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IIR Filter design & frequency-warping

Discussion in 'Electronics Homework Help' started by adim222, Mar 3, 2014.

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  1. adim222


    Mar 3, 2014
    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. Harald Kapp

    Harald Kapp Moderator Moderator

    Nov 17, 2011
    Welcome to the forum.

    This is homework and we won't do your homework. We'll be happy to guide you along in finding the solution yourself. We need some more information to be able to help:
    What have you done so far to find a solution?
    Which part(s) of the assignment need more explanation?
  3. adim222


    Mar 3, 2014
    My attempts on the Question


    Thanks for response, I am really struggling with DSP will be glad if I am assisted on the general understanding of the course, below is my attempt on question 1.

    I tried to perform the following

    Attempt on question 1
    For -1dB gain with frequency 7.5khz
    Ω= 2 * 3.142*Frequency/sample frequency = 2*3.142*7.5/44.1=1.069hz
    ω = 2*sample frequency *tan(Ω/2)= 139,9 Radian/sec
    For -96dB gain with frequency 12khz
    Ω= 2 * 3.142*Frequency/sample frequency =2*3,142*12/44.1=1.71hz
    ω = 2*sample frequency *tan(Ω/2)= 358 Radian/sec

    If the above is correct I dont know how to proceed for other questions

    Any help will be appreciated
    Last edited by a moderator: Mar 18, 2014
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