−T
2 d t = 3(t − T ) + 2T
∗ 2T ≤ t < 3T
tT 2T τ
y(t) =
2T∫
t−T
2 d t +
t−T∫
t−2T
3 d t = 2(3T − t) + 3T
∗ 3T ≤ t < 4T
tT 2T τ
y(t) =
2T∫
t−2T
3 d t = 3(4T − t)
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
43
∗ t ≥ 4T
tT 2T τ
No overlap: y(t) = 0
∗ In su mmary
y(t) =
0, t < 0
2t, 0 ≤ t < T
2T + 3(t − T ), T ≤ t < 2T
2(3T − t) + 3T, 2T ≤ t < 3T
3(4T − t), 3T ≤ t < 4T
0, t ≥ 4T
5
4
3
2
1
y(t)/ T
3 421 t/ T
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
44
2.3 P rope rtie s of Line a r T im e –Inv a ria nt S yste m s
L in earity an d time in v arian ce ⇒ complete ch aracterization by im-
pu lse respon se
y[k] =∞
∑
k= − ∞
x[k]h[n − k] = x[k] ∗ h[k]
y(t) =
∞∫
− ∞
x(τ )h(t − τ ) d τ = x(t) ∗ h(t)
Fu rth er properties based on an d in terms of impu lse respon se repre-
sen tation
2.3.1 C om m u ta tiv e , D istrib u tiv e , a nd A ssoc ia tiv e P rope rty
of C onvolu tion — Inte rconne ctions of LT I S yste m s
Commu ta tiv e property: order of th e sig n als to be con volv ed can be
ch an g ed
– C on tin u ou s– time case
x(t) ∗ h(t) = h(t) ∗ x(t) =
∞∫
− ∞
h(τ )x(t − τ ) d τ
– D iscrete– time case
x[n] ∗ h[n] = h[n] ∗ x[n] =∞
∑
k= − ∞
h[k]x[n − k]
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
Combine all the cases, the final expression for y(t) is:
43
∗ t ≥ 4T
tT 2T τ
No overlap: y(t) = 0
∗ In su mmary
y(t) =
0, t < 0
2t, 0 ≤ t < T
2T + 3(t − T ), T ≤ t < 2T
2(3T − t) + 3T, 2T ≤ t < 3T
3(4T − t), 3T ≤ t < 4T
0, t ≥ 4T
5
4
3
2
1
y(t)/ T
3 421 t/ T
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
44
2.3 P rope rtie s of Line a r T im e –Inv a ria nt S yste m s
L in earity an d time in v arian ce ⇒ complete ch aracterization by im-
pu lse respon se
y[k] =∞
∑
k= − ∞
x[k]h[n − k] = x[k] ∗ h[k]
y(t) =
∞∫
− ∞
x(τ )h(t − τ ) d τ = x(t) ∗ h(t)
Fu rth er properties based on an d in terms of impu lse respon se repre-
sen tation
2.3.1 C om m u ta tiv e , D istrib u tiv e , a nd A ssoc ia tiv e P rope rty
of C onvolu tion — Inte rconne ctions of LT I S yste m s
Commu ta tiv e property: order of th e sig n als to be con volv ed can be
ch an g ed
– C on tin u ou s– time case
x(t) ∗ h(t) = h(t) ∗ x(t) =
∞∫
− ∞
h(τ )x(t − τ ) d τ
– D iscrete– time case
x[n] ∗ h[n] = h[n] ∗ x[n] =∞
∑
k= − ∞
h[k]x[n − k]
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
43
∗ t ≥ 4T
tT 2T τ
No overlap: y(t) = 0
∗ In su mmary
y(t) =
0, t < 0
2t, 0 ≤ t < T
2T + 3(t − T ), T ≤ t < 2T
2(3T − t) + 3T, 2T ≤ t < 3T
3(4T − t), 3T ≤ t < 4T
0, t ≥ 4T
5
4
3
2
1
y(t)/ T
3 421 t/ T
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
44
2.3 P rope rtie s of Line a r T im e –Inv a ria nt S yste m s
L in earity an d time in v arian ce ⇒ complete ch aracterization by im-
pu lse respon se
y[k] =∞
∑
k= − ∞
x[k]h[n − k] = x[k] ∗ h[k]
y(t) =
∞∫
− ∞
x(τ )h(t − τ ) d τ = x(t) ∗ h(t)
Fu rth er properties based on an d in terms of impu lse respon se repre-
sen tation
2.3.1 C om m u ta tiv e , D istrib u tiv e , a nd A ssoc ia tiv e P rope rty
of C onvolu tion — Inte rconne ctions of LT I S yste m s
Commu ta tiv e property: order of th e sig n als to be con volv ed can be
ch an g ed
– C on tin u ou s– time case
x(t) ∗ h(t) = h(t) ∗ x(t) =
∞∫
− ∞
h(τ )x(t − τ ) d τ
– D iscrete– time case
x[n] ∗ h[n] = h[n] ∗ x[n] =∞
∑
k= − ∞
h[k]x[n − k]
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
Dr. H. Nguyen Page 66
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Convolution Integral Derivation: Summary
- h(t) -x(t) =
∫ ∞
−∞
x(τ)δ(t− τ)dτ y(t) =
∫ ∞
−∞
x(τ)h(t− τ)dτ
Linearity
- h(t) -x(τ)δ(t− τ) x(τ)h(t− τ)
Linearity
- h(t) -δ(t− τ) h(t− τ)
Time Invariance
- h(t) -δ(τ) h(t− τ)
Definition of h(t)
- h(t) -x(t) y(t)
LTI SystemInput Output
Dr. H. Nguyen Page 67
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Summary of Convolution Integral
- h(t) -x(t) y(t) = x(t) ∗ h(t) =
∫ ∞
−∞
x(τ)h(t− τ)dτ
• The convolution integral describes how the output y(t) is related to the input
signal x(t) and the impulse response h(t).
• Only two assumptions were made about the system:
– Linear
– Time Invariant
• Key points:
– The impulse response h(t) completely defines the behavior of a
continuous-time LTI system.
– If h(t) is known, then the output of a continuous-time LTI system can be
found for any input x(t) using the convolution integral.
Dr. H. Nguyen Page 68
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Properties of LTI Systems
1 Commutative property: The order of the signals to be convolved can be
interchanged:
y[n] = x[n] ∗ h[n] = h[n] ∗ x[n]
y(t) = x(t) ∗ h(t) = h(t) ∗ x(t)
Implications:
– The output of an LTI system with input x(t) and impulse response h(t) is
identical to the output of an LTI system with input h(t) and impulse
response x(t).
– Irrelevant whether h(t) or x(t) is reflected and shifted to compute the
convolution integral.
Dr. H. Nguyen Page 69
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
2 Distributive property:
x[n] ∗ (h1[n] + h2[n]) = x[n] ∗ h1[n] + x[n] ∗ h2[n]
x(t) ∗ (h1(t) + h2(t)) = x(t) ∗ h1(t) + x(t) ∗ h2(t)
Implication: A parallel combination of LTI systems can be replaced by a single
LTI system whose unit impulse response is the sum of the individual unit impulse
responses in the parallel combination.
45
– Implication s:
∗ ou tpu t of system with impu lse respon se h(t) to in pu t x(t)
iden tical to ou tpu t of system with impu lse respon se x(t) to
in pu t h(t)
∗ Irrelev an t wh eth er h(t) or x(t) is refl ected an d sh ifted for
con volu tion
Distributive property
– C on tin u ou s– time case
x(t) ∗ (h1(t) + h2(t)) = x(t) ∗ h1(t) + x(t) ∗ h2(t)
– D iscrete– time case
x[n] ∗ (h1[n] + h2[n]) = x[n] ∗ h1[n] + x[n] ∗ h2[n]
– Implication
Parallel con n ection of two LT I systems represen ted by sin g le
eq u iv alen t LT I system
h2(t)
h1(t)
h1(t) + h2(t)
≡
y(t)x(t)
x(t) y(t)
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
46
A sso cia tive Property
– C on tin u ou s– time case
x(t) ∗ (h1(t) ∗ h2(t)) = (x(t) ∗ h1(t)) ∗ h2(t)
– D iscrete– time case
x[n] ∗ (h1[n] ∗ h2[n]) = (x[n] ∗ h1[n]) ∗ h2[n]
– C on seq u en tly
y[n] = x[n] ∗ h1[n] ∗ h2[n]
– Implication s
∗ C h ose con v en ien t order of con volu tion
∗ C ascade con n ection of two LT I system represen ted by sin g le
LT I system
h1(t) h2(t)
≡
h1(t) ∗ h2(t) y(t)x(t)
x(t) y(t)
A ssociativ e + commu tativ e property ⇒ order in a cascade of LT I
systems irrelev an t
y(t) = (x(t) ∗ h1(t)) ∗ h2(t) = (x(t) ∗ h2(t)) ∗ h1(t)
≡
h2(t)
h2(t)
h1(t)
h1(t)
y(t)x(t)
x(t) y(t)
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
Dr. H. Nguyen Page 70
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
3 Associative property:
x[n] ∗ (h1[n] ∗ h2[n]) = (x[n] ∗ h1[n]) ∗ h2[n]
x(t) ∗ (h1(t) ∗ h2(t)) = (x(t) ∗ h1(t)) ∗ h2(t)
Implications:
– The order of convolution is not important.
– Cascade connection of two LTI system is represented by a single LTI system.
– Using both the associative and commutative properties ⇒ The order in a
cascade of LTI systems is irrelevant.
45
– Implication s:
∗ ou tpu t of system with impu lse respon se h(t) to in pu t x(t)
iden tical to ou tpu t of system with impu lse respon se x(t) to
in pu t h(t)
∗ Irrelev an t wh eth er h(t) or x(t) is refl ected an d sh ifted for
con volu tion
Distributive property
– C on tin u ou s– time case
x(t) ∗ (h1(t) + h2(t)) = x(t) ∗ h1(t) + x(t) ∗ h2(t)
– D iscrete– time case
x[n] ∗ (h1[n] + h2[n]) = x[n] ∗ h1[n] + x[n] ∗ h2[n]
– Implication
Parallel con n ection of two LT I systems represen ted by sin g le
eq u iv alen t LT I system
h2(t)
h1(t)
h1(t) + h2(t)
≡
y(t)x(t)
x(t) y(t)
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
46
A sso cia tive Property
– C on tin u ou s– time case
x(t) ∗ (h1(t) ∗ h2(t)) = (x(t) ∗ h1(t)) ∗ h2(t)
– D iscrete– time case
x[n] ∗ (h1[n] ∗ h2[n]) = (x[n] ∗ h1[n]) ∗ h2[n]
– C on seq u en tly
y[n] = x[n] ∗ h1[n] ∗ h2[n]
– Implication s
∗ C h ose con v en ien t order of con volu tion
∗ C ascade con n ection of two LT I system represen ted by sin g le
LT I system
h1(t) h2(t)
≡
h1(t) ∗ h2(t) y(t)x(t)
x(t) y(t)
A ssociativ e + commu tativ e property ⇒ order in a cascade of LT I
systems irrelev an t
y(t) = (x(t) ∗ h1(t)) ∗ h2(t) = (x(t) ∗ h2(t)) ∗ h1(t)
≡
h2(t)
h2(t)
h1(t)
h1(t)
y(t)x(t)
x(t) y(t)
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
45
– Implication s:
∗ ou tpu t of system with impu lse respon se h(t) to in pu t x(t)
iden tical to ou tpu t of system with impu lse respon se x(t) to
in pu t h(t)
∗ Irrelev an t wh eth er h(t) or x(t) is refl ected an d sh ifted for
con volu tion
Distributive property
– C on tin u ou s– time case
x(t) ∗ (h1(t) + h2(t)) = x(t) ∗ h1(t) + x(t) ∗ h2(t)
– D iscrete– time case
x[n] ∗ (h1[n] + h2[n]) = x[n] ∗ h1[n] + x[n] ∗ h2[n]
– Implication
Parallel con n ection of two LT I systems represen ted by sin g le
eq u iv alen t LT I system
h2(t)
h1(t)
h1(t) + h2(t)
≡
y(t)x(t)
x(t) y(t)
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
46
A sso cia tive Property
– C on tin u ou s– time case
x(t) ∗ (h1(t) ∗ h2(t)) = (x(t) ∗ h1(t)) ∗ h2(t)
– D iscrete– time case
x[n] ∗ (h1[n] ∗ h2[n]) = (x[n] ∗ h1[n]) ∗ h2[n]
– C on seq u en tly
y[n] = x[n] ∗ h1[n] ∗ h2[n]
– Implication s
∗ C h ose con v en ien t order of con volu tion
∗ C ascade con n ection of two LT I system represen ted by sin g le
LT I system
h1(t) h2(t)
≡
h1(t) ∗ h2(t) y(t)x(t)
x(t) y(t)
A ssociativ e + commu tativ e property ⇒ order in a cascade of LT I
systems irrelev an t
y(t) = (x(t) ∗ h1(t)) ∗ h2(t) = (x(t) ∗ h2(t)) ∗ h1(t)
≡
h2(t)
h2(t)
h1(t)
h1(t)
y(t)x(t)
x(t) y(t)
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
Dr. H. Nguyen Page 71
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Proofs (for continuous-time case only):
• Commutative property:
x(t) ∗ h(t) =
∫ ∞
−∞
x(τ)h(t− τ)dτλ=t−τ
= −∫ −∞
∞
x(t− λ)h(λ)dλ
=
∫ ∞
−∞
x(t− λ)h(λ)dλ = h(t) ∗ x(t)
• Distributive property:
x(t) ∗ (h1(t) + h2(t)) =
∫ ∞
−∞
x(τ)(h1(t− τ) + h2(t− τ))dτ
=
∫ ∞
−∞
x(τ)h1(t− τ)dτ +
∫ ∞
−∞
x(τ)h2(t− τ)dτ
= x(t) ∗ h1(t) + x(t) ∗ h2(t)
Dr. H. Nguyen Page 72
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• Associative property:
x(t) ∗ (h1(t) ∗ h2(t)) = x(t) ∗ (h2(t) ∗ h1(t))
= x(t) ∗∫ ∞
−∞
h2(τ)h1(t− τ)dτ
=
∫ ∞
−∞
x(λ)
∫ ∞
−∞
h2(τ)h1(t− λ− τ)dτdλ
=
∫ ∞
−∞
∫ ∞
−∞
x(λ)h1(t− λ− τ)dλh2(τ)dτ
=
[∫ ∞
−∞
x(λ)h1(t− λ)dλ
]
∗ h2(t)
= (x(t) ∗ h1(t)) ∗ h2(t)
Dr. H. Nguyen Page 73
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
LTI System’s Properties by the Impulse Response
• LTI systems with and without memory
– Recall that for memoryless systems, the output signal depends only on the
present value of the input signal.
– For discrete-time LTI systems:
y[n] =
∞∑
k=−∞
x[k]h[n− k]?= Kx[n], (K constant)
⇒ h[n] = Kδ[n] = h[0]δ[n]
– For continuous-time LTI systems:
y(t) =
∫ ∞
−∞
x(τ)h(t− τ)dτ?= Kx(t)
⇒ h(t) = Kδ(t) = h(0)δ(t)
– Identity systems (K = 1): x[n] = x[n] ∗ δ[n] and x(t) = x(t) ∗ δ(t).
Dr. H. Nguyen Page 74
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• Invertibility of LTI systems
– An Invertible LTI system has LTI inverse
– For discrete-time LTI systems:
h[n] ∗ hinv[n] = δ[n]
– For continuous-time LTI systems:
h(t) ∗ hinv(t) = δ(t)
Example: For an accumulator, h[n] = u[n], the output is:
y[n] =
∞∑
k=−∞
x[k]u[n− k] =
n∑
k=−∞
x[k]
Note that u[n]− u[n− 1] = h[n]− h[n− 1] = δ[n]
⇒ hinv[n] = δ[n]− δ[n− 1]
Dr. H. Nguyen Page 75
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• Causality of LTI systems
– Recall that for causal systems, the output at anytime depends on only past and
present values of the input.
– For discrete-time LTI systems:
y[n] =
∞∑
k=−∞
x[k]h[n− k] =
∞∑
k=−∞
h[k]x[n− k]?=
∞∑
k=0
h[k]x[n− k]
⇒ h[n] = 0, for n < 0
– Similarly, for continuous-time LTI system:
⇒ h(t) = 0, for t < 0
• Stability of LTI systems
– Recall that a system is stable if it produces bounded output for any bounded
input.
– For discrete-time LTI systems:
∗ Bounded input |x[n]| ≤ Bx, for all n
Dr. H. Nguyen Page 76
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
∗ Output:
|y[n]| =
∣∣∣∣∣
∞∑
k=−∞
h[k]x[n− k]
∣∣∣∣∣≤
∞∑
k=−∞
|h[k]||x[n− k]| ≤ Bx
∞∑
k=−∞
|h[k]|?≤ By
⇒ Sufficient condition for BIBO stability is that h[n] is absolutely summable:
∞∑
n=−∞
|h[n]| <∞
It can be shown that the above condition is also necessary.
– For continuous-time LTI systems, the sufficient and necessary condition for
BIBO stability is that h(t) is absolutely integrable:
∫ ∞
−∞
|h(τ)|dτ ≤ ∞
Example: For an integrator, h(t) = u(t), one has∫ ∞
−∞
|u(τ)|dτ =
∫ ∞
0
1dτ =∞
This system is therefore not stable.
Dr. H. Nguyen Page 77
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
LTI System’s Properties by the Impulse Response: Summary
Property Discrete-time LTI systems Continuous-time LTI systems
Memoryless h[n] = δ[n] h(t) = δ(t)
Invertibility h[n] ∗ hinv[n] = δ[n] h(t) ∗ hinv(t) = δ(t)
Causal h[n] = 0, for n < 0 h(t) = 0, for t < 0
Stability
∞∑
n=−∞
|h[n]| <∞∫
∞
−∞
|h(τ)|dτ <∞
Dr. H. Nguyen Page 78
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Fourier Representation of Signals and Systems
• The representation and analysis of LTI systems through the convolution
operation developed before are based on representing signals as linear
combinations of shifted impulses:
x[n] =
∞∑
k=−∞
x[k]δ[n− k]
x(t) =
∫ ∞
−∞
x(τ)δ(t− τ)dτ
• An alternative representation for signals and LTI systems is considered by using
complex exponentials as the basic signals. Such representations are known as
the continuous-time and discrete-time Fourier series and transform.
– Fourier series and transform can be used to construct broad and useful
classes of signals.
– They provide another convenient expression for the input-output relationship
of LTI systems.
– They allow for insightful characterization and analysis of signals and LTI
systems.
Dr. H. Nguyen Page 79
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
A Brief History
1748 Euler observed that if a vibrating string could be written as a sum of normal
modes, this expression is also true for future times.
1753 Bernoulli argued that all physical positions of a string could be written as this
type of sum.
1759 Lagrange criticized this representation based on the belief that it could not
represent signals with corners, so it was of limited use.
1807 Fourier claimed any periodic signal could be represented as a sum of sinusoids.
Many of his ideas were developed by others. Lacroix, Monge and Laplace were
in favor, but Lagrange fervently opposed.
1822 Fourier finally published a book.
1829 Dirichlet provided precise conditions under which periodic signals could be
represented.
1965 Fast Forier Transform (FFT) published independently by Cooley & Tukey.
1984 Grossman & Morlet introduced wavelets as a specialized field.
Dr. H. Nguyen Page 80
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Jean Baptiste Joseph Fourier (1768-1830)
Engineers need historical perspective. For background material on J.B.J. Fourier see,
for example,
www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fourier.html
Dr. H. Nguyen Page 81
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Fourier Series Representation of Periodic Signals
As mentioned before, it is advantageous in the study of LTI systems to represent
signals as linear combinations of basic signals that possess the following two
properties:
1. The set of basic signals can be used to construct a broad and useful class of
signals.
2. The response of an LTI system to each basic signal is simple enough in structure
so that the representation for the response of the system to any signal
constructed as a linear combination of the basic signals can be easily obtained.
Fourier representation and analysis are developed mainly from the fact that both of
the above properties are provided by the set of complex exponential signals (in
continuous and discrete time).
Recall that complex exponential signals are the signals of the form est in continuous
time and zn in discrete time, where s and z are complex numbers.
Dr. H. Nguyen Page 82
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Response of an LTI System to a Complex Exponential
• The importance of complex exponential signals in the study of LTI systems
stems from the fact that the response of an LTI system to a complex exponential
input is the same complex exponential with only a change in amplitude:
– Continuous-time: est −→ H(s)est
– Discrete-time: zn −→ H(z)zn
where the complex amplitude factor H(s) or H(z) is, in general, a function of
the complex variable s or z. As will be seen shortly, the amplitude factor H(s)
and H(z) are directly related to the impulse responses h(t) and h[n],
respectively.
• A signal for which the system output is a (possibly complex) constant times the
input is referred to as an eigenfunction of the system, and the amplitude factor
is referred to as the system’s eigenvalue.
• Thus, est and H(s) are the eigenfunction and eigenvalue of a continuous-time
LTI system. Similarly, zn and H(z) are the eigenfunction and eigenvalue of a
discrete-time LTI system.
Dr. H. Nguyen Page 83
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Derivation of the Continuous-Time Eigenvalue
For the input x(t) = est, the output can be found through the convolution integral:
y(t) = x(t) ∗ h(t) =
∫ +∞
−∞
x(t− τ)h(τ)dτ
=
∫ +∞
−∞
es(t−τ)h(τ)dτ = est
∫ +∞
−∞
h(τ)e−sτdτ
︸ ︷︷ ︸
H(s)
= H(s)est
where H(s)4=
∫ +∞
−∞
h(τ)e−sτdτ =
∫ +∞
−∞
h(t)e−stdt
is a complex constant whose value depends on the complex variable s and the
system’s impulse response h(t).
Fourier analysis only involves the variable s that is purely imaginary, i.e., s = jω.
This means that we consider only complex exponentials of the form ejωt. With this
restriction, one has:
ejωt −→[∫ +∞
−∞
h(t)e−jωtdt
]
ejωt = H(jω)ejωt
Dr. H. Nguyen Page 84
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Derivation of the Discrete-Time Eigenvalue
Similarly, the response of a discrete-time LTI system to a complex exponential input
can be computed through the convolution sum:
y[n] = x[n] ∗ h[n] =
+∞∑
k=−∞
x[n− k]h[k]
=
+∞∑
k=−∞
zn−kh[k] = zn+∞∑
k=−∞
h[k]z−k
︸ ︷︷ ︸
H(z)
= H(z)zn
where H(z)4=
+∞∑
k=−∞
h[k]z−k =
+∞∑
n=−∞
h[n]z−n is a complex constant whose value
depends on the complex variable z and the system’s impulse response h[n].
For the discrete-time case, Fourier analysis only involves variable z that has unit
magnitude, i.e., z = ejω. Thus we focus on complex exponentials of the form ejωn:
ejωn −→[
+∞∑
n=−∞
h[n]e−jωn
]
ejωn = H(ejω)ejωn
Dr. H. Nguyen Page 85
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Response to A Linear Combination of Complex Exponentials
By linearity and time-invariance properties of LTI systems, one easily obtains:
x(t) =∑
k
akeskt −→ y(t) =∑
k
ak
[H(sk)eskt
]
x[n] =∑
k
ak(zk)n −→ y[n] =∑
k
ak [H(zk)(zk)n]
• If the input to an LTI system can be expressed as a linear combination of
complex exponentials, then the output can also be represented as a linear
combination of the same complex exponential signals.
• Note that each coefficient in the representation of the output is obtained as the
product of the corresponding coefficient ak of the input and the system’s
eigenvalue H(sk) or H(zk) associated with the eigenfunction eskt or znk ,
respectively.
• But what types of signals can be represented in this form?
• Virtually all of the (periodic) signals that we are interested in!
• This is important and interesting idea.
Dr. H. Nguyen Page 86
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Fourier Series Representation of CT Periodic Signals
• Recall that a continuous-time signal x(t) is periodic if there exists a positive
constant T (T > 0) such that:
x(t + T ) = x(t) for all t
The fundamental period of x(t) is the minimum value of T for which the
above is satisfied. The fundamental period is often denoted as T0, while the
value ω0 = 2πT0
(rad/s) is referred to as the fundamental frequency.
• Note that when the fundamental frequency and/or the fundamental period is
clear from the context, the subscript 0 in ω0 and/or T0 might be dropped to
simplify notation.
• Introduced earlier are two basic periodic signals, namely the sinusoidal signal
x(t) = cos(ω0t) and the periodic complex exponential x(t) = ejω0t. Both of
these signals are periodic with fundamental frequency ω0 and fundamental
period T = 2π/ω0.
Dr. H. Nguyen Page 87
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• The set of harmonically related complex exponentials is:
φk(t) = ejkω0t = ejk(2π/T )t, k = 0,±1,±2, ...
where each of these signals has a fundamental frequency that is a multiple of ω0.
• The kth harmonic components φk(t) and φ−k(t) have a fundamental frequency
of |k|ω0 and a fundamental period of T/|k|.
• Observe that a linear combination of harmonically related exponentials is also
periodic with fundamental period T (i.e., fundamental frequency ω0 = 2π/T ):
x(t) =
∞∑
k=−∞
akejkω0t (1)
This is simply because each term in the above sum has one or more complete
cycles every T = 2πω0
seconds.
• The representation of a periodic signal as a linear combination of harmonically
related complex exponentials in the form of (1) is referred to as Fourier series
representation.
• The question is how to find the coefficients ak.
Dr. H. Nguyen Page 88
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Finding the CT Fourier Series Coefficients
Assume that the periodic signal x(t) with fundamental period T0 can be written as
x(t) =
+∞∑
k=−∞
akejkωt, where ω = 2π/T0.
Then: x(t)e−jnωt =+∞∑
k=−∞
akejkωte−jnωt
∫
T0
x(t)e−jnωtdt =
∫
T0
+∞∑
k=−∞
akej(k−n)ωtdt =
+∞∑
k=−∞
ak
∫
T0
ej(k−n)ωtdt
=
+∞∑
k=−∞
ak
∫
T0
[cos((k − n)ωt) + j sin((k − n)ωt)]dt = T0an
Thus an =1
T0
∫
T0
x(t)e−jnωtdt, or ak =1
T0
∫
T0
x(t)e−jkωtdt .
Remark : The notation∫
T0
implies that the integration is performed over any interval
of length T0.
Dr. H. Nguyen Page 89
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• To summarize, if a periodic signal x(t) has a Fourier series representation:
x(t) = x(t + T0) =
+∞∑
k=−∞
akejkωt (synthesis equation)
then the coefficients are given by
ak =1
T0
∫
T0
x(t)e−jkωtdt (analysis equation)
where ω = 2π/T0.
• The coefficients ak are called the spectral coefficients or Fourier series
coefficients of x(t).
• These complex coefficients measure the portions of the signal x(t) at each
harmonic of the fundamental component.
• The coefficient a0 is the dc (or constant) component of x(t), which is simply the
average value of x(t) over one period:
a0 =1
T0
∫
T0
x(t)dt
Dr. H. Nguyen Page 90
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Example: Consider the continuous-time periodic signal:
x(t) = 2 sin(2πt− 3) + sin(6πt) (2)
It is clear that the fundamental frequency is ω0 = 2π. Using Euler’s relation one has:
2 sin(2πt− 3) = 2 sin(ω0t− 3) =2
2j
(
ej(ω0t−3) − e−j(ω0t−3))
=e−3j
jejω0t − e3j
je−jω0t
sin(6πt) = sin(3ω0t) =1
2j
(ej3ω0t − e−j3ω0t
)=
1
2jej3ω0t − 1
2je−j3ω0t
Hence,
x(t) = − 1
2je−j3ω0t − e3j
je−jω0t +
e−3j
jejω0t +
1
2jej3ω0t (3)
Dr. H. Nguyen Page 91
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
From Equation (3), the nonzero Fourier series coefficients ak are identified to be:
a−3 = − 1
2j=
j
2=
1
2ej( π
2)
a−1 = −e3j
j= je3j = ej( π
2+3) = ej(3− 3π
2) = e−1.7124j
a1 =e−3j
j= −je−3j = ej[−( π
2+3)] = ej( 3π
2−3) = e1.7124j
a3 =1
2j= − j
2=
1
2ej(−π
2)
With ak = |ak|ejθk , then |ak| and θk are the magnitude and phase of ak,
respectively. Also, as a convention, the phase angles are always converted to the
range [−π, π] by adding (or subtracting) with multiples of 2π.
The magnitude and phase spectra of x(t) are sketched in Figure 3. Note that the
magnitude spectrum is even and the phase spectrum is odd. These are the common
properties for real-valued periodic signals (which will be discussed shortly).
Dr. H. Nguyen Page 92
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
−5 0 50
0.5
1Magnitude Spectrum
−5 0 5−2
−1
0
1
2Phase Spectrum
Normalized Frequency (ω/ω0)
Figure 3: Magnitude spectrum of x(t).
Dr. H. Nguyen Page 93
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Example: Consider the periodic square wave:
t
( )x t
0 1
1
t
( )x t
0
1
2
1 21−2−
t
( )x t
0
K
1T T 2T1T−T−2T−
a0 =1
T
∫ T/2
−T/2
x(t)dt = K2T1
T
ak =1
T
∫ T/2
−T/2
x(t)e−jkω0tdt =1
T
∫ T1
−T1
Ke−jkω0tdt
= − K
jkω0Te−jkω0t
∣∣∣∣
T1
−T1
=2K
kω0T
[ejkω0T1 − e−jkω0T1
2j
]
= Ksin(kω0T1)
kπ= K
2T1
Tsinc
(
k2T1
T
)
where ω0 = 2πT and sinc(x) = sin(πx)
πx .
Dr. H. Nguyen Page 94
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
−20 −15 −10 −5 0 5 10 15 20−0.5
0
0.5
T=4T1a k
−20 −15 −10 −5 0 5 10 15 20−0.5
0
0.5
T=8T1a k
−20 −15 −10 −5 0 5 10 15 20−0.2
0
0.2
T=16T1
a k
Normalized frequency (k=ω/ω0)
Figure 4: FS coefficients of the square wave for different ratios T/T1: Combined
magnitude and phase spectrum.
Dr. H. Nguyen Page 95
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
−20 −15 −10 −5 0 5 10 15 20−0.2
0
0.2
T=16T1
a k
−20 −15 −10 −5 0 5 10 15 200
0.1
0.2
|ak|
−20 −15 −10 −5 0 5 10 15 20−5
0
5
∠a k
Normalized frequency (k=ω/ω0)
Figure 5: FS coefficients of the square wave for T/T1 = 16: Combined magni-
tude/phase spectrum and separate magnitude and phase spectra.
Dr. H. Nguyen Page 96
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Properties of FS Coefficients for Real-Valued Signals
If x(t) is a real-valued periodic signal and can be represented as a Fourier series:
x(t) =
+∞∑
k=−∞
akejkωt ⇒ x∗(t) =+∞∑
k=−∞
a∗
ke−jkωt
Let l = −k, then one has
x∗(t) =
+∞∑
l=−∞
a∗
−lejlωt =
+∞∑
k=−∞
a∗
−kejkωt = x(t) =
+∞∑
k=−∞
akejkωt
Thus, by comparison of coefficients, it follows that
a∗
−k = ak, a−k = a∗
k
The above is known as the complex-conjugate symmetry of the Fourier series
coefficients of real-valued periodic signals. The complex-conjugate symmetry implies
that:
• The magnitude spectrum of the FS coefficients is even: |ak| = |a−k|.
• The phase spectrum of the FS coefficients is odd : ∠ak = −∠a−k.
Dr. H. Nguyen Page 97
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Alternative Forms for the FS of Real Periodic Signals
Since a−k = a∗k for real periodic signals, one can write:
x(t) =
∞∑
k=−∞
akejkωt = a0 +∞∑
k=1
akejkωt +∞∑
k=1
a−ke−jkωt
= a0 +
∞∑
k=1
[(akejkωt) + (akejkωt)∗
]
= a0 + 2
∞∑
k=1
Reakejkωt = a0 + 2
∞∑
k=1
Re|ak|ejθkejkωt
= a0 + 2
∞∑
k=1
|ak|Reej(kωt+θk)
= a0 + 2∞∑
k=1
Akcos(kωt + θk) (amplitude-phase form)
where Ak4= |ak| is the magnitude of ak and θk
4= ∠ak is the phase angle of ak.
The above representation is known as the amplitude-phase form of the Fourier series.
Dr. H. Nguyen Page 98
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Alternatively, one can also write:
x(t) =
∞∑
k=−∞
akejkωt
= a0 + 2
∞∑
k=1
Reakejkωt
= a0 + 2
∞∑
k=1
Reak cos(kωt) + jak sin(kωt)
= a0 + 2
∞∑
k=1
(Reak cos(kωt)− Imak sin(kωt))
= a0 + 2
∞∑
k=1
[Bk cos(kωt)− Ck sin(kωt)] (trigonometric form)
where Bk4= Reak and Ck
4= Imak.
The above is referred to as the trigonometric form of Fourier series.
Dr. H. Nguyen Page 99
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Alternative Forms of FS for Real Periodic Signals: Summary
Exponential: x(t) =∞∑
k=−∞
akejkωt
Amplitude-Phase: x(t) = a0 + 2
∞∑
k=1
Akcos(kωt + θk)
Trigonometric: x(t) = a0 + 2
∞∑
k=1
[Bk cos(kωt)− Ck sin(kωt)]
ak = Akejθk = Bk + jCk
Ak = |ak| =√
B2k + C2
k θk = ∠ak = tan−1
(Ck
Bk
)
Bk = Reak = Ak cos(θk) Ck = Imak = Ak sin(θk)
ak =1
T0
∫
T0
x(t)e−jkωtdt
Bk =1
T0
∫
T0
x(t) cos(kωt)dt, Ck = − 1
T0
∫
T0
x(t) sin(kωt)dt
Dr. H. Nguyen Page 100
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Convergence & Discontinuities
• It has been shown that if a periodic signal can be represented by a Fourier series
then one can compute the Fourier coefficients.
• It is not clear, however, if any periodic signal could be represented by a Fourier
series.
• In general, the Fourier series representation x(t) of a periodic signal x(t) might
not always equal to x(t) for all t:
x(t) =
∞∑
k=−∞
akejkωt, with ak =1
T0
∫
T0
x(t)e−jkωtdt
• In fact, if x(t) is discontinuous, x(t) and x(t) are not equal for all t. Intuitively,
this should make sense because how can a linear combination of continuous
signals (sinusoids) represent a discontinuous signal?
• The question is when a periodic signal x(t) does in fact have a Fourier series
representation, i.e., when the infinite FS converges to the original signal x(t)?
Dr. H. Nguyen Page 101
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• Consider the difference between the true signal and its FS representation:
e(t)4= x(t)− x(t)
• The Fourier series is said to converge if this error signal has zero energy over
one period:
E =
∫
T0
|e(t)|2dt = 0
• Note that, just because the zero error energy (E = 0), it does not imply
x(t) = x(t) for all t. It, however, does imply that any differences occur only at a
finite number of discrete (zero duration) points in time.
• In general, if t0 is a point of discontinuity, then:
x(t0) =1
2lim∆→0
[x(t0 + ∆) + x(t0 −∆)]
• At all other points the two signals x(t) and x(t) are equal.
• A sufficient condition for convergence is that the signal has a finite energy over a
single period:∫
T0
|x(t)|2dt <∞
Dr. H. Nguyen Page 102
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
– The above is true of all signals you could generate in the lab. Thus all of the
periodic signals generated by a function generator have equivalent FS
representations.
– If x(t) is a continuous signal, then it is safe to assume that x(t) = x(t). Note
that this is a stronger statement than merely stating that the FS converges.
Dirichlet Conditions for Convergence: An alternative sets of conditions developed by
Dirichlet guarantees that x(t) equals its FS representation, except at isolated values
of t for which x(t) is discontinuous. The Dirichlet conditions are as follows:
1. The signal x(t) must be absolutely integrable over any period:
∫
T0
|x(t)|dt <∞.
2. There must be a finite number of distinct maxima and minima during any single
period T0 of the signal.
3. In any finite interval of time, there are only finite number of discontinuities.
Furthermore, each of these discontinuities is finite.
• Figure 3.8 in textbook gives examples of signals that do not satisfy Dirichlet
condition(s).
• Note that the above are sufficient, but not necessary, conditions.
Dr. H. Nguyen Page 103
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Gibbs Phenomenon
To gain some understanding of how the Fourier series converges for a periodic signal
with discontinuities, consider approximating a given periodic signal x(t) by the finite
Fourier series (with 2N + 1 terms):
xN (t) =
N∑
k=−N
akejkωt
For discontinuous signals x(t), it is observed that:
• There are ripples at the vicinity of the discontinuity.
• Gibbs showed that the peak amplitude of these ripples does not decrease with
increasing N .
• Specifically, there is an overshoot of 9% of the height of the discontinuity, no
matter how large N becomes.
• In fact, as N increases, the ripples in the finite Fourier series approximation
become compressed toward the discontinuity, but for any finite value of N , the
peak amplitude of the ripples remain constant.
The above behavior is known as the Gibbs phenomenon.
Dr. H. Nguyen Page 104
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Example: Fourier series approximations of the periodic square wave.
−2 −1 0 1 2−0.5
0
0.5
1
1.5Fourier Series Approximation (N=5)
t (sec)−2 −1 0 1 2
−0.5
0
0.5
1
1.5Fourier Series Approximation (N=10)
t (sec)
−2 −1 0 1 2−0.5
0
0.5
1
1.5Fourier Series Approximation (N=25)
t (sec)−2 −1 0 1 2
−0.5
0
0.5
1
1.5Fourier Series Approximation (N=100)
t (sec)
Dr. H. Nguyen Page 105
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Example: Consider the continuous-time periodic waveform shown below:
t
( )x t
0 1
1
t
( )x t
0
1
2
1 21−2−
(a) Obviously, the fundamental period of x(t) is T0 = 2 and the fundamental
frequency is ω0 = 2πT0
= π.
(b) Next, we compute the trigonometric Fourier series coefficients Bk and Ck. To
this end, consider x(t) in one period, from 0 ≤ t ≤ 2. Then
x(t) =t
2, 0 ≤ t ≤ 2
Hence,
Bk =1
T0
∫
T0
x(t) cos(kω0t)dt =1
T0
∫
T0
t cos(kω0t)dt = 0
Dr. H. Nguyen Page 106
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
since t cos(kω0t) is an odd function. Similarly
Ck = − 1
T0
∫
T0
x(t) sin(kω0t)dt = −1
4
∫ 2
0
t sin(ω0kt)dt
= −1
4
[sin(ω0kt)
(ω0k)2− t
ω0kcos(ω0kt)
]∣∣∣∣
2
0
= −1
4× −2
πk=
1
2πk
The DC component of x(t) is simply
a0 =1
T0
∫
T0
x(t)dt =1
2
∫ 2
0
t
2dt =
t2
8
∣∣∣∣
2
0
=1
2
(c) The magnitude and phase spectrum for the FS coefficients of x(t) are plotted in
Figure 6. Observe that, since x(t) is a real function, it has an even amplitude
spectrum and an odd phase spectrum.
Dr. H. Nguyen Page 107
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
−5 0 5
0
0.2
0.4
0.6Magnitude Spectrum
−5 0 5−2
−1
0
1
2Phase Spectrum
Normalized Frequency (ω/ω0)
Figure 6: Magnitude and phase spectra of x(t).
Dr. H. Nguyen Page 108
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
(d) The following partial Fourier series approximation
x(t) = a0 + 2N∑
k=1
[Bk cos(kω0t)− Ck sin(kω0t)] of x(t) is also plotted below for
N = 5, 10, 50. Note the Gibbs phenomenon at the points of discontinuity.
−3 −2 −1 0 1 2 3
0
0.5
1N=5
−3 −2 −1 0 1 2 3
0
0.5
1N=10
−3 −2 −1 0 1 2 3
0
0.5
1N=50
t (sec)
Dr. H. Nguyen Page 109
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Properties of CT Fourier Series
The Fourier series representation possesses a number of useful properties. Many
properties are listed in Table 3.1 of textbook (p. 206). Though we only discuss a few
of these properties in class, you should be familiar with all of them.
1. Linearity : If x1(t) and x2(t) are periodic signals with period T and they have
Fourier series representations:
x1(t)FS←→ ak, x2(t)
FS←→ bk
then
y(t) = α1x1(t) + α2x2(t)FS←→ α1ak + α2bk
2. Time Shifting : If x(t) is a periodic signal with fundamental period T that has
Fourier series coefficients ak, then
y(t) = x(t− t0)FS←→ bk = e−jkωt0ak = e−jk(2π/T )t0ak
Observe that time-shifting only changes the phase spectrum, not the magnitude
spectrum: |bk| = |ak|.
Dr. H. Nguyen Page 110
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
3. Time Reversal : If x(t) is a periodic signal with fundamental period T that has
Fourier series coefficients ak, then
y(t) = x(−t)FS←→ bk = a−k
It follows from the above relationship that:
• If x(t) is even, then bk = a−k = ak, i.e., the FS coefficients are even.
• If x(t) is odd, then bk = a−k = −ak, i.e., the FS coefficients are odd.
Furthermore, if x(t) is a real-valued function, then combining the above
time-reversal property and the complex-conjugate symmetry (a−k = a∗k)
discussed before leads to the following conclusions:
• If x(t) is real and even, then a−k = ak = a∗k. This means the FS coefficients
are real and even. Furthermore, the phases of ak can only be 0, π or −π.
• If x(t) is real and odd, then bk = a−k = −ak = a∗k. This implies that the FS
coefficients are purely imaginary and odd. Furthermore, the phase of ak can
only be 0, π/2 or −π/2.
Dr. H. Nguyen Page 111
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
4. Multiplication: If x1(t) and x2(t) are periodic signals with period T and they
have Fourier series representations:
x1(t)FS←→ ak, x2(t)
FS←→ bk
then
y(t) = x1(t)x2(t)FS←→ ck =
∞∑
l=−∞
albk−l = ak ∗ bk
Observe that
∞∑
l=−∞
albk−l is precisely the discrete-time convolution sum of two
sequences ak and bk.
5. Conjugation and Conjugate Symmetry : If x(t) is a periodic signal with
fundamental period T that has Fourier series coefficients ak, then
y(t) = x∗(t)FS←→ bk = a∗
−k
It follows from the above relationship that:
• If x(t) is real, i.e., x(t) = x∗(t), then bk = a∗
−k = ak ⇒ the FS coefficients
are conjugate symmetric.
Dr. H. Nguyen Page 112
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• If x(t) is purely imaginary, i.e., x(t) = −x∗(t) then bk = a∗−k = −ak.
6. Parseval’s Relation: If the signal x(t) is a periodic signal with fundamental
period T that has Fourier coefficients ak, then its average power can be
computed as follows:1
T
∫
T
|x(t)|2dt =∞∑
k=−∞
|ak|2
Observe that the average power of the kth harmonic component is
1
T
∫
T
∣∣akejkωt
∣∣2dt =
1
T
∫
T
|ak|2dt = |ak|2
Thus Parseval’s relation simply states that the average total power of the signal
is the sum of the average powers in all of its harmonic components.
7. FS coefficients of symmetric signals: Recall that any signal x(t) can be written
as a sum of even and odd signals: x(t) = xe(t) + xo(t), where
xe(t) = x(t)+x(−t)2 and xo(t) = x(t)−x(−t)
2 . Consider the trigonometric form of
the Fourier series of a real periodic signal x(t):
x(t) = a0 + 2
∞∑
k=1
[Bk cos(kωt)− Ck sin(kωt)]
Dr. H. Nguyen Page 113
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
The coefficients Bk and Ck are determined by:
Bk =1
T0
∫
T0
x(t) cos(kωt)dt
=1
T0
∫ +T0/2
−T0/2
xe(t) cos(kωt)dt +1
T0
∫ +T0/2
−T0/2
xo(t) cos(kωt)dt
=1
T0
∫ +T0/2
−T0/2
xe(t) cos(kωt)dt =2
T0
∫ +T0/2
0
xe(t) cos(kωt)dt
and
Ck = − 1
T0
∫
T0
x(t) sin(kωt)dt
= − 1
T0
∫ +T0/2
−T0/2
xe(t) sin(kωt)dt− 1
T0
∫ +T0/2
−T0/2
xo(t) sin(kωt)dt
= − 1
T0
∫ +T0/2
−T0/2
xo(t) sin(kωt)dt = − 2
T0
∫ +T0/2
0
xo(t) sin(kωt)dt
It follows that:
• If x(t) is even, then xo(t) = 0 and therefore Ck = 0 and ak = Bk (real).
• If x(t) is odd, then xe(t) = 0 and hence Bk = 0 and ak = jCk (imaginary).
Dr. H. Nguyen Page 114
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Fourier Series Representation of Discrete-Time Periodic Signals
• Recall that the set of all discrete-time complex exponential signals that are
periodic with period N is given by
φk[n] = ejkωn = ejk 2π
Nn, k = 0,±1,±2, ...
where ejωn is called the fundamental component, ej2ωn is called the 2nd
harmonic component, and, in general, ejkωn is called the kth harmonic
component.
• Unlike continuous-time exponentials, there are only N distinct harmonics. This
is because:
φk+lN [n] = ej(k+lN) 2π
Nn = ejk 2π
Nn+jl2πn = ejk 2π
Nn + ejl2πn = ejk 2π
Nn = φk[n]
• Wish to represent general DT periodic signals in terms of linear combinations of
harmonically related exponentials φk[n]:
x[n] =∑
k
akφk[n] =∑
k
akejkωn =∑
k
akejk 2π
Nn (4)
• The above representation yields a signal x[n] that is periodic with fundamental
period N .
Dr. H. Nguyen Page 115
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• Since the signals (sequences) φk[n] are distinct only over a range of N
successive values of k, the summation in (4) needs only include terms in this
range. We use the notation < N > to indicate any range of k that contains N
successive integers. Thus the representation in (4) is rewritten as:
x[n] =∑
k=<N>
akφk[n] =∑
k=<N>
akejkωn =∑
k=<N>
akejk 2π
Nn (5)
• Equation (5) is referred to as the discrete-time Fourier series and the
coefficients ak as the Fourier series coefficients.
• Any finite-valued discrete-time periodic signal can be written exactly in this
form.
• The task now is to find the coefficients ak.
• In our derivation of the CT Fourier series coefficients we used the following
relation:∫
T
ejkωtdt =
∫
T
ejk(2π/T )tdt =
T k = 0,
0 k 6= 0
Dr. H. Nguyen Page 116
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
To solve for the DTFS coefficients, we need a similar relation:
∑
n=<N>
ejk(2π/N)n =
N k = lN,
0 k 6= lNfor any integer l.
The above essentially states that the sum over one period of the values of a
periodic complex exponential is zero, unless that complex exponential is a
constant.
• Now, since x[n] =∑
k=<N>
akejkωn, one has
∑
n=<N>
x[n]e−jlωn =∑
n=<N>
(∑
k=<N>
akejkωn
)
e−jlωn
=∑
k=<N>
ak
∑
n=<N>
ej(k−l)ωn = Nal
Thus al =1
N
∑
n=<N>
x[n]e−jlωn, or ak =1
N
∑
n=<N>
x[n]e−jkωn .
Dr. H. Nguyen Page 117
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• To summarize, the Fourier series representation of a discrete-time periodic signal
x[n] with fundamental period N is given by:
x[n] =∑
k=<N>
akejkωn =∑
k=<N>
akejk(2π/N)n
where ak =1
N
∑
n=<N>
x[n]e−jkωn =1
N
∑
n=<N>
x[n]e−jk(2π/N)n
– The above two equations are known as the discrete-time Fourier series pair.
The first equation is called the synthesis equation, whereas the second
equation is called the analysis equation.
– The coefficients ak are called the spectral coefficients or the Fourier series
coefficients of x[n].
• Since φk[n] = φk+Nl[n] (there are only N distinct discrete-time exponential
harmonics) and the DTFS sum is over any N consecutive terms, one has:
x[n] = a0φ0[n] + a1φ1[n] + . . . + aN−1φN−1[n]
= aNlφ0[n] + a1+Nlφ1[n] + . . . + a(N−1)+NlφN−1[n]
By comparing terms by terms, it can be concluded that ak = ak+Nl , i.e., the
values of ak repeat periodically with period N .
Dr. H. Nguyen Page 118
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Example: Consider the following discrete-time periodic square wave with period N :
83
Short-han d n otation
x[n]FS←→ ak
D en omin ation :
– ak: Fou rier series coeffi cien ts or spectral coeffi cien ts
D iff eren ces to con tin u ou s-time case
– D iscrete-time Fou rier series is finite
(R ecall from Section 1 .2 .2 : T here are on ly N distin ct discrete–
time complex expon en tial sig n als φk[n] = ejk(2π/N)n that are
periodic with period N (harmon ically related sig n als).)
– N o mathematical issu es with con v erg en ce — discrete–time Fou rier
series represen tation alway s exists
– ak = ak+N sin ce φk[n] = φk+N [n]
Remark :
T he set of coeffi cien ts
ak =1
N
N−1∑
n=0
x[n]e−jk(2π/N)n
is common ly referred to as the N -poin t d iscrete Fo u rier tra nsfo rm
(D FT ) of a fi n ite du ration sig n al x[n] with x[n] = 0 ou tside the
in terv al 0 ≤ n ≤ N . D u e to the existen ce of an extremely fast
alg orithm for the calcu lation of the D FT , called the fa st Fo u rier
tra nsfo rm (FFT ), the D FT (FFT ) is of u tmost importan ce in dig ital
sig n al processin g .
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
84
E x amp le:
– D iscrete–time periodic sq u are wave with period N
N1
1
N−N1 n
x[n]
– Fou rier series coeffi cien ts
ak =1
N
∑
n=〈N〉
x[n]e−jk(2π/N)n =1
N
N1∑
n=−N1
e−jk(2π/N)n
=
1
N
sin (2πk(N1 + 1/2)/N)
sin (πk/N), k 6= 0, ±N, ±2N, . . .
2N1 + 1
N, k = 0, ±N, ±2N, . . .
ak−→
k −→
– R emark : x[n] correspon ds to a finite n u mber of Fou rier coeffi -
cien ts ⇒ n o Gibbs phen omen on , n o con v erg en ce issu es
Lampe , S c h o b e r: S ig n als an d C ommu n icatio n s
Dr. H. Nguyen Page 119
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Example: Partial Fourier series for a discrete-time periodic square wave:
−15 −10 −5 0 5 10 15−1
−0.5
0
0.5
1
1.5Fourier Series Approximation (N=1)
t (sec)−15 −10 −5 0 5 10 15−1
−0.5
0
0.5
1
1.5Fourier Series Approximation (N=3)
t (sec)
−15 −10 −5 0 5 10 15−1
−0.5
0
0.5
1
1.5Fourier Series Approximation (N=4)
t (sec)−15 −10 −5 0 5 10 15−1
−0.5
0
0.5
1
1.5Fourier Series Approximation (N=5)
t (sec)
Dr. H. Nguyen Page 120
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Example: A discrete-time periodic signal x[n] is real-valued and has a fundamental
period N = 5 (i.e., the fundamental frequency is ω0 = 2πN ). The nonzero Fourier
series coefficients for x[n] are:
a0 = 2, a2 = a∗
−2 = 2ejπ/6, a4 = a∗
−4 = 2ejπ/3
Express x[n] in the form:
x[n] = α0 +
∞∑
k=1
αk cos(ωkn + φk)
Solution: We know that x[n] can be represented by a linear combination of the set
of harmonically-related complex exponentials:
x[n] =∑
<N>
akejkω0n
Dr. H. Nguyen Page 121
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Consider one period of ak with 0 ≤ k ≤ 4, then
a0 = 2
a1 = a1−N = a−4 = 2e−jπ/3
a2 = 2ejπ/6
a3 = a3−N = a−2 = 2e−jπ/6
a4 = 2ejπ/3
Hence,
x[n] =∑
<N>
akejkω0n =
4∑
k=0
akejk(2π/5)n
= 2 +(
a1ej(2π/5)n + a4e
j(8π/5)n)
+(
a2ej(4π/5)n + a3e
j(6π/5)n)
= 2 +(
a1ej(2π/5)n + a4e
−j(2π/5)n)
+(
a2ej(4π/5)n + a3e
−j(4π/5)n)
= 2 + 2(
ej[(2π/5)n−π/3] + e−j[(2π/5)n−π/3])
+2(
ej[(4π/5)n+π/6] + e−j[(4π/5)n+π/6])
= 2 + 4 cos(
2π
5n− π/3
)
+ 4 cos(
4π
5n + π/6
)
Dr. H. Nguyen Page 122
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Compare and Contrast DTFS with CTFS
x(t) =
+∞∑
k=−∞
akejkωt, ak =1
T
∫
T
x(t)e−jkωtdt, (where ω =2π
T)
x[n] =∑
k=<N>
akejkωn, ak =1
N
∑
n=<N>
x[n]e−jkωn, (where ω =2π
N)
Unlike the continuous-time Fourier series (CTFS):
• There are only N terms in the sum of the discrete-time Fourier series (DTFS)
(since there are only N distinct DT exponential harmonics)
• The finite DTFS sum can be obtained using any N consecutive Fourier series
coefficients ak.
• The DTFS coefficients ak form a discrete-time periodic signal with the same
fundamental period N as that of x[n].
• The DTFS always converges for all periodic signals such that |x[n]| <∞
• There is no Gibbs phenomenon with DTFS.
Dr. H. Nguyen Page 123
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
DTFS Coefficients and The Fast Fourier Transform (FFT)
ak =1
N
∑
n=<N>
x[n]e−jk 2π
Nn
︸ ︷︷ ︸
DTFS coefficients
, X(k) =N−1∑
n=0
x[n]e−jk 2π
Nn
︸ ︷︷ ︸
discrete Fourier transform (DFT)
• X(k) defined as above is commonly referred to as the N -point discrete Fourier
transform (DFT) of a finite duration signal x[n], where x[n] = 0 outside the
interval 0 ≤ n ≤ N − 1.
• Observe from the above two equations that the DFT of one period of x[n] is
proportional to the Fourier series coefficients: X(k) = Nak
• The Fast Fourier Transform (FFT) is just a fast algorithm for the calculation of
DFT:
– The direct approach to find each of the N coefficients ak would require N2
computations.
– The FFT enables this to be solved using N · log2 N computations.
• FFT was developed by Tukey and Cooley in 1965.
Dr. H. Nguyen Page 124
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Properties of Discrete-Time Fourier Series (DTFS)
• Many properties of DTFS are summarized in Table 3.2 in the textbook (page
221). You should be familiar with all of them.
• There are strong similarities between the properties of the discrete-time and
continuous-time Fourier series .
• As examples, several properties are listed below.
– Conjugate Symmetry for Real Signals: If x[n] is real, then a−k = a∗
k .
– Multiplication: If x1[n] and x2[n] are periodic signals with period N and they
have Fourier series representations:
x1[n]FS←→ ak, x2[n]
FS←→ bk
then y[n] = x1[n]x2[n] is periodic with period N and
y[n] = x1[n]x2[n]FS←→ ck =
∑
l=<N>
albk−l
The sum∑
l=<N>
albk−l is known as the periodic convolution sum between
the two periodic sequences ak and bk.
Dr. H. Nguyen Page 125
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
– Parseval’s Relation: If the signal x[n] is a periodic signal with fundamental
period N and it has FS representation x[n]FS←→ ak, then its average power
can be computed as follows:
1
N
∑
n=<N>
|x[n]|2 =∑
k=<N>
|ak|2
Observe that the average power of the kth harmonic component is
1
N
∑
n=<N>
∣∣∣akejk(2π/N)n
∣∣∣
2
= |ak|2
Thus Parseval’s relation simply states that the average total power of the
signal is the sum of the average powers in all of its harmonic components.
Dr. H. Nguyen Page 126
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Fourier Series and LTI Systems
x(t) - h(t) - y(t) x[n] - h[n] - y[n]
• Let x(t) and x[n] be continuous-time and discrete-time periodic signals at the
inputs of LTI systems with impulse responses h(t) and h[n], respectively. Then
the outputs of the LTI systems can be easily found as follows:
x(t) =
+∞∑
k=−∞
akejkωt −→ y(t) =+∞∑
k=−∞
akH(jkω)ejkωt
x[n] =∑
k=<N>
akejkωn −→ y[n] =∑
k=<N>
akH(ejkω)ejkωn
where H(jω) and H(ejω) are called the frequency responses of continuous-time
and discrete-time LTI systems, respectively. They are defined as:
H(jω) =
∫ ∞
−∞
h(t)e−jωtdt, and H(ejω) =
∞∑
n=−∞
h[n]e−jωn
Dr. H. Nguyen Page 127
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• Observations:
x(t) =
+∞∑
k=−∞
akejkωt −→ y(t) =+∞∑
k=−∞
akH(jkω)ejkωt
x[n] =∑
k=<N>
akejkωn −→ y[n] =∑
k=<N>
akH(ejkω)ejkωn
– The outputs y(t) and y[n] are also periodic with the same fundamental
frequencies as x(t) and x[n], respectively.
– If the set ak is the set of Fourier series coefficients for the input x(t), then
akH(jkω) is the set of the Fourier series coefficients for the output y(t):
x(t)FS←→ ak −→ y(t)
FS←→ akH(jkω)
Thus, the effect of the LTI system is to modify individually each of the
Fourier series coefficients of the input through multiplication by the value of
the frequency response at the corresponding frequency.
– For discrete-time case, the relationship between the Fourier series coefficients
of the input and output of an LTI system is exactly the same:
x[n]FS←→ ak −→ y[n]
FS←→ akH(
ejk(2π/N))
Dr. H. Nguyen Page 128
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Continuous-Time Fourier Transform: Overview
• Fourier series enables us to represent periodic signals as linear combinations of
harmonically related complex exponentials. Such representation can be used to
clearly describe the effect of LTI systems on periodic signals.
• This chapter extends such representation to continuous-time signals that are
not periodic (i.e., aperiodic signals).
• We will see that, a rather large class of signals, including all energy signals, can
also be represented through a linear combination of complex exponentials.
• The main difference compared to FS representations of periodic signals is that
the complex exponentials in the representation of aperiodic signals are
infinitesimally close in frequency ⇒ The representation in terms of linear
combination takes the form of an integral rather than the sum.
• In the FS representation of a periodic signal, as the period increases, the
fundamental frequency decreases and the harmonically related components
become closer in frequency.
• Fourier reasoned that any aperiodic signal can be viewed as a periodic signal with
an infinite period. As the period becomes infinite, the frequency components
form a continuum and the FS sum becomes an integral (i.e., Fourier transform).
Dr. H. Nguyen Page 129
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Derivation of Fourier Transform
• Consider an aperiodic signal x(t) that is of finite duration: x(t) = 0 if |t| > T1.
• Form a periodic signal x(t), with a period T , for which x(t) is one period.
• As T →∞ (or ω0 = 2πT → 0), x(t) is equal to x(t) for any finite value of t.
x(t)
t
…
t
1T1T− 0
1T1T− TT−T2− T2
( ) tjjX ωω e
( ) tjkjkX 0e0ωω
( ) 000eArea ωω ω tjkjkX=
0ωk0)1( ω+k
ω
0
0
)(~ tx
…
• We know that x(t) can be represented with a Fourier series:
x(t) =
+∞∑
k=−∞
akejkω0t, ak =1
T
∫ T/2
−T/2
x(t)e−jkω0tdt
Dr. H. Nguyen Page 130
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
The FS coefficients of x(t) is:
ak =1
T
∫ T/2
−T/2
x(t)e−jkω0tdt =1
T
∫ +∞
−∞
x(t)e−jkω0tdt =1
TX(jkω0)
where X(jω) =
∫ ∞
−∞
x(t)e−jωtdt is the envelope of Tak. Alternatively, Tak can be
obtained as the sample X(jω)|ω=kω0of the envelope function X(jω). Now:
x(t) =
+∞∑
k=−∞
1
TX(jkω0)e
jkω0t =1
2π
+∞∑
k=−∞
X(jkω0)ejkω0tω0
⇒ x(t) = limT→∞
x(t) = limω0→0
[
1
2π
+∞∑
k=−∞
X(jkω0)ejkω0tω0
]
=1
2π
∫ +∞
−∞
X(jω)ejωtdω
x(t)
t
…
t
1T1T− 0
1T1T− TT−T2− T2
( ) tjjX ωω e
( ) tjkjkX 0e0ωω
( ) 000eArea ωω ω tjkjkX=
0ωk0)1( ω+k
ω
0
0
)(~ tx
…
Dr. H. Nguyen Page 131
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Fourier Transform: Summary
• Fourier transform pair :
Fx(t) = X(jω)4=
∫ +∞
−∞
x(t)e−jωtdt : Fourier transform
(Fourier integral, spectrum)
F−1X(jω) = x(t)4=
1
2π
∫ +∞
−∞
X(jw)ejωtdω : inverse Fourier transform
• The synthesis equation represents an aperiodic signal as a linear combination of
complex exponentials.
• For a periodic signal, the complex exponentials in its FS respresentation have
amplitudes ak occurring at a discrete set of harmonically related frequencies
kω0, k = 0,±1,±2, . . .
• For aperiodic signals, the complex exponentials occur at a continuum of
frequencies and have “amplitude” X(jω)(dω/2π). Thus X(jω) should be
interpreted as the amplitude density
• X(jω) provides information for describing x(t) as a linear combination of
sinusoidal signals at different frequencies.
Dr. H. Nguyen Page 132
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Conditions for Convergence of Fourier Transforms
The Fourier transform of a signal x(t) exists if∫ +∞
−∞
|x(t)|2dt <∞
and any discontinuities are finite
• The above is true for all signals of finite amplitude and duration.
• Convergence does not imply that the inverse Fourier transform x(t) will be equal
to the original signal x(t) for all values of t.
• However, convergence implies that, although x(t) and x(t) may differ
significantly at individual values of t, there is no energy in their difference (i.e.,
the energy of the difference signal e(t) = x(t)− x(t) is zero.
• An alternative set of sufficient conditions (Dirichlet conditions) for the
convergence of Fourier transforms is stated in the textbook (page 290).
• Does a periodic signal have a Fourier transform? The answer is No, but we can
find one if we allow X(jω) to be expressed in terms of impulse functions ⇒ The
Fourier series and Fourier transform can be incorporated into a common
framework (convenient).
Dr. H. Nguyen Page 133
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Example: Transform of a Rectangular Pulse
x(t) =
1 |t| < T1
0 otherwise
FT⇐⇒ X(jω) =
∫ T1
−T1
e−jωtdt =2 sin(ωT1)
ω= 2T1sinc
(ωT1
π
)
−5 0 50
0.5
1x(
t)
t (sec)−20 −10 0 10 20
−0.5
0
0.5
1
T1=0.5
X(j
ω)
ω (rad/sec)
−5 0 50
0.5
1
x(t)
t (sec)−20 −10 0 10 20
−1
0
1
2
T1=1
X(j
ω)
ω (rad/sec)
−5 0 50
0.5
1
x(t)
t (sec)−20 −10 0 10 20
−5
0
5
10
T1=5
X(j
ω)
ω (rad/sec)
If the signal stretches in time, its spectrum compresses in frequency.
Dr. H. Nguyen Page 134
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Sinc Function
• The sinc function arises frequently in Fourier analysis: sinc(θ)4=
sin(πθ)
πθ
• sinc(0) = 1, sinc(n) = 0 if n is a nonzero integer.
•∫ ∞
−∞
sinc(t)dt = 1
−8 −6 −4 −2 0 2 4 6 8−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
sinc
(θ)
θ
Dr. H. Nguyen Page 135
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
More Examples
• Fourier transform of the unit impulse δ(t):
X(jω) =
∫ ∞
−∞
δ(t)e−jωtdt =
∫ ∞
−∞
δ(t)e−jω0dt =
∫ ∞
−∞
δ(t)dt = 1
The unit impulse has a FT consisting of equal contributions at all
frequencies.
Remark: In class, the above result was obtained by considering the FT of a
rectangular pulse over [−T1, T1] and with amplitude 12T1
, and then let T1 → 0.
• Fourier transform of a constant: Let x(t) = 1. The Fourier transform of x(t)
can be found as a limit of the FT of the rectangular pulse when T1 →∞.
Observe that the function 2T1sinc(
ωT1
π
)approaches an impulse at ω = 0 as
T1 →∞. Specifically, since∫ ∞
−∞
2T1sinc
(ωT1
π
)
dω = 2π, then
x(t) = 1FT⇐⇒ X(jω) = 2πδ(ω)
The above result is intuitively satisfying since x(t) only contains a DC
component at ω = 0.
Dr. H. Nguyen Page 136
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• Transform of a decaying exponential : Let x(t) = e−atu(t) where a > 0. Then
the Fourier transform of x(t) is
X(jω) =
∫ ∞
0
e−ate−jωtdt = − 1
a + jωe−(a+jω)t
∣∣∣∣
∞
0
=1
a + jω, a > 0
The magnitude and phase spectra of x(t) are:
|X(jω)| = 1√a2 + ω2
, ∠X(jω) = − tan−1(ω
a
)
−25 −20 −15 −10 −5 0 5 10 15 20 250
0.2
0.4
0.6
0.8
1
(a=1)
Mag
nitu
de o
f X
(jω
)
ω (rad/sec)
−25 −20 −15 −10 −5 0 5 10 15 20 25−2
−1
0
1
2
(a=1)
Phas
e of
X(j
ω)
ω (rad/sec)
Dr. H. Nguyen Page 137
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Inverse Fourier Transform of a Rectangular Pulse
X(jω) =
1 |w| < W
0 otherwise
FT⇐⇒ x(t) =1
2π
∫ W
−W
ejωtdω =sin(Wt)
πt
−10 −5 0 5 10
0
0.2
0.4
0.6
0.8
1X
(jω
)
ω−10 −5 0 5 10
−0.1
0
0.1
0.2
0.3
0.4
x(t)
t (sec)
−10 −5 0 5 10
0
0.2
0.4
0.6
0.8
1
X(j
ω)
ω−10 −5 0 5 10
−0.2
0
0.2
0.4
0.6
0.8
x(t)
t (sec)
If the signal’s spectrum compresses in frequency, the signal stretches in time.
Dr. H. Nguyen Page 138
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Fourier Transforms of Periodic Signals
• Consider a signal x(t) with Fourier transform X(jω) = 2πδ(ω − ω0). Then x(t)
can be found by performing the inverse Fourier transform:
x(t) =1
2π
∫ ∞
−∞
2πδ(ω − ω0)ejωtdω = ejω0t
• The above implies the following Fourier transform pair:
ejω0t FT⇐⇒ 2πδ(ω − ω0)
• Now, if x(t) is a periodic signal, it has a Fourier series representation:
x(t) =
∞∑
k=−∞
akejkω0t
By the linearity of integral, it is straightforward to verify that
x(t) =∞∑
k=−∞
akejkω0t FT⇐⇒ X(jω) =∞∑
k=−∞
2πakδ(ω − kω0)
Dr. H. Nguyen Page 139
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Properies of Fourier Transform
Linearity: If x1(t)FT⇐⇒ X1(jω) and x2(t)
FT⇐⇒ X2(jω), then:
a1x1(t) + a2x2(t)FT⇐⇒ a1X1(jω) + a2X2(jω)
This property follows directly from the linearity of integrals.
Time-Shifting:
x(t− t0)FT⇐⇒ e−jωt0X(jω)
Note that:
|X(jω)e−jωt0 | = |X(jω)|∠X(jω)e−jωt0 = ∠X(jω)− ωt0
• Thus, a shift in time does not affect the magnitude of the Fourier transform.
• The effect of a time shift is to introduce a phase shift −ωt0, which is a linear
function of the frequency ω.
Dr. H. Nguyen Page 140
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Conjugation and Conjugate Symmetry:
x∗(t)FT⇐⇒ X∗(−jω)
• If x(t) is real, then
x(t) = x∗(t)⇒ X(jω) = X∗(−jω)
– The magnitude spectrum is even: |X(jω)| = |X(−jω)|– The phase spectrum is odd: ∠X(jω) = −∠X(−jω)
• Because of this symmetry
– X(jω) for a real signal is often only plotted for positive frequencies.
– X(jω) for w < 0 can be inferred from these plots.
• If x(t) is both real and even, then X(jω) is a real and even function of ω. This
implies that the value of phase spectrum ∠X(jω) can only be 0, π or −π.
• If x(t) is both real and odd, then X(jω) is a purely imaginary and odd function
of ω. This implies that the value of phase spectrum ∠X(jω) can only be 0, π/2
or −π/2.
Dr. H. Nguyen Page 141
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Differentiation:
dx(t)
dt
FT⇐⇒ jωX(jω)
dnx(t)
dtnFT⇐⇒ (jω)nX(jω)
• This property can be easily derived by taking the derivative of both sides of the
synthesis equation. The key advantage is that ordinary differential equations
become algebraic in the frequency domain.
Integration:
∫ t
−∞
x(τ)dτFT⇐⇒ 1
jωX(jω) + πX(0)δ(ω)
• Differentiation in time = Multiplication by jω in frequency ⇒ Integration in
time = Division by jω in frequency. Also there is an impulse at ω = 0 to reflect
the possible existence of a DC (average) value.
Example: To see the useful application of the differentiation property of the Fourier
transform, consider finding the Fourier transform of the signal x(t) shown below.
Dr. H. Nguyen Page 142
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
t
1
1
0
x(t)
-1
-1
t10-1
-1
dx(t)/dt
)1( −tδ)1( +tδ
The first approach: Applying FT directly
X(jω) =
∫∞
−∞
x(t)e−jωtdt =
∫ 1
−1
(−t)[cos(jωt)− j sin(jωt)]dt
= −
∫ 1
−1
t cos(jωt)dt
︸ ︷︷ ︸
=0 since t cos(jωt) is an odd function
+j
∫ 1
−1
t sin(jωt)dt
= 2j
[sin(ωt)
ω2−
t cos(ωt)
ω
]∣∣∣∣
1
0
= 2j
[sin(ω)
ω2−
cos(ω)
ω
]
The second approach: Applying differentiation propertydx(t)
dt
FT←→
−2 sin(ω)
ω+ e−jω + ejω =
−2 sin(ω)
ω+ 2 cos(ω)
x(t)FT←→ X(jω) =
1
jω
[−2 sin(ω)
ω+ 2 cos(ω)
]
= 2j
[sin(ω)
ω2−
cos(ω)
ω
]
Dr. H. Nguyen Page 143
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Remark: In general, impulses arise when a function to be differentiated is
discontinuous. The impulse occurs at the point of the discontinuity with a strength
equal to the ‘size’ of the jump.
t
( )x t
0 T
A
t
( )u t
0
1
T− t
( )x t
0 T
A
T−
A−
t0
1 2
t0
1 2
1 2−
)(sign2
1t
)(exp2
1at−
1exp( )
2at−
t
( )x t
01t 2t
Upward-going impulse in thederivative at t1 of strengthequal to the size of the jump
Downward-going (i.e., negativestrength) impulse in thederivative at t2 of strength equalto the negative size of the jump
Dr. H. Nguyen Page 144
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Time & Frequency Scaling: x(at)FT⇐⇒ 1
|a|X(
jω
a
)
• This applies for both positive and negative values of a.
• If the signal is stretched out in time (|a| < 1), the Fourier transform is
compressed (high frequencies are moved to lower frequencies).
• If the signal is compressed in time (|a| > 1), the Fourier transform is expanded
(low frequencies are moved to higher frequencies).
• This time-frequency relationship can be seen in many examples of Fourier
transform pairs considered before.
−10 −5 0 5 10
0
0.2
0.4
0.6
0.8
1
X(j
ω)
ω−10 −5 0 5 10
−0.1
0
0.1
0.2
0.3
0.4
x(t)
t (sec)
−10 −5 0 5 10
0
0.2
0.4
0.6
0.8
1
X(j
ω)
ω−10 −5 0 5 10
−0.2
0
0.2
0.4
0.6
0.8
x(t)
t (sec)
Dr. H. Nguyen Page 145
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Parseval’s Relation: The energy of a signal x(t) is defined as:
Wx4=
∫ +∞
−∞
|x(t)|2dt
If x(t) represents the voltage across a 1Ω resistor, then Wx is the energy dissipated
by the resistor.
Parseval’s relation states that:
∫ +∞
−∞
|x(t)|2dt =1
2π
∫ +∞
−∞
|X(jω)|2dω
The important of this theorem is that it tells us to think of |X(jω)|2 as the energy
spectral density (i.e., how the signal’s energy is distributed over frequency).
A more general version of Parseval’s relation is
∫ +∞
−∞
x(t)y∗(t)dt =1
2π
∫ +∞
−∞
X(jω)Y ∗(jω)dω
where x(t) and y(t) are two arbitrary continuous-time signals.
Dr. H. Nguyen Page 146
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Example: To illustrate the application of Parseval’s relation, consider the following
problem: An engineer wishes to send a signal sinc(
Wtπ
)across a communications
channel, where W is a positive constant. The channel can be modeled as an ideal
low-pass filter with a transfer function
H(jω) =
1, ω ≤ 2π × 25× 103 rad/sec
0, otherwise
What range of values of W can the engineer guarantee that at least 95% of the
signal energy will reach the receiver?
Solution:
x(t) = sinc
(Wt
π
)
=sin(Wt)
Wt=
π
W
sin(Wt)
πt
Since
a(t) =sin(Wt)
πt
F⇐⇒ A(jω) =
1, ω ≤W
0, otherwise
Then
x(t) =π
W
sin(Wt)
πt
F⇐⇒ X(jω) =
πW , ω ≤W
0, otherwise
Dr. H. Nguyen Page 147
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
The energy of x(t) is
Ex =1
2π
∫ W
−W
|X(jω)|2dω =1
2π
∫ W
−W
( π
W
)2
dω =π
W
The Fourier transform of the output signal y(t) is Y (jω) = X(jω)H(jω). The
channel can be modeled as an ideal low-pass filter with a transfer function
H(jω) =
1, ω ≤ ωc
0, otherwise
where ωc = 2π × 25× 103 rad/sec is the cutoff frequency of the channel. Obviously
the effect of the channel on the input signal depends on the value of W :
• If W ≤ ωc then Y (jω) = X(jω) and therefore y(t) = x(t): The channel passes
the signal x(t) without any distortion and of course Ey = Ex.
• The more interesting situation is when W > ωc. In this case the transmitted
signal x(t) will be filtered by the channel and not all the energy of the
Dr. H. Nguyen Page 148
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
transmitted signal can be received at the receiver. Since
Y (jω) = X(jω)H(jω) =
πW , ω ≤ ωc
0, otherwise
the energy of the output signal y(t) is
Ey =1
2π
∫ ωc
−ωc
|Y (jω)|2dω =1
2π
∫ ωc
−ωc
( π
W
)2
dω =πωc
W 2
Thus if one requires Ey = 0.95Ex, then πωc
W 2 = 0.95 πW . It then follows that
W =ωc
0.95= 52.6π × 103 rad/sec
To conclude: 0 < W ≤ 52.6π × 103 rad/sec.
Dr. H. Nguyen Page 149
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
The Convolution Property
- h(t) -x(t) y(t) = x(t) ∗ h(t)
• Recall the response of an LTI system to a periodic input signal is:
+∞∑
k=−∞
akejkω0t −→+∞∑
k=−∞
akH(jkω0)ejkω0t
where H(jω) is the frequency response of the LTI system. It was defined as
H(jω) =
∫ ∞
−∞
h(t)e−jωtdt
• Thus the system’s frequency response is precisely the Fourier transform of the
system’s impulse response!
• The above says that the FS coefficients of the output are those of the input
multiplied by the frequency response of the system evaluated at the
corresponding harmonic frequencies.
• For aperiodic input signals, what is the effect of the LTI system in
frequency-domain (i.e., how does it change the spectrum of the input)?
Dr. H. Nguyen Page 150
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
• By the linearity property of LTI systems:
x(t) =1
2π
∫ +∞
−∞
X(jω)ejωtdω −→ y(t) =1
2π
∫ +∞
−∞
X(jω)[H(jω)ejωt]dω
since y(t) =1
2π
∫ +∞
−∞
Y (jω)ejωtdω
then Y (jω) = X(jω)H(jω)
• To summarize: x(t) ∗ h(t)FT⇐⇒ X(jω) ·H(jω)
• The Fourier transform maps the convolution of two signals into the product of
their Fourier transforms (a complicated convolution in the time-domain is
equivalent to a simple multiplication in the frequency-domain).
• For an LTI system, the frequency response captures the change in complex
amplitude of the Fourier transform of the input at each frequency ω.
• Since h(t) completely characterizes an LTI system, then so must H(jω).
• H(jω) is also known as the transfer function.
• |H(jω)| is the magnitude-frequency response: |Y (jω)| = |X(jω)||H(jω)|
• ∠H(jω) is the phase-frequency response: ∠Y (jω) = ∠X(jω) + ∠H(jω)
Dr. H. Nguyen Page 151
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
A formal proof of the convolution property :
Suppose y(t) = x(t) ∗ h(t). One can solve for Y (jω) in terms of X(jω) and H(jω)
as follows:
Y (jω) =
∫ +∞
−∞
(∫ +∞
−∞
x(τ)h(t− τ)dτ
)
e−jωtdt
=
∫ +∞
−∞
x(τ)
(∫ +∞
−∞
h(t− τ)e−jωtdt
)
dτ
=
∫ +∞
−∞
x(τ)
(∫ +∞
−∞
h(u)e−jω(u+τ)du
)
dτ
=
∫ +∞
−∞
x(τ)
(∫ +∞
−∞
h(u)e−jωudu
)
e−jωτdτ
= H(jω)
∫ +∞
−∞
x(τ)e−jωτdτ
= H(jω) ·X(jω)
= X(jω) ·H(jω)
Dr. H. Nguyen Page 152
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Example: This example illustrates the use of convolution property. It also shows the
importance of the phase-frequency response of an LTI system.
Consider the continuous-time LTI system with frequency response
H(jω) =a− jω
a + jω, a > 0
(a) The magnitude and phase responses of the system are:
|H(jω)| =√
a2 + ω2
√a2 + ω2
= 1
∠H(jω) = (− tan−1 ω
a)− (tan−1 ω
a) = −2 tan−1 ω
a
−10 −5 0 5 10
0
0.5
1
|H(jω)|
ω (rad/sec)−10 −5 0 5 10−4
−2
0
2
4∠H(jω)
ω (rad/sec)
Dr. H. Nguyen Page 153
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
(b) Using the table of Fourier transform on page 329 of textbook, the impulse
response of the system is found to be
H(jω) = −1 +2a
a + jω
FT⇐⇒ h(t) = −δ(t) + 2ae−atu(t)
(c) Let a = 1. We wish to find the output of the system when the input is
x(t) = cos(t/√
3) + cos(t) + cos(√
3t)
Since a = 1, then |H(jω)| = 1 and ∠H(jω) = −2 tan−1 ω. First, it is
convenient to find the output of an LTI system to the sinusoidal input
cos(ω0t + θ). To this end, recall that ej(ω0t+θ) is an eigenfunction of the LTI
systems:
ej(ω0t+θ) ⇒ H(jω0)ej(ω0t+θ) = |H(jω0)|ej[(ω0t+θ)+∠H(jω0)]
e−j(ω0t+θ) ⇒ H(−jω0)e−j(ω0t+θ) = |H(jω0)|e−j[(ω0t+θ)−∠H(jω0)]
where we have used the fact that
H(−jω0) = |H(−jω0)|ej∠H(−jω0) = |H(jω0)|e−j∠H(jω0). Thus
cos(ω0t + θ)⇒ |H(jω0)| cos(ω0t + θ + ∠H(jw0))
Dr. H. Nguyen Page 154
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Therefore,
y(t) = cos
(t√3− π
3
)
+ cos(
t− π
2
)
+ cos
(√3t− 2π
3
)
Plots of the input and output are shown below. Observe that since the system’s
phase-frequency response is not a linear function of ω, the output is not simply a
shifted version of the input. This example clearly shows that one cannot ignore
the phase-frequency response of an LTI system.
−20 −15 −10 −5 0 5 10 15 20−4
−2
0
2
4Input
t (sec)
−20 −15 −10 −5 0 5 10 15 20−4
−2
0
2
4Output
t (sec)
Dr. H. Nguyen Page 155
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Filtering
• In may applications, it is of interest to change the relative amplitudes of the
frequency components in a signal, or to eliminate some frequency components
entirely. Such processes are referred to as filtering.
• LTI systems that change the shape of the input spectrum are often referred to as
frequency-shaping filters.
• LTI systems that are designed to pass some frequencies essentially undistorted
and significantly attenuate or eliminate other frequencies are referred to as
frequency-selective filters.
• From the convolution property of the Fourier transform, it is evident that
filtering can be conveniently accomplished through the use of LTI systems with
an appropriate chosen frequency response H(jω).
• Frequency-domain method thus provides ideal tools to examine this very
important class of applications.
Dr. H. Nguyen Page 156
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Ideal Frequency-Selective Filters
0
1
ω
)( ωjH
cω
Lowpass
0
1
ω
)( ωjH
cω
Highpass
0
1
ω
)( ωjH
cω
Notch
0
1
ω
)( ωjH Bandpass
0
1
ω
)( ωjH Bandstop
1cω 2cω 1cω 2cω
The frequency components within the passband are passed without modification,
whereas the freq. components that fall into the stopband are completely eliminated.
• Lowpass filters pass low frequencies (ω < ωc).
• Highpass filters pass high frequencies (ω > ωc).
• Bandpass filters pass a range of frequencies (ωc1 < ω < ωc2).
• Bandstop filters pass two ranges of frequencies (ω < ωc1 and ω > ωc2).
• Notch filters pass all frequencies except ω ≈ ωc.
Dr. H. Nguyen Page 157
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Example: Impulse Response of the Ideal Lowpass Filter
H(jω) =
1, |w| < W
0, otherwise
FT⇐⇒ h(t) =1
2π
∫ W
−W
ejωtdω =sin(Wt)
πt
−10 −8 −6 −4 −2 0 2 4 6 8 10
0
0.2
0.4
0.6
0.8
1
H(j
ω)
ω
−10 −8 −6 −4 −2 0 2 4 6 8 10−0.2
0
0.2
0.4
0.6
0.8
h(t)
t (sec)
The above ideal LPF cannot be implemented in practice since it is
non-causal. Moreover, the oscillatory behavior in the filter’s impulse response
is highly undesirable.
Dr. H. Nguyen Page 158
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
An Example of Frequency-Shaping Filter: A Practical LPF
( )iv t
R( )i t
C ( )ov t
+ +
−−
vi(t) = Ri(t) +1
C
∫ t
−∞
i(τ)dτ, vo(t) =1
C
∫ t
−∞
i(τ)dτ
Taking the Fourier transform of both sides of the above equations (and ignoring any DC
signals in the circuit) gives
Vi(jω) = RI(jω) +1
C
I(jω)
jω, Vo(jω) =
1
C
I(jω)
jω
⇒ H(jω) =Y (jω)
X(jω)=
1
1 + jωRC=
1
1 + jω/ωc
where ωc = 1RC
is called the cutoff frequency.
The impulse response of the filter is
h(t) = ωce−ωctu(t) =
1
RCe−
t
RC u(t) (Note that the filter is causal)
Dr. H. Nguyen Page 159
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
The magnitude-frequency and phase-frequency spectra are
|H(jω)| =1
√
1 + (ω/ωc)2, ∠H(jω) = − tan−1(ω/ωc)
−10 −5 0 5 100
0.5
1Magnitude Frequency−response
ω/ωc
−10 −5 0 5 10−2
−1
0
1
2Phase frequency−response
ω/ωc
The RC lowpass filter is a relatively crude approximation of the ideal lowpass filter. Better
approximations can be obtained with more-complicated circuits.
Dr. H. Nguyen Page 160
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
More Properties of Fourier Transform
Multiplication:
x(t) · w(t)FT⇐⇒ 1
2πX(jω) ∗W (jω)
Frequency-Shifting:
ejω0tx(t)FT⇐⇒ X(j(ω − ω0))
• Multiplication by a complex exponential shifts the Fourier transform to the
specified frequency.
• This is the basis of amplitude modulation (AM) technique used in
communications.
• This is also a convenient method for multiplexing multiple bandlimited signals
into a single channel.
• The signal can then be recovered by bandpass filtering and AM demodulation
Dr. H. Nguyen Page 161
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Example: To illustrate the application of the frequency-shifting property, consider
finding the Fourier transform of a windowed cosine. Specifically, consider n cycles of
a cosine of frequency ω0 (rad/sec) that lasts only over the time interval −T/2 to
T/2 seconds, i.e., T = 2πω0
n. A typical plot of it is shown below.
−15 −10 −5 0 5 10 15−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
t (sec)
Such a signal can be written as x(t) = A cos(ω0t)w(t), where the “window” function
w(t) = u(t + T/2)− u(t− T/2).
The Fourier transform of Aw(t) is AW (jω) = AT sin(ωT/2)(ωT/2) . Using the
Dr. H. Nguyen Page 162
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
frequency-shifting property of the Fourier transform one has
x(t) cos(ω0t) =1
2
[x(t)e−jω0t + x(t)ejω0t
] FT⇐⇒ 1
2X(j(ω + ω0)) +
1
2X(j(ω − ω0))
The above states that multiplying a function x(t) by a cos(ω0t) shifts its spectrum
‘up’ and ‘down’ by the frequency ω0 and scales the amplitude by 2. Thus the result is
G(jωn) = nAπ
ω0
[sin(nπ(ωn − 1))
nπ(ωn − 1)+
sin(nπ(ωn + 1))
nπ(ωn + 1)
]
where ωn = ω/ω0 is the normalized frequency. A further normalization by (Aπ/ω0)
yields:
Gnorm(jωn) =G(jω)
(Aπ/ω0)= n
[sin(nπ(ωn − 1))
nπ(ωn − 1)+
sin(nπ(ωn + 1))
nπ(ωn + 1)
]
Plots of Gnorm(jωn) for different values of n are shown below. Note that as more
and more cycles are taken the Fourier transform (i.e., the spectrum density) becomes
more and more concentrated around ±ω0 (rad/sec). In the limit as n→∞ or
equivalently T →∞ the density becomes 2 impulses located at ±ω0 (rad/sec).
Dr. H. Nguyen Page 163
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
−40 −20 0 20 40−0.5
0
0.5
1
1.5
ωn
Am
plitu
de D
ensi
ty S
pect
rum
n=1
−40 −20 0 20 40−2
0
2
4
6
ωn
Am
plitu
de D
ensi
ty S
pect
rum
n=5
−40 −20 0 20 40−5
0
5
10
ωn
Am
plitu
de D
ensi
ty S
pect
rum
n=10
−40 −20 0 20 40−20
−10
0
10
20
30
40
ωn
Am
plitu
de D
ensi
ty S
pect
rum
n=50
Dr. H. Nguyen Page 164
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
Example: Figure (a) below shows a system with input signal x(t) and output signal
y(t). The input signal has Fourier transform X(jω) shown in Figure (b). Determine
and sketch the spectrum Y (jω) of y(t).
-3W 3W 5W-5W
1x(t)
cos(5Wt) cos(3Wt)
3W-3W
1y(t)
(a)
-2W -2W
1
(b)
)( ωjX
ω
After modulated with carrier frequency 5W and passing through band-pass filter, the
output signal is shown in Figure (c) (shaded area). That signal is modulated one
more time with carrier frequency 3W . The modulated signal is plotted in Figure (d).
Therefore, Y (jω), the output of low-pass filter with cut-off frequency 3W can be
seen in Figure (e).
Dr. H. Nguyen Page 165
EE351–Spectrum Analysis and Discrete Time Systems University of Saskatchewan
5W
1/2
ω
-3W 3W 5W-5W
1x(t)
cos(5Wt) cos(3Wt)
3W-3W
1y(t)
(a)
7W3W-5W -3W-7W
ω6W-6W 8W-8W 2W-2W
0
0
ω2W-2W 0
)( jwY
1/4
1/4
(c)
(d)
(e)
-2W 2W
1
(b)
)( ωjX
ω0
After the first modulation andpassing via band-pass filter
The second modulation
Output of the low-pass filter
Dr. H. Nguyen Page 166