Operation in System Hany Ferdinando Dept. of Electrical Engineering Petra Christian University.
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Operation in System
Hany FerdinandoDept. of Electrical EngineeringPetra Christian University
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Operation in system 2
General Overview
Convolution both in continuous- and discrete-time system
De-convolution in discrete-time system
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Operation in system 3
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
n hn means ‘input n gives output hn‘ c{n} c{hn} c{n±m} c{hn±m}
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Operation in system 4
Convolution (discrete)
For a signal uk, we can write
uj{dk-j} gives output uj{hk-j} Applying the superposition property,
the total response is merely the sum of the individual response of the uj{hk-j}
jjkjk
kkokkk
uu
uuuuu
...... 111122
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Operation in system 5
Convolution (discrete)
Thus the output sequence is
If we let m = k-j, then
kkj
jkjk huhuy *
kkm
mkmm
mmkk uhuhhuy *
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Operation in system 6
Convolution Operation
Calculate the sequence for every k
To find y1, we need h1-n, then the shifted h1-n is multiplied with u
For others k, the procedures are similar
,...,, 332211
j jjjjj
jjj huyhuyhuy
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Operation in system 7
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 for n ≥
0 … (get other exercises from the books)
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Operation in system 8
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…!!
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Operation in system 9
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
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Operation in system 10
Convolution Operation
The formula is Remark:
Inside the integral, ‘t’ is transformed into ‘’
h(t) h() and u(t) u() Get h(-) and shift it to the right to get
h(t-)
dtthuty )()()(
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Operation in system 11
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<t<1, 1<t<2, 2<t<3, etc.
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Operation in system 12
Examples and Exercises
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
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Operation in system 13
Deconvolution (discrete only)
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
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Operation in system 14
Deconvolution formulation
To find u:
To find h:0
1
0
h
huyu
k
mmkmk
k
0
1
0
u
uhyh
k
mmkmk
k
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Operation in system 15
Exercise
Input u = {1,½} and output y = (½)k for k ≥ 0. Find h!
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Operation in system 16
Next…
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)