Filter Bank: Insights into Computer Vision's Filter Bank Techniques

By Fouad Sabry

What is Filter Bank

A filter bank is an array of bandpass filters that is used in signal processing. Its purpose is to divide the input signal into several components, each of which carries a sub-band of the original signal. Attenuating the components in a new way and recombining them into a modified version of the original signal is one of the applications of a filter bank. A graphic equalizer is one example of this type of application. The result of analysis is referred to as a subband signal, and it contains as many subbands as there are filters in the filter bank. The process of decomposition that is carried out by the filter bank is referred to as analysis. Synthesis is the term used to describe the process of reconstruction, which refers to the act of reconstructing a complete signal that was produced by the filtering process.

How you will benefit

(I) Insights, and validations about the following topics:

Chapter 1: Filter Bank

Chapter 2: Discrete Fourier Transform

Chapter 3: Digital Filter

Chapter 4: Wavelet

Chapter 5: Modified Discrete Cosine Transform

Chapter 6: Finite Impulse Response

Chapter 7: Daubechies Wavelet

Chapter 8: Discrete Wavelet Transform

Chapter 9: Discrete-Time Fourier Transform

Chapter 10: Downsampling (Signal Processing)

(II) Answering the public top questions about filter bank.

(III) Real world examples for the usage of filter bank in many fields.

Who this book is for

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of Filter Bank.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: Apr 29, 2024

Book preview

Chapter 1: Filter bank

A filter bank (or filterbank) is a collection of bandpass filters used in signal processing to divide an input signal into many channels, each of which carries a discrete frequency band. A graphic equalizer makes use of a filter bank to attenuate individual signal components before recombining them in a way that alters the overall signal. Analyzing the signal in terms of its components in each sub-band is what the filter bank does when it decomposes it, and the resulting signal has as many sub-bands as there are filters in the bank. Synthesis refers to the process of re-creating a whole signal after filtering has been applied.

A group of receivers might be referred to as a filter bank in digital signal processing. However, receivers also down-convert the subbands to a low center frequency so that they can be re-sampled at a lower rate. It is possible to get the same effect by undersampling the bandpass subbands in some cases.

When some frequencies are more crucial than others in a signal, filter banks can be used to compress the signal. Decomposition allows for highly precise encoding of the crucial frequencies. A coding technique that retains even minute variations at these frequencies is necessary. However, minor frequencies can be off a little bit. If the finer (but less relevant) features can be sacrificed, a coarser coding scheme can be utilized.

The vocoder works by imposing the modulator's dynamic properties on the carrier signal by using the modulator's subband amplitude information to regulate the amplitude of the carrier signal's subbands. The modulator signal in this case is the voice.

Some filter banks operate almost exclusively in the time domain, dividing the signal into discrete frequency bands via a sequence of filters such as quadrature mirror filters or the Goertzel algorithm. The Fourier transform is used quickly in other filter banks (FFT).

Applying a series of FFTs to overlapping chunks of the input data stream yields a receiver bank.

Each segment is given its own weighting function (or window function) to regulate the filters' frequency responses.

The broader the form, the, the greater the frequency with which FFTs must be performed in order to meet Nyquist sampling requirements.

Given a constant segment length,, FFTs are performed at a rate proportional to the amount of overlap (and vice versa).

Also, More circular filter shapes, the less input bandwidth splitting filters are required,.

Removing extra steps (such as.

frequency decimation) quickly and easily by thinking about each weighted segment as a series of blocks, and just the total of the blocks is subjected to the FFT.

Weight overlap-add (WOLA) and weighted pre-sum fast Fourier transform (FFT) are two names for this technique.

(see § Sampling the DTFT)

When the length of the blocks is planned to be an integer multiple of the time between FFTs, we have a specific case. One or more polyphase filter structures, where the phases are recombined by an FFT rather than a simple summation, can then be used to characterize the FFT filter bank. The length (or depth) of the impulse response of a filter is equal to the number of blocks in a given segment. On a general purpose processor, the computational efficiency of the FFT and the polyphase structures are equivalent.

Simply upsampling each receiver's output at a rate proportional to the desired bandwidth, translating each channel to its new center frequency, and summing the streams of samples constitutes synthesis (i.e., the recombination of the outputs of many receivers). The upsampling-related interpolation filter is known as a synthesis filter in this setting. Each channel's net frequency response is calculated by multiplying the synthesis filter's response by the filter bank's response (analysis filter). It is preferable for there to be a constant value between the channel centers at every frequency in the sum of the frequency responses of adjacent channels. We call that state of affairs perfect reconstruction..

A filter bank, in the context of time-frequency signal processing, is a joint time-frequency domain representation of the signal in the form of a specific quadratic time-frequency distribution (TFD). It describes the class of quadratic (or bilinear) time-frequency distributions and is related to the Wigner-Ville distribution via a two-dimensional filtering. While the spectrogram is obtained by slicing the time domain and then performing a Fourier transform, the filter bank is obtained by slicing the frequency domain and then forming bandpass filters that are excited by the signal of interest. Both methods result in a quadratic TFD, but the filter bank and the spectrogram are the simplest to implement.

A signal is split up into multiple bands by using a multirate filter bank, which, depending on the width of the frequency bands, can be analyzed at varying speeds.

Downsampling (decimation) and upsampling are used in the implementation (expansion).

See Discrete-time Fourier transform § Properties and Z-transform § Properties for additional insight into the effects of those operations in the transform domains.

To put it another way, a narrow lowpass filter is one that has a relatively small passband. Substituting a lowpass antialiasing filter, decimator, interpolator, and lowpass anti-imaging filter for the time-invariant FIR filter yields a multirate narrow lowpass FIR filter. The decimator and interpolator turn the multirate system into a linear-phase filter with a variable time constant. The lowpass filter is made up of an interpolator polyphase filter and a decimator polyphase filter.

A filter bank divides the input signal x\left(n\right) into a set of signals x_{{1}}(n),x_{{2}}(n),x_{{3}}(n),... .

In this way each of the generated signals corresponds to a different region in the spectrum of x\left(n\right) .

It's possible that the zones will overlap during this procedure, on the basis of submissions).

The generated signals x_{{1}}(n),x_{{2}}(n),x_{{3}}(n),... can be generated via a collection of set of bandpass filters with bandwidths {\displaystyle {\rm {BW_{1},BW_{2},BW_{3},...}}} and center frequencies f_{{c1}},f_{{c2}},f_{{c3}},... (respectively).

By filtering and subsampling a single input signal, a multirate filter bank generates numerous versions of the signal at different rates.

Creating two or more distinct output signals from a same input signal, the usage of an analysis-synthesis tool.

The signal would split with the help of four filters H_{{k}}(z) 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.

Each band's transmission is split into its constituent frequencies, Our signals would look different.

The filter's synthesis function involves recreating the initial 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 F_{{k}}(z) for k = 0,1,2,3.

Finally, Adding the results from these four filters.

In addition to the conventional perfect reconstruction property, a discrete-time filter bank framework permits the incorporation of features dependent on the input signal.

Features from the realm of information theory, such as maximum energy compaction,, The ideal filter banks are designed with complete de-correlation of sub-band signals and other features for the specified input covariance/correlation structure.

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.

Multirate systems and filter banks primarily consist of multidimensional filtering, downsampling, and upsampling.

Both the analysis and synthesis sides make up a full filter bank.

An input signal is separated into subbands with distinct frequency ranges using the analysis filter bank.

In the synthesis phase, the individual subband signals are pieced back together to create a new signal.

The decimator and expander are two fundamental components.

For example, The input is split into four separate subbands, or directions, with each covering one of the frequency zones shown as a wedge.

In one-dimensional systems, Only samples that are multiples of M are kept by M-fold decimators; the rest are thrown away.

while in multi-dimensional systems the decimators are D × D nonsingular integer matrix.

It only takes into account samples that are on the lattice that the decimator produced.

Commonly used decimator is the quincunx decimator whose lattice is generated from the Quincunx matrix which is