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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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Downsampling a sequence $\scriptstyle x[n]$ by retaining only every M^th sample creates a new sequence  $\scriptstyle y[n]=x[nM].$  If the original sequence contains significant normalized frequency components in the region $\scriptstyle [0.5/M,\ 1-0.5/M]$ (cycles/sample), the downsampler should be preceded by a low-pass filter with cutoff frequency $\scriptstyle 0.5/M$.  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

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See also: Interpolation

Upsampling a sequence $\scriptstyle x[n]$ creates a new sequence where every L^th sample is taken from $\scriptstyle x[n]$ with all others zero. The embedded zeros create $\scriptstyle L-1$ aliases in the normalized frequency region $\scriptstyle [0.5/L,\ 1-0.5/L]$ (cycles/sample). To remove the aliases (and replace the embedded zeros), the upsampler should be followed by a low-pass filter with cutoff frequency $\scriptstyle 0.5/L$. 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. 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 $\scriptstyle L/M$ can be accomplished by first upsampling by $\scriptstyle L,$ then downsampling by $\scriptstyle M.$  A low pass filter with cutoff $\scriptstyle \min(0.5/L,\ 0.5/M)$ (cycles/sample) is placed between the upsampler and downsampler to prevent aliasing.

Notes

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See also

• Sampling (information theory)
• Aliasing
• Cascaded integrator-comb filter
• Quadrature mirror filter

References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534