What is a Filter Bank? In image or signal processing,

What is a Filter Bank?

In image or signal processing, a Filter Bank is defined as an arrangement of bandpass filters that split the input signal into a set of analysis signals, each one carrying a single frequency sub-band of the original signal.

In other words, a class of systems that generates scaling and wavelet function are known as filter banks. The below figure shows the simple structure of a filter bank. It is a set of bandpass filters with each filter centered at a different frequency.

A digital filter bank consists of multiple filters that share a common input or output. In digital signal processing, “filter bank” often refers to a set of receivers.

A filter bank is used to performed various tasks like bandwidth reduction, combinations of the common operations of spectral translation, spectral composition and decompositions of signals, and sample rate changes, etc. since most filter banks include various sampling rates, they are also referred to as multi-rate systems.

Filter banks are crucial in modern signal and image processing, particularly in audio and image coding.

How does a Filter Bank work?

A filter bank separates or splits the input signal into multiple components. The process of separating the input signals into multiple components is known as analysis. The output of the analysis is referred to as a sub-band of the original signal.

Now, the Filter bank attenuates the components of the signal differently and reconstructed them into an improved version of the original signal.  This reconstruction process is known as synthesis.

For example, if we have an input audio signal x(n), then filter banks separate this input audio signal into a set of analysis signals i.e., x₁(n), x₂(n), x₃(n), etc…, each of these set of analysis signals corresponds to a different region in the spectrum of the input signal x(n).

This set of analysis signals x₁(n), x₂(n), x₃(n)… can be obtained by filter banks with bandwidths BW₁, BW₂, BW₃… and centre frequencies f_c1, f_c2, f_c3… respectively.

The below figure shows the frequency response of a Filter Bank in which bands of a three-band filter bank do not overlap, but are lined up one after the other with adjacent band edges touching each other. These three bands span the frequency range from f_cl1 = 0 Hz to f_ch3 = f_max.

Analysis and Synthesis Filter Bank

There are two main types of filter banks. An analysis filter bank and a synthesis filter bank. An analysis filter bank is a set of analysis filters H_k(n) which splits an input signal into M sub-band signals X_k(n) and a synthesis filter bank is a set of M synthesis filters F_k(z) which combine M signal Y_k(n) into a reconstructed signal x^(n) as shown in the below figure.

The analysis filter bank decomposes the input signal to a different sub-band with a different frequency spectrum and the synthesis filter bank reconstructs the different sub-band signal and generates a modified version of the original signal.

Types of Filter Banks

There are several types of filter banks. Let’s explore each type.

DCT Filter Banks

DCT (Discrete Cosine Transform) Filter Banks is linear transform filter banks used to compress data in sets of discrete DCT blocks. DCT Filter Banks are similar to the DFT filter banks but it uses only real numbers.

The representation of the DCT decomposition as a filter bank or a bank of filters is shown in the below figure. This describes the simplification of the DCT as a filter bank and modifications of sub-bands for image modifications.

As shown, we apply DCT to blocks of 8*8 pixels.  we have one filter for each sub-band of the DCT. Here, 8 samples coming out from the filter are given to the down sampler. we have a block-wise representation that corresponds to down sampling by a factor of 8 for each sub-band.

This down sampler only passes every 8^th sample and blocks other samples hence, we get only one sample.

So, DCT works as a bank of filters as we get smaller sub-band images in sets of discrete DCT samples.

Advantages of a DCT Filter Bank

Some of the advantages of a DCT as filter banks include:

• We can represent a DCT Filter Bank as a square matrix multiplication since the square matrix is invertible hence, the DCT filter bank is invertible. So, the DCT filter bank can be used to obtain a synthesis filter bank for perfect reconstruction.
• The DCT is more efficient computationally.