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

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

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

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

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

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:

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

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

Noble Identities:

For example take an FIR Filter since

Similarly:

since

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

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

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

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

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

POLYPHASE Apply to Downsampling…

… apply Noble Identity

S/P Serial to Parallel (Buffer): Serial to Parallel (Buffer)

POLYPHASE Same for Upsampling…

NOBLE IDENTITY … apply Noble Identity

This is a Parallel to Serial (an Unbuffer): P/S Parallel to Serial (Unbuffer or Interlacer)