\\n\\n
Dr. Pletser’s experience includes 30 years of working with the European Space Agency as a Senior Physicist/Engineer and coordinating their parabolic flight campaigns, and he is the Guinness World Record holder for the most number of aircraft flown (12) in parabolas, personally logging more than 7,300 parabolas.
\\n\\nSeeing the 5,000th book published makes us at the same time proud, happy, humble, and grateful. This is a great opportunity to stop and celebrate what we have done so far, but is also an opportunity to engage even more, grow, and succeed. It wouldn't be possible to get here without the synergy of team members’ hard work and authors and editors who devote time and their expertise into Open Access book publishing with us.
\\n\\nOver these years, we have gone from pioneering the scientific Open Access book publishing field to being the world’s largest Open Access book publisher. Nonetheless, our vision has remained the same: to meet the challenges of making relevant knowledge available to the worldwide community under the Open Access model.
\\n\\nWe are excited about the present, and we look forward to sharing many more successes in the future.
\\n\\nThank you all for being part of the journey. 5,000 times thank you!
\\n\\nNow with 5,000 titles available Open Access, which one will you read next?
\\n\\nRead, share and download for free: https://www.intechopen.com/books
\\n\\n\\n\\n
\\n"}]',published:!0,mainMedia:null},components:[{type:"htmlEditorComponent",content:'
Preparation of Space Experiments edited by international leading expert Dr. Vladimir Pletser, Director of Space Training Operations at Blue Abyss is the 5,000th Open Access book published by IntechOpen and our milestone publication!
\n\n"This book presents some of the current trends in space microgravity research. The eleven chapters introduce various facets of space research in physical sciences, human physiology and technology developed using the microgravity environment not only to improve our fundamental understanding in these domains but also to adapt this new knowledge for application on earth." says the editor. Listen what else Dr. Pletser has to say...
\n\n\n\nDr. Pletser’s experience includes 30 years of working with the European Space Agency as a Senior Physicist/Engineer and coordinating their parabolic flight campaigns, and he is the Guinness World Record holder for the most number of aircraft flown (12) in parabolas, personally logging more than 7,300 parabolas.
\n\nSeeing the 5,000th book published makes us at the same time proud, happy, humble, and grateful. This is a great opportunity to stop and celebrate what we have done so far, but is also an opportunity to engage even more, grow, and succeed. It wouldn't be possible to get here without the synergy of team members’ hard work and authors and editors who devote time and their expertise into Open Access book publishing with us.
\n\nOver these years, we have gone from pioneering the scientific Open Access book publishing field to being the world’s largest Open Access book publisher. Nonetheless, our vision has remained the same: to meet the challenges of making relevant knowledge available to the worldwide community under the Open Access model.
\n\nWe are excited about the present, and we look forward to sharing many more successes in the future.
\n\nThank you all for being part of the journey. 5,000 times thank you!
\n\nNow with 5,000 titles available Open Access, which one will you read next?
\n\nRead, share and download for free: https://www.intechopen.com/books
\n\n\n\n
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The sampling operators could either be downsamplers or upsamplers. The downsamplers are called decimators while the upsamplers are called expanders. The technique of filter banks plays an important role in most digital systems that rely on signal processing for their operations. Using this technique, any signal feature can be reliably extracted and analyzed; hence filter banks have wide applications in digital signal processing systems. A filter bank as shown in Figure 1 [1, 2] consists of different parts, which collectively execute a desired function.
\nk-Channel filter bank [1, 2].
As can be seen in Figure 1, the filter bank is made of two sections: the analysis filter bank section (composed of analysis filters and downsamplers), and the synthesis filter bank section (composed of upsamplers and synthesis filters). In this chapter, we will discuss the analysis and synthesis filter bank sections, their responses to incoming signals, and how they work together in the derivation of a wavelet transform function.
\nThe analysis filter bank section is made up of the analysis filter banks, and downsamplers or decimators which together act on an input signal to perform a desired function through decomposition of the signal. In this section, we will analyze the mathematical relationship that exists between these two components. To have a thorough understanding of this relationship, it is important to briefly discuss these components separately.
\nThe filters that make up the analysis filter banks could either be low-pass filters, or high-pass filters. Each of these filters, as shown in Figure 2, allows the passage of only a particular frequency component of the input signal \n
Separation of input signals into sub-band frequencies by analysis filter bank.
It can be seen from the frequency responses that the output of the filters overlap each other. This is because in practice, the filters are not ideal. However, the overlapping condition can be improved through an optimized design of the filters. Mathematically, the effect of each of the filters in the filter bank on the input signal \n
where \n
The downsampler shown in Figure 1 downsamples an input signal by a factor of N. This implies that it only retains all the Nth samples in a given sequence. For example, if \n
Figure 3 shows the conceptual depiction of the relationship in Eq. (2).
\nDecimation by a factor of 2.
Mathematically, the output of the decimator in Figure 1 can be expressed as a product of the input sequence \n
The relationship in Eq. (3) will only select the kNth sample of \n
\n
Setting \n
\n
Substituting Eq. (5) into (3) yields:
\n\n
In terms of z-transformation, the relationship in Eq. (6) can be expressed as:
\n\n
Having looked at the analysis filters and downsamplers, we will now turn our attention to synthesis filter bank section of Figure 1.
\nThe synthesis filter bank section is made of the upsamplers and synthesis filter banks. These components work together to perform the opposite operation performed by the analysis filter bank section shown in Figure 1. In this section, we will make an analysis of the mathematical relationship that governs the operation of the synthesis filters and upsamplers.
\nSimilar to the analysis filter bank, the synthesis filter bank is made of low-pass and high-pass filters. The output of these filters as shown in Figure 1, are summed to a common output. In typical filter bank applications, the frequency responses of these filters are typically matched to those of the analysis filters shown in Figure 2. The mathematical expression for the effect each of these filters has on the corresponding input signal \n
\n
In Figure 2, the input to the synthesis filter bank is upsamplers or expanders. The next section gives a brief review of the upsamplers.
\nThe upsampler expands an input signal by a factor N. It does this by inserting zeros at every nth position in the sequence of the input signal. For example, if \n
Upsampling by a factor of 2.
Given an input signal \n
\n
Similar to the expression in Eq. (3), the z-transform of the expression in Eq. (9) which is an upsampler is stated as follows [8]:
\n\n
To be useful in wavelet designs, filter banks must be designed to have certain characteristics which guarantee that a signal at the input of a filter bank will be received accurately at the output of the filter bank. In the next section, we will examine the properties of filter banks and how these properties influence the design of wavelets.
\nIn wavelet designs, filter banks are required to possess three important properties which are fundamental to the realization of a wavelet function. These properties include: perfect reconstruction, orthogonality, and paraunitary condition.
\nThis property guarantees that the signal at the output of a given filter bank is a delayed version of the signal at the input of the filter bank. Perfect reconstruction is an important property of a filter bank because it cancels the effect of aliasing of the input signal at the output, caused by the downsamplers and upsamplers. To understand this point, consider a two-channel finite impulse response FIR filter bank shown in Figure 5.
\nTwo-channel FIR filter bank.
The output \n
where the signal_component and aliasing_component are defined as:
\n\n
To achieve perfect reconstruction, the following condition must be satisfied [1]:
\n\n
The relationships in Eqs. (11) and (13) are possible when the filter bank is constructed as a QMF (quadrature mirror filter) filter bank or CQF (conjugate quadrature filter) filter bank. Both QMF and CQF banks provide a mechanism by which complete cancellation of the aliasing component in Eq. (11) can be accomplished. Using QMF, aliasing cancellation can be achieved by constructing the filters in Figure 5 based on the following relationships [4, 5]:
\n\n
In Eq. (14), the synthesis filter \n
QMF two-channel FIR filter bank.
For the CQF bank, the coefficients of the analysis filter \n
\n
Based on the relationship in Eq. (15), the filter bank shown in Figure 6 for CQF will assume the structure shown in Figure 7.
\nCQF two-channel FIR filter bank.
Based on the structure of Figures 6 or 7, the output signal \n
\n
If we impose the condition that \n
\n
The relationship in Eq. (17) states that the output signal \n
Having looked at perfect reconstruction as a necessary property for a filter bank in wavelet design, we now look at orthogonality as also an essential property for a filter bank in the design of wavelets.
\nOrthogonality in a filter bank is a situation in which the synthesis filter bank is a transpose of the analysis filter bank. This is a useful property in the sense that it allows for the energy preservation of the signal being processed. This important property is achieved through the imposition of the orthogonality condition on both the analysis and filter bank sections while at the same time preserving the perfect reconstruction condition of the filter bank. The imposition of the orthogonality condition in a filter bank (see Figure 5) occurs when the following relationships are satisfied [11]:
\nwhere
\nand
\nIn Eq. (18), the inner product of the coefficients of the synthesis filter \n
Also, the low-pass analysis filter \n
where L denotes the length of the filter which must be even, and c is a constant with \n
The condition in Eq. (20) also describe the necessary requirement for a filter bank to be paraunitary (which we shall examine in the next section), i.e., the low-pass filter \n
where \n
then \n
The orthogonality condition for a filter bank can also be examined from a polyphase perspective. Consider the polyphase representation of the filter bank in Figure 5 as illustrated in Figure 8 [13].
\nPolyphase implementation of filter bank.
If \n
The conditions in Eqs. (20)–(22) hold true iff \n
where \n
We now look at the paraunitary condition of a filter bank, which is also a necessary property in filter bank implementation of wavelets.
\nIn the filter bank implementation of a wavelet transform, the paraunitary condition plays the critical role of guaranteeing the generation of orthonormal wavelets, and also perfect recovery of a decomposed signal. The paraunitary condition guarantees that recovered signal will suffer no phase or aliasing effect if a filter bank satisfies the paraunitary condition [14].
\nGiven a polyphase transfer function matrix \n
where the H superscript denotes the conjugated transpose, and I denotes the identity matrix. Paraunitary filter banks also have an attractive property of losslessness, which implies that for every frequency, the total signal power is conserved [16]. From this property [17], any \n
Cascade implementation of \n\nE\n\nz\n\n\n as FIR lossless unitary matrices separated by delays.
If the real-coefficient lossless matrix is denoted by \n
\n
where \n
\n
Having looked at the filter bank and its three important properties for the design of a wavelet, we will in the next section examine the application of these properties in the design of a wavelet.
\nThe filter bank design of a wavelet transform is usually implemented from the analysis filter bank segment to the synthesis filter bank segment.
\nGiven that the expression for a scaling function \n
where \n
which translates into:
\nwhen \n
In a similar consideration to Eq. (28), the wavelet function \n
where \n
which translates into:
\nwhen \n
In the synthesis filter bank, the reconstruction of the original coefficients of a signal can be achieved through the combination of the scaling and wavelet function coefficients at a coarse level of resolution. Given a signal at \n
For the next scale, Eq. (34) becomes:
\nSubstituting Eqs. (28) and (31) into Eq. (35) and after algebraic manipulations yields [14]:
\nIn the design of a wavelet system using filter banks, it is of utmost importance that the filters which will execute the filter bank system as shown in Figure 1, possess the properties discussed in Section 4. Owing to the fact that in a filter bank, all the filters can be derived from an initial filter \n
State chart for wavelet design procedure.
In the first state in Figure 10, the design problem formulation which can be achieved using trigonometric polynomial, takes the following into consideration [14]:
Compact support which guarantees that the wavelet is characterized by finite non-zero coefficient.
Paraunitary condition which guarantees the generation of orthonormal wavelets.
Flatness/k-regularity which guarantees the smoothness of the wavelet in both time and frequency domains.
The second state which involves conditioning the problem as a tractable problem involves, if necessary, transforming a non-linear formulation of the problem to a linear formulation, and then optimizing the problem using techniques like convex optimization. The generation of the filter coefficients using solvers in the third state of the machine involves techniques like spectral factorization. Through simulation in the fourth state of the chart, the generated coefficients can be verified whether or not they meet the design constraints. Using the QMF or CQF relationships in Eqs. (13) and (14), the other filters in the filter bank are generated in the fifth state of the chart.
\nIn this chapter, we have presented an analysis of the design of wavelets using filter bank technique. The chapter looked at the two major components of a filter bank which the analysis and the synthesis components. The properties of filter banks which are desirable in the design of wavelets were also investigated, alongside the mathematical description of these properties. The chapter also gave a brief mathematical description of the role the analysis and the synthesis filter banks play in the design of wavelets. Finally, the required general procedure for the design of wavelets was presented, showing the necessary steps to take in order to achieve an effective design.
\nThe major contribution of this chapter is the provision of a step by step analysis and procedure for the design of filter banks in a precise and concise manner.
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