Transcript Operation in System
Operation in System
Hany Ferdinando Dept. of Electrical Engineering Petra Christian University
General Overview
Convolution both in continuous- and discrete-time system De-convolution in discrete-time system Operation in system 2
Convolution (discrete)
The output of a system can be analyzed with impulse response sequence Impulse response is response of a system due to an impulse sequence as input d n h n means ‘input d n c{ d n } c{h n } c{ d n ±m } c{h n ±m } gives output h n ‘ Operation in system 3
Convolution (discrete)
For a signal u k , we can write
u k
...
2 d
k
2 1 d
k
1
u o
d
k
u
1 d
k
1 ...
u k
j
u j
d
k
j
u j {d k-j } gives output u j {h k-j } Applying the superposition property, the total response is merely the sum of the individual response of the u j {h k-j } Operation in system 4
Convolution (discrete)
Thus the output sequence is
y k
j
u j h k
j
u k
*
h k
If we let m = k-j, then
y k
m
u k
m h m
m
h m u k
m
h k
*
u k
Operation in system 5
Convolution Operation
Calculate the sequence for every k
y
1
j
u j h
1
j
,
y
2
j
u j h
2
j
,
y
3
j
u j h
3
j
,...
To find y 1 , we need h 1-n , then the shifted h 1-n is multiplied with u For others k, the procedures are similar Operation in system 6
Example and Exercise
h(n) = ( ½) n for n ≥ 0 (even) and 0 for n is odd, u(n) = {1,2} do it with u*h and h*u h(n) = {1,2,1}, u(n) = {1,2,1} h(n) = ( ½) n for n ≥ 0, u(n) = (¼) n 0 for n ≥ … (get other exercises from the books) Operation in system 7
Convolution (discrete)
If both signal is positive semi infinite, then we can use table matrix to calculate the convolution Place the values of one signal at the top row and the other at the left most column Be careful!! You have to verify the first result…!!
Operation in system 8
Convolution (continuous)
To derive procedure for convolution in continuous-time system is similar to that of discrete-time system The input is decomposed into a sum of impulse function, then express the output as a sum of the response resulting from individual impulses Operation in system 9
Convolution Operation
The formula is
y
(
t
)
u
( t )
h
(
t
t )
dt
Remark: Inside the integral, ‘t’ is transformed into ‘ t ’ h(t) h( t ) and u(t) u( t ) Get h( t ) and shift it to the right to get h(t t ) Operation in system 10
Convolution Operation
Always draw the signals before convolving them, this is important to get the limit for integration It is calculated based on the range, e.g. 0 Operation in system 11 Convolve h(t) = 1 for 0 ≤t≤2 with u(t) = t for t≥0 h(t) = 1 for 0 ≤t≤1 and -1 for 1≤t≤2, u(t) = t for 0 ≤t≤2 Operation in system 12 It is how to ‘undo’ convolution From the output y and input u relationship, one can derive the impulse response h , etc. Application: To find transfer function of a system To measure the linearity of unknown system Operation in system 13 To find u : To find h : u k y k m 1 k 0 u m h k m h 0 h k y k m 1 k 0 h m u k m u 0 Operation in system 14 Input u = {1, ½} and output y k ≥ 0. Find h ! = ( ½) k for Operation in system 15 The operations in system are discussed! Students have to exercise themselves in order to understand those operation well. The next topic is Fourier analysis. Read the Signals and Linear Systems by Robert A. Gabel (p. 239-255) or Modern Signals and System by Huibert Kwakernaak (p. 330-367) or Signals and System by Alan V. Oppenheim (p.161-179) Operation in system 16Examples and Exercises
Deconvolution (discrete only)
Deconvolution formulation
Exercise
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