downsampling

Downsampling is the process of reducing the sampling rate of a signal. This is usually done to reduce the data rate or the size of the data. The downsampling factor (commonly denoted by M) is usually an integer or a rational fraction greater than unity. This factor multiplies the sampling time or, equivalently, divides the sampling rate. For example, if compact disc audio was downsampled by a factor of 5/4 then the resulting sampling rate goes from 44,100 Hz to 35,280 Hz, which reduces the bit rate from 1,411,200 bit/s to 1,128,960 bit/s.

Sampling theorem satisfaction

By downsampling, the sampling rate is also reduced so the Shannon-Nyquist sampling theorem satisfaction must be maintained. If the sampling theorem is not satisfied then the resulting signal will have aliasing and to ensure that the sampling theorem is satisfied a low-pass filter is used as an anti-aliasing filter to reduce the bandwidth of the signal before the signal is downsampled. Note that the anti-aliasing filter must be a low-pass filter in downsampling. This unlike sampling from a continuous signal, which can be either a low-pass filter or a band-pass filter.

Downsampling process

Consider a discrete signal

f(k)

on a radian frequency digital frequency range.

Downsampling by integer factor

Let M denote the downsampling factor.

Filter the signal to ensure satisfaction of the sampling theorem. This filter should, theoretically, be the sinc filter with frequency cut off at $\frac{\pi}{2M}$ . Let the filtered signal be denoted $g(k)$ .
Decimate the data by picked out every $M^{th}$ sample: $h(k) = g(Mk)$ . It is in this step where data rate reduction occurs.

The first step calls for the use of a perfect low-pass filter, which is not implementable. When choosing a realizable low-pass filter this will have to be considered and aliasing effects it will have.

Downsampling by rational fraction

Let M/L denote the downsampling factor.

Upsample by a factor of L
Downsample by a factor of M

Note that upsampling requires an interpolation filter after increasing the data rate and that downsampling requires a filter before decimation. These two filters can be combined into a single filter. Also note that these two steps are generally not reversible. Downsampling results in a loss of data and, if performed first, could result in data loss if there is any data filtered out by the downsampler's low-pass filter. Since both interpolation and anti-aliasing filters are low-pass filters, the filter with the smallest bandwidth is more restrictive and, thus, can be used in place of both filters. Since the rational fraction M/L is greater than unity then

L < M

and the single low-pass filter should have cutoff at

\frac{\pi}{2M}

Downsampling

Sampling theorem satisfaction

Downsampling process

Downsampling by integer factor

Downsampling by rational fraction

See also