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Multistage Implementation

96. 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:.

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Multistage Implementation

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  1. 96 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:

  2. 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

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

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

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

  6. 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:

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

  8. 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

  9. Noble Identities:

  10. For example take an FIR Filter since

  11. Similarly:

  12. since

  13. Application: POLYPHASE Filters Decimator: take, for example, D=2 even odd

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

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

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

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

  18. Apply to Downsampling… POLYPHASE

  19. … apply Noble Identity

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

  21. Same for Upsampling… POLYPHASE

  22. … apply Noble Identity NOBLE IDENTITY

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

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