Explain frequency warping/wrapping effect in designing IIR filter using BLT method.
1 Answer

BIlinear Transformation Technique (BLT) is a technique for designing Discrete Time IIR filters from Continuous Time filters. BLT avoids the problem of aliasing encountered with the use of Impulse Invariance Technique (IIT). BLT maps the entire imaginary axis of the s-plane onto the unit circle in the Z-plane.

In BLT, the relation between the analog frequency ($\Omega$) and corresponding digital frequency ($\Omega$) is $\Omega=\frac{2}{T} \tan \frac{\omega}{2}$ or $\omega=2 \tan ^{-1} \frac{\Omega T}{2}$.

As "Tan inverse" is a non-linear function, it causes nonlinear compression of the frequency axis. This non-linear mapping which introduces a distortion in the frequency axis is called Frequency Warping.

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The design of DT filters using the BLT is useful only when this distortion can be tolerated or compensated for, as in the case of filters that approximate ideal piece wise constant magnitude response characteristics. If the critical frequencies of the CT filter are pre-warped according to above equation then, when the CT filter is transformed to the DT filter, the DT filter will meet the desired specifications.

Due to Frequency Warping, phase response of analog filter is not preserved but magnitude response can be preserved by pre-warping analog frequencies. Thus, a DT low-pass filter with a linear phase characteristic cannot be obtained by applying the BLT to a CT low pass filter with a linear phase characteristic.

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