DIGITAL SIGNAL PROCESSING

LECTURE 1
Fall 2010
2K8-5th Semester
Tahir Muhammad
tmuhammad_07@yahoo.com
Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and
Buck, ©1999-2000 Prentice Hall Inc.
Introduction

Digital Signal Processing Lecture 1 2

Examples

Digital Signal Processing Lecture 1 3

Examples

Digital Signal Processing Lecture 1 4

Examples

Digital Signal Processing Lecture 1 5

Applications

Digital Signal Processing Lecture 1 6

Why DSP?

Digital Signal Processing Lecture 1 7

Textbook and References
Textbook:
| Oppenheim, A.V., Schafer, R.W, "Discrete-Time
Signal Processing", 2nd Edition, Prentice-Hall,
1999.
1999

Reference Books:
| Digital Signal Processing (4th Edition), John G.
Proakis, Dimitris K Manolakis
| Vinay K. Ingle, John G. Proakis, “Digital
Digital Signal
Processing using MATLAB”, 2nd Ed., Thomson,
2007.

Digital Signal Processing Lecture 1 8

Course Outline
| Chapter
p # 1 [[Introduction]]
| Chapter # 2 [Discrete-Time Signals and Systems]

| Chapter # 3 [The Z-Transform]

| Chapter # 4 [Sampling of Continuous-Time Signals]

| Chapter # 6 [Structures for Discrete-Time Systems]

| Chapter # 7 [Filter Design Techniques]

| Chapter # 8 [The Discrete Fourier Transform]

Digital Signal Processing Lecture 1 9

Definitions of DSP
| Signal
g
y A function of independent variables such as time,
distance, position, temperature and pressure
y Signals are analog in nature(continuous) such as
human voice, electrical signal(voltage or current), radio
wave, optical, audio, and so on which contains a stream
of information or data.
data
y Or may be discrete such as temperature, stock, etc.

| Processing
y Operating in some fashion on signal to extract some
useful information

Digital Signal Processing Lecture 1 10

Definitions of DSP
| Digital
g Signal
g Processing
g
y Concerned with the representation of signals by sequence
of numbers or symbols and the processing of these
sequence
y The purpose of such processing may be to estimate
characteristic parameters or transform a signal

Digital Signal Processing Lecture 1 11

Characterization and classification of
signals
| Depending
p g on number of independent
p variables
y 1-D Signals : speech signal
y 2-D Signals : Image signal
y M-D
M D Signals
Si l : Video
Vid signal
i l
| Based on independent variables
y Continuous time signal: signal is defined at every
Continuous-time
instant of time
y Discrete-time signal: takes certain numerical values
at specified discrete instants of time,
time basically a
sequence of numbers

Digital Signal Processing Lecture 1 12

Types of Signals

Digital Signal Processing Lecture 1 13

Types of Signals

Digital Signal Processing Lecture 1 14

Definition of Discrete-time Signal &
System
| Define at equally
q y spaced
p discrete value of time
| Represented as a sequence of numbers

| The sequence is denoted as x[n]; where n is an

integer in the range of -∞ to ∞.
| A discrete time is represented as {x(n)}

{ ( )} = {{... 0.95, -0.2, 2.17, 1.1, 0.2, -3.67, 2.9 ...}}

{x(n)}

| Arrow indicate time index, n = 0

Digital Signal Processing Lecture 1 15

Definition of Discrete-time Signal &
System
| Define at equally
q y spaced
p discrete value of time
| Represented as a sequence of numbers

| The sequence is denoted as x[n]; where n is an

integer in the range of -∞ to ∞.
| A discrete time is represented as {x(n)}

{ ( )} = {{... 0.95, -0.2, 2.17, 1.1, 0.2, -3.67, 2.9 ...}}

{x(n)}

| Arrow indicate time index, n = 0

Digital Signal Processing Lecture 1 16

Definition of Discrete-time Signal &
System
| The discrete-time signal is obtained by
periodically
i di ll sampling li a continuous-time
i i signal
i l at
uniform time interval.
| The sampling interval or period is denoted as Ts.
Th the
Thus th sampling li frequency
f can be
b d defined
fi d as
reciprocal of Ts, namely,
Fs = 1 / Ts.
| When the analog is sampled at certain period of
time, the discrete-time signal can be written as
below :-
[ ] = xa[t]
x[n] [ ] = xa[nT
[ Ts],
] n = …,-2,-1,0,1,2,...
2 1012

Digital Signal Processing Lecture 1 17

Definition of Discrete-time Signal &
System
| Periodic Sampling
p g of an analog
g signal
g is shown
below:

Digital Signal Processing Lecture 1 18

Operation on Sequence
| If the input
p signal
g to the systems
y is DTS,, the
output of the systems will be DTS.

INPUT SYSTEM OUTPUT

x[n] y[n]

Digital Signal Processing Lecture 1 19

Operation on Sequence
| Product/modulation
w1[n] = x[n].y[n]
| Multiplication/scaling

w2[n] = Ax[n]
| Addition

w3[n] = x[n] + y[n]

Digital Signal Processing Lecture 1 20

Operation on Sequence
| Time shiftingg
w4[n] = x[n – N] , N is an integer
y If N > 0 ; it’s a delay operation ; is a unit delay
y If N < 0 ; its an advance operation
w5[n] = x[n + 1] ; is a unit advance
| Time reversal

w6[n] = x[- n]

Digital Signal Processing Lecture 1 21

SEQUENCE REPRESENTATION
| Unit sample/unit
p
impulse
δ[n] = {1, n = 0;
0 n≠0}
0,
y Unit sample shifted by
k samples is
δ[n k ] =
δ[n- {1,
{1 n = k;
0, n ≠ k}

Digital Signal Processing Lecture 1 22

 SEQUENCE REPRESENTATION
| Unit Step
p
µ[n] = {1, n ≥ 0;
0, n < 0 }
y Unit step shifted by k
samples is
µ[n - k] = {1,
{1 n ≥ k;
0, n < k }

Digital Signal Processing Lecture 1 23

Sequence Representation
| Unit sample
p and unit step
p are related as follows

∞ n
µ[n] = ∑ δ [n − m] = ∑ δ [k ]
m =0 k = −∞

δ [ n ] = µ [ n ] − µ [ n − 1]

Digital Signal Processing Lecture 1 24

Sequence Representation
| Sinusoidal

x[n] = A cos(ωo n + φ ), − ∞ < n < ∞

Digital Signal Processing Lecture 1 25

Sinusoidal

Digital Signal Processing Lecture 1 26

Digital Signal Processing Lecture 1 27
Sequence Representation
| Real Exponential
p
| x[n] = Aαn

Digital Signal Processing Lecture 1 28

Sequence Representation
| Complex
p Exponential
p
x
| [n] = Aеjωn; ω frequency of complex exponential sinusoid, A
is a constant

Digital Signal Processing Lecture 1 29

Introduction to LTI System
| Discrete-time Systems
y
y Function: to process a given input sequence to
generate an output sequence

Discrete-time
Discrete time system
x[n] y[n]
Input Output
sequence sequence

Fig: Example of a single-input, single-output system

Digital Signal Processing Lecture 1 30

Introduction to LTI System Classification
of Discrete-time System

Linear DTS
[ ]
x[n] [ ]
y[n]
= αx1[n] + βx2[n] = αy1[n] + βy2[n]
Digital Signal Processing Lecture 1 31
Introduction to LTI System
Classification of Discrete-time System

Digital Signal Processing Lecture 1 32

Classification of Discrete-time System
Time Invariant system

Digital Signal Processing Lecture 1 33

Introduction to LTI System
Classification of Discrete-time System
| Causal System
y
y Changes in output samples do not precede changes in
input samples
y y[no] depends only on x[n] for n ≤ no
y Example:
y[n] = x[n]-x[n-1]
y

Digital Signal Processing Lecture 1 34

Introduction to LTI System
Classification of Discrete-time System
| Stable System
y
y For every bounded input, the output is also bounded
(BIBO)
y Is the y[n] is the response to x[n],
x[n] and if
|x[n]| < Bx for all value of n
then
|y[n]| < By for all value of n

Where Bx and By are finite positive constant

Digital Signal Processing Lecture 1 35

Introduction to LTI System Impulse and
Step Response
| If the input
p to the DTS system
y is Unit Impulse
p
(δ[n]), then output of the system will be
Impulse Response (h[n]).

| If the input to the DTS system is Unit Step (µ[n]),

then output of the system will be
S
Step R
Response (s[n]).
( [ ])

Digital Signal Processing Lecture 1 36

Impulse Response

Digital Signal Processing Lecture 1 37

Input-Output Relationship
|A Linear time
time-invariant
invariant system satisfied
both the linearity and time invariance
properties.
| An LTI discrete-time system is
characterized by its impulse response
| Example:
x[n] = 0.5δ[n+2] + 1.5δ[n-1] - δ[n-4]
will result in
y[n] = 0.5h[n+2] + 1.5h[n-1] - h[n-4]

Digital Signal Processing Lecture 1 38

Input-Output Relationship
| [ ] can be expressed
x[n] p in the form
∞
x[ n ] = ∑ x[ k ]δ [ n − k ]
k = −∞
where x[k] denotes the kth sample of sequence {x[n]}
| The response to the LTI system is
∞ ∞
y[n] = ∑ x[k ]h [n − k ] = ∑ x[n − k ]h [k ]
k = −∞ k = −∞
or represented
t d as

y[n] = x[n] ∗ h [n]

Digital Signal Processing Lecture 1 39
Computation of Discrete Convolution

Digital Signal Processing Lecture 1 40

Input-output Relationship
| Properties of convolution
y Commutative

x1[n] ∗ x2 [n] = x2 [n] ∗ x1[n]

y Associative

x1[n]∗(x2[n] + x3[n]) = x1[n]∗ x2[n] + x1[n]∗ x3[n]

y Di t ib ti
Distributive
( x1[n] ∗ x2 [n]) ∗ x3 [n] = x1[n] ∗ ( x2 [n] ∗ x3 [n])

Digital Signal Processing Lecture 1 41

Properties of LTI Systems
| Stability
y if and only if, sum of magnitude of Impulse
Response, h[n] is finite

∞
S = ∑ | h [n] | < ∞
n = −∞

Digital Signal Processing Lecture 1 42

Properties of LTI Systems
| Causality
y
y if and only if Impulse Response,h[n] = 0 for all n < 0

Digital Signal Processing Lecture 1 43

Properties of LTI Systems

Digital Signal Processing Lecture 1 44