Tool for Digital Signal Processing
In signal processing, a filter bank (or filterbank) is an array of bandpass filters that separates the input signal into multiple components, each one carrying a sub-band of the original signal. One application of a filter bank is a graphic equalizer, 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, 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 the bandpass subbands.
Another application of filter banks is lossy 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 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 filters or the Goertzel algorithm 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 FFTs on overlapping "segments" of the input data stream. A weighting function (aka window function) is applied to each segment to control the shape of the frequency responses 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 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 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 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 formula_1 into a set of signals formula_2. In this way each of the generated signals corresponds to a different region in the spectrum of formula_1.
In this process it can be possible for the regions overlap (or not, based on application).
The generated signals formula_2 can be generated via a collection of set of bandpass filters with bandwidths formula_5 and center frequencies formula_6(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 formula_7 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 formula_8 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, and upsampling 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 which is defined by formula_9
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 formula_10, synthesis filters formula_11, and sampling matrices formula_12.
In the analysis side, we can define vectors in "formula_13" as
formula_14,
each index by two parameters: formula_15 and formula_16.
Similarly, for the synthesis filters formula_17 we can define formula_18.
Considering the definition of analysis/synthesis sides we can verify that formula_19 and for reconstruction part:
formula_20.
In other words, the analysis filter bank calculate the inner product of the input signal and the vector from analysis set. Moreover, the reconstructed signal in the combination of the vectors from the synthesis set, and the combination coefficients of the computed inner products, meaning that
formula_21
If there is no loss in the decomposition and the subsequent reconstruction, the filter bank is called "perfect reconstruction". (in that case we would have formula_22.
Figure shows a general multidimensional filter bank with "N" channels and a common sampling matrix "M".
The analysis part transforms the input signal formula_23 into "N" filtered and downsampled outputs formula_24 formula_25.
The synthesis part recovers the original signal from formula_26 by upsampling and filtering.
This kind of setup is used in many applications such as subband coding, multichannel acquisition, and discrete wavelet transforms.
Perfect reconstruction filter banks.
We can use polyphase representation, so input signal formula_23 can be represented by a vector of its polyphase components formula_28.
If we have set of polynomial vectors
formula_29
where formula_30 are polynomials.
The Module is analogous to the "span" of a set of vectors in linear algebra. The theory of Gröbner bases implies that the Module has a unique reduced Gröbner basis for a given order of power products in polynomials.
If we define the Gröbner basis as formula_31, it can be
obtained from formula_32 by a finite sequence of reduction
(division) steps.
Using reverse engineering, we can compute the basis vectors formula_33 in terms of the original vectors formula_34 through a formula_35 transformation matrix formula_36 as:
formula_37
Mapping-based multidimensional filter banks.
Designing filters with good frequency responses is challenging via Gröbner bases approach.
Mapping based design in popularly used to design nonseparable multidimensional filter banks with good frequency responses.
The mapping approaches have certain restrictions on the kind of filters; however, it brings many important advantages, such as efficient implementation via lifting/ladder structures.
Here we provide an example of two-channel filter banks in 2D with sampling matrix 
formula_38
We would have several possible choices of ideal frequency responses of the channel filter formula_39 and formula_40. (Note that the other two filters formula_41 and formula_42 are supported on complementary regions.)
All the frequency regions in Figure can be critically sampled by the rectangular lattice spanned by formula_43.
So imagine the filter bank achieves perfect reconstruction
with FIR filters. Then from the polyphase domain characterization it follows that the filters H1(z) and G1(z) are completely
specified by H0(z) and G0(z), respectively. Therefore, we need to design H0(x) and G0(z) which have desired frequency responses and satisfy the polyphase-domain conditions.
formula_44
There are different mapping technique that can be used to get above result.
Filter-bank design in the frequency domain.
When perfect reconstruction is not needed, the design problem can be simplified by working in frequency domain instead of using FIR filters.
Note that the frequency domain method is not limited to the design of nonsubsampled filter banks (read ).
Direct frequency-domain optimization.
Many of the existing methods for designing 2-channel filter banks are based on transformation of variable technique. For example, McClellan transform can be used to design 1-D 2-channel filter banks. Though the 2-D filter banks have many similar properties with the 1-D prototype, but it is difficult to extend to more than 2-channel cases.
In Nguyen, the authors talk about the design of multidimensional filter banks by direct optimization in the frequency domain. The method proposed here is mainly focused on the M-channel 2D filter banks design. The method is flexible towards frequency support configurations. 2D filter banks designed by optimization in the frequency domain has been used in Wei and Lu. In Nguyen's paper,