unit - 1

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: 𝑠1 𝑡 = 5𝑡 𝑠2 𝑡 = 20𝑡2

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: 𝑥(𝑡) = 10𝐶𝐶𝑠(2 ∗ 𝑝𝑝 ∗ 50 ∗ 𝑡) ≈ 10𝐶𝐶𝑠(341.4𝑡)

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

• Unit sample (impulse) sequence

• Unit step sequence

• Exponential sequences

=≠

=δ0n10n0

]n[

≥<

=0n10n0

]n[u

nA]n[x α=

-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 – time signal 𝑥[n] is a function of an independent variable that is

an integer.

Representation of discrete time signals:

1. Functional representation:

𝑥[𝑛] = �1, 𝑓𝐶𝑓 𝑛 = 1,34, 𝑓𝐶𝑓 𝑛 = 20, 𝑒𝑒𝑠𝑒 𝑤𝑤𝑒𝑓𝑒

2. Tabular representation:

n …. -2 -1 0 1 2 3 4 5 ….

x[n] …. 0 0 0 1 4 1 0 0 ..

Discrete time signals: Sequences 3. Sequence representation:

An infinite – duration signal or sequence with the time origin

𝑛 = 0 indicated by symbol ↑ is represented as

𝑥[𝑛] = … … . 0, 0, 1, 4, 1, 0, 0, … … .↑

A sequence x(n), which is zero for 𝑛 < 0, can be represented as

𝑥[𝑛] = 0, 1, 4, 1, 0, 0, … … .↑

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:

• Folding

• Shifting

• Time scaling

Folding:

This operation is done by replacing the independent variable

‘n’ by ‘-n’

Shifting:

A signal 𝑥[𝑛] 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 𝑥[𝑛 − 𝑘], where ‘k’ is an

integer.

• If ‘k’ is posotive, the signal is delayed by ‘k’ units.

• If ‘k’ is negative, the signal is advanced by ‘k’ units.

Scaling:

This involves to replace the independent variable ‘n’ by ‘kn’,

where ‘k’ is an integer. Scaling compresses or dilates a signal.

Problem: 1

A discrete time signal 𝑥 𝑛 is shown in Figure. Sketch and label each of

the following signals.

(a) 𝑥(𝑛 −2) (b) 𝑥(2𝑛) (c) 𝑥(−𝑛) (d) 𝑥(−𝑛 + 2)

Problem:2

A discrete – time signal 𝑥 𝑛 is defined as

𝑥 𝑛 = �1 + 𝑛

3 , −3 ≤ 𝑛 ≤ −1

1, 0 ≤ 𝑛 ≤ 30 , 𝑒𝑒𝑠𝑒𝑤𝑤𝑒𝑓𝑒

a) Determine its values and sketch the signal 𝑥 𝑛 .

b) Sketch the signals that result if we:

(i) First fold 𝑥 𝑛 and then delay the resulting signal by four samples.

(ii) First delay 𝑥 𝑛 by four samples and then fold the resulting signal.

c) Sketch the signal 𝑥 −𝑛 + 4 .

d) Compare the results in parts Q1(b) and (c) and derive a rule for obtaining the

signal𝑥 −𝑛 + 4 from 𝑥 𝑛 .

e) Express the signal 𝑥 𝑛 in terms of 𝛿 𝑛 and 𝑢 𝑛

Basic operations on Signals

The basic set of operations are

• Addition

• Multiplication

• Scaling of sequences

Amplitude scaling of a signal by a constant A is accomplished by

multiplying the value of every signal sample by A.

𝑦 𝑛 = 𝐴𝑥 𝑛 −∞ < 𝑛 < ∞

The sum of two signals 𝑥1 𝑛 𝑎𝑛𝑎 𝑥2 𝑛 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.

𝑦 𝑛 = 𝑥1 𝑛 + 𝑥2 𝑛 , −∞ < 𝑛 < ∞

Basic operations on Signals

The product of two signals 𝑥1 𝑛 𝑎𝑛𝑎 𝑥2 𝑛 is a signal y(n), whose value

at any instant is equal to the product he values of those two signals at

that instant.

𝑦 𝑛 = 𝑥1 𝑛 𝑥2 𝑛 , −∞ < 𝑛 < ∞

Problem: 3

Using the discrete – time signal 𝑥1 𝑛 𝑎𝑛𝑎 𝑥2 𝑛 shown in Figure represent

each of the following signals by a graph and by a sequence of numbers.

a) 𝑦1 𝑛 = 𝑥1 𝑛 + 𝑥2 𝑛

b) 𝑦2 𝑛 = 2 𝑥1 𝑛

c) 𝑦3 𝑛 = 𝑥1 𝑛 𝑥2 𝑛

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:

𝑥(𝑛) a 𝑦 𝑛 = 𝑎𝑥(𝑛) +

𝑥1(𝑛)

𝑥2(𝑛)

𝑦 𝑛 = 𝑥1 𝑛 + 𝑥2(𝑛)

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

𝑦 𝑛 → Τ 𝑥(𝑛)

Common system properties:

Static VS Dynamic

Time - invariant VS Time – 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 ‘n’

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

𝑦 𝑛 = Τ 𝑥 𝑛

Example:

Determine whether the following systems are static or dynamic:

a) 𝑦 𝑛 = 𝑥 𝑛 𝑥 𝑛 − 1

b) 𝑦 𝑛 = 𝑥2 𝑛 + 𝑥(𝑛)

Classification of discrete time systems

Time – invariant vs Time – variant Systems Time-invariant system: input-output characteristics do not change with

time

A system is time-invariant if

𝑥 𝑛 ⟶ 𝑦 𝑛 ⇒ 𝑥 𝑛 − 𝑘 ⟶ 𝑦(𝑛 − 𝑘)

For every input𝑥 𝑛 and every time shift k

Τ Τ

Example: Determine if the system shown in Figure is time-invariant or time –

variant.

Classification of discrete time systems

Linear vs Nonlinear Systems Linear system: obeys superposition principle

A system is said to be linear if

Τ 𝑎1𝑥1 𝑛 + 𝑎2𝑥2 𝑛 = 𝑎1Τ 𝑥1 𝑛 + 𝑎2Τ[𝑥2 𝑛 ]

For any arbitrary input sequence 𝑥1 𝑛 𝑎𝑛𝑎 𝑥2 𝑛 , and any arbitrary

conditions 𝑎1 𝑎𝑛𝑎 𝑎2

Example:

Determine whether the following systems are linear or non linear:

a) 𝑦 𝑛 = 𝑥 𝑛 + 1𝑥 𝑛−1

b) 𝑦 𝑛 = 𝑥2 𝑛

c) 𝑦 𝑛 = 𝑛𝑥(𝑛)

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

𝑦 𝑛 = Τ 𝑥 𝑛 , 𝑥 𝑛 − 1 , 𝑥 𝑛 − 2 … … … . .

For all n

Example:

Test whether the following systems are causal or non causal:

a) 𝑦 𝑛 = 𝐴𝑥 𝑛 + 𝐵

b) 𝑦 𝑛 = 𝑎𝑥 𝑛 + 𝑏𝑥(𝑛 − 1)

Classification of discrete time systems

Stable vs Unstable Systems Bounded input –Bounded output (BIBO) stable: every bounded input

produces a bounded output

A system is BIBO stable if

𝑥 𝑛 ≤ 𝑀𝑥 < ∞ ⟹ 𝑦 𝑛 ≤ 𝑀𝑦 < ∞

for all n

Block Diagram Representation of Discrete Time Systems

A signal Multiplier:

A unit Delay:

+ 𝑥1(𝑛)

𝑥2(𝑛)

𝑦 𝑛 = 𝑥1 𝑛 𝑥2(𝑛)

𝑥(𝑛) 𝑦 𝑛 = 𝑥(𝑛 − 1) 𝑧−1 𝑥(𝑛) 𝑦 𝑛 = 𝑥(𝑛 + 1)

𝑧

A unit Advance:

Problem: 4

Using basic building blocks, sketch the block diagram representation of the

discrete – time system described by the input-output relation.

𝑦 𝑛 =14𝑦 𝑛 − 1 +

12𝑥 𝑛 +

12𝑥(𝑛 − 1)

Impulse Response

T [ ] x(n)=δ(n)

h(n)=T[δ(n)]

0 0

0 0 5 5

δ(n) h(n)

δ(n-5) h(n-5)

Convolution Sum

)(*)()()()( nhnxknhkxnyk

=−= ∑∞

−∞=

convolution

T [ ] δ(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 −−=

<≥

=000

)(nna

nhn

y(n)=? 0 1 2 3 4 5 6

Example

)()()(*)()( knhkxnhnxnyk

−== ∑∞

−∞=

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(0−k)

Example )()()(*)()( knhkxnhnxny

k−== ∑

∞

−∞=

0 1 2 3 4 5 6 k

x(k)

0 1 2 3 4 5 6 k h(0−k)

0 1 2 3 4 5 6 k h(1−k)

compute y(0)

compute y(1)

How to computer y(n)?

Example

)()()(*)()( knhkxnhnxnyk

−== ∑∞

−∞=

0 1 2 3 4 5 6 k

x(k)

0 1 2 3 4 5 6 k h(0−k)

0 1 2 3 4 5 6 k h(1−k)

compute y(0)

compute y(1)

How to computer y(n)?

Two conditions have to be considered.

n<N and n≥N.

Example )()()(*)()( knhkxnhnxny

k−== ∑

∞

−∞=

1

1

1

)1(

00 111)( −

−

−

+−

=

−

=

−

−−

=−

−=== ∑∑ a

aaa

aaaaanynn

nn

k

knn

k

kn

n < N

n ≥ N

11

1

0

1

0 111)( −

−

−

−−

=

−−

=

−

−−

=−−

=== ∑∑ aaa

aaaaaany

NnnNn

N

k

knN

k

kn

Example )()()(*)()( knhkxnhnxny

k−== ∑

∞

−∞=

1

1

1

)1(

00 111)( −

−

−

+−

=

−

=

−

−−

=−

−=== ∑∑ a

aaa

aaaaanynn

nn

k

knn

k

kn

n < N

n ≥ N

11

1

0

1

0 111)( −

−

−

−−

=

−−

=

−

−−

=−−

=== ∑∑ 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 𝑦(𝑛) of the signals by tabulation method.

𝑥 𝑛 = 1, 1, 0, 1, 1↑ and 𝑤 𝑛 = 1,−2,−3, 4

↑

Solution:

The values of 𝑥 𝑛 𝑎𝑛𝑎 𝑤 𝑛 𝑐𝑎𝑛 𝑏𝑒 𝑤𝑓𝑝𝑡𝑡𝑒𝑛 𝑎𝑠 𝑓𝐶𝑒𝑒𝐶𝑤𝑠:

𝒙 −𝟐 = 𝟏 𝒉 −𝟑 = 𝟏

𝒙 −𝟏 = 𝟏 𝒉 −𝟐 = −𝟐

𝒙 𝟎 = 𝟎 𝒉 −𝟏 = −𝟑

𝒙 𝟏 = 𝟏 𝒉 𝟎 = 𝟒

𝒙 𝟐 = 𝟏

Do by yourself

Determine the convolution 𝑦(𝑛) of the signals by analytical method.

𝑥 𝑛 = �13𝑛, 0 ≤ 𝑛 ≤ 6

0, 𝑒𝑒𝑠𝑒𝑤𝑤𝑒𝑓𝑒

𝑤 𝑛 = �1, −2 ≤ 𝑛 ≤ 2 0, 𝑒𝑒𝑠𝑒𝑤𝑤𝑒𝑓𝑒