 Multirate digital signal processing

Multirate 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.
Contents
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 noninteger rate changes.
Downsampling
Main article: Downsampling See also: Decimation (signal processing)
Downsampling a sequence x[n] by retaining only every Mth sample creates a new sequence x_{d}[n] = x[nM]. If the original sequence contains frequency components above π / M, the downsampler should be preceded by a lowpass filter with cutoff frequency π / M. In this application, such an antialiasing filter is referred to as a decimation filter and the combined process of filtering and downsampling is called decimation.
Upsampling
Main article: Upsampling See also: Interpolation
Upsampling a sequence x[n] creates a new sequence x_{e}[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 lowpass filter with gain L and cutoff frequency π / L. In this application, such an antialiasing 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 to denote downsampling by a factor M and to denote upsampling by a factor L, we have
and
Polyphase Decomposition
The polyphase decomposition of a filter
H(z) = ∑ h_{n}z ^{− n} n is represented by
where
H_{k}(z^{M}) = ∑ h_{k + nM}z ^{− nM}. n An important application of polyphase filters is in decimation, where the downsampling following the decimation filter can be moved before the subfilters H_{k}(z^{M}), allowing each subfilter to be calculated at the lower sampling rate as H_{k}(z) (per the Nobel identities). Similarly, for interpolation, the upsampling can be moved after the subfilters, which are calculated as H_{k}(z).i
See also
 Sampling (information theory)
 Aliasing
 Cascaded integratorcomb filter
 Quadrature mirror filter
References
 Crochiere, Ronald E.; Rabiner, Lawrence R. (1983). Multirate Digital Signal Processing. PrenticeHall. ISBN 0136051626.
 Oppenheim, Alan V.; Schafer, Ronald W. (1999). DiscreteTime Signal Processing (2nd ed.). PrenticeHall. ISBN 0137549202.
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