In
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
, a filter bank (or filterbank) is an array of
bandpass filter
A band-pass filter or bandpass filter (BPF) is a device that passes frequencies within a certain range and rejects ( attenuates) frequencies outside that range.
Description
In electronics and signal processing, a filter is usually a two-p ...
s that separates the input signal into multiple components, each one carrying a single
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
sub-band of the original signal. One application of a filter bank is a
graphic equalizer
Equalization, or simply EQ, in sound recording and reproduction is the process of adjusting the volume of different frequency bands within an audio signal. The circuit or equipment used to achieve this is called an equalizer.
Most hi-fi eq ...
, which can attenuate the components differently and recombine them into a modified version of the original signal. The process of decomposition performed by the filter bank is called ''analysis'' (meaning analysis of the signal in terms of its components in each sub-band); the output of analysis is referred to as a subband signal with as many subbands as there are filters in the filter bank. The reconstruction process is called ''synthesis'', meaning reconstitution of a complete signal resulting from the filtering process.
In
digital signal processing
Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are ...
, the term ''filter bank'' is also commonly applied to a bank of receivers. The difference is that receivers also
down-convert the subbands to a low center frequency that can be re-sampled at a reduced rate. The same result can sometimes be achieved by
undersampling
In signal processing, undersampling or bandpass sampling is a technique where one samples a bandpass-filtered signal at a sample rate below its Nyquist rate (twice the upper cutoff frequency), but is still able to reconstruct the signal.
When ...
the bandpass subbands.
Another application of filter banks is
signal
In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The '' IEEE Transactions on Signal Processing' ...
compression when some frequencies are more important than others. After decomposition, the important frequencies can be coded with a fine resolution. Small differences at these frequencies are significant and a
coding scheme that preserves these differences must be used. On the other hand, less important frequencies do not have to be exact. A coarser coding scheme can be used, even though some of the finer (but less important) details will be lost in the coding.
The
vocoder
A vocoder (, a portmanteau of ''voice'' and ''encoder'') is a category of speech coding that analyzes and synthesizes the human voice signal for audio data compression, multiplexing, voice encryption or voice transformation.
The vocoder was ...
uses a filter bank to determine the amplitude information of the subbands of a modulator signal (such as a voice) and uses them to control the amplitude of the subbands of a carrier signal (such as the output of a guitar or synthesizer), thus imposing the dynamic characteristics of the modulator on the carrier.
Some filter banks work almost entirely in the time domain, using a series of filters such as
quadrature mirror filter In digital signal processing, a quadrature mirror filter is a filter whose magnitude response is the mirror image around \pi/2 of that of another filter. Together these filters, first introduced by Croisier et al., are known as the quadrature mirror ...
s or the
Goertzel algorithm
The Goertzel algorithm is a technique in digital signal processing (DSP) for efficient evaluation of the individual terms of the discrete Fourier transform (DFT). It is useful in certain practical applications, such as recognition of dual-tone mult ...
to divide the signal into smaller bands.
Other filter banks use a fast Fourier transform (FFT).
FFT filter banks
A bank of receivers can be created by performing a sequence of
FFT
A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in the ...
s on overlapping ''segments'' of the input data stream. A weighting function (aka
window function
In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval, normally symmetric around the middle of the inte ...
) is applied to each segment to control the shape of the
frequency response
In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of sy ...
s of the filters. The wider the shape, the more often the FFTs have to be done to satisfy the
Nyquist sampling criteria. For a fixed segment length, the amount of overlap determines how often the FFTs are done (and vice versa). Also, the wider the shape of the filters, the fewer filters that are needed to span the input bandwidth. Eliminating unnecessary filters (i.e. decimation in frequency) is efficiently done by treating each weighted segment as a sequence of smaller ''blocks'', and the FFT is performed on only the sum of the blocks. This has been referred to as ''weight overlap-add (WOLA)'' and ''weighted pre-sum FFT''. (see )
A special case occurs when, by design, the length of the blocks is an integer multiple of the interval between FFTs. Then the FFT filter bank can be described in terms of one or more polyphase filter structures where the phases are recombined by an FFT instead of a simple summation. The number of blocks per segment is the impulse response length (or ''depth'') of each filter. The computational efficiencies of the FFT and polyphase structures, on a general purpose processor, are identical.
Synthesis (i.e. recombining the outputs of multiple receivers) is basically a matter of
upsampling
In digital signal processing, upsampling, expansion, and interpolation are terms associated with the process of resampling in a multi-rate digital signal processing system. ''Upsampling'' can be synonymous with ''expansion'', or it can describe a ...
each one at a rate commensurate with the total bandwidth to be created, translating each channel to its new center frequency, and summing the streams of samples. In that context, the interpolation filter associated with upsampling is called ''synthesis filter''. The net frequency response of each channel is the product of the synthesis filter with the frequency response of the filter bank (''analysis filter''). Ideally, the frequency responses of adjacent channels sum to a constant value at every frequency between the channel centers. That condition is known as ''perfect reconstruction''.
Filter banks as time–frequency distributions
In
time–frequency signal processing, a filter bank is a special quadratic
time–frequency distribution (TFD) that represents the signal in a joint
time–frequency domain. It is related to the
Wigner–Ville distribution
The Wigner quasiprobability distribution (also called the Wigner function or the Wigner–Ville distribution, after Eugene Wigner and Jean-André Ville) is a quasiprobability distribution. It was introduced by Eugene Wigner in 1932 to study quan ...
by a two-dimensional filtering that defines the class of
quadratic (or bilinear) time–frequency distributions. The filter bank and the spectrogram are the two simplest ways of producing a quadratic TFD; they are in essence similar as one (the spectrogram) is obtained by dividing the time domain into slices and then taking a Fourier transform, while the other (the filter bank) is obtained by dividing the frequency domain in slices forming bandpass filters that are excited by the signal under analysis.
Multirate filter bank
A multirate filter bank divides a signal into a number of subbands, which can be analysed at different rates corresponding to the bandwidth of the frequency bands. The implementation makes use of
downsampling (decimation) and
upsampling (expansion). See and for additional insight into the effects of those operations in the transform domains.
Narrow lowpass filter
One can define a narrow lowpass filter as a
lowpass filter
A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter ...
with a narrow passband.
In order to create a multirate narrow lowpass FIR filter, one can replace the time-invariant FIR filter with a lowpass antialiasing filter and a decimator, along with an interpolator and lowpass anti-imaging filter.
In this way, the resulting multirate system is a time-varying linear-phase filter via the decimator and interpolator.
The lowpass filter consists of two polyphase filters, one for the decimator and one for the interpolator.
A filter bank divides the input signal
into a set of signals
. In this way each of the generated signals corresponds to a different region in the spectrum of
.
In this process it can be possible for the regions overlap (or not, based on application).
The generated signals
can be generated via a collection of set of bandpass filters with bandwidths
and center frequencies
(respectively).
A multirate filter bank uses a single input signal and then produces multiple outputs of the signal by filtering and subsampling.
In order to split the input signal into two or more signals, an analysis-synthesis system can be used.
The signal would split with the help of four filters
for ''k'' =0,1,2,3 into 4 bands of the same bandwidths (In the analysis bank) and then each sub-signal is decimated by a factor of 4.
In each band by dividing the signal in each band, we would have different signal characteristics.
In synthesis section the filter will reconstruct the original signal:
First, upsampling the 4 sub-signals at the output of the processing unit by a factor of 4 and then filter by 4 synthesis filters
for ''k'' = 0,1,2,3.
Finally, the outputs of these four filters are added.
Statistically optimized filter bank (Eigen filter bank)
A discrete-time filter bank framework allows inclusion of desired input signal dependent features in the design in addition to the more traditional perfect reconstruction property. The information theoretic features like maximized energy compaction, perfect de-correlation of sub-band signals and other characteristics for the given input covariance/correlation structure are incorporated in the design of optimal filter banks. These filter banks resemble the signal dependent
Karhunen–Loève transform (KLT) that is the optimal block transform where the length L of basis functions (filters) and the subspace dimension M are the same.
Multidimensional filter banks
Multidimensional filtering,
downsampling In digital signal processing, downsampling, compression, and decimation are terms associated with the process of ''resampling'' in a multi-rate digital signal processing system. Both ''downsampling'' and ''decimation'' can be synonymous with ''com ...
, and
upsampling
In digital signal processing, upsampling, expansion, and interpolation are terms associated with the process of resampling in a multi-rate digital signal processing system. ''Upsampling'' can be synonymous with ''expansion'', or it can describe a ...
are the main parts of
multirate systems and filter banks.
A complete filter bank consists of the analysis and synthesis side.
The analysis filter bank divides an input signal to different subbands with different frequency spectra.
The synthesis part reassembles the different subband signals and generates a reconstructed signal.
Two of the basic building blocks are the decimator and expander. For example, the input divides into four directional sub bands that each of them covers one of the wedge-shaped frequency regions. In 1D systems, M-fold decimators keep only those samples that are multiples of M and discard the rest. while in multi-dimensional systems the decimators are ''D'' × ''D'' nonsingular integer matrix. it considers only those samples that are on the lattice generated by the decimator. Commonly used decimator is the quincunx decimator whose lattice is generated from the
Quincunx matrix
In mathematics, the matrix
: \begin
1 & -1 \\
1 & 1
\end
is sometimes called the quincunx matrix. It is a 2×2 Hadamard matrix, and its rows form the basis of a diagonal square lattice consisting of the integer points whose coordinate ...
which is defined by
The quincunx lattice generated by quincunx matrix is as shown; the synthesis part is dual to the analysis part.
Filter banks can be analyzed from a frequency-domain perspective in terms of subband decomposition and reconstruction. However, equally important is
Hilbert-space interpretation of filter banks, which plays a key role in geometrical signal representations.
For generic ''K''-channel filter bank, with analysis filters
, synthesis filters
, and sampling matrices
.
In the analysis side, we can define vectors in ''
'' as
: