5264:
of non-contiguous channels. Appropriate filter design is necessary to reduce performance degradation caused by the filter bank. In order to obtain universally applicable designs, mild assumptions can be made about waveform format, channel statistics and the coding/decoding scheme. Both heuristic and optimal design methodologies can be used, and excellent performance is possible with low complexity as long as the transceiver operates with a reasonably large oversampling factor. A practical application is OFDM transmission, where they provide very good performance with small additional complexity.
3291:
3090:
94:
214:. The filter bank and the spectrogram are the two simplest ways of producing a quadratic TFD; they are in essence similar as one (the spectrogram) is obtained by dividing the time domain into slices and then taking a Fourier transform, while the other (the filter bank) is obtained by dividing the frequency domain in slices forming bandpass filters that are excited by the signal under analysis.
2975:) 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:
763:
5116:
by optimization in the frequency domain has been used in Wei and Lu. In Nguyen's paper, the proposed method is not limited to two-channel 2D filter banks design; the approach is generalized to M-channel filter banks with any critical subsampling matrix. According to the implementation in the paper, it can be used to achieve up to 8-channel 2D filter banks design.
142:. For a fixed segment length, the amount of overlap determines how often the FFTs are done (and vice versa). Also, the wider the shape of the filters, the fewer filters that are needed to span the input bandwidth. Eliminating unnecessary filters (i.e. decimation in frequency) is efficiently done by treating each weighted segment as a sequence of smaller
3098:
construction, angular resolution and perfect reconstruction. In the general M-dimensional case, the ideal frequency supports of the MDFB are hypercube-based hyperpyramids. The first level of decomposition for MDFB is achieved by an N-channel undecimated filter bank, whose component filters are M-D "hourglass"-shaped filter aligned with the w
789:
blocks are the decimator and expander. For example, the input divides 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
3141:
5247:
subbands with wedge-shaped frequency partition (see Figure). The original construction of the DFB involves modulating the input signal and using diamond-shaped filters. Moreover, in order to obtain the desired frequency partition, a complicated tree expanding rule has to be followed. As a result, the
4903:
So imagine the filter bank achieves perfect reconstruction with FIR filters. Then from the polyphase domain characterization it follows that the filters H1(z) and G1(z) are completely specified by H0(z) and G0(z), respectively. Therefore, we need to design H0(x) and G0(z) which have desired frequency
2927:
With the fast development of communication technology, signal processing system needs 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
250:
with a narrow passband. In order to create a multirate narrow lowpass FIR filter, one can replace the time-invariant FIR filter with a lowpass antialiasing filter and a decimator, along with an interpolator and lowpass anti-imaging filter. In this way, the resulting multirate system is a time-varying
5111:
Many of the existing methods for designing 2-channel filter banks are based on transformation of variable technique. For example, McClellan transform can be used to design 1-D 2-channel filter banks. Though the 2-D filter banks have many similar properties with the 1-D prototype, but it is difficult
5263:
Filter banks are important elements for the physical layer in wideband wireless communication, where the problem is efficient base-band processing of multiple channels. A filter-bank-based transceiver architecture eliminates the scalability and efficiency issues observed by previous schemes in case
5251:
The first advantage of DFB is that not only it is not a redundant transform but also it offers perfect reconstruction. Another advantage of DFB is its directional-selectivity and efficient structure. This advantage makes DFB an appropriate approach for many signal and image processing usage. (e.g.,
5115:
In Nguyen, the authors talk about the design of multidimensional filter banks by direct optimization in the frequency domain. The method proposed here is mainly focused on the M-channel 2D filter banks design. The method is flexible towards frequency support configurations. 2D filter banks designed
2935:
The simplest approach to design a multi-dimensional filter bank 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. However, the region of support
161:
A special case occurs when, by design, the length of the blocks is an integer multiple of the interval between FFTs. Then the FFT filter bank can be described in terms of one or more polyphase filter structures where the phases are recombined by an FFT instead of a simple summation. The number of
89:
uses a filter bank to determine the amplitude information of the subbands of a modulator signal (such as a voice) and uses them to control the amplitude of the subbands of a carrier signal (such as the output of a guitar or synthesizer), thus imposing the dynamic characteristics of the modulator on
3301:
In Charo, a multivariate polynomial matrix-factorization algorithm is introduced and discussed. The most common problem is the multidimensional filter banks for perfect reconstruction. This paper talks about the method to achieve this goal that satisfies the constrained condition of linear phase.
3097:
M-dimensional directional filter banks (MDFB) are a family of filter banks that can achieve the directional decomposition of arbitrary M-dimensional signals with a simple and efficient tree-structured construction. It has many distinctive properties like: directional decomposition, efficient tree
788:
A complete filter bank consists of the analysis and synthesis side. 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
749:
A discrete-time filter bank framework allows inclusion of desired input signal dependent features in the design in addition to the more traditional perfect reconstruction property. The information theoretic features like maximized energy compaction, perfect de-correlation of sub-band signals and
3148:
Oversampled filter banks are multirate filter banks where the number of output samples at the analysis stage is larger than the number of input samples. It is proposed for robust applications. One particular class of oversampled filter banks is nonsubsampled filter banks without downsampling or
2931:
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.
2939:
A filter bank consists of an analysis stage and a synthesis stage. Each stage consists of a set of filters in parallel. The filter bank design is the design of the filters in the analysis and synthesis stages. The analysis filters divide the signal into overlapping or non-overlapping subbands
1658:
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
3152:
For IIR oversampled filter bank, perfect reconstruction have been studied in
Wolovich and Kailath. in the context of control theory. While for FIR oversampled filter bank we have to use different strategy for 1-D and M-D. FIR filter are more popular since it is easier to implement. For 1-D
2940:
depending on the application requirements. The synthesis filters should be designed to reconstruct the input signal back from the subbands when the outputs of these filters are combined. Processing is typically performed after the analysis stage. These filter banks can be designed as
4128:
3153:
oversampled FIR filter banks, the
Euclidean algorithm plays a key role in the matrix inverse problem. However, the Euclidean algorithm fails for multidimensional (MD) filters. For MD filter, we can convert the FIR representation into a polynomial representation. And then use
5563:
3132:) and superscript (Li) means the levels of decomposition for the ith level filter bank. Note that, starting from the second level, we attach an IRC filter bank to each output channel from the previous level, and hence the entire filter has a total of 2 output channels.
5207:
In this paper, the authors proposed that the FIR filter with 128 taps be used as a basic filter, and decimation factor is computed for RJ matrices. They did simulations based on different parameters and achieve a good quality performances in low decimation factor.
4660:
The mapping approaches have certain restrictions on the kind of filters; however, it brings many important advantages, such as efficient implementation via lifting/ladder structures. Here we provide an example of two-channel filter banks in 2D with sampling matrix
3313:
This approach based on multivariate matrix factorization can be used in different areas. The algorithmic theory of polynomial ideals and modules can be modified to address problems in processing, compression, transmission, and decoding of multidimensional signals.
862:
The quincunx lattice generated by quincunx matrix is as shown; the synthesis part is dual to the analysis part. Filter banks can be analyzed from a frequency-domain perspective in terms of subband decomposition and reconstruction. However, equally important is
3755:
1815:
3080:
Linear phase PR filters are very useful for image processing. This two-channel filter bank is relatively easy to implement. But two channels sometimes are not enough. Two-channel filter banks can be cascaded to generate multi-channel filter banks.
655:(respectively). A multirate filter bank uses 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, an analysis-synthesis system can be used.
81:
scheme that preserves these differences must be used. On the other hand, less important frequencies do not have to be exact. A coarser coding scheme can be used, even though some of the finer (but less important) details will be lost in the coding.
2735:
2916:
2908:
173:
each one at a rate commensurate with the total bandwidth to be created, translating each channel to its new center frequency, and summing the streams of samples. In that context, the interpolation filter associated with upsampling is called
97:
Depiction of the implementation and operation of a weighted overlap add (WOLA) filter bank. Wrap-around of a circular input buffer is used to offset phase discontinuities, caused by lack of a true time reference for the
Fourier transform
47:(meaning analysis of the signal in terms of its components in each sub-band); the output of analysis is referred to as a subband signal with as many subbands as there are filters in the filter bank. The reconstruction process is called
5090:
3305:
According to the description of the paper, some new results in factorization are discussed and being applied to issues of multidimensional linear phase perfect reconstruction finite-impulse response filter banks. The basic concept of
3165:
Nonsubsampled filter banks are particular oversampled filter banks without downsampling or upsampling. The perfect reconstruction condition for nonsubsampled FIR filter banks leads to a vector inverse problem: the analysis filters
1652:
698:=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.
2366:
2228:
5790:
858:
4645:
1217:
3912:
5284:
implies that it preserves the information within its passband, and suppresses the information (or noise) outside the passband. When the FFT rate is not sufficient for that, the design is typically called
1433:
2923:
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.
5846:
W.-S. Lu, A. Antoniou, and H. Xu, "A direct method for the design of 2-D nonseparable diamond-shaped filter banks", IEEE Transactions on
Circuits and Systems II, vol. 45, no. 8, pp. 1146–1150, Aug
77:
when some frequencies are more important than others. After decomposition, the important frequencies can be coded with a fine resolution. Small differences at these frequencies are significant and a
701:
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 filter by 4 synthesis filters
797:
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
1521:
222:
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. The implementation makes use of
4371:
4291:
579:
3486:
3298:
As multidimensional filter banks can be represented by multivariate rational matrices, this method is a very effective tool that can be used to deal with the multidimensional filter banks.
1061:
997:
933:
501:
375:
3556:
1107:
43:, which can attenuate the components differently and recombine them into a modified version of the original signal. The process of decomposition performed by the filter bank is called
1665:
653:
2573:
4201:
1288:
750:
other characteristics for the given input covariance/correlation structure are incorporated in the design of optimal filter banks. These filter banks resemble the signal dependent
4210:
of a set of vectors in linear algebra. The theory of Gröbner bases implies that the Module has a unique reduced Gröbner basis for a given order of power products in polynomials.
3280:
3222:
2833:
3904:
3826:
2897:
2578:
406:
283:
2423:
2012:
1253:
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1877:
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4798:
4762:
4726:
2777:
2465:
4511:
4472:
3425:
3399:
5750:
5612:
4446:
4410:
2505:
735:
692:
1957:
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4899:
2048:
1327:
4907:
3544:
3515:
3373:
3344:
251:
linear-phase filter via the decimator and interpolator. The lowpass filter consists of two polyphase filters, one for the decimator and one for the interpolator.
2093:
1914:
182:). Ideally, the frequency responses of adjacent channels sum to a constant value at every frequency between the channel centers. That condition is known as
1529:
3835:
Gröbner bases can be used to characterizing perfect reconstruction multidimensional filter banks, but it first need to extend from polynomial matrices to
211:
2233:
3149:
upsampling. The perfect reconstruction condition for an oversampled filter bank can be stated as a matrix inverse problem in the polyphase domain.
2936:
for the filter banks might not be separable. In that case designing of filter bank gets complex. In most cases we deal with non-separable systems.
2098:
5364:
5310:
Sarangi, Susanta; Sahidullah, Md; Saha, Goutam (September 2020). "Optimization of data-driven filterbank for automatic speaker verification".
5101:
When perfect reconstruction is not needed, the design problem can be simplified by working in frequency domain instead of using FIR filters.
195:
804:
5668:
Hyung-Ju, Park (1995). "A computational theory of
Laurent polynomial rings and multidimensional FIR systems" (University of California).
4519:
3106:
respectively axes. After that, the input signal is further decomposed by a series of 2-D iteratively resampled checkerboard filter banks
5914:
Park, Sang-Il; Smith, Mark JT & Mersereau, Russell M (1999). "A new directional filter bank for image analysis and classification".
4123:{\displaystyle \mathrm {Module} \left\{h_{1}(z),...,h_{N}(z)\right\}{\stackrel {\rm {def}}{=}}\{c_{1}(z)h_{1}(z)+...+c_{N}(z)h_{N}(z)\}}
864:
5566:" IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, Volume 46 Issue 12, pp. 1475–1486, Dec, 1999
2968:
When it is necessary to reconstruct the divided signal back to the original one, perfect-reconstruction (PR) filter banks may be used.
2948:(FIR). In order to reduce the data rate, downsampling and upsampling are performed in the analysis and synthesis stages, respectively.
1116:
6030:
166:) of each filter. The computational efficiencies of the FFT and polyphase structures, on a general purpose processor, are identical.
5878:." IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, Vol. 45 Issue 3, pp. 436–441. Mar, 1998.
754:(KLT) that is the optimal block transform where the length L of basis functions (filters) and the subspace dimension M are the same.
178:. The net frequency response of each channel is the product of the synthesis filter with the frequency response of the filter bank (
6011:
5931:
5396:
1332:
5391:
B. Boashash, editor, "Time-Frequency Signal
Analysis and Processing – A Comprehensive Reference", Elsevier Science, Oxford, 2003;
3842:
The Gröbner-basis computation can be considered equivalently as
Gaussian elimination for solving the polynomial matrix equation
223:
5248:
frequency regions for the resulting subbands do not follow a simple ordering as shown in Figure 9 based on the channel indices.
751:
199:
5576:
231:
155:
66:
the subbands to a low center frequency that can be re-sampled at a reduced rate. The same result can sometimes be achieved by
5901:
3317:
The general multidimensional filter bank (Figure 7) can be represented by a pair of analysis and synthesis polyphase matrices
5514:
5376:
770:
5517:" in Proc. SPIE Conf. Wavelet Applications Signal Image Processing XI, San Diego, CA, pp. 591424–1-591424-12, July 2005
1441:
207:
4657:
Mapping based design in popularly used to design nonseparable multidimensional filter banks with good frequency responses.
5255:
Directional Filter Banks can be developed to higher dimensions. It can be use in 3-D to achieve the frequency sectioning.
4296:
4216:
3157:
and Gröbner bases to get the framework and the reconstruction condition of the multidimensional oversampled filter banks.
5916:
1999 IEEE International
Conference on Acoustics, Speech, and Signal Processing. Proceedings. ICASSP99 (Cat. No.99CH36258)
5777:
3832:
In the multidimensional case with multivariate polynomials we need to use the theory and algorithms of Gröbner bases.
506:
3750:{\displaystyle F(z)=\sum _{k\in \mathbf {Z} ^{d}}fz^{k}=\sum _{k\in \mathbf {Z} ^{d}}fz_{1}^{k_{1}}...z_{d}^{k_{d}}}
3434:
1002:
938:
874:
414:
288:
203:
2941:
1810:{\displaystyle {\hat {x}}=\sum _{1\leq k\leq K,m\in \mathbf {Z} ^{2}}\langle x,\varphi _{k,m}\rangle \psi _{k,m}}
1067:
55:
5765:
Flexible design of multidimensional perfect reconstruction FIR 2-band filters using transformations of variables
2956:
Below are several approaches on the design of multidimensional filter banks. For more details, please check the
5423:
103:
5835:
A new approach to the design of multidimensional nonseparable two-channel orthonormal filterbanks and wavelets
3290:
3140:
584:
2928:
these goals. Filter banks can be used in various areas, such as image coding, voice coding, radar and so on.
6035:
3089:
2945:
2513:
5216:
Bamberger and Smith proposed a 2D directional filter bank (DFB). The DFB is efficiently implemented via an
4136:
3546:
are the z-transform of the polyphase components of the analysis and synthesis filters. Therefore, they are
1258:
5103:
Note that the frequency domain method is not limited to the design of nonsubsampled filter banks (read ).
867:
interpretation of filter banks, which plays a key role in geometrical signal representations. For generic
123:
63:
3227:
3169:
5744:
5681:
5606:
2785:
2730:{\displaystyle {\hat {x}}(z){\stackrel {\rm {def}}{=}}({\hat {X}}_{0}(z),...,{\hat {X}}_{|M|-1}(z))^{T}}
5981:
5865:", TENCON 99. Proceedings of the IEEE Region 10 Conference. Vol.1 pp. 637–641, Conference in 1999.
782:
3845:
3767:
2838:
1820:
If there is no loss in the decomposition and the subsequent reconstruction, the filter bank is called
5329:
5252:
Laplacian pyramid, constructed the contourlets, sparse image representation, medical imaging, etc.).
380:
257:
5125:
In Lee's 1999 paper, the authors talk about the multidimensional filter bank design using a reverse
4874:
All the frequency regions in Figure can be critically sampled by the rectangular lattice spanned by
5545:
3761:
Laurent polynomial matrix equation need to be solve to design perfect reconstruction filter banks:
2372:
1961:
1226:
5148:
1827:
5937:
5803:
5669:
5650:
5345:
5319:
4839:
4803:
4767:
4731:
4665:
3836:
3154:
2055:
377:. In this way each of the generated signals corresponds to a different region in the spectrum of
135:
107:
2740:
2428:
5437:
5426:
Proc. SPIE Visual
Communications and Image Processing, pp. 86–94, vol. 1605, Boston, Nov. 1991.
4477:
4451:
3404:
3378:
6040:
6007:
5927:
5627:
5486:." Multidimensional Systems and Signal Processing. Volume 20 Issue 1, pp. 3–24. Mar. 2009
5483:
5392:
5372:
4415:
4379:
2474:
704:
661:
111:
74:
40:
20:
5725:"Nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for R^n"
5708:
McClellan, James (1973). "The design of two-dimensional digital filters by transformations".
5496:
5085:{\displaystyle H_{0}(z_{1},z_{2})G_{0}(z_{1},z_{2})+H_{0}(-z_{1},z_{2})G_{0}(-z_{1},z_{2})=2}
408:. In this process it can be possible for the regions overlap (or not, based on application).
138:
of the filters. The wider the shape, the more often the FFTs have to be done to satisfy the
102:
Some filter banks work almost entirely in the time domain, using a series of filters such as
5919:
5732:
5642:
5594:
5337:
3307:
1923:
5955:
5575:
Adams, William W., and
Philippe Loustaunau. "An introduction to Gröbner bases, volume 3 of
5470:
5223:
4877:
2017:
1296:
5694:
5499:", IEEE Transactions on Image Processing. Volume 16 Issue 4, pp. 918–931. April, 2007
5134:
4728:
We would have several possible choices of ideal frequency responses of the channel filter
4655:
Designing filters with good frequency responses is challenging via Gröbner bases approach.
3520:
3491:
3349:
3320:
798:
131:
93:
36:
32:
5888:
5821:
5968:
5875:
5862:
5834:
5764:
5548:" IEEE Transactions on Signal Processing, Vol.46 Issue 5, pp. 1245–1255. May, 1998.
5333:
169:
Synthesis (i.e. recombining the outputs of multiple receivers) is basically a matter of
146:, and the FFT is performed on only the sum of the blocks. This has been referred to as
62:
is also commonly applied to a bank of receivers. The difference is that receivers also
5956:
The contourlet transform: an efficient directional multiresolution image representation
5778:
Discrete wavelet transform implementation in Fourier domain for multidimensional signal
5484:
Multidimensional perfect reconstruction filter banks: an approach of algebraic geometry
2069:
2051:
1890:
247:
1647:{\displaystyle {\hat {x}}=\sum _{1\leq k\leq K,m\in \mathbf {Z} ^{2}}c_{k}\psi _{k,m}}
238:
for additional insight into the effects of those operations in the transform domains.
6024:
5982:
A Filter-Bank Transceiver Architecture for Massive Non-Contiguous Carrier Aggregation
5824:" IEEE International Symposium onCircuits and Systems, pp. 1090–1093. May, 2005.
5673:
5473:" IEEE International Symposium on Circuits and Systems, pp. 643–646., May, 1993.
5349:
5126:
3069:
The input signal can be perfectly reconstructed if the alias term is cancelled and T(
2050:
by upsampling and filtering. This kind of setup is used in many applications such as
78:
67:
5941:
5654:
51:, meaning reconstitution of a complete signal resulting from the filtering process.
5535:
Kailath, Thomas. Linear systems. Vol. 1. Englewood Cliffs, NJ: Prentice-Hall, 1980.
5526:
Wolovich, William A. Linear multivariable systems. New York: Springer-Verlag, 1974.
774:
139:
4687:
5598:
235:
5923:
3124:
operates on 2-D slices of the input signal represented by the dimension pair (n
5837:", IEEE Signal Processing Letters, vol. 7, no. 11, pp. 327–330, Nov 2000.
5646:
5589:
Buchberger, Bruno (1985). "An algorithmic method in polynomial ideal theory".
5341:
2361:{\displaystyle y(z){\stackrel {\rm {def}}{=}}(Y_{0}(z),...,Y_{|N|-1}(z))^{T}.}
778:
762:
503:
can be generated via a collection of set of bandpass filters with bandwidths
227:
170:
35:
that separates the input signal into multiple components, each one carrying a
2223:{\displaystyle x(z){\stackrel {\rm {def}}{=}}(X_{0}(z),...,X_{|M|-1}(z))^{T}}
5902:
A filter bank for the directional decomposition of images: Theory and design
5804:
An orthogonal family of quincunx wavelets with continuously adjustable order
5093:
There are different mapping technique that can be used to get above result.
2915:
2907:
1438:
Considering the definition of analysis/synthesis sides we can verify that
5724:
3224:
are given and FIR, and the goal is to find a set of FIR synthesis filters
5438:"Multidimensional filter banks and multiscale geometric representations"
110:
to divide the signal into smaller bands. Other filter banks use a fast
86:
5863:
The design of multidimensional filter bank using Reverse Jacket matrix
5736:
5200:. In the 1999 paper, the authors generalize the reverse jacket matrix
3488:
is the absolute value of the determinant of the sampling matrix. Also
5289:. And in that case, it is not necessary for the segments to overlap.
853:{\displaystyle {\begin{bmatrix}\;\;\,1&1\\-1&1\end{bmatrix}}}
5793:." Applied and Computational Harmonic Analysis 28.2 (2010): 171-188.
5767:." Image Processing, IEEE Transactions on 2, no. 4 (1993): 466-480.
5324:
4640:{\displaystyle b_{i}(z)=\sum _{j=1}^{N}W_{ij}(z)h_{j}(z),i=1,...,K}
2779:
denotes ith polyphase component of the jth synthesis filter Gj(z).
5958:." Image Processing, IEEE Transactions on 14.12 (2005): 2091-2106.
3289:
3139:
3088:
2914:
2906:
761:
92:
5971:." Computer Vision and Image Understanding 113.1 (2009): 101-112.
1212:{\displaystyle \varphi _{k,m}{\stackrel {\rm {def}}{=}}h_{k}^{*}}
741:= 0,1,2,3. Finally, the outputs of these four filters are added.
5710:
Proc. 7th Annu. Princeton Conf. Information Sciences and Systems
5806:." Image Processing, IEEE Transactions on 14.4 (2005): 499-510.
122:
A bank of receivers can be created by performing a sequence of
1428:{\displaystyle \psi _{k,m}{\stackrel {\rm {def}}{=}}g_{k}^{*}}
39:
of the original signal. One application of a filter bank is a
5904:." IEEE Transactions, Signal Processing 40.4 (1992): 882-893.
5891:", Accepted IEEE Trans. on CAS-II, pp. 39–47, Jan. 2000.
5579:" American Mathematical Society, Providence, RI 24(47), 1994.
3073:) equal to a monomial. So the necessary condition is that T'(
5876:
On the Reverse Jacket matrix for weighted Hadamard transform
5802:
Feilner, Manuela, Dimitri Van De Ville, and Michael Unser. "
4376:
Using reverse engineering, we can compute the basis vectors
5822:
Multidimensional filter banks design by direct optimization
5564:
Multidimensional FIR filter bank design using Gröbner bases
5145:
is closely related to its inverse. The correct formula is:
2095:
can be represented by a vector of its polyphase components
1879:. Figure shows a general multidimensional filter bank with
5628:"Gröbner bases and multidimensional FIR multirate systems"
5626:
Park, Hyungju; Kalker, Ton & Vetterli, Martin (1997).
5497:
Multidimensional directional filter banks and surfacelets
5371:. Englewood Cliffs, NJ: Prentice-Hall. pp. 313–323.
3085:
Multidimensional directional filter banks and surfacelets
210:
by a two-dimensional filtering that defines the class of
134:) is applied to each segment to control the shape of the
5791:
3D curvelet transforms and astronomical data restoration
5204:
using Hadamard matrices and weighted Hadamard matrices.
5969:
Vessel enhancement filter using directional filter bank
4904:
responses and satisfy the polyphase-domain conditions.
2014:. The synthesis part recovers the original signal from
745:
Statistically optimized filter bank (Eigen filter bank)
5780:." Journal of Electronic Imaging 11.3 (2002): 338-346.
1516:{\displaystyle c_{k}=\langle x,\varphi _{k,m}\rangle }
813:
162:
blocks per segment is the impulse response length (or
6004:
Multirate signal processing for communication systems
5731:(Institute of Electrical and Electronics Engineers).
5471:
Considerations in multidimensional filter bank design
5226:
5151:
4910:
4880:
4842:
4806:
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4734:
4668:
4522:
4480:
4454:
4418:
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4219:
4139:
3915:
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3352:
3323:
3230:
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2841:
2788:
2743:
2581:
2516:
2477:
2431:
2375:
2236:
2101:
2072:
2066:
We can use polyphase representation, so input signal
2020:
1964:
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1261:
1229:
1119:
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807:
707:
664:
658:
The signal would split with the help of four filters
587:
509:
417:
383:
291:
260:
130:
of the input data stream. A weighting function (aka
4373:
by a finite sequence of reduction (division) steps.
4366:{\displaystyle \left\{h_{1}(z),...,h_{N}(z)\right\}}
4286:{\displaystyle \left\{b_{1}(z),...,b_{N}(z)\right\}}
3077:) is generally symmetric and of an odd-by-odd size.
2964:
Multidimensional perfect-reconstruction filter banks
212:
quadratic (or bilinear) time–frequency distributions
5457:
A wavelet tour of signal processing: the sparse way
5220:-level tree-structured decomposition that leads to
5889:A New Reverse Jacket Matrix and Its Fast Algorithm
5239:
5180:
5084:
4893:
4864:
4828:
4792:
4756:
4720:
4639:
4505:
4466:
4440:
4404:
4365:
4285:
4195:
4122:
3898:
3820:
3749:
3538:
3509:
3480:
3419:
3393:
3367:
3338:
3274:
3216:
2891:
2827:
2771:
2729:
2567:
2499:
2459:
2417:
2360:
2222:
2087:
2042:
2006:
1951:
1908:
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1646:
1515:
1427:
1321:
1282:
1247:
1211:
1101:
1055:
991:
927:
852:
729:
686:
647:
573:
495:
400:
369:
277:
2899:which means that G(z) is a left inverse of H(z).
1063:. In the analysis side, we can define vectors in
232:Discrete-time Fourier transform § Properties
5986:IEEE Journal on Selected Areas in Communications
1887:. The analysis part transforms the input signal
574:{\displaystyle {\rm {BW_{1},BW_{2},BW_{3},...}}}
5789:Woiselle, Arnaud, J-L. Starck, and J. Fadili. "
3161:Multidimensional nonsubsampled FIR filter banks
2510:Similarly, for the output signal we would have
5635:Multidimensional Systems and Signal Processing
5562:Charoenlarpnopparut, Chalie, and N. K. Bose. "
3481:{\displaystyle M{\stackrel {\rm {def}}{=}}|M|}
2782:The filter bank has perfect reconstruction if
1056:{\displaystyle \left\{M_{k}\right\}_{k=1}^{K}}
992:{\displaystyle \left\{g_{k}\right\}_{k=1}^{K}}
928:{\displaystyle \left\{h_{k}\right\}_{k=1}^{K}}
496:{\displaystyle x_{1}(n),x_{2}(n),x_{3}(n),...}
370:{\displaystyle x_{1}(n),x_{2}(n),x_{3}(n),...}
6006:. Upper Saddle River, NJ: Prentice Hall PTR.
8:
5820:Nguyen, Truong T., and Soontorn Oraintara. "
5776:Laligant, Olivier, and Frederic Truchetet. "
5749:: CS1 maint: multiple names: authors list (
5723:Kovacevic, Vetterli, Jelena, Martin (1992).
5611:: CS1 maint: DOI inactive as of July 2024 (
4117:
4023:
3269:
3231:
3211:
3173:
1779:
1736:
1510:
1467:
871:-channel filter bank, with analysis filters
246:One can define a narrow lowpass filter as a
202:(TFD) that represents the signal in a joint
190:Filter banks as time–frequency distributions
5900:Bamberger, Roberto H., and Mark JT Smith. "
4651:Mapping-based multidimensional filter banks
1102:{\displaystyle \ell ^{2}(\mathbf {Z} ^{d})}
5097:Filter-bank design in the frequency domain
817:
816:
5816:
5814:
5812:
5515:Multidimensional oversampled filter banks
5323:
5231:
5225:
5172:
5159:
5150:
5067:
5054:
5038:
5025:
5012:
4996:
4980:
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4885:
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4309:
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4007:
4002:
4000:
3999:
3979:
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3708:
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3685:
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3522:
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3436:
3406:
3380:
3351:
3322:
3263:
3238:
3229:
3205:
3180:
3171:
3136:Multidimensional oversampled filter banks
2855:
2847:
2846:
2840:
2805:
2804:
2787:
2748:
2742:
2721:
2695:
2687:
2686:
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2250:
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2235:
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2188:
2180:
2179:
2145:
2123:
2122:
2117:
2115:
2114:
2100:
2071:
2025:
2019:
1963:
1931:
1925:
1892:
1847:
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1829:
1786:
1758:
1728:
1723:
1697:
1670:
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1667:
1623:
1604:
1592:
1587:
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1534:
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1407:
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1340:
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1304:
1298:
1274:
1269:
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1228:
1191:
1178:
1173:
1154:
1153:
1148:
1146:
1145:
1124:
1118:
1090:
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1016:
1004:
983:
972:
952:
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919:
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888:
876:
818:
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806:
712:
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608:
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551:
535:
519:
511:
510:
508:
466:
444:
422:
416:
382:
340:
318:
296:
290:
259:
5558:
5556:
5554:
5544:Cvetkovic, Zoran, and Martin Vetterli. "
5424:"Statistically Optimized PR-QMF Design,"
5112:to extend to more than 2-channel cases.
4872:are supported on complementary regions.)
648:{\displaystyle f_{c1},f_{c2},f_{c3},...}
5763:Tay, David BH, and Nick G. Kingsbury. "
5729:IEEE Transactions on Information Theory
5509:
5507:
5505:
5469:Chen, Tsuhan, and P. P. Vaidyanathan. "
5363:Crochiere, R.E.; Rabiner, L.R. (1983).
5302:
5273:
3906:. If we have set of polynomial vectors
3294:Multidimensional M-channel filter banks
3144:Multidimensional synthesis filter banks
254:A filter bank divides the input signal
198:, a filter bank is a special quadratic
73:Another application of filter banks is
5857:
5855:
5853:
5742:
5690:
5679:
5604:
3093:Multidimensional analysis filter banks
2568:{\displaystyle {\hat {x}}(z)=G(z)y(z)}
2471:-th polyphase component of the filter
1883:channels and a common sampling matrix
4196:{\displaystyle c_{1}(z),...,c_{N}(z)}
1293:Similarly, for the synthesis filters
1283:{\displaystyle m\in \mathbf {Z} ^{2}}
7:
5422:H. Caglar, Y. Liu and A.N. Akansu,
5107:Direct frequency-domain optimization
5954:Do, Minh N., and Martin Vetterli. "
5369:Multirate Digital Signal Processing
4800:. (Note that the other two filters
3275:{\displaystyle \{G_{1},...,G_{N}\}}
3217:{\displaystyle \{H_{1},...,H_{N}\}}
2062:Perfect reconstruction filter banks
5874:Lee, Seung-Rae, and Moon Ho Lee. "
4213:If we define the Gröbner basis as
4015:
4012:
4009:
3932:
3929:
3926:
3923:
3920:
3917:
3457:
3454:
3451:
2828:{\displaystyle x(z)={\hat {x}}(z)}
2619:
2616:
2613:
2265:
2262:
2259:
2130:
2127:
2124:
1377:
1374:
1371:
1161:
1158:
1155:
548:
544:
532:
528:
516:
512:
16:Tool for Digital Signal Processing
14:
5861:Lee, Moon Ho, and Ju Yong Park. "
5482:Zhang, Lei, and Anamitra Makur. "
4412:in terms of the original vectors
1920:filtered and downsampled outputs
3899:{\displaystyle G(z)H(z)=I_{|M|}}
3821:{\displaystyle G(z)H(z)=I_{|M|}}
3638:
3588:
3548:multivariate Laurent polynomials
3058:are decomposition filters, and F
2892:{\displaystyle I_{|M|}=G(z)H(z)}
2054:, multichannel acquisition, and
1724:
1588:
1270:
1086:
196:time–frequency signal processing
5980:S. Stefanatos and F. Foukalas "
5591:Multidimensional Systems Theory
5577:Graduate Studies in Mathematics
5192:is the nĂ—n identity matrix and
4206:The Module is analogous to the
3550:, which have the general form:
2835:for any input, or equivalently
401:{\displaystyle x\left(n\right)}
278:{\displaystyle x\left(n\right)}
5073:
5044:
5031:
5002:
4986:
4960:
4947:
4921:
4859:
4853:
4823:
4817:
4787:
4781:
4751:
4745:
4604:
4598:
4585:
4579:
4539:
4533:
4500:
4494:
4435:
4429:
4399:
4393:
4355:
4349:
4321:
4315:
4275:
4269:
4241:
4235:
4190:
4184:
4156:
4150:
4114:
4108:
4095:
4089:
4061:
4055:
4042:
4036:
3991:
3985:
3957:
3951:
3890:
3882:
3870:
3864:
3858:
3852:
3812:
3804:
3792:
3786:
3780:
3774:
3691:
3653:
3609:
3603:
3569:
3563:
3533:
3527:
3504:
3498:
3474:
3466:
3431:is the number of channels and
3362:
3356:
3333:
3327:
2903:Multidimensional filter design
2886:
2880:
2874:
2868:
2856:
2848:
2822:
2816:
2810:
2798:
2792:
2766:
2760:
2718:
2714:
2708:
2696:
2688:
2680:
2655:
2649:
2637:
2627:
2600:
2594:
2588:
2562:
2556:
2550:
2544:
2535:
2529:
2523:
2494:
2488:
2454:
2448:
2412:
2406:
2400:
2394:
2385:
2379:
2346:
2342:
2336:
2324:
2316:
2292:
2286:
2273:
2246:
2240:
2211:
2207:
2201:
2189:
2181:
2157:
2151:
2138:
2111:
2105:
2082:
2076:
2037:
2031:
1943:
1937:
1903:
1897:
1863:
1859:
1853:
1840:
1834:
1824:. (in that case we would have
1804:
1798:
1776:
1770:
1748:
1742:
1687:
1681:
1675:
1641:
1635:
1616:
1610:
1551:
1545:
1539:
1507:
1501:
1479:
1473:
1461:
1455:
1422:
1400:
1358:
1352:
1316:
1310:
1223:each index by two parameters:
1206:
1184:
1142:
1136:
1096:
1081:
1028:
1022:
964:
958:
900:
894:
724:
718:
681:
675:
478:
472:
456:
450:
434:
428:
352:
346:
330:
324:
308:
302:
1:
5495:Lu, Yue M., and Minh N. Do. "
2418:{\displaystyle y(z)=H(z)x(z)}
2007:{\displaystyle j=0,1,...,N-1}
1523:and for reconstruction part:
1248:{\displaystyle 1\leq k\leq K}
758:Multidimensional filter banks
236:Z-transform § Properties
5988:, 35(1), Jan. 2017, 215–227.
5918:. pp. 1417–1420 vol.3.
5181:{\displaystyle HH^{T}=I_{n}}
3066:are reconstruction filters.
1872:{\displaystyle x={\hat {x}}}
6002:Harris, Fredric J. (2004).
5599:10.1007/978-94-009-5225-6_6
4865:{\displaystyle G_{1}(\xi )}
4829:{\displaystyle H_{1}(\xi )}
4793:{\displaystyle G_{0}(\xi )}
4757:{\displaystyle H_{0}(\xi )}
4721:{\displaystyle D_{1}=\left}
2737:. Also G is a matrix where
2056:discrete wavelet transforms
200:time–frequency distribution
6057:
5924:10.1109/ICASSP.1999.756247
4293:, it can be obtained from
2772:{\displaystyle G_{i,j}(z)}
2460:{\displaystyle H_{i,j}(z)}
771:Multidimensional filtering
6031:Digital signal processing
5455:Mallat, Stephane (2008).
5342:10.1016/j.dsp.2020.102795
5312:Digital Signal Processing
4506:{\displaystyle W_{ij}(z)}
4467:{\displaystyle K\times N}
3420:{\displaystyle M\times N}
3394:{\displaystyle N\times M}
2942:Infinite impulse response
224:downsampling (decimation)
208:Wigner–Ville distribution
148:weight overlap-add (WOLA)
140:Nyquist sampling criteria
104:quadrature mirror filters
56:digital signal processing
5546:Oversampled filter banks
5212:Directional filter banks
4441:{\displaystyle h_{j}(z)}
4405:{\displaystyle b_{i}(z)}
2500:{\displaystyle H_{i}(z)}
999:, and sampling matrices
752:Karhunen–Loève transform
730:{\displaystyle F_{k}(z)}
687:{\displaystyle H_{k}(z)}
156:§ Sampling the DTFT
5967:Truc, Phan TH, et al. "
5647:10.1023/A:1008299221759
5513:J. Zhou and M. N. Do, "
5259:Filter-bank transceiver
2946:Finite impulse response
581:and center frequencies
206:. It is related to the
70:the bandpass subbands.
5689:Cite journal requires
5601:(inactive 2024-07-12).
5241:
5182:
5122:Reverse Jacket Matrix
5086:
4895:
4866:
4830:
4794:
4758:
4722:
4641:
4565:
4507:
4474:transformation matrix
4468:
4442:
4406:
4367:
4287:
4197:
4124:
3900:
3822:
3751:
3540:
3511:
3482:
3421:
3395:
3369:
3340:
3295:
3276:
3218:
3145:
3094:
2920:
2912:
2893:
2829:
2773:
2731:
2569:
2501:
2461:
2419:
2362:
2224:
2089:
2044:
2008:
1953:
1952:{\displaystyle y_{j},}
1910:
1873:
1822:perfect reconstruction
1811:
1648:
1517:
1429:
1323:
1284:
1249:
1213:
1103:
1057:
993:
929:
854:
781:are the main parts of
767:
731:
688:
649:
575:
497:
411:The generated signals
402:
371:
285:into a set of signals
279:
228:upsampling (expansion)
184:perfect reconstruction
99:
5412:. Wiley-Interscience.
5410:Digital Filter Design
5242:
5240:{\displaystyle 2^{l}}
5183:
5087:
4896:
4894:{\displaystyle D_{1}}
4867:
4831:
4795:
4759:
4723:
4642:
4545:
4508:
4469:
4443:
4407:
4368:
4288:
4198:
4125:
3901:
3823:
3752:
3541:
3512:
3483:
3422:
3396:
3370:
3341:
3293:
3277:
3219:
3143:
3115:(i=2,3,...,M), where
3092:
2918:
2910:
2894:
2830:
2774:
2732:
2570:
2502:
2462:
2420:
2363:
2225:
2090:
2045:
2043:{\displaystyle y_{j}}
2009:
1954:
1911:
1874:
1812:
1649:
1518:
1430:
1324:
1322:{\displaystyle g_{k}}
1285:
1250:
1214:
1104:
1058:
994:
930:
855:
765:
732:
689:
650:
576:
498:
403:
372:
280:
242:Narrow lowpass filter
218:Multirate filter bank
204:time–frequency domain
96:
5833:D. Wei and S. Guo, "
5224:
5196:is the transpose of
5149:
4908:
4878:
4840:
4804:
4768:
4732:
4666:
4520:
4478:
4452:
4416:
4380:
4297:
4217:
4137:
3913:
3846:
3768:
3557:
3539:{\displaystyle G(z)}
3521:
3510:{\displaystyle H(z)}
3492:
3435:
3405:
3379:
3368:{\displaystyle G(z)}
3350:
3339:{\displaystyle H(z)}
3321:
3228:
3170:
2839:
2786:
2741:
2579:
2514:
2475:
2429:
2373:
2234:
2099:
2070:
2018:
1962:
1924:
1891:
1828:
1666:
1530:
1442:
1333:
1297:
1259:
1227:
1117:
1068:
1003:
939:
935:, synthesis filters
875:
805:
801:which is defined by
766:The quincunx lattice
705:
662:
585:
507:
415:
381:
289:
258:
152:weighted pre-sum FFT
5641:(Springer): 11–30.
5436:Do, Minh N (2011).
5334:2020DSP...10402795S
5141:, the transpose of
3746:
3715:
3310:is given in Adams.
3286:Using Gröbner bases
2952:Existing approaches
1399:
1183:
1052:
988:
924:
136:frequency responses
5408:Parks, TW (1987).
5237:
5178:
5082:
4891:
4862:
4826:
4790:
4754:
4718:
4712:
4637:
4503:
4464:
4438:
4402:
4363:
4283:
4193:
4120:
3896:
3837:Laurent polynomial
3818:
3747:
3725:
3694:
3649:
3599:
3536:
3507:
3478:
3417:
3391:
3365:
3336:
3296:
3272:
3214:
3155:Algebraic geometry
3146:
3095:
2921:
2913:
2889:
2825:
2769:
2727:
2565:
2497:
2457:
2415:
2358:
2220:
2085:
2040:
2004:
1949:
1906:
1869:
1807:
1735:
1644:
1599:
1513:
1425:
1385:
1319:
1280:
1245:
1209:
1169:
1099:
1053:
1006:
989:
942:
925:
878:
850:
844:
785:and filter banks.
768:
727:
684:
645:
571:
493:
398:
367:
275:
108:Goertzel algorithm
100:
5737:10.1109/18.119722
5459:. Academic press.
5442:Signal Processing
5287:spectrum analyzer
4203:are polynomials.
4020:
3625:
3575:
3462:
2813:
2683:
2640:
2624:
2591:
2526:
2369:So we would have
2270:
2135:
2088:{\displaystyle x}
1909:{\displaystyle x}
1866:
1693:
1678:
1557:
1542:
1382:
1166:
783:multirate systems
112:Fourier transform
75:lossy compression
41:graphic equalizer
31:) is an array of
21:signal processing
6048:
6017:
5989:
5978:
5972:
5965:
5959:
5952:
5946:
5945:
5911:
5905:
5898:
5892:
5885:
5879:
5872:
5866:
5859:
5848:
5844:
5838:
5831:
5825:
5818:
5807:
5800:
5794:
5787:
5781:
5774:
5768:
5761:
5755:
5754:
5748:
5740:
5720:
5714:
5713:
5705:
5699:
5698:
5692:
5687:
5685:
5677:
5665:
5659:
5658:
5632:
5623:
5617:
5616:
5610:
5602:
5586:
5580:
5573:
5567:
5560:
5549:
5542:
5536:
5533:
5527:
5524:
5518:
5511:
5500:
5493:
5487:
5480:
5474:
5467:
5461:
5460:
5452:
5446:
5445:
5433:
5427:
5420:
5414:
5413:
5405:
5399:
5389:
5383:
5382:
5360:
5354:
5353:
5327:
5307:
5290:
5278:
5246:
5244:
5243:
5238:
5236:
5235:
5187:
5185:
5184:
5179:
5177:
5176:
5164:
5163:
5091:
5089:
5088:
5083:
5072:
5071:
5059:
5058:
5043:
5042:
5030:
5029:
5017:
5016:
5001:
5000:
4985:
4984:
4972:
4971:
4959:
4958:
4946:
4945:
4933:
4932:
4920:
4919:
4900:
4898:
4897:
4892:
4890:
4889:
4871:
4869:
4868:
4863:
4852:
4851:
4835:
4833:
4832:
4827:
4816:
4815:
4799:
4797:
4796:
4791:
4780:
4779:
4763:
4761:
4760:
4755:
4744:
4743:
4727:
4725:
4724:
4719:
4717:
4713:
4678:
4677:
4646:
4644:
4643:
4638:
4597:
4596:
4578:
4577:
4564:
4559:
4532:
4531:
4512:
4510:
4509:
4504:
4493:
4492:
4473:
4471:
4470:
4465:
4447:
4445:
4444:
4439:
4428:
4427:
4411:
4409:
4408:
4403:
4392:
4391:
4372:
4370:
4369:
4364:
4362:
4358:
4348:
4347:
4314:
4313:
4292:
4290:
4289:
4284:
4282:
4278:
4268:
4267:
4234:
4233:
4202:
4200:
4199:
4194:
4183:
4182:
4149:
4148:
4129:
4127:
4126:
4121:
4107:
4106:
4088:
4087:
4054:
4053:
4035:
4034:
4022:
4021:
4019:
4018:
4006:
4001:
3998:
3994:
3984:
3983:
3950:
3949:
3935:
3905:
3903:
3902:
3897:
3895:
3894:
3893:
3885:
3827:
3825:
3824:
3819:
3817:
3816:
3815:
3807:
3756:
3754:
3753:
3748:
3745:
3744:
3743:
3733:
3714:
3713:
3712:
3702:
3690:
3689:
3665:
3664:
3648:
3647:
3646:
3641:
3621:
3620:
3598:
3597:
3596:
3591:
3545:
3543:
3542:
3537:
3516:
3514:
3513:
3508:
3487:
3485:
3484:
3479:
3477:
3469:
3464:
3463:
3461:
3460:
3448:
3443:
3426:
3424:
3423:
3418:
3400:
3398:
3397:
3392:
3374:
3372:
3371:
3366:
3345:
3343:
3342:
3337:
3281:
3279:
3278:
3273:
3268:
3267:
3243:
3242:
3223:
3221:
3220:
3215:
3210:
3209:
3185:
3184:
2898:
2896:
2895:
2890:
2861:
2860:
2859:
2851:
2834:
2832:
2831:
2826:
2815:
2814:
2806:
2778:
2776:
2775:
2770:
2759:
2758:
2736:
2734:
2733:
2728:
2726:
2725:
2707:
2706:
2699:
2691:
2685:
2684:
2676:
2648:
2647:
2642:
2641:
2633:
2626:
2625:
2623:
2622:
2610:
2605:
2593:
2592:
2584:
2574:
2572:
2571:
2566:
2528:
2527:
2519:
2506:
2504:
2503:
2498:
2487:
2486:
2466:
2464:
2463:
2458:
2447:
2446:
2424:
2422:
2421:
2416:
2367:
2365:
2364:
2359:
2354:
2353:
2335:
2334:
2327:
2319:
2285:
2284:
2272:
2271:
2269:
2268:
2256:
2251:
2229:
2227:
2226:
2221:
2219:
2218:
2200:
2199:
2192:
2184:
2150:
2149:
2137:
2136:
2134:
2133:
2121:
2116:
2094:
2092:
2091:
2086:
2049:
2047:
2046:
2041:
2030:
2029:
2013:
2011:
2010:
2005:
1958:
1956:
1955:
1950:
1936:
1935:
1915:
1913:
1912:
1907:
1878:
1876:
1875:
1870:
1868:
1867:
1862:
1848:
1816:
1814:
1813:
1808:
1797:
1796:
1769:
1768:
1734:
1733:
1732:
1727:
1680:
1679:
1671:
1653:
1651:
1650:
1645:
1634:
1633:
1609:
1608:
1598:
1597:
1596:
1591:
1544:
1543:
1535:
1522:
1520:
1519:
1514:
1500:
1499:
1454:
1453:
1434:
1432:
1431:
1426:
1412:
1411:
1398:
1393:
1384:
1383:
1381:
1380:
1368:
1363:
1351:
1350:
1328:
1326:
1325:
1320:
1309:
1308:
1289:
1287:
1286:
1281:
1279:
1278:
1273:
1254:
1252:
1251:
1246:
1218:
1216:
1215:
1210:
1196:
1195:
1182:
1177:
1168:
1167:
1165:
1164:
1152:
1147:
1135:
1134:
1108:
1106:
1105:
1100:
1095:
1094:
1089:
1080:
1079:
1062:
1060:
1059:
1054:
1051:
1046:
1035:
1031:
1021:
1020:
998:
996:
995:
990:
987:
982:
971:
967:
957:
956:
934:
932:
931:
926:
923:
918:
907:
903:
893:
892:
859:
857:
856:
851:
849:
848:
736:
734:
733:
728:
717:
716:
693:
691:
690:
685:
674:
673:
654:
652:
651:
646:
632:
631:
616:
615:
600:
599:
580:
578:
577:
572:
570:
569:
556:
555:
540:
539:
524:
523:
502:
500:
499:
494:
471:
470:
449:
448:
427:
426:
407:
405:
404:
399:
397:
376:
374:
373:
368:
345:
344:
323:
322:
301:
300:
284:
282:
281:
276:
274:
176:synthesis filter
118:FFT filter banks
33:bandpass filters
6056:
6055:
6051:
6050:
6049:
6047:
6046:
6045:
6021:
6020:
6014:
6001:
5998:
5996:Further reading
5993:
5992:
5979:
5975:
5966:
5962:
5953:
5949:
5934:
5913:
5912:
5908:
5899:
5895:
5886:
5882:
5873:
5869:
5860:
5851:
5845:
5841:
5832:
5828:
5819:
5810:
5801:
5797:
5788:
5784:
5775:
5771:
5762:
5758:
5741:
5722:
5721:
5717:
5707:
5706:
5702:
5688:
5678:
5667:
5666:
5662:
5630:
5625:
5624:
5620:
5603:
5588:
5587:
5583:
5574:
5570:
5561:
5552:
5543:
5539:
5534:
5530:
5525:
5521:
5512:
5503:
5494:
5490:
5481:
5477:
5468:
5464:
5454:
5453:
5449:
5435:
5434:
5430:
5421:
5417:
5407:
5406:
5402:
5390:
5386:
5379:
5362:
5361:
5357:
5309:
5308:
5304:
5299:
5294:
5293:
5279:
5275:
5270:
5261:
5227:
5222:
5221:
5214:
5203:
5191:
5168:
5155:
5147:
5146:
5135:Hadamard matrix
5109:
5102:
5099:
5092:
5063:
5050:
5034:
5021:
5008:
4992:
4976:
4963:
4950:
4937:
4924:
4911:
4906:
4905:
4902:
4881:
4876:
4875:
4873:
4843:
4838:
4837:
4807:
4802:
4801:
4771:
4766:
4765:
4735:
4730:
4729:
4711:
4710:
4705:
4699:
4698:
4693:
4682:
4669:
4664:
4663:
4662:
4656:
4653:
4588:
4566:
4523:
4518:
4517:
4481:
4476:
4475:
4450:
4449:
4419:
4414:
4413:
4383:
4378:
4377:
4339:
4305:
4304:
4300:
4295:
4294:
4259:
4225:
4224:
4220:
4215:
4214:
4174:
4140:
4135:
4134:
4098:
4079:
4045:
4026:
3975:
3941:
3940:
3936:
3911:
3910:
3876:
3844:
3843:
3798:
3766:
3765:
3735:
3704:
3681:
3656:
3636:
3612:
3586:
3555:
3554:
3519:
3518:
3490:
3489:
3433:
3432:
3403:
3402:
3377:
3376:
3348:
3347:
3319:
3318:
3288:
3259:
3234:
3226:
3225:
3201:
3176:
3168:
3167:
3163:
3138:
3131:
3127:
3123:
3114:
3105:
3101:
3087:
3065:
3061:
3057:
3053:
3045:
3037:
3029:
3021:
3009:
3001:
2993:
2985:
2966:
2954:
2905:
2842:
2837:
2836:
2784:
2783:
2744:
2739:
2738:
2717:
2673:
2630:
2577:
2576:
2512:
2511:
2478:
2473:
2472:
2432:
2427:
2426:
2371:
2370:
2368:
2345:
2310:
2276:
2232:
2231:
2210:
2175:
2141:
2097:
2096:
2068:
2067:
2064:
2021:
2016:
2015:
1960:
1959:
1927:
1922:
1921:
1889:
1888:
1849:
1826:
1825:
1782:
1754:
1722:
1664:
1663:
1619:
1600:
1586:
1528:
1527:
1485:
1445:
1440:
1439:
1403:
1336:
1331:
1330:
1300:
1295:
1294:
1268:
1257:
1256:
1225:
1224:
1187:
1120:
1115:
1114:
1084:
1071:
1066:
1065:
1012:
1011:
1007:
1001:
1000:
948:
947:
943:
937:
936:
884:
883:
879:
873:
872:
843:
842:
837:
828:
827:
822:
809:
803:
802:
799:Quincunx matrix
760:
747:
708:
703:
702:
665:
660:
659:
620:
604:
588:
583:
582:
547:
531:
515:
505:
504:
462:
440:
418:
413:
412:
387:
379:
378:
336:
314:
292:
287:
286:
264:
256:
255:
244:
220:
192:
180:analysis filter
132:window function
126:on overlapping
120:
17:
12:
11:
5:
6054:
6052:
6044:
6043:
6038:
6036:Linear filters
6033:
6023:
6022:
6019:
6018:
6012:
5997:
5994:
5991:
5990:
5973:
5960:
5947:
5932:
5906:
5893:
5887:Moon Ho Lee, "
5880:
5867:
5849:
5839:
5826:
5808:
5795:
5782:
5769:
5756:
5715:
5700:
5691:|journal=
5660:
5618:
5581:
5568:
5550:
5537:
5528:
5519:
5501:
5488:
5475:
5462:
5447:
5428:
5415:
5400:
5384:
5377:
5355:
5301:
5300:
5298:
5295:
5292:
5291:
5272:
5271:
5269:
5266:
5260:
5257:
5234:
5230:
5213:
5210:
5201:
5189:
5175:
5171:
5167:
5162:
5158:
5154:
5108:
5105:
5098:
5095:
5081:
5078:
5075:
5070:
5066:
5062:
5057:
5053:
5049:
5046:
5041:
5037:
5033:
5028:
5024:
5020:
5015:
5011:
5007:
5004:
4999:
4995:
4991:
4988:
4983:
4979:
4975:
4970:
4966:
4962:
4957:
4953:
4949:
4944:
4940:
4936:
4931:
4927:
4923:
4918:
4914:
4888:
4884:
4861:
4858:
4855:
4850:
4846:
4825:
4822:
4819:
4814:
4810:
4789:
4786:
4783:
4778:
4774:
4753:
4750:
4747:
4742:
4738:
4716:
4709:
4706:
4704:
4701:
4700:
4697:
4694:
4692:
4689:
4688:
4685:
4681:
4676:
4672:
4652:
4649:
4648:
4647:
4636:
4633:
4630:
4627:
4624:
4621:
4618:
4615:
4612:
4609:
4606:
4603:
4600:
4595:
4591:
4587:
4584:
4581:
4576:
4573:
4569:
4563:
4558:
4555:
4552:
4548:
4544:
4541:
4538:
4535:
4530:
4526:
4502:
4499:
4496:
4491:
4488:
4484:
4463:
4460:
4457:
4437:
4434:
4431:
4426:
4422:
4401:
4398:
4395:
4390:
4386:
4361:
4357:
4354:
4351:
4346:
4342:
4338:
4335:
4332:
4329:
4326:
4323:
4320:
4317:
4312:
4308:
4303:
4281:
4277:
4274:
4271:
4266:
4262:
4258:
4255:
4252:
4249:
4246:
4243:
4240:
4237:
4232:
4228:
4223:
4192:
4189:
4186:
4181:
4177:
4173:
4170:
4167:
4164:
4161:
4158:
4155:
4152:
4147:
4143:
4131:
4130:
4119:
4116:
4113:
4110:
4105:
4101:
4097:
4094:
4091:
4086:
4082:
4078:
4075:
4072:
4069:
4066:
4063:
4060:
4057:
4052:
4048:
4044:
4041:
4038:
4033:
4029:
4025:
4017:
4014:
4011:
4005:
3997:
3993:
3990:
3987:
3982:
3978:
3974:
3971:
3968:
3965:
3962:
3959:
3956:
3953:
3948:
3944:
3939:
3934:
3931:
3928:
3925:
3922:
3919:
3892:
3888:
3884:
3879:
3875:
3872:
3869:
3866:
3863:
3860:
3857:
3854:
3851:
3830:
3829:
3814:
3810:
3806:
3801:
3797:
3794:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3759:
3758:
3742:
3738:
3732:
3728:
3724:
3721:
3718:
3711:
3707:
3701:
3697:
3693:
3688:
3684:
3680:
3677:
3674:
3671:
3668:
3663:
3659:
3655:
3652:
3645:
3640:
3635:
3632:
3628:
3624:
3619:
3615:
3611:
3608:
3605:
3602:
3595:
3590:
3585:
3582:
3578:
3574:
3571:
3568:
3565:
3562:
3535:
3532:
3529:
3526:
3506:
3503:
3500:
3497:
3476:
3472:
3468:
3459:
3456:
3453:
3447:
3440:
3416:
3413:
3410:
3390:
3387:
3384:
3364:
3361:
3358:
3355:
3335:
3332:
3329:
3326:
3287:
3284:
3271:
3266:
3262:
3258:
3255:
3252:
3249:
3246:
3241:
3237:
3233:
3213:
3208:
3204:
3200:
3197:
3194:
3191:
3188:
3183:
3179:
3175:
3162:
3159:
3137:
3134:
3129:
3125:
3119:
3110:
3103:
3099:
3086:
3083:
3063:
3059:
3055:
3051:
3043:
3035:
3027:
3019:
3007:
2999:
2991:
2983:
2965:
2962:
2953:
2950:
2919:2D filter bank
2911:1D filter bank
2904:
2901:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2864:
2858:
2854:
2850:
2845:
2824:
2821:
2818:
2812:
2809:
2803:
2800:
2797:
2794:
2791:
2768:
2765:
2762:
2757:
2754:
2751:
2747:
2724:
2720:
2716:
2713:
2710:
2705:
2702:
2698:
2694:
2690:
2682:
2679:
2672:
2669:
2666:
2663:
2660:
2657:
2654:
2651:
2646:
2639:
2636:
2629:
2621:
2618:
2615:
2609:
2602:
2599:
2596:
2590:
2587:
2564:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2540:
2537:
2534:
2531:
2525:
2522:
2496:
2493:
2490:
2485:
2481:
2456:
2453:
2450:
2445:
2442:
2439:
2435:
2414:
2411:
2408:
2405:
2402:
2399:
2396:
2393:
2390:
2387:
2384:
2381:
2378:
2357:
2352:
2348:
2344:
2341:
2338:
2333:
2330:
2326:
2322:
2318:
2313:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2283:
2279:
2275:
2267:
2264:
2261:
2255:
2248:
2245:
2242:
2239:
2217:
2213:
2209:
2206:
2203:
2198:
2195:
2191:
2187:
2183:
2178:
2174:
2171:
2168:
2165:
2162:
2159:
2156:
2153:
2148:
2144:
2140:
2132:
2129:
2126:
2120:
2113:
2110:
2107:
2104:
2084:
2081:
2078:
2075:
2063:
2060:
2052:subband coding
2039:
2036:
2033:
2028:
2024:
2003:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1948:
1945:
1942:
1939:
1934:
1930:
1905:
1902:
1899:
1896:
1865:
1861:
1858:
1855:
1852:
1845:
1842:
1839:
1836:
1833:
1818:
1817:
1806:
1803:
1800:
1795:
1792:
1789:
1785:
1781:
1778:
1775:
1772:
1767:
1764:
1761:
1757:
1753:
1750:
1747:
1744:
1741:
1738:
1731:
1726:
1721:
1718:
1715:
1712:
1709:
1706:
1703:
1700:
1696:
1692:
1689:
1686:
1683:
1677:
1674:
1656:
1655:
1643:
1640:
1637:
1632:
1629:
1626:
1622:
1618:
1615:
1612:
1607:
1603:
1595:
1590:
1585:
1582:
1579:
1576:
1573:
1570:
1567:
1564:
1560:
1556:
1553:
1550:
1547:
1541:
1538:
1512:
1509:
1506:
1503:
1498:
1495:
1492:
1488:
1484:
1481:
1478:
1475:
1472:
1469:
1466:
1463:
1460:
1457:
1452:
1448:
1424:
1421:
1418:
1415:
1410:
1406:
1402:
1397:
1392:
1388:
1379:
1376:
1373:
1367:
1360:
1357:
1354:
1349:
1346:
1343:
1339:
1329:we can define
1318:
1315:
1312:
1307:
1303:
1277:
1272:
1267:
1264:
1244:
1241:
1238:
1235:
1232:
1221:
1220:
1208:
1205:
1202:
1199:
1194:
1190:
1186:
1181:
1176:
1172:
1163:
1160:
1157:
1151:
1144:
1141:
1138:
1133:
1130:
1127:
1123:
1098:
1093:
1088:
1083:
1078:
1074:
1050:
1045:
1042:
1039:
1034:
1030:
1027:
1024:
1019:
1015:
1010:
986:
981:
978:
975:
970:
966:
963:
960:
955:
951:
946:
922:
917:
914:
911:
906:
902:
899:
896:
891:
887:
882:
847:
841:
838:
836:
833:
830:
829:
826:
823:
821:
815:
814:
812:
759:
756:
746:
743:
726:
723:
720:
715:
711:
683:
680:
677:
672:
668:
644:
641:
638:
635:
630:
627:
623:
619:
614:
611:
607:
603:
598:
595:
591:
568:
565:
562:
559:
554:
550:
546:
543:
538:
534:
530:
527:
522:
518:
514:
492:
489:
486:
483:
480:
477:
474:
469:
465:
461:
458:
455:
452:
447:
443:
439:
436:
433:
430:
425:
421:
396:
393:
390:
386:
366:
363:
360:
357:
354:
351:
348:
343:
339:
335:
332:
329:
326:
321:
317:
313:
310:
307:
304:
299:
295:
273:
270:
267:
263:
248:lowpass filter
243:
240:
219:
216:
191:
188:
119:
116:
15:
13:
10:
9:
6:
4:
3:
2:
6053:
6042:
6039:
6037:
6034:
6032:
6029:
6028:
6026:
6015:
6013:0-13-146511-2
6009:
6005:
6000:
5999:
5995:
5987:
5983:
5977:
5974:
5970:
5964:
5961:
5957:
5951:
5948:
5943:
5939:
5935:
5933:0-7803-5041-3
5929:
5925:
5921:
5917:
5910:
5907:
5903:
5897:
5894:
5890:
5884:
5881:
5877:
5871:
5868:
5864:
5858:
5856:
5854:
5850:
5843:
5840:
5836:
5830:
5827:
5823:
5817:
5815:
5813:
5809:
5805:
5799:
5796:
5792:
5786:
5783:
5779:
5773:
5770:
5766:
5760:
5757:
5752:
5746:
5738:
5734:
5730:
5726:
5719:
5716:
5711:
5704:
5701:
5696:
5683:
5675:
5671:
5664:
5661:
5656:
5652:
5648:
5644:
5640:
5636:
5629:
5622:
5619:
5614:
5608:
5600:
5596:
5592:
5585:
5582:
5578:
5572:
5569:
5565:
5559:
5557:
5555:
5551:
5547:
5541:
5538:
5532:
5529:
5523:
5520:
5516:
5510:
5508:
5506:
5502:
5498:
5492:
5489:
5485:
5479:
5476:
5472:
5466:
5463:
5458:
5451:
5448:
5443:
5439:
5432:
5429:
5425:
5419:
5416:
5411:
5404:
5401:
5398:
5397:0-08-044335-4
5394:
5388:
5385:
5380:
5374:
5370:
5366:
5359:
5356:
5351:
5347:
5343:
5339:
5335:
5331:
5326:
5321:
5317:
5313:
5306:
5303:
5296:
5288:
5283:
5277:
5274:
5267:
5265:
5258:
5256:
5253:
5249:
5232:
5228:
5219:
5211:
5209:
5205:
5199:
5195:
5173:
5169:
5165:
5160:
5156:
5152:
5144:
5140:
5136:
5132:
5128:
5127:jacket matrix
5123:
5121:
5117:
5113:
5106:
5104:
5096:
5094:
5079:
5076:
5068:
5064:
5060:
5055:
5051:
5047:
5039:
5035:
5026:
5022:
5018:
5013:
5009:
5005:
4997:
4993:
4989:
4981:
4977:
4973:
4968:
4964:
4955:
4951:
4942:
4938:
4934:
4929:
4925:
4916:
4912:
4886:
4882:
4856:
4848:
4844:
4820:
4812:
4808:
4784:
4776:
4772:
4748:
4740:
4736:
4714:
4707:
4702:
4695:
4690:
4683:
4679:
4674:
4670:
4658:
4650:
4634:
4631:
4628:
4625:
4622:
4619:
4616:
4613:
4610:
4607:
4601:
4593:
4589:
4582:
4574:
4571:
4567:
4561:
4556:
4553:
4550:
4546:
4542:
4536:
4528:
4524:
4516:
4515:
4514:
4497:
4489:
4486:
4482:
4461:
4458:
4455:
4432:
4424:
4420:
4396:
4388:
4384:
4374:
4359:
4352:
4344:
4340:
4336:
4333:
4330:
4327:
4324:
4318:
4310:
4306:
4301:
4279:
4272:
4264:
4260:
4256:
4253:
4250:
4247:
4244:
4238:
4230:
4226:
4221:
4211:
4209:
4204:
4187:
4179:
4175:
4171:
4168:
4165:
4162:
4159:
4153:
4145:
4141:
4111:
4103:
4099:
4092:
4084:
4080:
4076:
4073:
4070:
4067:
4064:
4058:
4050:
4046:
4039:
4031:
4027:
4003:
3995:
3988:
3980:
3976:
3972:
3969:
3966:
3963:
3960:
3954:
3946:
3942:
3937:
3909:
3908:
3907:
3886:
3877:
3873:
3867:
3861:
3855:
3849:
3840:
3838:
3833:
3808:
3799:
3795:
3789:
3783:
3777:
3771:
3764:
3763:
3762:
3740:
3736:
3730:
3726:
3722:
3719:
3716:
3709:
3705:
3699:
3695:
3686:
3682:
3678:
3675:
3672:
3669:
3666:
3661:
3657:
3650:
3643:
3633:
3630:
3626:
3622:
3617:
3613:
3606:
3600:
3593:
3583:
3580:
3576:
3572:
3566:
3560:
3553:
3552:
3551:
3549:
3530:
3524:
3501:
3495:
3470:
3445:
3438:
3430:
3414:
3411:
3408:
3388:
3385:
3382:
3359:
3353:
3330:
3324:
3315:
3311:
3309:
3308:Gröbner bases
3303:
3299:
3292:
3285:
3283:
3264:
3260:
3256:
3253:
3250:
3247:
3244:
3239:
3235:
3206:
3202:
3198:
3195:
3192:
3189:
3186:
3181:
3177:
3160:
3158:
3156:
3150:
3142:
3135:
3133:
3122:
3118:
3113:
3109:
3091:
3084:
3082:
3078:
3076:
3072:
3067:
3049:
3041:
3033:
3025:
3017:
3013:
3005:
2997:
2989:
2981:
2976:
2974:
2969:
2963:
2961:
2959:
2951:
2949:
2947:
2943:
2937:
2933:
2929:
2925:
2917:
2909:
2902:
2900:
2883:
2877:
2871:
2865:
2862:
2852:
2843:
2819:
2807:
2801:
2795:
2789:
2780:
2763:
2755:
2752:
2749:
2745:
2722:
2711:
2703:
2700:
2692:
2677:
2670:
2667:
2664:
2661:
2658:
2652:
2644:
2634:
2607:
2597:
2585:
2559:
2553:
2547:
2541:
2538:
2532:
2520:
2508:
2491:
2483:
2479:
2470:
2451:
2443:
2440:
2437:
2433:
2409:
2403:
2397:
2391:
2388:
2382:
2376:
2355:
2350:
2339:
2331:
2328:
2320:
2311:
2307:
2304:
2301:
2298:
2295:
2289:
2281:
2277:
2253:
2243:
2237:
2215:
2204:
2196:
2193:
2185:
2176:
2172:
2169:
2166:
2163:
2160:
2154:
2146:
2142:
2118:
2108:
2102:
2079:
2073:
2061:
2059:
2057:
2053:
2034:
2026:
2022:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1974:
1971:
1968:
1965:
1946:
1940:
1932:
1928:
1919:
1900:
1894:
1886:
1882:
1856:
1850:
1843:
1837:
1831:
1823:
1801:
1793:
1790:
1787:
1783:
1773:
1765:
1762:
1759:
1755:
1751:
1745:
1739:
1729:
1719:
1716:
1713:
1710:
1707:
1704:
1701:
1698:
1694:
1690:
1684:
1672:
1662:
1661:
1660:
1638:
1630:
1627:
1624:
1620:
1613:
1605:
1601:
1593:
1583:
1580:
1577:
1574:
1571:
1568:
1565:
1562:
1558:
1554:
1548:
1536:
1526:
1525:
1524:
1504:
1496:
1493:
1490:
1486:
1482:
1476:
1470:
1464:
1458:
1450:
1446:
1436:
1419:
1416:
1413:
1408:
1404:
1395:
1390:
1386:
1365:
1355:
1347:
1344:
1341:
1337:
1313:
1305:
1301:
1291:
1275:
1265:
1262:
1242:
1239:
1236:
1233:
1230:
1203:
1200:
1197:
1192:
1188:
1179:
1174:
1170:
1149:
1139:
1131:
1128:
1125:
1121:
1113:
1112:
1111:
1109:
1091:
1076:
1072:
1048:
1043:
1040:
1037:
1032:
1025:
1017:
1013:
1008:
984:
979:
976:
973:
968:
961:
953:
949:
944:
920:
915:
912:
909:
904:
897:
889:
885:
880:
870:
866:
865:Hilbert-space
860:
845:
839:
834:
831:
824:
819:
810:
800:
796:
792:
786:
784:
780:
776:
772:
764:
757:
755:
753:
744:
742:
740:
721:
713:
709:
699:
697:
678:
670:
666:
656:
642:
639:
636:
633:
628:
625:
621:
617:
612:
609:
605:
601:
596:
593:
589:
566:
563:
560:
557:
552:
541:
536:
525:
520:
490:
487:
484:
481:
475:
467:
463:
459:
453:
445:
441:
437:
431:
423:
419:
409:
394:
391:
388:
384:
364:
361:
358:
355:
349:
341:
337:
333:
327:
319:
315:
311:
305:
297:
293:
271:
268:
265:
261:
252:
249:
241:
239:
237:
233:
229:
225:
217:
215:
213:
209:
205:
201:
197:
189:
187:
185:
181:
177:
172:
167:
165:
159:
157:
153:
149:
145:
141:
137:
133:
129:
125:
117:
115:
113:
109:
105:
95:
91:
90:the carrier.
88:
83:
80:
76:
71:
69:
68:undersampling
65:
61:
57:
52:
50:
46:
42:
38:
34:
30:
26:
22:
6003:
5985:
5976:
5963:
5950:
5915:
5909:
5896:
5883:
5870:
5842:
5829:
5798:
5785:
5772:
5759:
5745:cite journal
5728:
5718:
5709:
5703:
5682:cite journal
5663:
5638:
5634:
5621:
5607:cite journal
5590:
5584:
5571:
5540:
5531:
5522:
5491:
5478:
5465:
5456:
5450:
5441:
5431:
5418:
5409:
5403:
5387:
5368:
5358:
5315:
5311:
5305:
5286:
5281:
5276:
5262:
5254:
5250:
5217:
5215:
5206:
5197:
5193:
5142:
5138:
5130:
5124:
5119:
5118:
5114:
5110:
5100:
4659:
4654:
4375:
4212:
4207:
4205:
4132:
3841:
3834:
3831:
3760:
3547:
3428:
3316:
3312:
3304:
3300:
3297:
3282:satisfying.
3164:
3151:
3147:
3120:
3116:
3111:
3107:
3096:
3079:
3074:
3070:
3068:
3047:
3039:
3031:
3023:
3015:
3011:
3003:
2995:
2987:
2979:
2977:
2972:
2970:
2967:
2960:references.
2957:
2955:
2938:
2934:
2930:
2926:
2922:
2781:
2509:
2468:
2467:denotes the
2065:
1917:
1884:
1880:
1821:
1819:
1657:
1437:
1292:
1222:
1064:
868:
861:
794:
790:
787:
775:downsampling
769:
748:
738:
700:
695:
657:
410:
253:
245:
221:
193:
183:
179:
175:
168:
163:
160:
151:
147:
143:
127:
121:
101:
84:
72:
64:down-convert
59:
53:
48:
44:
28:
24:
18:
3050:)), where H
60:filter bank
58:, the term
25:filter bank
6025:Categories
5444:: 157–264.
5378:0136051626
5325:2007.10729
5318:: 102795.
5297:References
4448:through a
3839:matrices.
779:upsampling
171:upsampling
29:filterbank
5674:116370718
5350:220665533
5280:The term
5188:, where I
5137:of order
5048:−
5006:−
4857:ξ
4821:ξ
4785:ξ
4749:ξ
4547:∑
4459:×
3634:∈
3627:∑
3584:∈
3577:∑
3427:, where
3412:×
3386:×
2944:(IIR) or
2811:^
2701:−
2681:^
2638:^
2589:^
2524:^
2329:−
2230:. Denote
2194:−
1999:−
1864:^
1784:ψ
1780:⟩
1756:φ
1737:⟨
1720:∈
1708:≤
1702:≤
1695:∑
1676:^
1621:ψ
1584:∈
1572:≤
1566:≤
1559:∑
1540:^
1511:⟩
1487:φ
1468:⟨
1417:−
1396:∗
1338:ψ
1266:∈
1240:≤
1234:≤
1201:−
1180:∗
1122:φ
1073:ℓ
832:−
49:synthesis
6041:Wavelets
5942:18149121
5655:18427023
3375:of size
2958:ORIGINAL
2575:, where
2425:, where
128:segments
45:analysis
37:sub-band
5330:Bibcode
3018:)=1/2(H
2982:)=1/2(H
230:. See
154:. (see
114:(FFT).
106:or the
87:vocoder
6010:
5940:
5930:
5672:
5653:
5395:
5375:
5348:
5282:filter
5129:. Let
4133:where
3102:,...,w
3014:)); T(
2971:Let H(
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