7. A digital filter with a 3dB bandwidth of 0.25π is to be designed from the analog filter whose system response is H(S) = Ωc s + Ωc Use bilinear transformation and obtain H(z).
Question
- A digital filter with a 3dB bandwidth of 0.25π is to be designed from the analog filter whose system response is H(S) = Ωc s + Ωc Use bilinear transformation and obtain H(z).
Solution
The bilinear transformation is a method used to convert an analog filter into a digital filter. The transformation is given by the equation:
s = 2/T * (1 - z^-1) / (1 + z^-1)
where T is the sampling period.
Step 1: Substitute s in H(s) with the bilinear transformation equation:
H(s) = Ωc / (s + Ωc)
Substituting s, we get:
H(z) = Ωc / (2/T * (1 - z^-1) / (1 + z^-1) + Ωc)
Step 2: Simplify the equation:
H(z) = Ωc * (1 + z^-1) / (2/T + Ωc - Ωc*z^-1)
Step 3: Normalize the frequency:
The 3dB bandwidth of the digital filter is 0.25π, which corresponds to a normalized frequency of 0.25. The cutoff frequency Ωc of the analog filter should be chosen such that it corresponds to the same frequency after the bilinear transformation. This is given by the equation:
Ωc = 2/T * tan(π*0.25)
Substitute Ωc in H(z) to get the final system response of the digital filter.
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