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Overview of Multirate Signal Processing

Module by: Douglas L. Jones

Digital transformation of the sampling rate of signals, or signal processing with different sampling rates in the system.

Applications

  1. Sampling-rate conversion: CD to DAT format change, for example.
  2. Improved D/A, A/D conversion: oversampling converters; which reduce performance requirements on anti-aliasing or reconstruction filters
  3. FDM channel modulation and processing: bandwidth of individual channels is much less than the overall bandwidth
  4. Subband coding of speech and images: Eyes and ears are not as sensitive to errors in higher frequency bands, so many coding schemes split signals into different frequency bands and quantize higher-frequency bands with much less precision.

General Rate-Changing Procedure

This procedure is motivated by an analog-based method: one conceptually simple method to change the sampling rate is to simply convert a digital signal to an analog signal and resample it! (Figure 1)

Figure 1
Figure 1 (imag001.png)
H aa Ω=1if|Ω|<π T 1 0otherwise H aa Ω 1 Ω T 1 0 h aa t=sinπ T 1 tπ T 1 t h aa t T 1 t T 1 t Recall the ideal D/A:
x a ′ t=∑n=-∞∞ x 0 nsinπt-n T 0 T 0 πt-n T 0 T 0 x a ′ t n x 0 n t n T 0 T 0 t n T 0 T 0 (1)
The problems with this scheme are:
  1. A/D, D/A, filters cost money
  2. imperfections in these devices introduce errors

Digital implementation of rate-changing according to this formula has three problems:

  1. Infinite sum: The solution is to truncate. Consider sinct≈0 sinc t 0 for t< t 1 t t 1 , t> t 2 t t 2 : Then m T 1 -n T 0 < t 1 m T 1 n T 0 t 1 and m T 1 -n T 0 > t 2 m T 1 n T 0 t 2 which implies N 1 =⌈m T 1 - t 2 T 0 ⌉ N 1 m T 1 t 2 T 0 N 2 =⌊m T 1 - t 1 T 0 ⌋ N 2 m T 1 t 1 T 0 x 1 m=∑n= N 1 N 2 x 0 n sinc T ′ m T 1 -n T 0 x 1 m n N 1 N 2 x 0 n sinc T ′ m T 1 n T 0

    note:

    This is essentially lowpass filter design using a boxcar window: other finite-length filter design methods could be used for this.
  2. Lack of causality: The solution is to delay by max{|N|} N samples. The mathematics of the analog portions of this system can be implemented digitally.
    x 1 m= h aa t* x a ′ t|t=m T 1 =∫-∞∞∑n=-∞∞ x 0 nsinπm T 1 -τ-n T 0 T 0 πm T 1 -τ-n T 0 T 0 sinπτ T 1 πτ T 1 dτ x 1 m t m T 1 h aa t x a ′ t τ n x 0 n m T 1 τ n T 0 T 0 m T 1 τ n T 0 T 0 τ T 1 τ T 1 (2)
    x 1 m=∑n=-∞∞ x 0 nsinπ T ′ m T 1 -n T 0 π T ′ m T 1 -n T 0 | T ′ =max{ T 0 T 1 }=∑n=-∞∞ x 0 n sinc T ′ m T 1 -n T 0 x 1 m T ′ T 0 T 1 n x 0 n T ′ m T 1 n T 0 T ′ m T 1 n T 0 n x 0 n sinc T ′ m T 1 n T 0 (3)
    So we have an all-digital formula for exact digital-to-digital rate changing!
  3. Cost of computing sinc T ′ m T 1 -n T 0 sinc T ′ m T 1 n T 0 : The solution is to precompute the table of sinct sinc t values. However, if T 1 T 0 T 1 T 0 is not a rational fraction, an infinite number of samples will be needed, so some approximation will have to be tolerated.

    note:

    Rate transformation of any rate to any other rate can be accomplished digitally with arbitrary precision (if some delay is acceptable). This method is used in practice in many cases. We will examine a number of special cases and computational improvements, but in some sense everything that follows are details; the above idea is the central idea in multirate signal processing.

Useful references for the traditional material (everything except PRFBs) are Crochiere and Rabiner, 1981 and Crochiere and Rabiner, 1983. A more recent tutorial is Vaidyanathan; see also Rioul and Vetterli. References to most of the original papers can be found in these tutorials.

References

  1. R.E. Crochiere and L.R. Rabiner. (1981, March). Interpolation and Decimation of Digital Signals: A Tutorial Review. Proc. IEEE, 69(3), 300-331.
  2. R.E. Crochiere and L.R. Rabiner. (1983). Multirate Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall.
  3. P.P Vaidyanathan. (1990, January). Multirate Digital Filters, Filter Banks, Polyphase Networks, and Applications: A Tutorial. Proc. IEEE, 78(1), 56-93.
  4. O. Rioul and M. Vetterli. (1991, October). Wavelets and Signal Processing. IEEE Signal Processing Magazine, 8(4), 14-38.

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