Transcript
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LTI SYSTEMS
CONVOLUTION AND IMPULSE RESPONSE
The Lengths of Input and Output Sequences
Two interpretations of its computation
LINEAR BUT TIME-VARYING SYTEMS
FINITE/INFINITE IMPULSE RESPONSE SYSTEMS (FIR, IIR)
PROPERTIES OF LTI SYSTEMS
Convolution is commutative
Convolution is associative
Cascading LTI Systems
Parallel LTI Systems
BIBO Stability
Causal LTI Systems
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LTI System
T{.}
y n x nT
x n y n
IMPULSE RESPONSE AND CONVOLUTION
Linearity 1 2 1 2a x n b x n a x n b x n T T T
Time-invariance 0 0 x n y n x n n y n n T T .
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An input signal can be written as
[] ⋯ + [1][ + 1] + [0][] + [1][ 1] + ⋯
[][ ]=−
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Using the linearity and time-invariance of the system the output of the LTI system
can be written as
[] []ℎ[ ]=−
ℎ[][ ]
=−
where ℎ[] is the “impulse response” of the LTI system.
This operation is called convolution of
[], and
ℎ[].
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Derivation [] []
{ [][ ]
=− }
[][ ]=−
[]ℎ[ ]
=−
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Convolution is shown by a ′ ∗ ′,
[] [] ∗ ℎ[]
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It is easy to show that
[]ℎ[ ]=− ℎ[][ ]=−
i.e., [] ∗ ℎ[] ℎ[] ∗ []
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Note that
[] []ℎ[ ]
=−
ℎ[][ ]=− +
[] ∗ ℎ[]
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Ex:
[] ℎ[]⏟ [] + ℎ[ 1]⏟ [] or equivalently
[] [] + [ 1] + [ 2]
[]
……
ℎ[]
… …
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Ex: The output can be considered as the superposition of responses to individual
samples of the input.
The response to [2]
The response to [0]
The response to
[3]
The complete response
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To compute one output sample at a time:
For example, to get [3]
1) multiply the two sequences [] and ℎ[3 ]
2) Add the sample values of this product
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Ex:
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THE LENGTHS OF INPUT AND OUTPUT SEQUENCES
Suppose that the input signal and the impulse response have finite durations:
Input
[] : length
Imp. Resp. ℎ[] : length
Then the length of the output signal is + 1.
Output [] : length + 1
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Ex:
Input
[] : length is 6 samples
Imp. Resp. ℎ[] : length is 3 samples
Output [] : length is 8 samples
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CONVOLUTION IN MATLAB
>> help conv
conv Convolution and polynomial multiplication.
C = conv(A, B) convolves vectors A and B.
The resulting vector is lengthMAX([LENGTH(A)+LENGTH(B)-1,LENGTH(A),LENGTH(B)]).
If A and B are vectors of polynomial coefficients,
convolving them is equivalent to multiplying the two
polynomials.
C = conv(A, B, SHAPE) returns a subsection of the
convolution with
size
specified by SHAPE:'full' - (default) returns the full convolution,
'same' - returns the central part of the convolution
that is the same size as A.
'valid' - returns only those parts of the convolution
that are computed without the zero-padded edges.
LENGTH(C)is MAX(LENGTH(A)-MAX(0,LENGTH(B)-1),0).
Class support for inputs A,B:
float: double, single
See also deconv, conv2, convn, filter and,
in the signal Processing Toolbox, xcorr, convmtx.
Overloaded methods:
cvx/conv
gf/conv
gpuArray/conv
Reference page in Help browser
doc conv
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Exercise: Compare “conv” and “filter” commands
Exercise for enthusiastic readers: Study “deconv” command.
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Play and experiment with code1_LN3.m
clear all close all
h = [2 -3 1]; x = [1 0 0 0 -2];
% h = [2 -3 1 -1 4]; % x = rand(1,5); % uniformly distributed numbers from
[0,1] % x = 2*(rand(1,5)-0.5); % uniformly distributed numbers from
[-1,1]%
% x = rand(1,randi(7,1)) % x = rand(1,randi(7,1)) % uniformly distributed numbers from[0,1], signal length is also random y = conv(x,h)
nh = 0:length(h)-1; nx = 0:length(x)-1; ny = 0:length(y)-1;
mini = min([h x y]);
maxi = max([h x y]);
figure subplot(3,1,1); stem(nh,h,'linewidth',2); v = axis; v = [v(1)-1 length(y) mini maxi]; axis(v) subplot(3,1,2); stem(nx,x,'r','linewidth',2) v = axis;
v = [v(1)-1 length(y) mini maxi]; axis(v) subplot(3,1,3); stem(ny,y,'k','linewidth',2) v = axis; v = [v(1)-1 length(y) mini maxi]; axis(v)
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and also with code2_LN3.m
clear all close all
h = [2 -3 1 ] x = [1 0 0 0 -2] y = conv(h,x) yy = filter(h,1,x) yyy = filter(x,1,h)
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LINEAR BUT TIME-VARYING SYTEMS
If the system is linear but time-varying, the derivation on the first page yields
[] []ℎ[]=−
where ℎ[] is the response of the system to an impulse at time k , i.e, [ ].
(time-varying impulse response)
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LTV systems are used, for example, to model mobile communication channels.
In mobile communication channel impulse response changes due to the motion of
the transmitter and receiver, and also due to the change or moving objects in the
environment.
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FINITE/INFINITE IMPULSE RESPONSE SYSTEMS
If ℎ[] has a finite number of samples, i.e.,
ℎ[] 0 , < , > , <
then the system is said to be a Finite Impulse Response (FIR) system, otherwise an
Infinite Impulse Response (IIR) system.
FIR
IIR
n
h n
n
h n
1 N
2 N
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Ex:
FIR or IIR ?
[] 0.3 [] 2 [ 4]
[] 0.7 [ 1] []
[] 0.4 [] 2 [ 4] + 5
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PROPERTIES OF LTI SYSTEMS
CONVOLUTION IS COMMUTATIVE
[] ∗ ℎ[] ℎ[] ∗ []
[] ∗ ℎ[] []ℎ[ ]=−
Let
[ ]ℎ[]=−
ℎ[] ∗ []
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CONVOLUTION IS ASSOCIATIVE
([] ∗ []) ∗ [] [] ∗ ([] ∗ [])
Prove!
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CASCADING LTI SYSTEMS
[] ∗ ℎ[]
[] ([] ∗ ℎ[]) ∗ ℎ[]
[] ∗ (ℎ[] ∗ ℎ[])
[] ∗ (ℎ[] ∗ ℎ[])
([] ∗ ℎ[]) ∗ ℎ[]
If the systems are LTI, the order of cascade can be changed!
h1 h2 x n
h2 h1
x n
y n
n y
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Ex: Consider the following two systems described by
[] [] + [ 1] + [ 2]
and
[] [ ]=
[] + [ 1] + [ 2] + ⋯
They are LTI (check!), so their order is arbitrary in their cascade connection.
Show that their impulse responses are
ℎ[] [] + [ 3]
[] + [ 1] + [ 2]
ℎ[] []
What is the impulse response of their cascade?
Can you write a recursive description for the second system?
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Ex: Let two systems be described by
[] [ 2]
and
[] []
Their impulse responses
ℎ[] [ 2] and
ℎ[] [] ‼!
However, one of them is nonlinear so their order cannot be changed in their
cascade.
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PARALLEL LTI SYSTEMS
is equivalent to
h1[n]+h2[n] x n y n
h2[n]
h1[n]
x n y n+
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BIBO STABILITY
An LTI system is BIBO stable iff
|ℎ[]|=−
where is a finite constant.
i.e., impulse response is absolutely summable.
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Proof:
Sufficiency:
|[]| ℎ[][ ]
=− |ℎ[]| [ ]⏟ ≤
=−
|ℎ[]|
=−
so if ∑ |ℎ[]|=− then |[]|
Necessity: (by contradiction)
Assume that the impulse response is not absolutely summable, i.e. n
h n
|ℎ[]|=− → ∞
Also let
[] ℎ[]|ℎ[]| ℎ[] ≠ 0
0 ℎ[] 0
so that [] is bounded.
Now consider
[0] ℎ[][]=− ℎ[] ℎ[]|ℎ[]|
=− |ℎ[]|
=− → ∞
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FIR (LTI) SYSTEMS ARE ALWAYS STABLE
Why?
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CAUSAL LTI SYSTEMS
An LTI system is causal iff
ℎ[] 0 < 0
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