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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.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

Figure 1
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
Time Domain (fig2.png)
Figure 3
Frequency Domain
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
Figure 4 (fig4.png)

Example 1

2 point smoother

Figure 5
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
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
Figure 7 (fig7.png)
ie: smoothing ≡ lowpass filtering

Example 2

Figure 8
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
Figure 9 (fig9.png)
Figure 10
Figure 10 (fig10.png)
Figure 11
Figure 11 (fig11.png)
Figure 12
Figure 12 (fig12.png)
Figure 13
Figure 13 (fig13.png)

Example 3: Edge Detector

Figure 14
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
High Pass Filter (fig15.png)

Example 4: Circular Shift

Figure 16
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
Alll Pass Filter (fig17.png)
That is, delay ≡ "linear phase shift." (slope =-m2πN m 2 N

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