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'''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 ===
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{{Main|Downsampling}}
 
Downsampling a sequence <math>\scriptstyle x[n]</math> by retaining only every M<sup>th</sup> sample creates a new sequence &nbsp;<math>\scriptstyle y[n] = x[nM].</math>&nbsp; If the original sequence contains significant [[Normalized frequency (digital signal processing)|normalized frequency]] components in the region <math>\scriptstyle [0.5/M,\ 1-0.5/M]</math> (''cycles/sample''), the downsampler should be preceded by a low-pass [[Digital filter|filter]] with [[cutoff frequency]] <math>\scriptstyle 0.5/M</math>.<ref group="note">Realizable low-pass filters have a "skirt", where the response diminishes from near unity to near zero. So in practice the cutoff frequency is placed far enough below the theoretical cutoff that the filter's skirt is contained below the theoretical cutoff.</ref>&nbsp; In this application, such an [[anti-aliasing filter]] is referred to as a ''decimation filter'', and the combined process of filtering ([[convolution]]) and downsampling is called '''decimation'''.
 
=== Upsampling ===
{{Main|Upsampling}}
:''See also: [[Interpolation#Interpolation_in_digital_signal_processing|Interpolation]]''
 
Upsampling a sequence <math>\scriptstyle x[n]</math> creates a new sequence where every L<sup>th</sup> sample is taken from <math>\scriptstyle x[n]</math> with all others zero. The embedded zeros create <math>\scriptstyle L-1</math> aliases in the normalized frequency region <math>\scriptstyle [0.5/L,\ 1-0.5/L]</math> (''cycles/sample'').  To remove the aliases (and replace the embedded zeros), the upsampler should be followed by a [[low-pass filter]] with cutoff frequency <math>\scriptstyle 0.5/L</math>.  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'''. Depending on the frequency content, a realizable filter may not be able to attenuate the first alias to an imperceptable level, and the result will be some perceptible distortion.
 
=== Fractional rate changes ===
Changing the sampling rate of a signal by a rational fraction <math>\scriptstyle L/M</math> can be accomplished by first upsampling by <math>\scriptstyle L,</math> then downsampling by <math>\scriptstyle M.</math>&nbsp; A low pass filter with cutoff <math>\scriptstyle \min(0.5/L,\ 0.5/M)</math> (''cycles/sample'') is placed between the upsampler and downsampler to prevent [[aliasing]].
 
== Notes ==
{{reflist|group=note}}
 
== See also ==
* [[Sampling (information theory)]]
* [[Aliasing]]
* [[Cascaded integrator-comb filter]]
* [[Quadrature mirror filter]]
 
== References ==
*{{cite book | title = Multirate Digital Signal Processing | first1 = Ronald E. | last1 = Crochiere | first2 = Lawrence R. | last2 = Rabiner | publisher = Prentice-Hall | year = 1983 | isbn = 0-13-605162-6 }}
*{{cite book | title = Discrete-Time Signal Processing | edition = 2nd | first1 = Alan V. | last1 = Oppenheim | first2 = Ronald W. | last2 = Schafer | publisher = Prentice-Hall | year = 1999 | isbn = 0-13-754920-2 }}
 
[[Category:Digital signal processing]]

Latest revision as of 21:02, 28 December 2014


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