An ideal C/D converter is followed by a discrete-time LPF which is followed by a

Question

An ideal C/D converter is followed by a discrete-time LPF which is followed by an ideal D/C converter. The sampling rate is 8000 samples per second. The input analog signal is bandlimited to 3900 Hz. The overall frequency response is required to satisfy the following: Gain > -1.33 dB for frequencies less than 1, 333.33 Hz Gain < -7.5 dB for frequencies greater than 2, 133.33 Hz a) Generate specifications on the discrete-time filter and design the LPF using the BLT and the Butterworth design technique. b) Obtain the transfer functions of both the analog and the discrete-time filters. c) Verify, by obtaining it from the transfer function and plotting it, that the frequency response of the designed discrete-time filter satisfies the specifications. d) Give reason(s) as to why one cannot simply use the direct mapping from analog to digital frequency as given by the sampling theorem but the frequency warping relationships of the BLT have to be used instead.

Explanation / Answer

The Immanent Difference between Analog and Digital Filters One thing you should know about any digital filter, not only those ones I'm talking about: By nature, the frequency responses of any digital (or, to be more precise: "sampled") system is mirrored at half of the sampling rate (FS/2). As a consequence, at FS/2 any such frequency response is horizontal. This does not fit to the frequency responses of almost any analog filter. Thus: A digital filter can never match the frequency responses of any analog model exactly, it can only approximate it. The closer the corner frequency FC to FS/2 is, the bigger the differences between both become (but often rather as an advantage than a drawback for the digital filter). The filters described here are bilinear transformed ones. Without the bilinear transformation, alias effects within the frequency responses would occur, and this shall be avoided. Just a little background information about the bilinear transformation which modifies (or distorts, resp.) the filter's frequency axis in a way that for low frequencies the frequency axes (and thus the frequency responses) of both, the analog and the digital filter are almost identical the corner frequencies of both filters are identical the analog filter's amplitude response for f against infinite is mapped to FS/2, so that the digital filter's amplitude response is mirrored at FS/2 and repeated all n

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