Digital Signal Processing

Chapter 2
Discrete Time Signals & Systems

Digital Signal Processing – Dr. Lohrasbi – Spring 2016.

Ch.2 – Discrete Time Signals 2

Discrete Time Signal Representation

Concept of Finite sequence versus Infinite sequence

Ch.2 – Elementary Discrete Signals 3

Elementary Discrete Time Signals

Unit Impulse

Unit Step

Unit Ramp

Exponential Signal
Ch.2 – Amplitude & Phase Representation 4

Amplitude and Phase Representation

Point: The mathematical method we use for

analyzing discrete time signals and systems
depends to the characteristics of signals.
Ch.2 – Classification of Discrete Signals 5

a) Energy signals and Power signals

Classification of Discrete Time Signals b) Periodic and aperiodic signals
c) Symmetric (even) and Anisometric (odd) signals

Energy of Signals ??? For special interval:

Average Power of Signals ???

Energy and Power Signals

If the energy of a signal be finite then the signal is called Energy signal ➔ P = 0
If the power of a signal be finite and nonzero then the signal is called Power signal

Point: If E is infinite ➔ P may be finite or infinite.

 Ch.2 – Classification of Discrete Signals 6

Periodic and Aperiodic Signals

Example:

Point: Periodic signals have infinite energy and are power signal.

Symmetric and Antisymmetic signals

Signal Manipulation with FD & TD

Signal Scaling
 Ch.2 – Discrete Time Systems 7

Discrete Time Systems

We call the input signal is Transformed/operator/processing to the output signal. The system which is
worked on Discrete time signals are called discrete time system.

Mathematical relation between the

What is the input and output description of the system?
input and output of the system.

Examples of Discrete
Time Systems

Some system depend on current value

some on past some on future…
 Ch.2 – Block Diagram of DTS 8

Concept of Initial Condition

Initially Relaxed: If the system had no excitation for n < n0 then the initial condition is zero
➔ y(n) = 0 for n < n0 and the system is initially relaxed.

Block Diagram of DT Systems Unit Advance

Signal Adder

Unit Delay

Constant Multiplier
Signal Multiplier
 Ch.2 – Classification of DTS 9

Classification of DT System

A system is called static (Memoryless) if its output at any time instant n depends at
Static VS Dynamic most on the input at the same time and not on the past or future samples.

Time invariant VS Time invariant (shift invariant): if Input-Output

Time variant characteristics do not change with time

Linear VS Nonlinear A system is called linear if and only if for any two input signals we have

Extension to ➔
 Ch.2 – Classification of DTS 10

Examples of some important systems:

A system is called causal if its output at any time instant n depends only on
Casual Vs Noncausal
presents and past inputs.

Stable Vs Unstable A relaxed system is said to be bounded input-bounded output (BIBO) stable
if and only if every bounded input produces a bounded outputs.
 Ch.2 – Interconnection of DTS 11

Interconnection of DT Systems Parallel System

Cascade System

Point 1) If T1 and T2 be linear and time invariant then : Tc is

time invariant and T1T2 = T2T1

Point 2) It is important to note that in general →

Decomposition of input signals into the weighted sum of signals ➔

Our focus will be on elementary exponential input signals ➔

Ch.2 – Discrete LTI System 12

Analysis of Discrete Time Linear Time – Invariant System

General Form of LTI System

(Difference Equation) →

Convolution formula provide us with a means for computing the response of a relaxed,
Responses of LTI Systems linear Time invariant system for any arbitrary input signal.

Note that if the system be a variant system then we will need different h(n) for
different delays while for LTI systems you only a single h(n). ➔ Folding, Shifting,
Multiplication, Summation.
 Ch.2 – Discrete LTI System 13

Properties of Convolution & Interconnection of LTI Systems

Identity and shifting ➔

Commutative Law ➔

Associative Law ➔

So, any LTI system can be decomposed into the cascade interconnection of subsystem.

Distributive Law ➔

Generalization →
 Ch.2 – Causal & Stable LTI System 14

Causal LTI System

Causality can be discuss over the impulse response of the system. ➔

This is important for any Real time signal

processing. The reason is that the output must be
dependent to the only input signal:

An LTI system is causal if and only if its impulse response is zero for negative values of n.

Convolution for causal LTI system ➔

Stability of LTI System

An LTI system is stable if its impulse response absolutely summable ➔

 Ch.2 – LTI FIR & IIR Systems 15

LTI Finite and Infinite Impulse Response Systems

LTI system can be divided into the Finite duration Impulse response and Infinite duration Impulse response.

Causal FIR System Causal IIR System

The convolution sum and the impulse response of the systems gives idea about the implementation of the system.

FIR systems ➔ The realization involves additions, multiplications and finite number of memory locations.
IIR systems ➔ The realization involves infinite number of multiplication, additions and memories.

What should we do with IIR systems?

Those describes by difference equations are fine.
 Ch.2 – Recursive & Nonrecursive Systems 16

Recursive System
Cumulative Average of system.

Two multiplication, one addition, one memory ➔ recursive system: output depends on the past output values.

Square Root Algorithm

Ch.2 – Recursive & Nonrecursive Systems 17

Nonrecursive System

Causal LTI FIR System

Difference between recursive and non recursive system:

1) Feedback loop with delay

2) The output of nonrecursive can be computed in any

order like y(100) y(24) y(12) but the output of
recursive need to be calculated in order like: y(0) ,
y(1), y(2) , …
Ch.2 – LCCDE LTI Systems 18

LCCDE LTI Systems

Linear Time Invariant System Characterized by Constant Coefficient Difference Equation

is a subclass of recursive and nonrecursive system

The response has two parts including the

effect of initial response and input signal.

If a_k = 0 ➔ special case: Moving Average (MA) ➔ View the more recent M+1 inputs
Ch.2 – Solution of LCCDE Systems 19

Solution of LCCDE LTI Systems

Homogeneous Solution of LCCDE ➔

Particular Solution of LCCDE ➔

Point: Zero input response is due to the memory of system. Depends on the
nature of the system and initial condition.

Law1: A recursive system described by the linear difference equation above

satisfy the linearity conditions and is linear and time invariant.

Law2: The recursive system described above is stable if for every bounded input and
bounded initial condition, the total system response is bounded.
 Ch.2 – Example of 1st Order LCCDE 20

Example: First order difference equation

This system has constant coefficient versus the cumulative averaging system

First order difference equation is the

simplest version of recursive system
in the general class of LCCDE.

Forced response or zero state response :

The impulse response is :

Zero input response (natural response or free response) ➔

x(n) = 0 for all n but we have initial value and nonrelaxed
 Ch.2 – Implementation of DTS 21

Implementation of Discrete Time System

Cost, Hardware limitation, size limitation, power requirement.

Structures for realization of LIT systems;

Example : 1st order system ➔

Direct I : use separate delay ➔ Two LTI system in

Cascade (Non recursive)

Direct II: Change the order of the system

(recursive case):
Ch.2 – Implementation of DTS 22

Generalization to Direct I: Generalization Direct II:

It needs less delay compare to the last one

M+N delay and M+N+1 multiplications M+N+1 Multiplication , Max(M,N) delay
 Ch.2 – Implementation of DTS 23

Example : 2nd order system ➔

Second order system is important as it can be used as building block for realization of the system:
 Ch.2 – Correlation of DT signals 24

Correlation of Discrete Time Signals

Measure the degree to which the two signals are similar and extract information for some
special applications in radar, sonar, geology, digital communications, and other.

Similarities to Convolution

Examples radar and digital communication:

Ch.2 – Correlation of DT signals 25

Cross Correlation of Signals

Auto Correlation of Signals

 Ch.2 – Correlation of DT signals 26

Properties of the Autocorrelation & Crosscorrelation Sequences

Energy of signal :

Normalized sequences
Ch.2 – Correlation of DT signals 27

Correlation of Periodic Signals

If x(n) and y(n) be power signals then ➔

For Periodic Signal with Period N : ➔ results are periodic with normalization factor 1/N
Ch.2 – Correlation of LTI System 28

Correlation of LTI Systems

Input output relationship in correlation domain for LTI system

Input-Output Auto Correlation

Input-Output Cross Correlation

Autocorrelation exist if the system be stable.

 Digital Signal Processing
Chapter 3
The z-Transform and Its Application
to the Analysis of LTI Systems

Digital Signal Processing – Dr. Lohrasbi – Spring 2016.

 Ch.3 – The z-Transform 30

Direct Z-Transform of Discrete Time Signal

Convert time domain → Complex Plane

Region of Convergence (ROC)

What is the z-transform
existence condition? The set of all values of z for which
x(z) attains a finite value.

Example : The ROC of finite duration

signals are the entire z plan.

Example : Calculate the z-transform of

Ch.3 – The z-Transform 31

ROC for X(z) is equivalent to determining the

range of values of r for which the sequence
is absolutely summable.

ROC Analysis??

Convergence requires both sum to be finite

 Ch.3 – The z-Transform 32

Three Important Examples Analysis

Example 1.

Example 2.

Ambiguity of z-Transform?????

The ambiguity can be resolved only if in addition to closed-form

expression, the ROC be specified.
 Ch.3 – The z-Transform 33

Example 3.

X(z) doesn’t exist X(z) is available

Point: If there is a ROC for an infinite-duration two-sided signal,

it is a ring (annular region) in the z-plane.
Ch.3 – The z-Transform 34

Infinite & Finite Duration Signals

If there is a ROC for an infinite-duration two-sided signal, it is a ring (annular

region) in the z-plane. The ROC of a signal depends both on its duration
(finite or infinite) and on whether it is causal, anti-causal, or two-sided.

Signals are called right-sided, left sided, finite duration two-sided signals.

Two sided or bilateral z transform to

distinguish it from one sided or unilateral.
 Ch.3 – The z-Transform Properties 35

The Z-Transform Properties

The power of z-transform for discrete time signals is due to its powerful properties.

Linearity:

Point: When we combine several z transform, the ROC of overall transform is,
at least, the intersection of the ROC of the individual transforms.

Time Shifting:

The properties of linearity and time shifting make it more powerful

tool for analysis of discrete time LTI systems.
 Ch.3 – The z-Transform Properties 36

Z-Domain Scaling

Polar format ➔

Time Reversal

Z-Domain
Differentiation

Point: Both transforms have the same ROCs.

 Ch.3 – The z-Transform Properties 37

Convolution Law

Point: The ROC of X(z) is, at least, the intersection of that for X1(z) and X2(z).

Point: Convolution is an important property as it convert the convolution of two

signals in time domain into the multiplication of their transforms.

Multiplication Law

Point: The ROC of X(z) can be obtained as follow:

 Ch.3 – The z-Transform 38

Correlation Law

Point: The ROC of X(z) is, at least, the intersection of that for X1(z) and X2(z-1).

Parseval’s Relation

Initial Value Theorem If x(n) is causal [i.e, x(n) = 0 for all n < 0], then we have

Proof: Since x(n) is causal →

 Ch.3 – Important z-Transforms 39

Important Z-Transforms
Signal Z-transform ROC

Point: Rational z-transforms are

encountered not only as z-transform
of various important signals but also
in the characterization of discrete
time LTI systems described by
constant coefficient difference
equations (LCCDE systems).
 Ch.3 – Rational z-Transform 40

Rational Z-Transforms Analysis

Rational z-Transform
Representation →

finite zeros
where
Rational z-Transform
Pole-Zero Modeling Form→
finite poles

Point: We have different number of finite and infinite zeros and poles depending on M and N.

Point: A z-transform can be shown in pole-zero plot by circle and cross for
them correspondingly.
Ch.3 – Rational z-Transforms 41

Pole – Zero Pattern and a Graph of |X(z)|

 Ch.3 – Pole location and Time Domain Relation 42

Pole Location & Time Domain Behavior of Causal Signals

Ch.3 – Pole location and Time Domain Relation 43
Point: Pair of complex conjugate poles corresponds to
causal signals with oscillatory behavior
Ch.3 – Pole location and Time Domain Relation
44
 Ch.3 – Pole location and Time Domain Relation 45

Double pair of complex conjugate poles

on the unit circle for causal signal →

Conclusion.1: For causal real signals with simple real poles or simple complex
conjugate poles inside or on the unit circle, are always bounded in the amplitude.

Conclusion.2 : A signal with a pole or a complex conjugate pair of poles near the origin
decays more rapidly than one associated with a pole near (but inside) the unit circle.
Hence the time behavior of signal depends strongly to the location of poles.

Point : Whatever was mentioned for causal signal is true for causal LTI systems. So, if a
pole of system be outside of the unit circle, the impulse response will be unbounded and
thus the system is unstable.
 Ch.3 – The Inverse of z-Transform 46

Inverse of Z-Transform

Contour Integration Power Series Expansion Partial Fraction Expansion

Inverse of Z-Transform with Contour Integration

The procedure for transforming from the z-domain

to the time domain is called inverse z-transform.

Apply the
Cauchy integral theorem
 Ch.3 – The Inverse of z-Transform 47

Inverse of Z-Transform with Power Series Expansion

Inverse of Z-Transform with Partial Fraction Expansion

Ch.3 – Pole location and Time Domain Relation 48

Distinct Poles

Real Poles ➔

Complex Poles ➔

Multiple Order Poles