IIR Filter design & frequency-warping

Discussion in 'Homework Help' started by sutton185, Apr 21, 2013.

  1. sutton185

    Thread Starter New Member

    Jan 6, 2009
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    Hi all,

    Currently struggling with this assignment. Any explained solutions would be greatly appreciated!

    Thanks in advance...

    In this simulation we are going to explore the design of a recursive digital filter which might be usedvin 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: the filter’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.
     
  2. sutton185

    Thread Starter New Member

    Jan 6, 2009
    4
    0
    Any help would be appreciated!

    I've found lots of differing information on equations for a standard frequency-warping formula. What should I be applying in this case?
     
  3. tshuck

    Well-Known Member

    Oct 18, 2012
    3,531
    675
    I'm not quite sure what your question is.

    Using the bilinear transform maps the s-plane over a unit circle in the z-domain, warping it by nature of the transform.

    I believe the question is asking for you to use the basic transform, without regard to warping...
     
  4. sutton185

    Thread Starter New Member

    Jan 6, 2009
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    0
    But question 1. begins... Using the standard frequency-warping formula...
     
  5. tshuck

    Well-Known Member

    Oct 18, 2012
    3,531
    675
    Yes, the standard bilinear transform has the effect of warping the frequency...
     
  6. sutton185

    Thread Starter New Member

    Jan 6, 2009
    4
    0
    So I used these formulae for Q.1

    Ω=2 × π ×frequency⁄(sampling frequency)

    ω=2 ×sampling frequency × tan(Ω/2)

    These for Q.2

    G_dB (f)=-10 × log(1+(f/f_3dB )^2N )

    (〖log〗_10 (〖10〗^((G_dB (f))/(-10))-1))/(2〖log〗_10 (f)-〖2log〗_10 (f_3dB))=N

    Yielding a filter order of 17.66 and a -3dB of 8629.984 Hz for the analogue frequency warped prototype.

    How do I work out Q.3? I've tried substituting in the new analogue -3db value and a new sampling rate 24KHz but I'm getting about 5KHz as a result for the digital frequency warped -3dB point. This doesn't seem right considering we were designing for 7.5KHz at -1dB and 12KHz at -96dB.

    Any ideas anyone???
     
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