Downsampling
Encyclopedia
In signal processing, downsampling (or "subsampling") 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 at 44,100 Hz is downsampled to 22,050 Hz before broadcasting over FM radio
, the bit rate
is reduced in half, from 1,411,200 bit/s to 705,600 bit/s, assuming that each sample retains its size of 16 bits. The audio was therefore downsampled by a factor of 2.
. 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; the overall process (low-pass filter, then downsample) is called decimation
.
Note that the anti-aliasing filter must be a low-pass filter in downsampling. This is different from sampling
a continuous signal
, where either a low-pass filter or a band-pass filter
may be used.
Remark: A bandpass signal, i.e. a band-limited signal whose minimum frequency is different from zero, can be downsampled avoiding superposition of the spectra if certain conditions are satisfied (see e.g. http://spazioscuola.altervista.org/UndersamplingAR/UndersamplingARnv.htm).
on a radian frequency digital frequency
range.
The first step calls for the use of a perfect low-pass filter, which is not implementable for real-time signals.
When choosing a realizable low-pass filter this will have to be considered along with the aliasing effects it will have. 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.
Note that a proper upsampling design requires an interpolation filter after increasing the data rate and that a proper downsampling design requires a filter before eliminating some samples. These two low-pass 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 can therefore be used in place of both filters. When the rational fraction M/L is greater than unity then
and the single low-pass filter should have cutoff at 
.
Sample rate conversion
Sample rate conversion is the process of converting a signal from one sampling rate to another, while changing the information carried by the signal as little as possible...
of a signal. This is usually done to reduce the data rate
Data signaling rate
In telecommunication, data signaling rate , also known as gross bit rate, is the aggregate rate at which data pass a point in the transmission path of a data transmission system.Notes:#The DSR is usually expressed in bits per second....
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 at 44,100 Hz is downsampled to 22,050 Hz before broadcasting over FM radio
FM broadcasting
FM broadcasting is a broadcasting technology pioneered by Edwin Howard Armstrong which uses frequency modulation to provide high-fidelity sound over broadcast radio. The term "FM band" describes the "frequency band in which FM is used for broadcasting"...
, the bit rate
Bit rate
In telecommunications and computing, bit rate is the number of bits that are conveyed or processed per unit of time....
is reduced in half, from 1,411,200 bit/s to 705,600 bit/s, assuming that each sample retains its size of 16 bits. The audio was therefore downsampled by a factor of 2.
Maintaining the sampling theorem criterion
Since downsampling reduces the sampling rate, we must be careful to make sure the Shannon-Nyquist sampling theorem criterion is maintained. If the sampling theorem is not satisfied then the resulting digital signal will have aliasingAliasing
In signal processing and related disciplines, aliasing refers to an effect that causes different signals to become indistinguishable when sampled...
. To ensure that the sampling theorem is satisfied, a low-pass filter
Low-pass filter
A low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...
is used as an anti-aliasing filter
Anti-aliasing filter
An anti-aliasing filter is a filter used before a signal sampler, to restrict the bandwidth of a signal to approximately satisfy the sampling theorem....
to reduce the bandwidth of the signal before the signal is downsampled; the overall process (low-pass filter, then downsample) is called decimation
Decimation (signal processing)
In digital signal processing, decimation is a technique for reducing the number of samples in a discrete-time signal. The element which implements this technique is referred to as a decimator.Decimation is a two-step process:...
.
Note that the anti-aliasing filter must be a low-pass filter in downsampling. This is different from sampling
Sampling (signal processing)
In signal processing, sampling is the reduction of a continuous signal to a discrete signal. A common example is the conversion of a sound wave to a sequence of samples ....
a continuous signal
Continuous signal
A continuous signal or a continuous-time signal is a varying quantity whose domain, which is often time, is a continuum . That is, the function's domain is an uncountable set. The function itself need not be continuous...
, where either a low-pass filter or a band-pass filter
Band-pass filter
A band-pass filter is a device that passes frequencies within a certain range and rejects frequencies outside that range.Optical band-pass filters are of common usage....
may be used.
Remark: A bandpass signal, i.e. a band-limited signal whose minimum frequency is different from zero, can be downsampled avoiding superposition of the spectra if certain conditions are satisfied (see e.g. http://spazioscuola.altervista.org/UndersamplingAR/UndersamplingARnv.htm).
Downsampling process
Consider a discrete signalDiscrete signal
A discrete signal or discrete-time signal is a time series consisting of a sequence of qualities...
on a radian frequency digital frequencyDigital frequency
Digital frequency is the analogue for discrete signals as frequency is to continuous signals.Since a discrete signal is a sequence it contains no direct information as to determine the frequency of the corresponding continuous signal.Just like in frequency, a digital frequency can have values in...
range.
Downsampling by integer factor
Let M denote the downsampling factor.- Filter the signal to ensure that the sampling theorem is satisfied. This filter should, theoretically, be the sinc filterSinc filterIn signal processing, a sinc filter is an idealized filter that removes all frequency components above a given bandwidth, leaves the low frequencies alone, and has linear phase...
with frequency cutoff at
. Let the filtered signal be denoted 
. - Reduce the data by picking out every
sample: 
. Data rate reduction occurs in this step.
The first step calls for the use of a perfect low-pass filter, which is not implementable for real-time signals.
When choosing a realizable low-pass filter this will have to be considered along with the aliasing effects it will have. 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.
Downsampling by rational fraction
Let M/L denote the downsampling factor.- UpsampleUpsamplingUpsampling is the process of increasing the sampling rate of a signal. For instance, upsampling raster images such as photographs means increasing the resolution of the image....
by a factor of L - Downsample by a factor of M
Note that a proper upsampling design requires an interpolation filter after increasing the data rate and that a proper downsampling design requires a filter before eliminating some samples. These two low-pass 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 can therefore be used in place of both filters. When the rational fraction M/L is greater than unity then
and the single low-pass filter should have cutoff at .See also
- OversamplingOversamplingIn signal processing, oversampling is the process of sampling a signal with a sampling frequency significantly higher than twice the bandwidth or highest frequency of the signal being sampled...
- Sampling (information theory)
- Signal (information theory)
- Antialiasing
- PosterizationPosterizationPosterization of an image entails conversion of a continuous gradation of tone to several regions of fewer tones, with abrupt changes from one tone to another. This was originally done with photographic processes to create posters...