Frequency Warping
To determine the frequency response of a continuous-time filter, the transfer function is evaluated at which is on the axis. Likewise, to determine the frequency response of a discrete-time filter, the transfer function is evaluated at which is on the unit circle, . When the actual frequency of is input to the discrete-time filter designed by use of the bilinear transform, it is desired to know at what frequency, for the continuous-time filter that this is mapped to.
This shows that every point on the unit circle in the discrete-time filter z-plane, is mapped to a point on the axis on the continuous-time filter s-plane, . That is, the discrete-time to continuous-time frequency mapping of the bilinear transform is
and the inverse mapping is
The discrete-time filter behaves at frequency the same way that the continuous-time filter behaves at frequency . Specifically, the gain and phase shift that the discrete-time filter has at frequency is the same gain and phase shift that the continuous-time filter has at frequency . 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 or ), .
One can see that the entire continuous frequency range
is mapped onto the fundamental frequency interval
The continuous-time filter frequency corresponds to the discrete-time filter frequency and the continuous-time filter frequency correspond to the discrete-time filter frequency
One can also see that there is a nonlinear relationship between and 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 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.
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 frequency if the following transform is substituted into the continuous filter transfer function. This is a modified version of Tustin's transform shown above. However, note that this transform becomes the above transform as . That is to say, the above transform causes the digital filter response to match the analog filter response at DC.
The main advantage of the warping phenomenon is the absence of aliasing distortion of the frequency response characteristic, such as observed with Impulse invariance. It is necessary, however, to compensate for the frequency warping by pre-warping the given frequency specifications of the continuous-time system. These pre-warped specifications may then be used in the bilinear transform to obtain the desired discrete-time system.
Read more about this topic: Bilinear Transform
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