Downsampling (signal processing)

[1][2] When the process is performed on a sequence of samples of a signal or a continuous function, it produces an approximation of the sequence that would have been obtained by sampling the signal at a lower rate (or density, as in the case of a photograph).

[a] But in signal processing, decimation by a factor of 10 actually means keeping only every tenth sample.

For example, if compact disc audio at 44,100 samples/second is decimated by a factor of 5/4, the resulting sample rate is 35,280.

[3][4] Rate reduction by an integer factor M can be explained as a two-step process, with an equivalent implementation that is more efficient:[5] Step 2 alone creates undesirable aliasing (i.e. high-frequency signal components will copy into the lower frequency band and be mistaken for lower frequencies).

When the anti-aliasing filter is an IIR design, it relies on feedback from output to input, prior to the second step.

The calculation performed by a decimating FIR filter for the nth output sample is a dot product:[b] where the h[•] sequence is the impulse response, and K is its length.

In the case M=2, h[•] can be designed as a half-band filter, where almost half of the coefficients are zero and need not be included in the dot products.

In other words, the input stream is demultiplexed and sent through a bank of M filters whose outputs are summed.

For completeness, we now mention that a possible, but unlikely, implementation of each phase is to replace the coefficients of the other phases with zeros in a copy of the h[•] array, process the original x[•] sequence at the input rate (which means multiplying by zeros), and decimate the output by a factor of M. The equivalence of this inefficient method and the implementation described above is known as the first Noble identity.

The purpose of the anti-aliasing filter is to ensure that the reduced periodicity does not create overlap.

Fig 1: These graphs depict the spectral distributions of an oversampled function and the same function sampled at 1/3 the original rate. The bandwidth, B, in this example is just small enough that the slower sampling does not cause overlap (aliasing). Sometimes, a sampled function is resampled at a lower rate by keeping only every M th sample and discarding the others, commonly called "decimation". Potential aliasing is prevented by lowpass-filtering the samples before decimation. The maximum filter bandwidth is tabulated in the bandwidth units used by the common filter design applications.