Editing Bilinear transform (section)

== Frequency warping ==
To determine the frequency response of a continuous-time filter, the [[transfer function]] <math> H_a(s) </math> is evaluated at <math>s = j \omega_a </math> which is on the <math> j \omega </math> axis.  Likewise, to determine the frequency response of a discrete-time filter, the transfer function <math> H_d(z) </math> is evaluated at <math>z = e^{ j \omega_d T} </math> which is on the unit circle, <math> |z| = 1 </math>.  The bilinear transform maps the <math> j \omega </math> axis of the ''s''-plane (which is the domain of <math> H_a(s) </math>) to the unit circle of the ''z''-plane, <math> |z| = 1 </math> (which is the domain of <math> H_d(z) </math>), but it is '''not''' the same mapping <math> z = e^{sT} </math> which also maps the <math> j \omega </math> axis to the unit circle.  When the actual frequency of <math> \omega_d </math> is input to the discrete-time filter designed by use of the bilinear transform, then it is desired to know at what frequency, <math> \omega_a </math>, for the continuous-time filter that this <math> \omega_d </math> is mapped to.

:<math>H_d(z) = H_a \left( \frac{2}{T} \frac{z-1}{z+1}\right) </math>

:{|
|-
|<math>H_d(e^{ j \omega_d T}) </math>
|<math>= H_a \left( \frac{2}{T} \frac{e^{ j \omega_d T} - 1}{e^{ j \omega_d T} + 1}\right) </math>
|-
|
|<math>= H_a \left( \frac{2}{T} \cdot \frac{e^{j \omega_d T/2} \left(e^{j \omega_d T/2} - e^{-j \omega_d T/2}\right)}{e^{j \omega_d T/2} \left(e^{j \omega_d T/2} + e^{-j \omega_d T/2 }\right)}\right) </math>
|-
|
|<math>= H_a \left( \frac{2}{T} \cdot \frac{\left(e^{j \omega_d T/2} - e^{-j \omega_d T/2}\right)}{\left(e^{j \omega_d T/2} + e^{-j \omega_d T/2 }\right)}\right) </math>
|-
|
|<math>= H_a \left(j \frac{2}{T} \cdot \frac{ \left(e^{j \omega_d T/2} - e^{-j \omega_d T/2}\right) /(2j)}{\left(e^{j \omega_d T/2} + e^{-j \omega_d T/2 }\right) / 2}\right) </math>
|-
|
|<math>= H_a \left(j \frac{2}{T} \cdot \frac{ \sin(\omega_d T/2) }{ \cos(\omega_d T/2) }\right) </math>
|-
|
|<math>= H_a \left(j \frac{2}{T} \cdot \tan \left( \omega_d T/2 \right) \right) </math>
|}

This shows that every point on the unit circle in the discrete-time filter z-plane, <math>z = e^{ j \omega_d T}</math> is mapped to a point on the <math>j \omega</math> axis on the continuous-time filter s-plane, <math>s = j \omega_a</math>. That is, the discrete-time to continuous-time frequency mapping of the bilinear transform is

:<math> \omega_a = \frac{2}{T} \tan \left( \omega_d \frac{T}{2} \right) </math>

and the inverse mapping is

:<math> \omega_d = \frac{2}{T} \arctan \left( \omega_a \frac{T}{2} \right). </math>

The discrete-time filter behaves at frequency <math>\omega_d</math> the same way that the continuous-time filter behaves at frequency <math> (2/T) \tan(\omega_d T/2) </math>.  Specifically, the gain and phase shift that the discrete-time filter has at frequency <math>\omega_d</math> is the same gain and phase shift that the continuous-time filter has at frequency <math>(2/T) \tan(\omega_d T/2)</math>.  This means that every feature, every "bump" that is visible in the frequency response of the continuous-time filter is also visible in the discrete-time filter, but at a different frequency.  For low frequencies (that is, when <math>\omega_d \ll 2/T</math> or <math>\omega_a \ll 2/T</math>), then the features are mapped to a ''slightly'' different frequency; <math>\omega_d \approx \omega_a </math>.

One can see that the entire continuous frequency range

: <math> -\infty < \omega_a < +\infty </math>

is mapped onto the fundamental frequency interval

: <math> -\frac{\pi}{T} < \omega_d < +\frac{\pi}{T}. </math>

The continuous-time filter frequency <math> \omega_a = 0 </math> corresponds to the discrete-time filter frequency <math> \omega_d = 0 </math> and the continuous-time filter frequency <math> \omega_a = \pm \infty </math> correspond to the discrete-time filter frequency <math> \omega_d = \pm \pi / T. </math>

One can also see that there is a nonlinear relationship between <math> \omega_a </math> and <math> \omega_d.</math>  This effect of the bilinear transform is called '''frequency warping'''. The continuous-time filter can be designed to compensate for this frequency warping by setting <math> \omega_a = \frac{2}{T} \tan \left( \omega_d \frac{T}{2} \right) </math> for every frequency specification that the designer has control over (such as corner frequency or center frequency).  This is called '''pre-warping''' the filter design.

It is possible, however, to compensate for the frequency warping by pre-warping a frequency specification <math> \omega_0 </math> (usually a resonant frequency or the frequency of the most significant feature of the frequency response) of the continuous-time system.  These pre-warped specifications may then be used in the bilinear transform to obtain the desired discrete-time system. When designing a digital filter as an approximation of a continuous time filter, the frequency response (both amplitude and phase) of the digital filter can be made to match the frequency response of the continuous filter at a specified frequency <math> \omega_0 </math>, as well as matching at DC, if the following transform is substituted into the continuous filter transfer function.<ref>{{cite book |last=Astrom |first=Karl J. |date=1990 |title=Computer Controlled Systems, Theory and Design |edition=Second |publisher=Prentice-Hall |page=212 |isbn=0-13-168600-3}}</ref> This is a modified version of Tustin's transform shown above.

:<math>s \leftarrow \frac{\omega_0}{\tan\left(\frac{\omega_0 T}{2}\right)} \frac{z - 1}{z + 1}.</math>

However, note that this transform becomes the original transform

:<math>s \leftarrow \frac{2}{T} \frac{z - 1}{z + 1}</math>

as <math> \omega_0 \to 0 </math>.

The main advantage of the warping phenomenon is the absence of aliasing distortion of the frequency response characteristic, such as observed with [[Impulse invariance]].