lecture5 time domain system analysis
DESCRIPTION
Signals and systems lecture notesTRANSCRIPT
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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
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Queensland University of Technology
CRICOS No. 00213J
Announcements
• Assignment 1A (due 9th April by 5pm)
• Assignment 1B released next week
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Queensland University of Technology
CRICOS No. 00213J
• Topic Today:
– Discrete-Time System Response
– Convolution
Overview
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CRICOS No. 00213J 3/25/2015 4 3/25/2015 4
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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Queensland University of Technology
CRICOS No. 00213J 3/25/2015 5 3/25/2015 5
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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Queensland University of Technology
CRICOS No. 00213J 3/25/2015 8 3/25/2015 8
A Convolution Sum Example
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Queensland University of Technology
CRICOS No. 00213J 3/25/2015 9 3/25/2015 9
A Convolution Sum Example
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Queensland University of Technology
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A Convolution Sum Example
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Queensland University of Technology
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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
%
% Generate a discrete-time vector for y
%
ny = (nx(1) + nh(1)) + (0:(length(nx) + length(nh) - 2)) ;
%
% 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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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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Queensland University of Technology
CRICOS No. 00213J
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Queensland University of Technology
CRICOS No. 00213J
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Queensland University of Technology
CRICOS No. 00213J
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Queensland University of Technology
CRICOS No. 00213J
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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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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.
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Queensland University of Technology
CRICOS No. 00213J
Next Week
• Semester Break
• Assignment 1A due Thursday 9th April by 5pm (submit online)