In this simulation we are going to explore the design of a recursive digital filter which might be used in a digital radio station. The sampling frequency used in digital radio (DAB) is either 48 kHz or 24 kHz and for the purposes of this simulation, we assume 24kHz. However, the sampling rate of the audio coming from a CD is 44.1 kHz so the radio station would need to numerically re-sample CD audio data at the DAB rate. Before this can be done, all frequencies above half the new sampling frequency would first have to be filtered out to prevent aliasing when the change in sampling frequency is subsequently carried out. For the purposes of this experiment, we will assume this anti-alias filtering is to be done using a recursive low pass filter derived from the Butterworth analogue prototype (using the bilinear transformation).
We wish the digital filter to have a gain of -96dB at 12 kHz (i.e. half the sampling frequency we will be changing to after the filter has done its work) so that any residual components will be below the quantisation noise of the 16-bit representation used by CD. However, this one point of reference, as it stands, is not enough to design the filter because there are two parameters to be determined: thefilter's order and its -3 dB "corner" frequency. We therefore need another point on its amplitude response curve. For the purposes of this simulation we choose that the gain of the digital filter at 7.5 kHz will be -1 dB.
Because we are using the bilinear transformation to design the digital filter, we first need to design the frequency-warped analogue prototype. As preparation for the simulation(s), the student is required to carry out the following:
1. Use the standard frequency-warping formula to determine the frequencies at which the analogue prototype must have the gains of -1 dB and -96 dB (remember, at this point the sampling frequency is still 44.1kHz).
2. Using these results and the formula for the amplitude response of a Butterworth filter (see below), determine the order and -3 dB frequency of the warped analogue prototype.
3. Use the frequency warping formula to calculate the -3 dB frequency of the resulting digital filter.
4. Prepare a Microsoft-Word document which shows and explains the mathematical logic of stages 1-3 above and gives the numerical results obtained.
Note The amplitude response of a Butterworth filter can be written as:
where N is the filter order and f3 is the -3 dB frequency. You can confirm the latter by calculating GdB(f3).