analysis and synthesis filter

Filter Bank: What is it? (DCT, Polyphase, Gabor, Mel And FBMC)

What is a Filter Bank?

How does a filter bank work, analysis and synthesis filter bank, types of filter banks, dct filter banks, advantages of a dct filter bank, polyphase filter banks, gabor filter banks, mel filter banks, filter bank multicarrier (fbmc), dft filter banks.

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

Uniform DFT Filter Bank

Advantages of filter bank, applications of filter banks.

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Polyphase Analysis Filter Bank

I have a question regarding the polyphase DFT filter bank implementation in this page. ( https://cnx.org/contents/Peqc-TK2@16/Uniformally-Modulated-DFT-Filterbank )

In figure 2, to analyze the kth filter bank branch, the signal $x(n)$ was multiplied with $e^\dfrac{j2\pi kn}{M}$ . But to downconvert a signal to baseband and low pass filter it, the signal should be multiplied with $e^ -\dfrac{j2\pi kn}{M}$ . I made this change and continued derivation, I ended with getting IDFT block instead of getting DFT block as shown in Figure 7.

I referred multiple books and papers for this. Everywhere I found a DFT block in the block diagram. But, I couldn't find where I am doing wrong. Please help me in knowing the correct block diagram.

• filter-bank

• $\begingroup$ To imply that the negative must be used to down-convert also implies that the signal only exists in the positive frequency axis. If it is a real signal it will be equally positive and negative so either sign would “down-convert” the signal. $\endgroup$ –  Dan Boschen Commented Jan 6, 2019 at 14:12
• $\begingroup$ This analysis and synthesis filter bank algorithm, I am using in generating a wideband signal from $M$ narrowband signals and to extract the narrowband signals from the wideband signal. Since these signals are frequency division multiplexed, the wideband signal is a complex signal. $\endgroup$ –  Narendra Commented Jan 6, 2019 at 14:23
• 1 $\begingroup$ Hint: Look at the delay lines in a analysis/synthesis cascade. You are delaying different polyphases by a different amount of samples. $\endgroup$ –  Uroc327 Commented Jan 6, 2019 at 15:40

2 Answers 2

With every polyphase filter bank I have worked with, the first block in the analysis phase is an IFFT, and the block in the synthesis phase is a DFT. These operations essentially cancel one other, so it should be fairly intuitive. Think of the DFTs acting as sinusoids modulators in this case rather than an operation to convert to the frequency domain. By having complementary DFTs, the modulation is essentially performed in analysis and then removed in synthesis. From looking at the figure in the text you linked, it appears that they’re may be a misprint, at least in comparison to my industry experience implementing this types of filter banks.

The following is a great, compact and concise resource on polyphase filter banks, multi-rate identities, and DFT based filter banks: https://apps.dtic.mil/dtic/tr/fulltext/u2/a457390.pdf

• $\begingroup$ Wouldn’t it work either way, or is there a reason the first must be an IFFT? $\endgroup$ –  Dan Boschen Commented Jan 6, 2019 at 14:09
• 1 $\begingroup$ You’re right, it would work either way since multiplication is commutative. To my knowledge, a popular text by P.P Vaidyanathan popularized the structure of IDFT/DFT, and so it’s become sort of the standard order $\endgroup$ –  vintagevogue Commented Jan 6, 2019 at 14:20
• $\begingroup$ I think IFFT and FFT blocks can be interchanged in the analysis and synthesis filter banks. But, in the derivation, I am ending up getting IFFT blocks in both the filter banks. ( in.mathworks.com/help/dsp/ref/… ) ( in.mathworks.com/help/dsp/ref/channelizer.html ). In the derivation available on these pages, the last block diagrams are correct, but the derivation is wrong $\endgroup$ –  Narendra Commented Jan 6, 2019 at 14:27
• $\begingroup$ I have read the 4th chapter 'Fundamentals of multirate signal processing' from the book 'FILTER BANK TRANSCEIVERS FOR OFDM AND DMT SYSTEMS' by P. P. Vaidyanathan, but I couldn't find the correct solution in this book $\endgroup$ –  Narendra Commented Jan 6, 2019 at 14:31
• $\begingroup$ I have included the derivation part I have done in my question. Please look into it once and let me know if I am correct or not. $\endgroup$ –  Narendra Commented Jan 6, 2019 at 15:06

First of all, every suggestion and comment made me think once again and I could answer my question by myself. Thanks to all.

Considering $M$ signals $x_0(n)$ , $x_1(n)$ ,..... $x_{M-1}(n)$ , the prototype LPF $H(z)$ ,

$$H(z) =\sum_{l=0}^{M-1}z^{-l}S_l(z^M)=\sum_{l=0}^{M-1}z^{l}G_l(z^M) $$ where $s_l(n) = h(Mn+l)$ is the Inverse Z transform of $S_l(z)$ and $g_l(n) = h(Mn-l)$ is the Inverse Z transform of $G_l(z)$ , the final block diagram looks like

This design worked correctly. To verify this, I have sent 4 QPSK symbols across each channel. I chose the prototype LPF, $H(z) = 1+z^{-1}+z^{-2}+z^{-3}+z^{-4}+z^{-5}+z^{-6}++z^{-7}$ , so that each Polyphase filter $S_k(z) = 1$ , $G_k(z) = z^{-1}$ $ \forall k = 0,1,..7$ . This turns out to be an OFDM system. At the analysis filter bank output, I got the same QPSK symbols back but delayed by one sample because of polyphase filters in the analysis stage.

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Home > Books > Wavelet Transform and Complexity

Analysis of Wavelet Transform Design via Filter Bank Technique

Submitted: 03 December 2018 Reviewed: 07 February 2019 Published: 27 September 2019

DOI: 10.5772/intechopen.85051

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Wavelet Transform and Complexity

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The technique of filter banks has been extensively applied in signal processing in the last three decades. It provides a very efficient way of signal decomposition, characterization, and analysis. It is also the main driving idea in almost all frequency division multiplexing technologies. With the advent of wavelets and subsequent realization of its wide area of application, filter banks became even more important as it has been proven to be the most efficient way a wavelet system can be implemented. In this chapter, we present an analysis of the design of a wavelet transform using the filter bank technique. The analysis covers the different sections which make up a filter bank, i.e., analysis filters and synthesis filters, and also the upsamplers and downsamplers. We also investigate the mathematical properties of wavelets, which make them particularly suitable in the design of wavelets. The chapter then focuses attention to the particular role the analysis and the synthesis filters play in the design of a wavelet transform using filter banks. The precise procedure by which the design of a wavelet using filter banks can be achieved is presented in the last section of this chapter, and it includes the mathematical techniques involved in the design of wavelets.

• filter bank
• perfect reconstruction
• orthogonality
• paraunitary condition

Author Information

Peter yusuf dibal *.

• Telecommunication Engineering Department, Federal University of Technology Minna, Nigeria

Elizabeth Onwuka

James agajo.

• Computer Engineering Department, Federal University of Technology Minna, Nigeria

Caroline Alenoghena

*Address all correspondence to: yoksa77@gmail

1. Introduction

Filter banks can be defined as the cascaded arrangement of filters, i.e., low-pass, high-pass, and band-pass filters connected by sampling operators in such a manner as to achieve the decomposition and recomposition of a signal from a spectrum perspective. 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.

k-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.

2. Analysis filter bank section

The 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.

2.1 Analysis filter bank

The 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 y n . Thus, specific features of the input signal embedded at different frequencies can be individually extracted and investigated using the analysis filter bank [ 3 , 4 ]. The k-channel filter bank in Figure 2 separates the frequencies of the input signal in the manner presented.

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 y n can be stated as follows:

where U i z is the z-transform of the result from the convolution operation between the z-transform of the input signal Y Z and the z-transform of the filter H i Z . The output U i z in Figure 2 is fed into the corresponding downsampler of Figure 1 . In the next section, we will analyze the downsampler and state the mathematical operation it performs on a given signal.

2.2 Downsampler/decimator

The downsampler shown in Figure 1 downsamples an input signal by a factor of N . This implies that it only retains all the N th samples in a given sequence. For example, if N = 2 , then the downsampler will retain all even samples in a given sequence. Given an input signal x n , the downsampler with a factor of 2 will downsample the signal as:

Figure 3 shows the conceptual depiction of the relationship in Eq. (2) .

Decimation by a factor of 2.

Mathematically, the output of the decimator in Figure 1 can be expressed as a product of the input sequence u i n and the sequence of unit impulses which are N samples apart, i.e.,

The relationship in Eq. (3) will only select the kN th sample of u i n , and the Fourier series expansion of the impulse series can be expressed as [ 5 ]:

∑ k ∈ ℤ δ n − kN = 1 N ∑ k = 0 N − 1 e − j 2 πkn / N E4

Setting W N = e − j 2 π / N and n = 1 , the relationship in Eq. (4) becomes:

∑ k ∈ ℤ δ n − kN = 1 N ∑ k = 0 N − 1 W N − k E5

Substituting Eq. (5) into (3) yields:

v i n = 1 N ∑ k = 0 N − 1 u i n W N − k E6

In terms of z-transformation, the relationship in Eq. (6) can be expressed as:

V i Z = 1 N ∑ k = 0 N − 1 U i Z 1 N W N − k E7

Having looked at the analysis filters and downsamplers, we will now turn our attention to synthesis filter bank section of Figure 1 .

3. Synthesis filter bank section

The 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.

3.1 Synthesis filter bank

Similar 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 w i n is as stated below [ 6 ]:

P 0 Z = W 0 Z G 0 Z P 1 Z = W 1 Z G 1 Z P 2 Z = W 2 Z G 2 Z P k − 1 Z = W k − 1 Z G M − 1 Z E8

In Figure 2 , the input to the synthesis filter bank is upsamplers or expanders. The next section gives a brief review of the upsamplers.

3.2 Upsampler/expander

The 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 = 2 , then the upsampler will insert a zero between every two adjacent samples in a given sequence as shown in Figure 4 .

Upsampling by a factor of 2.

Given an input signal v i n in Figure 1 , an upsampler with a factor of 2 will upsample the signal using the relationship [ 7 ]:

w i n = ∑ k ∈ ℤ v i n δ n − kN , ∀ k ∈ ℤ E9

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 ]:

W i Z = 1 N ∑ k = 0 N − 1 V i Z N W N − k E10

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.

4. Properties of filter banks for wavelet design

In 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.

4.1 Perfect reconstruction

This 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 .

Two-channel FIR filter bank.

The output y ̂ n is derived using Eqs. (6) and (10) as follows in terms of the signal component and aliasing component as:

where the signal_component and aliasing_component are defined as:

signal _ component = 1 2 F 0 z H 0 z + F 1 z H 1 z X z aliasing _ component = 1 2 F 0 z H 0 − z + F 1 z H 1 − z X − z E12

To achieve perfect reconstruction, the following condition must be satisfied [ 1 ]:

F 0 z H 0 z + F 1 z H 1 z = 2 z − 1 F 0 z H 0 − z + F 1 z H 1 − z = 0 E13

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 ]:

F 0 z = H 0 z H 1 z = H 0 − z F 1 z = − H 1 z E14

In Eq. (14) , the synthesis filter F 0 z has the same coefficients as the analysis filter H 0 z ; the analysis filter H 1 z has the same coefficients as the analysis filter H 0 z , but every other value is negated; the synthesis filter F 1 z is a negative copy of the analysis filter H 1 z . For example, if the analysis filter H 0 z has coefficients p , q , r , s , then the filter bank in Figure 5 will assume the structure shown in Figure 6 .

QMF two-channel FIR filter bank.

For the CQF bank, the coefficients of the analysis filter H 1 z are a reversed version of the analysis filter H 0 z with every other value negated. The synthesis filters F 0 z and F 1 z are a reversed versions of the analysis filters H 0 z and H 1 z , respectively. These relationships can be stated mathematically as follows [ 10 ]:

H 1 z = z − 1 H 0 − z − 1 F 0 z = H 1 − z F 1 z = − H 0 − z E15

Based on the relationship in Eq. (15) , the filter bank shown in Figure 6 for CQF will assume the structure shown in Figure 7 .

CQF two-channel FIR filter bank.

Based on the structure of Figures 6 or 7 , the output signal y ̂ n is related to the input signal y n by the expression:

y ̂ n = pp + qq + rr + ss y n − 3 E16

If we impose the condition that pp + qq + rr + ss = 1 , then Eq. (16) becomes:

y ̂ n = y n − 3 E17

The relationship in Eq. (17) states that the output signal y ̂ n is delayed version of the input signal y n by three samples. We leave the verification of Eq. (16) as an exercise for the reader.

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.

4.2 Orthogonality

Orthogonality 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 ]:

In Eq. (18) , the inner product of the coefficients of the synthesis filter F 0 z and the analysis filter H 1 z must be zero and the inner product of the coefficients of the synthesis filter F 1 z and the analysis filter H 0 z must also be zero for the orthogonality condition to hold.

Also, the low-pass analysis filter H 0 z is related to the other three filters through the following expressions [ 12 ]:

where L denotes the length of the filter which must be even, and c is a constant with c = 1 ; H ˜ 0 − z is the flipped and conjugated version of H 0 z , H ˜ 0 z is the conjugated version of H 0 z , and H ˜ 1 z is the conjugated version of H 1 z .

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 H 0 z satisfy the following power symmetry of halfband condition [ 8 , 9 ]:

where P z = H 0 z H ̂ 0 z . If the low-pass filter H 0 z satisfies the required symmetry condition:

then P z is said to be a real filter. The implication of the constraint in Eq. (21) is that H 1 z and F 1 z be antisymmetric filters, and F 0 z is a symmetric filter. The relationships in Eqs. (20) – (22) give the necessary and sufficient condition for the characterization of a filter bank with orthogonality and symmetry.

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 ].

Polyphase implementation of filter bank.

If E z in Figure 8 is type-I analysis polyphase matrix, and R z is type-II synthesis polyphase matrix, then [ 13 ]:

The conditions in Eqs. (20) – (22) hold true iff E z and R z satisfy the following:

where k = L / 2 , with the first and second condition in Eq. (24) relating to the filter bank orthogonality condition, and the last represents the filter bank symmetry.

We now look at the paraunitary condition of a filter bank, which is also a necessary property in filter bank implementation of wavelets.

4.3 Paraunitary condition

In 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 ].

Given a polyphase transfer function matrix E z , the paraunitary condition is established by the matrix iff [ 15 ]:

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 M × M real-coefficient lossless matrix with N − 1 degree can be realized using the structure shown in Figure 9 [ 18 ].

Cascade implementation of E z as FIR lossless unitary matrices separated by delays.

If the real-coefficient lossless matrix is denoted by E z ; then the matrix is said to have a special case of lossless degree of one iff it can be characterized by the relationship [ 18 ]:

E z = I − vv + + z − 1 vv + R E26

where R is an arbitrary M × M unitary matrix and v is an M × 1 column vector with unit norm. From Eq. (26) , the paraunitary condition for a filter bank is obtained as follows [ 18 ]:

I − v k v k + + v k v k + z E k z = E k − 1 z E27

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.

5. Filter bank design of a wavelet transform

The filter bank design of a wavelet transform is usually implemented from the analysis filter bank segment to the synthesis filter bank segment.

5.1 Analysis filter bank in wavelet transform design

Given that the expression for a scaling function φ n is the series sum of the shifted versions of φ 2 n , then according to [ 15 , 16 ], φ n can be represented as:

where h k denotes the scaling coefficients. If n is transformed such that n → 2 α n − β , then the relationship in Eq. (28) becomes [ 14 ]:

which translates into:

when k = m − 2 β .

In a similar consideration to Eq. (28) , the wavelet function ψ n can be represented as [ 19 ]:

where g k denotes the wavelet coefficients. Also, if n is transformed such that n → 2 α n − β , then the relationship in Eq. (31) becomes [ 14 ]:

5.2 Synthesis filter bank in wavelet transform design

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 α + 1 scaling space f n ∈ V α + 1 , then according to [ 16 , 17 ], the reconstruction is derived as follows:

For the next scale, Eq. (34) becomes:

Substituting Eqs. (28) and (31) into Eq. (35) and after algebraic manipulations yields [ 14 ]:

6. Wavelet transform design procedure using filter banks

In 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 H 0 as described in Eq. (13) , then this initial filter must be designed in such a manner that the relationships in Sections 5.1 and 5.2 are realized. To this end, the following steps as shown in the state diagram in Figure 10 are necessary.

State chart for wavelet design procedure.

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.

7. Conclusion

In 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.

The 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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• 19. Burrus CS, Gopinath RA, Guo H. Introduction to Wavelets and Wavelet Transform: A Primer. Upper Saddle River: Prentice Hall; 1998

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• > Network Analysis and Practice
• > Filter synthesis

Book contents

• Frontmatter
• 1 Electric charge, field and potential
• 2 Electric current, resistance and electromotive force
• 3 Direct-current networks
• 4 Capacitance, inductance and electrical transients
• 5 Introduction to the steady-state responses of networks to sinusoidal sources
• 6 Transformers in networks
• 7 Alternating-current instruments and bridges
• 8 Attenuators and single-section filters
• 9 Multiple-section filters and transmission lines
• 10 Signal analysis of nonlinear and active networks
• 11 Fourier and Laplace transform techniques
• 12 Filter synthesis
• Mathematical background appendices

12 - Filter synthesis

Published online by Cambridge University Press:  05 June 2012

Introduction

An ideal filter would perfectly transmit signals at all desired frequencies and completely reject them at all other frequencies. In the particular case of an ideal low-pass filter, for example, the modulus of the transfer function, | J |, would behave as shown in figure 12.1( a ). Up to a certain critical pulsatance ω c , | J | would be unity but above this pulsatance, | J | would be zero. Any practical filter can only approximate to such an ideal, of course.

In section 8.2 it was pointed out how | J | 2 for a simple single-section L–R or C–R filter comprising just one reactive component only reaches a maximum rate of fall-off outside the pass band of 20 dB per decade of frequency compared with an infinite rate of fall-off for an ideal filter. Remember that the significance of | J | 2 is that it indicates the power in the load for a fixed amplitude of input signal. Increasing the number of reactive components in the filter stage to two, as in the simple low-pass L–C filter of figure 8.7( a ), causes | J | 2 to reach a maximum rate of fall-off outside the pass band of 40 dB per decade of frequency. With n reactive components in the filter stage, the maximum rate of fall-off of | J | 2 outside the pass band becomes 20 n dB per decade of frequency and the filter is accordingly said to be of nth order .

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• Book: Network Analysis and Practice
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• Chapter DOI: https://doi.org/10.1017/CBO9781139171816.013

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Seismic synthesis of the Al Haouz earthquake of September 8th, 2023 by integrating gravimetric and aeromagnetic data from the western High Atlas in Morocco

• Original Article
• Published: 12 September 2024

Cite this article

• Abderrahime Nouayti   ORCID: orcid.org/0000-0002-3460-3550 1 ,
• Lahcen El Moudnib 1 ,
• Driss Khattach 2 ,
• Martin Zeckra 3 ,
• Nordine Nouayti 1 ,
• Omar Saadi 4 ,
• Khalid Elhairechi 5 ,
• Brahim Oujane 1 &
• Hafid Iken 1  

On September 8, 2023, a magnitude 6.8 earthquake struck the Western High Atlas, approximately 70 km southwest of Marrakech, causing significant devastation and tragic human losses. This seismic event has generated increased scientific interest to understand its origins and impact, particularly through seismic, magnetic, gravity, and geological data analysis. In this study, we adopt an integrated methodological approach, using a GIS environment to highlight deep geological structures likely responsible for the Al Haouz earthquake and subsequent post-seismic activity. By applying various filters on the gravity and magnetic data (reduction to the pole RTP, vertical and horizontal derivatives, maxima of horizontal gradient, and upward continuation), we characterize the subsurface geological features within the study area. In addition, the seismic data allowed the analysis of the focal mechanism, which helped to identify the fault responsible for the earthquake and to compare its characteristics with those of other focal mechanisms of historical earthquakes in the High Atlas. The results show that the main earthquake (6.8 Mw) occurred in a fragile geological area, surrounded by two resistant formations composed mainly of a Precambrian and Paleozoic basement (Tichka and Siroua massifs). The model carried out highlights several faults-oriented NE-SW, E-W, and ENE-WSW. Therefore, the analysis and interpretation of all the data of the focal mechanism, the geological data, and the structural context of the disaster area indicate that the Tizi N’Test fault could be the origin of the Al Haouz earthquake. Furthermore, a significant spatial correlation was identified between the recognized fractures and the distribution of epicenters, notably with the Tizi N’Test fault. This fault exhibits a clustering of epicenters on either side, suggesting triggered of the seismic aftershocks, along with the reactivation of other faults. Therefore, the geodynamic model provides major advances to understand local seismic activities and constitutes an essential reference for future studies. It also contributes to the management of seismic risks, thus strengthening the safety and resilience of populations in high-risk areas.

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Geophysics and Natural Hazards Laboratory, Scientific Institute; Geophysics, Natural Patrimony and Green Chemistry Research Center “GEOPAC”, Mohammed V University in Rabat, Rabat, Morocco

Abderrahime Nouayti, Lahcen El Moudnib, Nordine Nouayti, Brahim Oujane & Hafid Iken

Laboratory of Applied Geosciences, Faculty of Sciences, University Mohamed First, Oujda, 60000, Morocco

Driss Khattach

Seismological Station Bensberg, University of Ciologne, Cologne, Germany

Martin Zeckra

Laboratory of Water and Environmental Engineering, Al Hoceima National School of Applied Sciences, PO Box 03, Ajdir Al Hoceima, Morocco

Territory Team: Dynamics, Planning and Sustainable Development (TEA2D), FPK of Khouribga, Sultan Moulay Slimane University, Beni Mellal, Morocco

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Nouayti, A., Moudnib, L.E., Khattach, D. et al. Seismic synthesis of the Al Haouz earthquake of September 8th, 2023 by integrating gravimetric and aeromagnetic data from the western High Atlas in Morocco. Model. Earth Syst. Environ. (2024). https://doi.org/10.1007/s40808-024-02148-3

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Analysis and synthesis filters for oversampled wavelet filter banks

Description

df = dtfilters( name ) returns the decomposition (analysis) filters corresponding to name . These filters are used most often as input arguments to dddtree and dddtree2 .

[ df , rf ] = dtfilters( name ) returns the reconstruction (synthesis) filters corresponding to name .

collapse all

Filters for Complex Dual-Tree Wavelet Transform

Obtain valid filters for the complex dual-tree wavelet transform. The transform uses Farras nearly symmetric filters for the first stage and Kingsbury Q-shift filters with 10 taps for subsequent stages.

Load the noisy Doppler signal. Obtain the filters for the first and subsequent stages of the complex dual-tree wavelet transform. Demonstrate perfect reconstruction using the complex dual-tree wavelet transform.

Filters for Double-Density Wavelet Transform

Obtain valid filters for the double-density wavelet transform.

Load the noisy Doppler signal. Obtain the filters for the double-density wavelet transform. The double-density wavelet transform uses the same filters at all stages. Demonstrate perfect reconstruction using the double-density wavelet transform.

Dual-Tree and Double-Density Wavelet Transforms Using Filter Names and Filters

Load a 1-D signal.

"dwt" - Critically sampled discrete wavelet transform

The critically sampled discrete wavelet transform can be applied to 1-D and 2-D data. The filter can be any valid orthogonal or biorthogonal wavelet name, or "farras" .

Specify a valid filter name. Use dtfilters to obtain the corresponding decomposition filters. Confirm the decomposition filters are returned as a two-column matrix.

Use dddtree to obtain two wavelet decompositions of the 1-D signal. Use the filter name for the first decomposition, and the filters for the second decomposition.

Confirm the wavelet coefficients in the decompositions are equal.

Confirm the filters in both decompositions are equal.

"ddt" - Double-density wavelet transform

The double-density wavelet transform can be applied to 1-D and 2-D data. Valid filter names for the double-density wavelet transform are "filters1" , "filters2" , and "doubledualfilt" .

Use dtfilters to obtain the filters corresponding to "filters1" . Inspect the filters. Confirm the decomposition filters are returned as a three-column matrix.

Use dtfilters to obtain the filters corresponding to "doubledualfilt" . Inspect the filters. Confirm the decomposition filters are returned as 1-by-2 cell array consisting of three-column matrices.

"realdt" - Real oriented dual-tree wavelet transform

The real oriented dual-tree wavelet transform can only be applied to 2-D data. Valid filter names are:

Any orthogonal or biorthogonal wavelet name, but only as a first-stage filter.

"dtfP" , where P can equal 1, 2, 3, 4, or 5.

"FSfarras" , but only as a first-stage filter.

"qshiftN" , where N can equal 6, 10, 14, 16, or 18, for stages > 1.

Obtain a 2-D image.

Use dtfilters to obtain the decomposition filters corresponding to "dtf1" . Confirm the filters are returned as 1-by-2 cell array consisting of 1-by-2 cell arrays.

Obtain the filters corresponding to "FSfarras" and "qshift6" . Confirm the filters are returned as 1-by-2 cell array consisting of two-column matrices.

Confirm the dtf filters are equal to the fs and qs filters.

Use dddtree2 to obtain two realdt decompositions of the image. Use the filter name " dtf1" for the first decomposition, and the filters fs and qs for the second decomposition. Confirm the wavelet coefficients in both decompositions are equal.

"cplxdt" - Complex oriented dual-tree wavelet transform

The complex oriented dual-tree wavelet transform can be applied to 1-D and 2-D data. Valid filter names are:

Use dtfilters to obtain the decompositions filters corresponding to the db4 orthogonal wavelet and the Kingsbury Q-shift filter with 14 taps.

Use dddtree and the filters to obtain the complex oriented dual-tree wavelet decomposition of the 1-D signal.

Demonstrate perfect reconstruction.

"realdddt" - Real double-density dual-tree wavelet transform

The real double-density dual-tree wavelet transform can only be applied to 2-D data. Valid filter names are:

Use dtfilters to obtain the decomposition filters corresponding to "dddtf1" . Confirm the filters are returned as 1-by-2 cell array consisting of 1-by-2 cell arrays.

Use dddtree2 to obtain two wavelet decompositions of the image. Use the filter name for the first decomposition, and the filters for the second decomposition. Confirm the decompositions are equal.

"cplxdddt" - Complex double-density dual-tree wavelet transform

The complex double-density dual-tree wavelet transform can be applied to 1-D and 2-D data. Valid filter names are:

Use dtfilters to obtain the decomposition filters corresponding to "self1" . Confirm the filters are returned as 1-by-2 cell array consisting of 1-by-2 cell arrays.

Use dddtree to obtain two wavelet decompositions of the 1-D signal. Use the filter name for the first decomposition, and the filters for the second decomposition. Confirm the decompositions are equal.

Input Arguments

Name — filter name "dtf1" | "dddtf1" | "self1" | "self2" | ....

Filter name, specified as a character vector or string scalar. Valid entries for name are:

Any valid orthogonal or biorthogonal wavelet name. See wfilters for details. An orthogonal or biorthogonal wavelet is only valid when the filter bank type is "dwt" , or when you use the filter as the first stage in a complex dual-tree transform, "realdt" or "cplxdt" .

An orthogonal or biorthogonal wavelet filter is not a valid filter if you have a double-density, "ddt" or dual-tree double-density, "realdddt" or "cplxdddt" , filter bank. An orthogonal or biorthogonal wavelet filter is not a valid filter for complex dual-tree filter banks for stages greater than 1.

"dtf P " — With P equal to 1, 2, 3, 4, or 5 returns the first-stage Farras filters ( "FSfarras" ) and Kingsbury Q-shift filters ( "qshift N " ) for subsequent stages. This input is only valid for a dual-tree transform, "realdt" or "cplxdt" . Setting P = 1, 2, 3, 4, or 5 specifies the Kingsbury Q-shift filters with N = 6, 10, 14, 16, or 18 taps, respectively.

"dddtf1" — Returns the filters for the first and subsequent stages of the double-density dual-tree transform. This input is only valid for the double-density dual-tree transforms, "realdddt" and "cplxdddt" .

"self1" — Returns 10-tap filters for the double-density wavelet transform. This option is only valid for double-density wavelet transforms, "realdddt" , and "cplxdddt" .

"self2" — Returns 16-tap filters for the double-density wavelet transform. This option is only valid for double-density wavelet transforms, "realdddt" , and "cplxdddt" .

"filters1" — Returns 6-tap filters for the double-density wavelet transform, "ddt" .

"filters2" — Returns 12-tap filters for the double-density wavelet transform, "ddt" .

"farras" — Farras nearly symmetric filters for a two-channel perfect reconstruction filter bank. This option is meant to be used for one-tree transforms and is valid only for an orthogonal critically sampled wavelet transform, "dwt" . The output of dtfilters is a two-column matrix. The first column of the matrix is a scaling (lowpass) filter, and the second column is a wavelet (highpass) filter.

"FSfarras" — Farras nearly symmetric first-stage filters intended for a dual-tree wavelet transform. With this option, the output of dtfilters is a cell array with two elements, one for each tree. Each element is a two-column matrix. The first column of the matrix is a scaling (lowpass) filter, and the second column is a wavelet (highpass) filter.

"qshift N " — Kingsbury Q-shift N-tap filters with N = 6, 10, 14, 16, or 18. The Kingsbury Q-shift filters are used most commonly in dual-tree wavelet transforms for stages greater than 1.

"doubledualfilt" — Filters for one stage of the double-density dual-tree wavelet transforms, "realdddt" or "cplxdddt" . This option can also be used in the double-density wavelet transform, "ddt" .

This table can help you decide which filter to choose:

Output Arguments

Df — decomposition (analysis) filters matrix | cell array.

Decomposition (analysis) filters, returned as a matrix or cell array of matrices.

rf — Reconstruction (synthesis) filters matrix | cell array

Reconstruction (synthesis) filters, returned as a matrix or cell array of matrices.

[1] Abdelnour, A. F., and I. W. Selesnick. “Design of 2-Band Orthogonal near-Symmetric CQF.” In 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221) , 6:3693–96. Salt Lake City, UT, USA: IEEE, 2001. https://doi.org/10.1109/ICASSP.2001.940644.

[2] Kingsbury, Nick. “Complex Wavelets for Shift Invariant Analysis and Filtering of Signals.” Applied and Computational Harmonic Analysis 10, no. 3 (May 2001): 234–53. https://doi.org/10.1006/acha.2000.0343.

[3] Selesnick, Ivan W., and A. Farras Abdelnour. “Symmetric Wavelet Tight Frames with Two Generators.” Applied and Computational Harmonic Analysis 17, no. 2 (September 2004): 211–25. https://doi.org/10.1016/j.acha.2004.05.003.

[4] Selesnick, I.W. “The Double-Density Dual-Tree DWT.” IEEE Transactions on Signal Processing 52, no. 5 (May 2004): 1304–14. https://doi.org/10.1109/TSP.2004.826174.

Version History

Introduced in R2013b

dddtree | dddtree2

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