unit - 1
DESCRIPTION
DSPTRANSCRIPT
Unit - 1
Fall 2015
Signals - Introduction Signal: Anything that carries some information can be called as
signal. A signal is also defined as any physical quantity that
varies with time, space or any other independent variable
or variables. Eg: π 1 π‘ = 5π‘ π 2 π‘ = 20π‘2
Examples of signals:
1. Speech signal
2. ECG signal
Types of signals Types:
1. Continuous time signals
2. Discrete time signals
Continuous Signal or Analog Signal
Eg 1: ECG signal
Analog Signals are defined for all time values
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-10
-8
-6
-4
-2
0
2
4
6
8
10x(t) Vs time t
time t in Seconds
valu
e of
x(t)
Amplitude=10, frequency =341.4 rad /sec. or 50Hz
Eg 2: π₯(π‘) = 10πΆπΆπ (2 β ππ β 50 β π‘) β 10πΆπΆπ (341.4π‘)
Discrete Signals
Defined for only discrete values of time
8a.m 9 a.m 10a.m 11a.m 12 a.m
20o C 22o C 25o C 25o C 25o C
Eg: value of temperature measured at every hour inside a room.
Basic Sequences
β’ Unit sample (impulse) sequence
β’ Unit step sequence
β’ Exponential sequences
=β
=Ξ΄0n10n0
]n[
β₯<
=0n10n0
]n[u
nA]n[x Ξ±=
-10 -5 0 5 10 0
0.5
1
1.5
-10 -5 0 5 10 0
0.5
1
1.5
-10 -5 0 5 10 0
0.5
1
Discrete time signals: Sequences
A discrete β time signal π₯[n] is a function of an independent variable that is
an integer.
Representation of discrete time signals:
1. Functional representation:
π₯[π] = οΏ½1, ππΆπ π = 1,34, ππΆπ π = 20, πππ π π€π€πππ
2. Tabular representation:
n β¦. -2 -1 0 1 2 3 4 5 β¦.
x[n] β¦. 0 0 0 1 4 1 0 0 ..
Discrete time signals: Sequences 3. Sequence representation:
An infinite β duration signal or sequence with the time origin
π = 0 indicated by symbol β is represented as
π₯[π] = β¦ β¦ . 0, 0, 1, 4, 1, 0, 0, β¦ β¦ .β
A sequence x(n), which is zero for π < 0, can be represented as
π₯[π] = 0, 1, 4, 1, 0, 0, β¦ β¦ .β
4. Graphical representation:
Simple Manipulations of discrete time Signals
When a signal is processed, the signal undergoes many manipulations
involving both the independent and dependent variable. Some of
these are:
β’ Folding
β’ Shifting
β’ Time scaling
Folding:
This operation is done by replacing the independent variable
βnβ by β-nβ
Shifting:
A signal π₯[π] may be shifted in time i.e; the signal can be
either advanced in time axis or delayed in time axis. The
shifted signal is represented by π₯[π β π], where βkβ is an
integer.
β’ If βkβ is posotive, the signal is delayed by βkβ units.
β’ If βkβ is negative, the signal is advanced by βkβ units.
Scaling:
This involves to replace the independent variable βnβ by βknβ,
where βkβ is an integer. Scaling compresses or dilates a signal.
Problem: 1
A discrete time signal π₯ π is shown in Figure. Sketch and label each of
the following signals.
(a) π₯(π β2) (b) π₯(2π) (c) π₯(βπ) (d) π₯(βπ + 2)
Problem:2
A discrete β time signal π₯ π is defined as
π₯ π = οΏ½1 + π
3 , β3 β€ π β€ β1
1, 0 β€ π β€ 30 , πππ ππ€π€πππ
a) Determine its values and sketch the signal π₯ π .
b) Sketch the signals that result if we:
(i) First fold π₯ π and then delay the resulting signal by four samples.
(ii) First delay π₯ π by four samples and then fold the resulting signal.
c) Sketch the signal π₯ βπ + 4 .
d) Compare the results in parts Q1(b) and (c) and derive a rule for obtaining the
signalπ₯ βπ + 4 from π₯ π .
e) Express the signal π₯ π in terms of πΏ π and π’ π
Basic operations on Signals
The basic set of operations are
β’ Addition
β’ Multiplication
β’ Scaling of sequences
Amplitude scaling of a signal by a constant A is accomplished by
multiplying the value of every signal sample by A.
π¦ π = π΄π₯ π ββ < π < β
The sum of two signals π₯1 π πππ π₯2 π is a signal y(n), whose value at
any instant is equal to the sum of the values of those two signals at
that instant.
π¦ π = π₯1 π + π₯2 π , ββ < π < β
Basic operations on Signals
The product of two signals π₯1 π πππ π₯2 π is a signal y(n), whose value
at any instant is equal to the product he values of those two signals at
that instant.
π¦ π = π₯1 π π₯2 π , ββ < π < β
Problem: 3
Using the discrete β time signal π₯1 π πππ π₯2 π shown in Figure represent
each of the following signals by a graph and by a sequence of numbers.
a) π¦1 π = π₯1 π + π₯2 π
b) π¦2 π = 2 π₯1 π
c) π¦3 π = π₯1 π π₯2 π
Some basic building blocks are used to represent a discrete time
systems.
1. An adder
2. A Constant Multiplier
3. A signal multiplier
4. A unit delay element
5. A unit advance element
Block Diagram Representation of Discrete Time Systems
An adder: A Constant Multiplier:
π₯(π) a π¦ π = ππ₯(π) +
π₯1(π)
π₯2(π)
π¦ π = π₯1 π + π₯2(π)
Signal Processing is a method of extracting
information from the signal which in turn
depends on the type of signal and the nature of
information it carries.
Signal Processing
What is a system?
A system is formally defined as an entity that manipulates one or
more signals to accomplish a function, thereby yielding new signals.
system output signal
input signal
Some Interesting Systems Communication system
Control systems
Remote sensing system
Perspectival view of Mount Shasta (California), derived from a pair of stereo radar images acquired from orbit with the shuttle Imaging Radar (SIR-B).
(Courtesy of Jet Propulsion Laboratory.)
Biomedical system(biomedical signal processing)
Classification of discrete time systems
π¦ π β Ξ€ π₯(π)
Common system properties:
Static VS Dynamic
Time - invariant VS Time β variant
Linear VS Nonlinear
Casual VS Non-causal
Stable VS unstable
Classification of discrete time systems
Why is this so important?
mathematical techniques developed to analyze systems are often
contingent upon the general characteristics of the systems being
considered.
For a system to possess a given property, the property must hold for
every possible input signal to the system.
Classification of discrete time systems
Static Vs Dynamic Systems
A discrete-time system is called static or memory less if its output at any instant βnβ
depends at most on the input sample at the same time, but not on past or future
samples of the input. Otherwise the system is dynamic.
A system is static, if and only if
π¦ π = Ξ€ π₯ π
Example:
Determine whether the following systems are static or dynamic:
a) π¦ π = π₯ π π₯ π β 1
b) π¦ π = π₯2 π + π₯(π)
Classification of discrete time systems
Time β invariant vs Time β variant Systems Time-invariant system: input-output characteristics do not change with
time
A system is time-invariant if
π₯ π βΆ π¦ π β π₯ π β π βΆ π¦(π β π)
For every inputπ₯ π and every time shift k
Ξ€ Ξ€
Example: Determine if the system shown in Figure is time-invariant or time β
variant.
Classification of discrete time systems
Linear vs Nonlinear Systems Linear system: obeys superposition principle
A system is said to be linear if
Ξ€ π1π₯1 π + π2π₯2 π = π1Ξ€ π₯1 π + π2Ξ€[π₯2 π ]
For any arbitrary input sequence π₯1 π πππ π₯2 π , and any arbitrary
conditions π1 πππ π2
Example:
Determine whether the following systems are linear or non linear:
a) π¦ π = π₯ π + 1π₯ πβ1
b) π¦ π = π₯2 π
c) π¦ π = ππ₯(π)
Classification of discrete time systems
Casual vs Non-casual Systems Causal system: output of system at any time n depends only on present
and past inputs
A system is said to be casual if
π¦ π = Ξ€ π₯ π , π₯ π β 1 , π₯ π β 2 β¦ β¦ β¦ . .
For all n
Example:
Test whether the following systems are causal or non causal:
a) π¦ π = π΄π₯ π + π΅
b) π¦ π = ππ₯ π + ππ₯(π β 1)
Classification of discrete time systems
Stable vs Unstable Systems Bounded input βBounded output (BIBO) stable: every bounded input
produces a bounded output
A system is BIBO stable if
π₯ π β€ ππ₯ < β βΉ π¦ π β€ ππ¦ < β
for all n
Block Diagram Representation of Discrete Time Systems
A signal Multiplier:
A unit Delay:
+ π₯1(π)
π₯2(π)
π¦ π = π₯1 π π₯2(π)
π₯(π) π¦ π = π₯(π β 1) π§β1 π₯(π) π¦ π = π₯(π + 1)
π§
A unit Advance:
Problem: 4
Using basic building blocks, sketch the block diagram representation of the
discrete β time system described by the input-output relation.
π¦ π =14π¦ π β 1 +
12π₯ π +
12π₯(π β 1)
Impulse Response
T [ ] x(n)=Ξ΄(n)
h(n)=T[Ξ΄(n)]
0 0
0 0 5 5
Ξ΄(n) h(n)
Ξ΄(n-5) h(n-5)
Convolution Sum
)(*)()()()( nhnxknhkxnyk
=β= ββ
ββ=
convolution
T [ ] Ξ΄(n) h(n)
x(n) y(n)
A linear shift-invariant system is completely characterized by its impulse response.
Characterize a System
h(n) x(n) x(n)*h(n)
Useful equations to compute convolution
35
For geometric series
For arithmetic series
Example
0 1 2 3 4 5 6
)()()( Nnununx ββ=
<β₯
=000
)(nna
nhn
y(n)=? 0 1 2 3 4 5 6
Example
)()()(*)()( knhkxnhnxnyk
β== ββ
ββ=
0 1 2 3 4 5 6 k
x(k)
0 1 2 3 4 5 6 k h(k)
0 1 2 3 4 5 6 k h(0βk)
Example )()()(*)()( knhkxnhnxny
kβ== β
β
ββ=
0 1 2 3 4 5 6 k
x(k)
0 1 2 3 4 5 6 k h(0βk)
0 1 2 3 4 5 6 k h(1βk)
compute y(0)
compute y(1)
How to computer y(n)?
Example
)()()(*)()( knhkxnhnxnyk
β== ββ
ββ=
0 1 2 3 4 5 6 k
x(k)
0 1 2 3 4 5 6 k h(0βk)
0 1 2 3 4 5 6 k h(1βk)
compute y(0)
compute y(1)
How to computer y(n)?
Two conditions have to be considered.
n<N and nβ₯N.
Example )()()(*)()( knhkxnhnxny
kβ== β
β
ββ=
1
1
1
)1(
00 111)( β
β
β
+β
=
β
=
β
ββ
=β
β=== ββ a
aaa
aaaaanynn
nn
k
knn
k
kn
n < N
n β₯ N
11
1
0
1
0 111)( β
β
β
ββ
=
ββ
=
β
ββ
=ββ
=== ββ aaa
aaaaaany
NnnNn
N
k
knN
k
kn
Example )()()(*)()( knhkxnhnxny
kβ== β
β
ββ=
1
1
1
)1(
00 111)( β
β
β
+β
=
β
=
β
ββ
=β
β=== ββ a
aaa
aaaaanynn
nn
k
knn
k
kn
n < N
n β₯ N
11
1
0
1
0 111)( β
β
β
ββ
=
ββ
=
β
ββ
=ββ
=== ββ aaa
aaaaaany
NnnNn
N
k
knN
k
kn
0
1
2
3
4
5
0 5 10 15 20 25 30 35 40 45 50
Convolution computation in tabular
form
Problem: 5
Compute the convolution π¦(π) of the signals by tabulation method.
π₯ π = 1, 1, 0, 1, 1β and π€ π = 1,β2,β3, 4
β
Solution:
The values of π₯ π πππ π€ π πππ ππ π€πππ‘π‘ππ ππ ππΆπππΆπ€π :
π βπ = π π βπ = π
π βπ = π π βπ = βπ
π π = π π βπ = βπ
π π = π π π = π
π π = π
Do by yourself
Determine the convolution π¦(π) of the signals by analytical method.
π₯ π = οΏ½13π, 0 β€ π β€ 6
0, πππ ππ€π€πππ
π€ π = οΏ½1, β2 β€ π β€ 2 0, πππ ππ€π€πππ