Multi-rate digital signal processing
Encyclopedia
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.
Downsampling a sequence
by retaining only every 
th sample creates a new sequence 
. If the original sequence contains frequency
components above
, the downsampler should be preceded by a low-pass filter
with cutoff frequency
. In this application, such an anti-aliasing filter
is referred to as a decimation filter and the combined process of filtering and downsampling is called decimation.
Upsampling a sequence
creates a new sequence 
where every 
th sample is taken from 
with all others zero. The upsampled sequence contains 
replicas of the original signal's spectrum. To restore the original spectrum, the upsampler should be followed by a low-pass filter with gain 
and cutoff frequency 
. 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.
can be accomplished by first upsampling by 
, then downsampling by 
. A low pass filter with cutoff 
is placed between the upsampler and downsampler to prevent aliasing.
to denote downsampling by a factor 
and 
to denote upsampling by a factor 
, we have
and
is represented by
where
An important application of polyphase filters is in decimation, where the downsampling
following the decimation filter can be moved before the subfilters 
, allowing each subfilter to be calculated at the lower sampling rate as 
(per the Nobel identities). Similarly, for interpolation, the upsampling 
can be moved after the subfilters, which are calculated as 
.i
Digital signal processing
Digital signal processing is concerned with the representation of discrete time signals by a sequence of numbers or symbols and the processing of these signals. Digital signal processing and analog signal processing are subfields of signal processing...
systems which include sample rate conversion
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...
. 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
- See also: Decimation (signal processing)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:...
Downsampling a sequence
by retaining only every th sample creates a new sequence . If the original sequence contains 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...
components above
, the downsampler should be preceded by a low-pass filterDigital filter
In electronics, computer science and mathematics, a digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other major type of electronic filter, the analog filter, which is...
with cutoff frequency
. In this application, such an anti-aliasing filterAnti-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....
is referred to as a decimation filter and the combined process of filtering and downsampling is called decimation.
Upsampling
- See also: Interpolation
Upsampling a sequence
creates a new sequence where every th sample is taken from with all others zero. The upsampled sequence contains replicas of the original signal's spectrum. To restore the original spectrum, the upsampler should be followed by a low-pass filter with gain and cutoff frequency . 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.Fractional rate changes
Changing the sampling rate of a signal by a rational fraction can be accomplished by first upsampling by , then downsampling by . A low pass filter with cutoff is placed between the upsampler and downsampler to prevent aliasing.Noble identities
The Noble identities describe the effect of interchanging sampling rate changes and filtering. Using to denote downsampling by a factor and to denote upsampling by a factor , we haveand
Polyphase Decomposition
The polyphase decomposition of a filter is represented bywhere
An important application of polyphase filters is in decimation, where the downsampling
following the decimation filter can be moved before the subfilters , allowing each subfilter to be calculated at the lower sampling rate as (per the Nobel identities). Similarly, for interpolation, the upsampling can be moved after the subfilters, which are calculated as .iSee also
- Sampling (information theory)
- AliasingAliasingIn signal processing and related disciplines, aliasing refers to an effect that causes different signals to become indistinguishable when sampled...
- Cascaded integrator-comb filterCascaded Integrator-Comb FilterIn digital signal processing, a cascaded integrator-comb is an optimized class of finite impulse response filter combined with an interpolator or decimator.A CIC filter consists of one or more integrator and comb filter pairs...
- Quadrature mirror filterQuadrature mirror filterIn digital signal processing, a quadrature mirror filter is a filter most commonly used to implement a filter bank that splits an input signal into two bands...