lecture5 time domain system analysis

Queensland University of Technology

CRICOS No. 00213J

Lecture 5: Time-Domain System Analysis

ENB342 – Signal, Systems and Transforms

Karla Ziri-Castro Queensland University of Technology


CRICOS No. 00213J

Announcements

• Assignment 1A (due 9th April by 5pm)

• Assignment 1B released next week

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• Topic Today:

– Discrete-Time System Response

– Convolution

Overview

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System Response • Once the response to a unit impulse is known, the

response of any LTI system to any arbitrary excitation can be found

• Any arbitrary excitation is simply a sequence of amplitude-scaled and time-shifted impulses

• Therefore the response is simply a sequence of amplitude-scaled and time-shifted impulse responses


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Simple System Response Example


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More Complicated System Response Example

System

Excitation

System

Impulse

Response

System

Response


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The Convolution Sum

The response y n to an arbitrary excitation x n is of the form

y n L x 1 h n 1 x 0 h n x 1 h n 1 L

where h n is the impulse response. This can be written in

a more compact form

y n x m h n m m

called the convolution sum.


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A Convolution Sum Example


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Convolution Sum Properties Convolution is defined mathematically by

y n x n h n x m h n m m

The following properties can be proven from the definition.

x n A n n0 Ax n n0 Let y n x n h n then

y n n0 x n h n n0 x n n0 h n

y n y n 1 x n h n h n 1 x n x n 1 h n and the sum of the impulse strengths in y is the product of

the sum of the impulse strengths in x and the sum of the

impulse strengths in h.


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Convolution Sum Properties

Commutativity

x n y n y n x n Associativity

x n y n z n x n y n z n Distributivity

x n y n z n x n z n y n z n


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Numerical Convolution


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nx = -2:8 ; nh = 0:12 ; % Set time vectors for x and h

x = usD(nx-1) - usD(nx-6) ; % Compute values of x

h = tri((nh-6)/ 4) ; % Compute values of h

y = conv(x,h) ; % Compute the convolution of x with h

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% Generate a discrete-time vector for y

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ny = (nx(1) + nh(1)) + (0:(length(nx) + length(nh) - 2)) ;

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% Graph the results

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subplot(3,1,1) ; stem(nx,x,'k','filled') ;

xlabel('n') ; ylabel('x') ; axis([-2,20,0,4]) ;

subplot(3,1,2) ; stem(nh,h,'k','filled') ;

xlabel('n') ; ylabel('h') ; axis([-2,20,0,4]) ;

subplot(3,1,3) ; stem(ny,y,'k','filled') ;

xlabel('n') ; ylabel('y') ; axis([-2,20,0,4]) ;


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General Procedure 1. Construct a table for x[n] and h[n].

2. Write the definition and simplify it.

3. Evaluate y[n] for n = 0; 1; ;N1 + N2 - 2.

Note: • In step 3: We start at n = 0, and so the number of

points between n = 0; 1; ;N1 + N2 - 2 is N1 + N2 - 1.

• The mathematics will take care of the ‘flipping’ and

‘shifting’.

Perform Linear Convolution by hand


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Stability and Impulse Response

It can be shown that a discrete-time BIBO-stable system

has an impulse response that is absolutely summable.

That is,

h n n

is finite.


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System Interconnections

The cascade connection of two systems can be viewed as

a single system whose impulse response is the convolution

of the two individual system impulse responses. This is a

direct consequence of the associativity property of

convolution.


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System Interconnections

The parallel connection of two systems can be viewed as

a single system whose impulse response is the sum

of the two individual system impulse responses. This is a

direct consequence of the distributivity property of

convolution.


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Take Home Points

• Convolution: – Every LTI system is completely characterized by its impulse response.

– The response of any LTI system to an arbitrary input signal can be found by convolving the input signal with its impulse response.

• System Interconnections: – The impulse response of a cascade connection of LTI systems is the

convolution of the individual impulse responses

– The impulse response of a parallel connection of LTI systems is the sum of the individual impulse responses

• Stability: – A discrete-time LTI system is BIBO stable if its impulse response is

absolutely summable.


CRICOS No. 00213J

Next Week

• Semester Break

• Assignment 1A due Thursday 9th April by 5pm (submit online)

lecture5 time domain system analysis

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