Download - 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
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ny = (nx(1) + nh(1)) + (0:(length(nx) + length(nh) - 2)) ;
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% Graph the results
%
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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CRICOS No. 00213J
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)