Frequency warping

• The bilinear transformation method has the following important features: t A stable analog filter gives a stable digital filter. t The maxima and minima of the amplitude response in the analog filter are preserved in the digital filter. As a consequence, – the pass band ripple, and – the minimum stop band attenuation of the analog filter are preserved in the digital filter frame.

• To determine the frequency response of a continuous-time filter, the transfer function $H_a(s)$ is evaluated at $s=j\omega$ which is on the $j\omega$ axis. Likewise, to determine the frequency response of a discrete-time filter, the transfer function $H_d(z)$ is evaluated at $z=e^{j\omega T}$ which is on the unit circle, $|z|=1$ .

• When the actual frequency of $\omega$ is input to the discrete-time filter designed by use of the bilinear transform, it is desired to know at what frequency,$\omega_a$ , for the continuous-time filter that this $\omega$ is mapped to.

$$H_d(z)=H_a(\dfrac 2T\dfrac {z-1}{z+1}) $$

$H_d(e^{j\omega T}= H_a(\dfrac 2T\dfrac {e^{j\omega T}-1}{e^{j\omega T}+1}) \\= H_a(\dfrac 2T. \dfrac {e^{j\omega T/2}(e^{j\omega T/2}-e^{-j\omega T/2})}{e^{j\omega T/2}(e^{j\omega T/2}+e^{-j\omega T/2})}) \\ =H_a (\dfrac 2T. \dfrac {(e^{j\omega T/2}-e^{-j\omega T/2})}{(e^{j\omega T/2}+e^{-j\omega T/2})}) \\ =H_a(j\dfrac 2T. \dfrac {(e^{j\omega T/2}-e^{-j\omega T/2})/(2j)}{(e^{j\omega T/2}+e^{-j\omega T/2})/2}) \\ =H_a (j\dfrac 2T .\dfrac{\sin (\omega T/2)}{\cos(\omega T/2)}) \\ =H_a(j\dfrac 2T.\tan (\omega T/2))$

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

$\omega_a=\dfrac 2T\tan (\omega\dfrac T2)$

and the inverse mapping is

$\omega=\dfrac 2T arc\tan (\omega_a\dfrac T2)$

• The discrete-time filter behaves at frequency the same way that the continuous-time filter behaves at frequency $(2/T)\tan (\omega T/2)$ . Specifically, the gain and phase shift that the discrete-time filter has at frequency $\omega$ is the same gain and phase shift that the continuous-time filter has at frequency $(2/T)\tan (\omega T/2)$ . 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 $\omega \ll 2/T$ or $\omega_a \ll2/T ), \omega \approx \omega_a .$

One can see that the entire continuous frequency range

$-\infty \lt \omega_a \lt + \infty$

is mapped onto the fundamental frequency interval

$-\dfrac \pi T \lt \omega \lt +\dfrac \pi T \\ \omega=\pm \pi/T.\omega_a = \pm\infty$

One can also see that there is a nonlinear relationship between $\omega_a$ and $\omega$ This effect of the bilinear transform is called frequency warping