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Frequency Response of an LSI System

Module by: Richard Baraniuk

Summary: An introduction to the frequency response of an LSI system.

Figure 1
fig1.png
H H: circulent matrix

h h: H H's 0th column

H H: DFT of h h; H=FHh H F H h contain equivalent info!

Effect of LSI system on an input x x is easy to describe in the Fourier domain (frequency domain)...

Figure 2
Time Domain
fig2.png
Figure 3
Frequency Domain
fig3.png
ie: Yk=HkXk Y k H k X k ; 0≤k≤N-1 0 k N 1

ie: pointwise multiplication

Figure 4
fig4.png

Example 1

2 point smoother

Figure 5
fig5.png
Compute frequency response H H if LSI system...

Hk=1N∑n=0N-1hnⅇ-ⅈ2πNkn=1N∑n=0N-112ⅇ-ⅈ2πNkn=1N1+ⅇⅈ2πNk=1Nⅇ-ⅈ2πNk2+ⅇⅈ2πNk2ⅇⅈ2πNk2=2NcosπNkⅇⅈπNk H k 1 N n N 1 0 h n 2 N k n 1 N n N 1 0 1 2 2 N k n 1 N 1 2 N k 1 N 2 N k 2 2 N k 2 2 N k 2 2 N N k N k (1)
|Hk|=2N|cosπNk| H k 2 N N k (2)
Figure 6
fig6.png
where 0 is low frequency, π is high frequency, 2π 2 is low frequency, and w k =2πnk w k 2 n k

Therefore it is a lowpass filter

Figure 7
fig7.png
ie: smoothing ≡ lowpass filtering

Example 2

Figure 8
fig8.png
ie: h h is a "square box".

The Frequency response is (using answer from test 1):

Hk=1NsinMπNksinπNkⅇ-ⅈπNM-1k H k 1 N M N k N k N M 1 k (3)
Where sinMπNksinπNk M N k N k is the "Dirichlet Kernel."

H0= H 0 ?

Note:

sin0=0 0 0 L'Hopitâl's rule to the rescue...
Hk|k=0=ddknumerator|k=0ddkdenominator|k=0=1N-MπNcosMπNk|k=0-πNcosπNk|k=0ⅇ0=1NM=MN k 0 H k k 0 k numerator k 0 k denominator 1 N k 0 M N M N k k 0 N N k 0 1 N M M N (4)
Figure 9
fig9.png
Figure 10
fig10.png
Figure 11
fig11.png
Figure 12
fig12.png
Figure 13
fig13.png

Example 3: Edge Detector

Figure 14
fig14.png
Hk=1N∑n=0N-1hnⅇ-ⅈ2πNkn=1N1-ⅇ-ⅈ2πNkn=1Nⅇⅈ2πNk2-ⅇ-ⅈ2πNk2ⅇ-ⅈ2πNk2=1N2ⅈsinπNkⅇ-ⅈπNk H k 1 N n 0 N 1 h n 2 N k n 1 N 1 2 N k n 1 N 2 N k 2 2 N k 2 2 N k 2 1 N 2 N k N k (5)
Figure 15
High Pass Filter
fig15.png

Example 4: Circular Shift

Figure 16
fig16.png
Hk=1N∑n=0N-1hnⅇ-ⅈ2πNkn=1Nⅇ-ⅈ2πNkm H k 1 N n 0 N 1 h n 2 N k n 1 N 2 N k m (6)
|Hk|=1N H k 1 N , ∡Hk=-2πNkm ∡ H k 2 N k m
Figure 17
Alll Pass Filter
fig17.png
That is, delay ≡ "linear phase shift." (slope =-m2πN m 2 N

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