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Multistage Implementation Problem: There are cases in which the filter requirements call for a digital filter of high complexity, in terms of number of stages. Example: a signal has a bandwidth of 450Hz and it is sampled at 96kHz. We want to resample it at 1kHz: 96

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Solution: for an FIR filter designed by the window method, the order of the filter is determined by the size of the transition region. With a Hamming Window the order is determined by the equation which yields Disdvantage: a lot of computations at a high freq. rate

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Multistage Implementation: we decimate the signal in several stages. D LPF One Stage Implementation: we decimate in one shot.

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See the last stage first: Passband: Stopband:, since This filter clears everything above.

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Problem: we can design the low pass filters in a clever way, by taking into consideration that the spectrum is bandlimited. passstop aliased

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Problem: we can design the low pass filters in a clever way, by taking into consideration that the spectrum is bandlimited. Specs for : pass: stop:

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Example: same problem we saw before: Use Multistage. Pass Band[0, 450] Hz Stop Band> 500 Hz Sampling Freq.96 kHz mult./sec

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Efficient Multirate Implementation Goal: we want to determine an efficient implementation of a multirate system. For example in Decimation and Interpolation: you have to compute only, ie. one every D samples. most of the values of s(mD) are zero

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Noble Identities:

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For example take an FIR Filter since

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Similarly:

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since

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Application: POLYPHASE Filters Decimator: take, for example, D=2 evenodd

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Therefore this system becomes: Filtering at High Sampling Rate Filtering at Low Sampling rate

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Similarly: Filtering at High Sampling Rate Filtering at Low Sampling Rate

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Example: Consider the Filter/Decimator structure shown below, with This can be written as and implemented in Polyphase form:

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Given any integer N: Example: take N=3 General Polyphase Decomposition

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POLYPHASE Apply to Downsampling…

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… apply Noble Identity

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S/P 12345 1 2 6 3 4 5 6 Serial to Parallel (Buffer): Serial to Parallel (Buffer)

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POLYPHASE Same for Upsampling…

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NOBLE IDENTITY … apply Noble Identity

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This is a Parallel to Serial (an Unbuffer): P/S 1 2 3 4 5 6 123456 Parallel to Serial (Unbuffer or Interlacer)

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