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

Queensland University of Technology

CRICOS No. 00213J

Announcements

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

• Assignment 1B released next week

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CRICOS No. 00213J

• 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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A Convolution Sum Example

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A Convolution Sum Example

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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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Numerical Convolution

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

%

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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Numerical Convolution

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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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CRICOS No. 00213J

Queensland University of Technology

CRICOS No. 00213J

Queensland University of Technology

CRICOS No. 00213J

Queensland University of Technology

CRICOS No. 00213J

Queensland University of Technology

CRICOS No. 00213J

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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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CRICOS No. 00213J

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.

Queensland University of Technology

CRICOS No. 00213J

Next Week

• Semester Break

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