# Multi-rate digital signal processing

Multi-rate digital signal processing

Multi-rate signal processing studies digital signal processing systems which include sample rate conversion. Multirate signal processing techniques are necessary for systems with different input and output sample rates, but may also be used to implement systems with equal input and output rates.

## Changing the sampling rate

The process of changing the sampling rate of a signal (resampling) is called downsampling if the sample rate is decreased and upsampling if the sample rate is increased. Integer rate changes are far more common than non-integer rate changes.

### Downsampling

Downsampling a sequence x[n] by retaining only every Mth sample creates a new sequence xd[n] = x[nM]. If the original sequence contains frequency components above π / M, the downsampler should be preceded by a low-pass filter with cutoff frequency π / M. In this application, such an anti-aliasing filter is referred to as a decimation filter and the combined process of filtering and downsampling is called decimation.

### Upsampling

Upsampling a sequence x[n] creates a new sequence xe[n] where every Lth sample is taken from x[n] with all others zero. The upsampled sequence contains L replicas of the original signal's spectrum. To restore the original spectrum, the upsampler should be followed by a low-pass filter with gain L and cutoff frequency π / L. In this application, such an anti-aliasing filter is referred to as an interpolation filter and the combined process of upsampling and filtering is called interpolation.

### Fractional rate changes

Changing the sampling rate of a signal by a rational fraction L / M can be accomplished by first upsampling by L, then downsampling by M. A low pass filter with cutoff min(π / L,π / M) is placed between the upsampler and downsampler to prevent aliasing.

## Noble identities

The Noble identities describe the effect of interchanging sampling rate changes and filtering. Using $(\downarrow M)$ to denote downsampling by a factor M and $(\uparrow L)$ to denote upsampling by a factor L, we have

$(\downarrow M) H(z) = H(z^M) (\downarrow M)$

and

$H(z) (\uparrow L) = (\uparrow L) H(z^L).$

## Polyphase Decomposition

The polyphase decomposition of a filter

 H(z) = ∑ hnz − n n

is represented by

$H(z) = \sum_{k = 0}^{M-1} H_k(z^M) z^{-k},$

where

 Hk(zM) = ∑ hk + nMz − nM. n

An important application of polyphase filters is in decimation, where the downsampling $(\downarrow M)$ following the decimation filter can be moved before the subfilters Hk(zM), allowing each subfilter to be calculated at the lower sampling rate as Hk(z) (per the Nobel identities). Similarly, for interpolation, the upsampling $(\uparrow M)$ can be moved after the subfilters, which are calculated as Hk(z).i

## References

• Crochiere, Ronald E.; Rabiner, Lawrence R. (1983). Multirate Digital Signal Processing. Prentice-Hall. ISBN 0-13-605162-6.
• Oppenheim, Alan V.; Schafer, Ronald W. (1999). Discrete-Time Signal Processing (2nd ed.). Prentice-Hall. ISBN 0-13-754920-2.

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