Filter bank
[1] 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 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.
A bank of receivers can be created by performing a sequence of FFTs on overlapping segments of the input data stream.
(see § Sampling the DTFT) A special case occurs when, by design, the length of the blocks is an integer multiple of the interval between FFTs.
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.
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.
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.
However, equally important is Hilbert-space interpretation of filter banks, which plays a key role in geometrical signal representations.
[7] Figure shows a general multidimensional filter bank with N channels and a common sampling matrix M. The analysis part transforms the input signal
In order to reduce the data to be processed, save storage and lower the complexity, multirate sampling techniques were introduced to achieve these goals.
In order to reduce the data rate, downsampling and upsampling are performed in the analysis and synthesis stages, respectively.
The symmetry or anti-symmetry of a polynomial determines the linear phase property of the corresponding filter and is related to its size.
It has many distinctive properties like: directional decomposition, efficient tree construction, angular resolution and perfect reconstruction.
The perfect reconstruction condition for an oversampled filter bank can be stated as a matrix inverse problem in the polyphase domain.
For 1-D oversampled FIR filter banks, the Euclidean algorithm plays a key role in the matrix inverse problem.
[15] And then use Algebraic geometry and Gröbner bases to get the framework and the reconstruction condition of the multidimensional oversampled filter banks.
According to the description of the paper, some new results in factorization are discussed and being applied to issues of multidimensional linear phase perfect reconstruction finite-impulse response filter banks.
The algorithmic theory of polynomial ideals and modules can be modified to address problems in processing, compression, transmission, and decoding of multidimensional signals.
The general multidimensional filter bank (Figure 7) can be represented by a pair of analysis and synthesis polyphase matrices
[18][19] The Gröbner-basis computation can be considered equivalently as Gaussian elimination for solving the polynomial matrix equation
[20][21] 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.
[22] When perfect reconstruction is not needed, the design problem can be simplified by working in frequency domain instead of using FIR filters.
[26] In Nguyen,[26] the authors talk about the design of multidimensional filter banks by direct optimization in the frequency domain.
They did simulations based on different parameters and achieve a good quality performances in low decimation factor.
Moreover, in order to obtain the desired frequency partition, a complicated tree expanding rule has to be followed.
Filter banks are important elements for the physical layer in wideband wireless communication, where the problem is efficient base-band processing of multiple channels.
A filter-bank-based transceiver architecture eliminates the scalability and efficiency issues observed by previous schemes in case of non-contiguous channels.
In order to obtain universally applicable designs, mild assumptions can be made about waveform format, channel statistics and the coding/decoding scheme.
Both heuristic and optimal design methodologies can be used, and excellent performance is possible with low complexity as long as the transceiver operates with a reasonably large oversampling factor.
