In image or signal processing, instruments often use Discrete Fourier Transform (DFT) on input signals

Polyphase Filter Banks

A polyphase filter bank is a multi-rate filter structure combined with a DFT to extracts sub-bands from an input signal. It is simply a computational structure for applying resampling and filtering to a signal.

In image or signal processing, instruments often use Discrete Fourier Transform (DFT) on input signals. DFT has two main drawbacks: leakage and scalloping loss.

The Polyphase Filter Banks is a technique used to overcome the drawbacks of DFT. The Polyphase Filter Banks are used to creates a flat response across the channel, also provide excellent suppression of out-of-band signals.

A polyphase filter bank implements the DFT to modulate a prototype filter and computes summation and filtered time-domain data from the number of DFT stages.

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Mathematically, the weighting(window) function can be written as,

$\begin{align*} y(n) = \sum_{p=0}^{P-1} x(n+pN)h(n+pN) \end{align*}$

Where h(n+pN) is the sub-filter coefficients that correspond to P-tap polyphase sub-filters. The N such polyphase sub-filters together with the number of DFT stage are collectively called a “Polyphase Filter Bank (PFB)”. The structure of a polyphase filter bank with P=3 taps and N sub-filters and the frequency response of a Polyphase Filter Bank is shown in the below image.

Frequency Response of a Polyphase Filter Bank

Gabor Filter Banks

In image or signal processing, Gabor Filter banks are defined as an array of functions for generating and filtering images using a bank of Gabor filters in a frequency domain.

Gabor Filter banks is a linear filter bank that is used for texture analysis, which usually means that it analyses whether there is any specific frequency band in the image in particular directions in a localized region around the region of analysis.

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Gabor filter is a convolution filter representing a combination of sinusoidal and gaussian components. The sinusoidal component provides directions and the gaussian component provides weights. The Gabor filter has a real and imaginary component that represents orthogonal directions.

Mathematically, it can be expressed as,

$\begin{align*} g(x,y;\lambda,\theta,\psi,\sigma,\gamma) = exp(\frac{{-x^'^2}{ +\gamma^2 y^'^2}}{2\sigma^2}) exp(i(\frac{2 \pi x^'}{\lambda}+{\psi})) \end{align*}$

So, Gabor is a convolution filter where we can change wavelength, orientation, phase offset, aspect ratio, etc. By changing these we can extract any number of Gabor filters.

By using real parts of a variety of different Gabor filter kernels we can filter the images. These Gabor filter kernels contain functions to extract amplitudes, phases, real and imaginary parts. The image response for different Gabor filter kernels at different frequencies and a frequency response of a Gabor filter bank is shown in the below images.

Image Response of Different Gabor Filter Kernels — Image Response of a Different Filter Kernels

Frequency Response of a Gabor Filter Banks — Frequency Response of a Gabor Filter Bank

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Gabor Filter Banks can compensate DC component also feature extraction and convolution can be done by using only one function. Frequency and orientation representations of the Gabor filter banks are similar to the human visual system.

Mel Filter Banks

Mel Filter Banks is a triangular filter bank that works similar to the human ears perception of sound which is more discriminative at lower frequencies and less discriminative at higher frequencies.

Mel Filter Banks are used to provide a better resolution at low frequencies and less resolution at high frequencies. It captures the energy at each critical frequency band and gives approximates the spectrum shape. The below block diagram shows how Mel Filter Banks works to obtain Mel-Frequency Cepstral Coefficients.

Diagram of a Mel Filter Bank — Block Diagram of How Mel Filter Bank Works

As shown, a speech signal is applied to a pre-emphasis filter, this filter sectionalized speech signal into (overlapping) frames then a window function is applied to each frame. Note that a Pre-emphasis filter is useful for 1. to balance the frequency spectrum as high frequencies usually have smaller magnitudes compared to lower frequencies, 2. to improve Signal-to-Noise Ratio (SNR).

Now, we need to separate the speech signal into short-time frames. To do that a Discrete Fourier Transform (DFT) on each frame (especially a Short-Time Fourier Transform) is applied and calculate the energy or power spectrum. Hence, we get an approximation of the frequency spectrum of the speech signal by cascading adjacent frames.

Now, to obtain Mel-Frequency Cepstral Coefficients (MFCC), an Inverse Discrete Fourier Transform (IDFT) or a Discrete Cosine Transform (DCT) is applied to the Mel Filter Banks, thus we get several MFCCs coefficients while the rest are discarded.

Then, the final step to computing Mel filter Banks is applying triangular filters on a Mel-scale to extract frequency bands. We can convert Mel(m) and frequency(f)(Hertz) by using the following equations.

$\begin{align*} Mel(m) = 2595 log_1_0(1+\frac{f}{700}) \end{align*}$

$\begin{align*} Frequency(f) = 700 (10^\frac{m}{2595} - 1) \end{align*}$

Each filter in the Mel Filter Bank is a triangular filter having a frequency response of 1 at the centre frequency and decreases linearly towards 0 till it reaches the centre frequencies of the two adjacent filters.

Mathematically, the Mel spectrum can be expressed as,

$\begin{align*} {S^~}(l) = \sum_{k=0}^\frac{N}{2} S(k)M_l(k) \end{align*}$

For each Mel spectrum, we get a triangular frequency response. Hence, the set of Mel spectrum i.e., m₁, m₂, m_3,……,m_M we get a whole range of triangular frequency responses. The frequency response of a Mel Filter bank is shown in the below image.

Filter Bank Multicarrier (FBMC)

FBMC stands for Filter Bank Multicarrier is defined as a form of multi-carrier modulation that transmits the data by splitting it into several components and transmit each of these components through separate carries signals. FBMC divides the frequency spectrum into narrow subchannels.

One of the limitations of OFDM is that the spectral localization of the subcarriers is weak. This results in spectral leakage and hence it interferes with unsynchronized signals.

FBMC is an alternative transmission method that is used to overcome the limitations of OFDM by using high-quality filters that reject both ingress and egress noises.

The comparison of frequency response between OFDM Vs FBMC is shown in the below graph. As shown, when carriers are modulated in an OFDM system, side-lobes are spread out either side but with the FBMC system these side-lobes are removed and we get cleaner carriers.

Comparison of a Frequency Response Between OFDM and FBMC — OFDM Vs FBMC

In FBMC systems, FIR filters are preferred over IIR filters, one reason behind that FIR filters have a linear phase whereas IIR does not. FBMC can provide higher data rates within a given radio spectrum band.

DFT Filter Banks

DFT Filter Banks is Fourier transform filter bank which can be constructed using length M FIR bandpass filters where each bandpass filter is implemented as a demodulator followed by a running-sum lowpass filter.

DFT Filter Banks are similar to the DCT filter banks but it uses complex numbers whereas DCT filter banks use only real numbers.

The below system diagram shows the N-channel DFT filter bank constructed using length M FIR running-sum lowpass filters.

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The k^th channel computes from the above system diagram are given by,

(1) $\begin{equation*} H_k(z) = \sum_{m=n-(M-1)}^{M} x(m) e^-^j^{\omega_k}^m \end{equation*}$

As we know that the length M Discrete Fourier Transform is defined as,

(2) $\begin{equation*} X(k) = \sum_{n=0}^{M-1} x(n) e^\frac{^-^2^\pi^n^k}{M} \end{equation*}$

Where, X(k) is in the frequency domain

x(n) is in the time domain

Now, compare equation (1) and (2), we can see that the output of the filter bank i.e., $H_k(z)$ (where, k=0,1, 2…, M-1) is the DFT of the input signal when n=M-1, it is given by

(3) $\begin{equation*} X(k) = Z_k (M-1) \end{equation*}$

In other words, the output of the filter bank at time n=M-1 is equal to the DFT of the first M samples of x(n) (where, n=0,1, 2…, M-1). Hence, we can say that the output of the all-filter bank channels at time M-1 is the DFT of the input signal from time 0 through M-1.

Hence, the DFT system is like a filter bank with analysis filters $H_k(z)$ . The frequency response of a DFT filter bank is shown in the below figure.

Uniform DFT Filter Bank

An analysis filter bank with M (M>1) filters is said to be a uniform DFT filter bank if all the filters are obtained from $H_0(z)$ according to $H_k(z) = H_0(zw^k)$ , where, 0<=k<=M-1. Here, $H_0(z)$ is called a prototype filter.

Note that, $H_k(e^j^\omega) = H_0 e^j^(^\omega^-^\frac{2\pi k}{M}^)$ which means that the frequency response of $H_k(z)$ is a uniformly shifted version of $H_0(e^j^\omega)$ . The below figure the typical frequency response of a uniform DFT filter bank where $H_k(z)$ is taken to be a low-pass filter.

Frequency Response of a Uniform DFT Filter Bank

Advantages of Filter Bank

Some of the advantages of a filter bank include:

High resolution with low spectral leakage
More stable peaks
The more accurate and consistent noise floor
By using filter banks, we can set channelizers to get different resolutions at different frequency bands. This is used in audio spectral analysis.

Applications of Filter Banks

Some of the applications of the filter banks include:

The filter bank is used to compress the signal when some particular frequencies are more important than other frequencies. They are used in speech processing, image compression, communications systems, antenna systems, analog voice privacy systems, and in the digital audio industry.
The filter bank can be used as a graphic equalizer, which can attenuate the components of the signal differently and reconstructed them into an improved version of the original signal.
Filter banks are used for trans multiplexers, sub-band adaptive filtering, vocoder, etc. Here, the vocoder uses a filter bank to determine and control the amplitude of the sub-bands of the carrier signal.
DCT filter Banks are used in JPEG image compression and modifications of sub-bands for image modifications.
DCT Filter banks are used in audio signal processing, digital audio, digital radio, speech processing, etc.
FBMC (Filter Bank Multicarrier) can be used as a tool for spectrum sensing.
In document image processing, Gabor filter banks are used for identifying the script of a word in a multilingual document.
Gabor filter banks are widely used for decomposing an image into oriented sub bands such as texture analysis, edge and texture detection, feature extraction, object and shape recognition, scene segmentation, etc.

Polyphase Filter Banks

Gabor Filter Banks

Mel Filter Banks

Filter Bank Multicarrier (FBMC)

DFT Filter Banks

Uniform DFT Filter Bank

Advantages of Filter Bank

Applications of Filter Banks

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