Frequency transformation on IIR filters.
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There are two methods of Frequency Transformation in Infinite Impulse Response (IIR) filters.

Impulse Invariance Technique (IIT):

In IIT the impulse response of the CT system is sampled to produce the impulse response of the DT system.

Frequency Response of DT system $H(\omega)=F_{s} \sum_{k=-\infty}^{\infty} H_{a}\left[(\omega-2 \pi k) F_{s}\right]$

The frequency response $\mathrm{H}(\omega)$ of the DT system is a sum of shifted copies of the frequency response $H_{a}(\omega)$ of the CT system. If the CT system is band-limited to a frequency less than the Nyquist frequency $F_{s}$ of the sampling, then $\mathrm{H}(\omega)$ will be approximately cqual to $H_{a}(\omega)$ for frequencies below the $F_{s}$.

In IIT, the frequency of Digital filter $\omega=\Omega T_{s}=\frac{\Omega}{F_{s}},$ where $\Omega$ is the analog frequency.,

The relation between $\mathrm{CT}$ and $\mathrm{DT}$ frequency is linear. So, except for aliasing, the shape of the frequency response is preserved. The mapping of points from the s-plane to the $z-$ plane is given by the relation $z=e^{s T} .$ In IIT there is many to one mapping of poles from s-plane to z-plane.

Bilinear Transformation Technique (BLT):

BLT is a conformal mapping which converts imaginary axis of s-plane into unit circle in z-plane. It is one to one mapping between s-plane and $z-$ plane. There is no aliasing effect in BLT.

In $\mathrm{BLT}$ , the relation between the analog frequency ($\Omega$) and corresponding digital frequency $(\omega)$ is

$\Omega=\frac{2}{T} \tan \frac{\omega}{2}$ or $\omega=2 \tan ^{-1}\left(\frac{\Omega T}{2}\right)$

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

So, the design of discrete-time 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 piecewise constant magnitude response characteristics.

Due to Frequency Warping, phase response of analog filter cannot be preserved but magnitude response can be preserved by pre-warping analog frequencies.

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