DSP Fundamentals Explained

DIGITAL SIGNAL PROCESSING
CHAPTER 2
Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and
Buck, ©1999-2000 Prentice Hall Inc.

Introduction
Digital Signal Processing

Examples

Applications

Why DSP?

Textbook and References
Textbook:
 Oppenheim, A.V., Schafer, R.W, "Discrete-Time
Signal Processing", 2nd Edition, Prentice-Hall,
1999.
Reference Books:
 Digital Signal Processing (4th Edition), John G.
Proakis, Dimitris K Manolakis
 Vinay K. Ingle, John G. Proakis, “Digital Signal
Processing using MATLAB”, 2nd Ed., Thomson,
2007.

Course Outline
 Chapter # 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]

Definitions of DSP
 Signal
● A function of independent variables such as time,
distance, position, temperature and pressure
● 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.
● Or may be discrete such as temperature, stock, etc.
 Processing
● Operating in some fashion on signal to extract some
useful information

Definitions of DSP
 Digital SignalProcessing
● Concerned with the representation of signals by sequence
of numbers or symbols and the processing of these
sequence
● The purpose of such processing may be to estimate
characteristic parameters or transform a signal

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

Types of Signals

Definition of Discrete-time Signal &
System
 Define at equally spaced 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)}
{x(n)} = {... 0.95, -0.2, 2.17, 1.1, 0.2, -3.67, 2.9 ...}
 Arrow indicate time index, n = 0

System
 The discrete-time signal is obtained by periodically
sampling a continuous-time signal at uniform time
interval.
 The sampling interval or period is denoted as Ts.
Thus the sampling frequency can be defined 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 :-
x[n] = xa[t] = xa[nTs], n = …,-2,-1,0,1,2,...

System
 Periodic Sampling of an analog signal is shown
below:

Operation on Sequence
 If the input signal to the systems is DTS, the
output of the systems will be DTS.
INPUT
x[n]
OUTPUT
y[n]
SYSTEM

 Product/modulation
w1[n] = x[n].y[n]
 Multiplication/scaling
w2[n] = Ax[n]
 Addition
w3[n] = x[n] + y[n]

 Time shifting
w4[n] = x[n – N] , N is an integer
● If N > 0 ; it’s a delay operation ; is a unit delay
● If N < 0 ; its an advance operation
w5[n] = x[n + 1] ; is a unit advance
 Time reversal
w6[n] = x[- n]

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

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

Sequence Representation
 Unit sample and unit step are related as follows
 n
[n]  [n  m]  [k]
m0 k 
[n]  [n]  [n 1]

 Sinusoidal
),   n  
x[n]  Acos(on 

Sinusoidal

 Real Exponential
 x[n] = Aαn
28

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

Introduction to LTI System
 Discrete-time Systems
● Function: to process a given input sequence to
generate an output sequence
Discrete-time system
x[n]
Input
sequenc
e
y[n]
Output
sequenc
e
Fig: Example of a single-input, single-output system

Introduction to LTI System Classification of
Discrete-time System
Linear DTS
x[n]
= αx1[n] + βx2[n]
y[n]
= αy1[n] + βy2[n]

Classification of Discrete-time System

Classification of Discrete-time System Time
Invariant system

 Causal System
● Changes in output samples do not precede changes in
input samples
● y[no] depends only on x[n] for n ≤ no
● Example:
y[n] = x[n]-x[n-1]

 Stable System
● For every bounded input, the output is also bounded
(BIBO)
● Is the y[n] is the response to 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 positiveconstant

Introduction to LTI System Impulse and
Step Response
 If the input to the DTS system is Unit Impulse
(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
Step Response (s[n]).

Impulse Response

Input-Output Relationship
 A Linear time-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]

Input-Output Relationship
 x[n] can be expressed in the form

x[n]   x[k] [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 as
y[n]  x[n]h[n]

Computation of Discrete Convolution

Input-output Relationship
 Properties of convolution
● Commutative
x1[n] x2[n]  x2[n] x1[n]
● Associative
x1[n](x2[n]x3[n])x1[n]x2[n]x1[n]x3[n]
● Distributive
(x1[n] x2[n]) x3[n]  x1[n](x2[n] x3[n])

Properties of LTI Systems
 Stability
● if and only if, sum of magnitude of Impulse
Response, h[n] is finite

S  | h[n] |  
n

 Causality
● if and only if Impulse Response,h[n] = 0 for all n < 0

DSP Fundamentals Explained

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