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 sampling frequency or sampling rate defines the number of samples per second taken from a continuous signal to make a discrete signal. ... In information theory, a signal is the sequence of states of a communications channel that encodes a message. ... In telecommunication, data signaling rate (DSR) is the aggregate rate at which data pass a point in the transmission path of a data transmission system. ...

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 is 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. The Compact Disc logo was inspired by that of the previous Compact Cassette. ... The hertz (symbol: Hz) is the SI unit of frequency. ... In telecommunications and computing, bit rate (sometimes written bitrate) is the frequency at which bits are passing a given (physical or metaphorical) point. It is quantified using the bit per second (bit/s) unit. ...

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. The Nyquist-Shannon sampling theorem is the fundamental theorem in the field of information theory, in particular telecommunications. ... In statistics, signal processing, and related disciplines, aliasing is an effect that causes different continuous signals to become indistinguishable (or aliases of one another) when sampled. ... A low-pass filter is a filter that passes low frequencies well, but attenuates (or reduces) frequencies higher than the cutoff frequency. ... An anti-aliasing filter is commonly used in conjuction with digital signal processing and is a filter to restrict the bandwidth to approximately satisfy the Shannon-Nyquist sampling theorem. ...

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. Sampling may refer to: Digital sampling of audio Sampling (information theory) Sampling (music) Sampling (signal processing) Sampling (statistics) This is a disambiguation page: a list of articles associated with the same title. ... A continuous signal or a continuous time signal is a varying quantity (a signal) that can be, or is expressed, as a continuous function of an independent variable, usually time. ... The frequency axis of this symbolic diagram would be logarithmically scaled. ...

Remark: A bandpass signal, i.e. a band-limited signal whose minimum frequency is different from zero, can be downsampled avoiding superposition of the spectrum if we satisfy certain conditions (see e.g. [1]).

Downsampling process

Consider a discrete signal f(k) on a radian frequency digital frequency range. A discrete signal is a signal that has been sampled from a continuous signal. ... Digital frequency is the analogue for discrete signals as frequency is to continuous signals. ...

Downsampling by integer factor

Let M denote the downsampling factor.

1. Filter the signal to ensure satisfaction of the sampling theorem. This filter should, theoretically, be the sinc filter with frequency cut off at . Let the filtered signal be denoted g(k).
2. Reduce the data by picking out every M^th sample: h(k) = g(Mk). Data rate reduction occurs in this step.

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. 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. In signal processing, the sinc filter strips high-frequency data from a signal. ...

Downsampling by rational fraction

Let M/L denote the downsampling factor.

1. Upsample by a factor of L
2. 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. Upsampling is the process of increasing the sampling rate of a signal. ...

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 .

See also

• Oversampling
• Sampling (information theory)
• Signal (information theory)
• Upsampling