This article provides a short survey of the concepts, principles and applications of Multirate filter banks and Multidimensional Directional filter banks.
Multirate systems Linear time-invariant systems typically operate at a single sampling rate, which means that we have the same sampling rate at input and output. In other words, in an LTI system, the sampling rate would not change in the system.
Systems that use different sampling rates at different stages are called multirate systems. The multirate system can have different sampling rates based on desire. Also multirate systems can provide different sampling rates without destroying the signal components. In Figure 1, you can see a block diagram of a two channel multirate system.
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. One important fact in multirate filtering is that the signal should be filtered before decimation, otherwise aliasing and frequency folding would occur.
Multirate filter designs Multirate filter design makes use of properties of decimation and interpolation (or expansion) in the design implementation of the filter. Decimation or downsampling by a factor of essentially means keeping every sample of a given sequence.
Decimation, interpolation, and modulation Generally speaking, using decimation is very common in multirate filter designs. In the second step, after using decimation, interpolation will be used to restore the sampling rate. The advantage of using decimators and interpolator is that they can reduce the computations when resulting in a lower sampling rate.
Decimation by a factor of can be mathematically defined as:Mitra, Sanjit Kumar, and Yonghong Kuo. Digital signal processing: a computer-based approach. New York: McGraw-Hill, 2006.
or equivalently, .
Expansion or upsampling by a factor of M means that we insert M-1 zeros between each sample of a given signal or a sequence. The expansion by a factor of M can be mathematically explained as:
or equivalently, .
Modulation is needed for different kinds of filter designs. For instance, in many communication applications we need to modulate the signal to baseband. After using lowpass filtering for the baseband signal, we use modulation and change the baseband signal to the center frequency of the bandpass filter. Here we provide two examples of designing multirate narrow lowpass and narrow bandpass filters.
Narrow lowpass filter We can define a narrow lowpass filter as a lowpass filter with a narrow passband. In order to create a multirate narrow lowpass FIR filter, we need to replace the time invariant FIR filter with a lowpass antialiasing filter and use a decimator along with an interpolator and lowpass anti-imaging filter. In this way the resulting multirate system would be a time varying linear phase filter via the decimator and interpolator. This process is explained in block diagram form where Figure 2 (a) is replaced by Figure 2(b). The lowpass filter consists of two polyphase filters, one for the decimator and one for the interpolator.
Multirate filter bank Filter banks has different usage in many areas, such as signal and image compression, and processing. The main usage of using filter banks is that in this way we can divide the signal or system to several separate frequency domains.
A filter bank divides the input signal into a set of signals . In this way each of the generated signals corresponds to a different region in the spectrum of . In this process it can be possible for the regions overlap (or not, based on application). Figure 4 shows an example of a three-band filter bank. The generated signals can be generated via a collection of set of bandpass filters with bandwidths and center frequencies (respectively). A multirate filter bank use 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 (see Figure 5) an analysis-synthesis system can be used . In figure 5, only 4 sub-signals are used.
The signal would split with the help of four filters 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 filtere by 4 synthesis filters for k = 0,1,2,3. Finally, the outputs of these four filters are added.
Multidimensional filter banks Multidimensional Filtering, downsampling, and upsampling are the main parts of multidimensional multirate systems and filter banks.
A complete filter bank consists of the analysis and synthesis sides. 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. As illustrated in Figure 6, the input gets divided 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 X 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 . A quincunx lattice that is generated by this matrix is shown in figure.
It is important to analyze filter banks from a frequency domain perspective in terms of subband decomposition and reconstruction. However, equally important is a Hilbert space interpretation of filter banks, which plays a key role in geometrical signal representations. For a generic K-channel filter bank, with analysis filters , synthesis filters , and sampling matrices . In the analysis side, we can define vectors in as
, each index by two parameters: and .
Similarly, for the synthesis filters we can define .
Considering the definition of analysis/synthesis sides we can verify that
and for reconstruction part .
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
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 .
Multidimensional filter banks design 1-D filter banks have been well developed until today. However, many signals, such as image, video, 3D sound, radar, sonar, are multidimensional, and require the design of multidimensional filter banks.
With the fast development of communication technology, signal processing systems need more room to store data during the processing, transmission and reception. In order to reduce the data to be processed, save storage and lower the complexity, multirate sampling techniques were introduced to achieve these goals. Filter banks can be used in various areas, such as image coding, voice coding, radar and so on.
Many 1D filter issues were well studied and researchers proposed many 1D filter bank design approaches. But there are still many multidimensional filter bank design problems that need to be solved. Some methods may not well reconstruct the signal; some methods are complex and hard to implement.
Design of separable filter bank The simplest approach to design a multi-dimensional filter banks is to cascade 1D filter banks in the form of a tree structure where the decimation matrix is diagonal and data is processed in each dimension separately. Such systems are referred to as separable systems.
Design of non-separable multidimensional filter banks Below are several approaches on the design of multidimensional filter banks.
2-channel multidimensional perfect reconstruction (PR) filter banks
In real life, we always want to reconstruct the divided signal back to the original one, which makes PR filter banks very important. Let H(z) be the transfer function of a filter. The size of the filter is defined as the order of corresponding polynomial in every dimension. The symmetry or anti-symmetry of a polynomial determines the linear phase property of the corresponding filter and is related to its size. Like the 1D case, the aliasing term A(z) and transfer function T(z) for a 2 channel filter bank are:
A(z)=1/2(H0(-z) F0 (z)+H1 (-z) F1 (z)); T(z)=1/2(H0 (z) F0 (z)+H1 (z) F1 (z)), where H0 and H1 are decomposition filters, and F0 and F1 are reconstruction filters.
The input signal can be perfectly reconstructed if the alias term is cancelled and T(z) equal to a monomial. So the necessary condition is that T'(z) is generally symmetric and of an odd-by-odd size. Linear phase PR filters are very useful for image processing. This 2-Channel filter bank is relatively easy to implement. But 2 channels sometimes are not enough for use. 2-channel filter banks can be cascaded to generate multi-channel filter banks.
To understand the working of 2-Channel Multidimensional filter banks we must first understand the design process of a simple 2D two-channel filter banks. In particular the diamond filter banks are of special interest in some image coding applications. The decimation matrix M for the diamond filter bank is usually the quincunx matrix which is briefly discussed in the above sections. For a two-channel system, there are only four filters, two analysis filters and two synthesis filters. So in some designs, two or three filters are chosen so that there is no aliasing and the remaining filters are then optimized to achieve approximate reconstruction. Design of 2D filters are more complex than 1D filters. So we usually use appropriate mapping techniques to achieve perfect reconstruction.. A polyphase mapping method is proposed to design an IIR analysis filter. For filter banks with FIR filter types, several 1D to 2D transformations have been considered. For example, the McClellan transformation is used to achieve the FIR filter banks.
There has also been some interest in quadrant filters as shown. The decimation matrix for a quadrant filter bank is given by D =
The fan filters are shifted versions of the diamond filters and hence the diamond filter banks are designed and can be shifted by ( π 0 ) in the frequency domain to obtain a fan filter. Filter banks in which the filters have parallelogram support are also of some importance. Several parallelogram supports for analysis and synthesis filters are also shown. These filters can be derived from the diamond filters by using uni-modular transformation.
Tree-structured filter banks
For any given subband analysis filter bank, we can split it into further subbands as shown in figure 8. By repeating this operation we can actually build a tree-structured analysis bank. Example of a 1D tree structured filter bank is the one that results in an octave stacking of the passbands. In the 2D case, tree structures based on simple two-channel modules can offer sophisticated band-splitting schemes, especially if we combine the various configurations shown above. The directional filter bank which will be discussed below is one such example.
Multidimensional directional filter banks