Underwater sound double-expansion channel parameter estimation method based on Newton orthogonal matching pursuit

Abstract

The invention discloses an underwater sound double-expansion channel parameter estimation method based on Newton orthogonal matching pursuit. The method comprises the following steps: firstly, realizing frame synchronization by using a cross-correlation peak of an LFM signal and a received signal; a coarse estimation value of a Doppler coefficient is obtained on a discrete time frequency grid through an ambiguity function method by utilizing the self-correlation characteristic of m-sequence pilot frequency, multipath time delay of a channel is detected by utilizing a matched filtering peak value of a signal after frequency offset compensation, and a coarse estimation result of the channel is obtained; executing Newton iteration optimization in a continuous parameter space by adopting a Newton orthogonal matching pursuit algorithm to realize super-resolution off-grid estimation of the Doppler coefficient of the first data block; and finally, proposing an iterative block tracking strategy, initializing the compensation of the rear block through the estimated value of the front block, and updating the time-varying Doppler coefficient by using the balanced signal feedback to realize the dynamic tracking of the time-varying Doppler coefficient. According to the invention, the Doppler estimation root-mean-square error is obviously reduced, and the calculation efficiency is high.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of underwater acoustic communication, in particular to a method for estimating parameters of underwater acoustic double spread channel based on Newtonized Orthogonal Matching Pursuit (NOMP), which is suitable for single carrier underwater acoustic communication system in time-varying Doppler environment. BACKGROUND

[0002] As the main underwater information transmission means at present, underwater acoustic communication technology shows great potential in many fields such as ocean environment monitoring, underwater Internet of Things, underwater search and rescue and underwater development. Due to the complexity of underwater environment and the particularity of sound wave propagation, underwater acoustic channel is considered to be one of the most challenging media in the field of communication. Overcoming Doppler spread and multipath spread is two key challenges in underwater acoustic communication, which is of great significance to the reliability and efficiency of underwater acoustic signal transmission. The channel with both multipath delay spread and Doppler spread is called double spread channel.

[0003] Traditional methods for estimating the delay and Doppler of double spread channel, such as grid search technology, face the trade-off between resolution and computational complexity. The estimation accuracy is proportional to the accuracy of the estimation grid, but finer grid division will also bring huge computational complexity. SUMMARY

[0004] In view of the shortcomings of the prior art, the present application provides a method for estimating parameters of underwater acoustic double spread channel based on Newtonized Orthogonal Matching Pursuit, which can break through the problem that double spread channel estimation is limited by the accuracy of grid division, realize off-grid parameter estimation, and improve the accuracy of channel estimation.

[0005] The purpose of the present application is achieved by the following technical solutions:

[0006] A method for estimating parameters of underwater acoustic double spread channel based on Newtonized Orthogonal Matching Pursuit, the method comprising the following steps:

[0007] Step one: using the cross-correlation method, using a copy of the local LFM frame header which is the same as the transmitted signal to perform correlation operation with the received signal, determining the signal starting time according to the correlation peak, realizing signal synchronization; then sampling the synchronized signal, down-converting to baseband, and passing through a low-pass filter to obtain the processed signal;

[0008] Step two: dividing the processed signal in the time domain into blocks, using the m sequence of the coherent data packet as the pilot sequence, dividing the grid in the preset frequency range with a fixed division step, using the sharp autocorrelation characteristics of the m sequence, and using the ambiguity function method to coarsely estimate the Doppler coefficient of the processed signal obtained in step one;

[0009] Step three: resample the processed signal obtained in step one according to the coarse estimation result of the Doppler coefficient obtained in step two, so as to realize preliminary compensation of the processed signal obtained in step one to eliminate the main Doppler shift; then perform matched filtering between the resampled signal and the local m sequence, and estimate different multipath time delays according to different correlation peaks.

[0010] Step four: based on the time delay estimated in step three and the coarse estimation result of the Doppler coefficient obtained in step two, use the Newton orthogonal matching pursuit method to realize secondary convergence and fine correction of the Doppler coefficient, realize off-grid estimation of the Doppler coefficient, and obtain the corrected Doppler coefficient.

[0011] Step five: take the corrected Doppler coefficient as the initial value of the Doppler coefficient of the second data block of the processed signal divided by block, perform frequency offset compensation and channel equalization of the second data block, obtain the self signal of the equalized second data block, and estimate the real time-varying Doppler coefficient of the second data block; in this way, the real time-varying Doppler coefficient of the εth data block is used as the initial value of the Doppler coefficient of the ε+1th data block, frequency offset compensation and channel equalization of the ε+1th data block are performed, and the real time-varying Doppler coefficient of the ε+1th data block is re-estimated according to the equalization result of the ε+1th data block, so as to realize tracking estimation of the time-varying Doppler coefficient.

[0012] Further, the coherent data packet includes an m sequence with sharp autocorrelation characteristics of a near-impulse function and a subsequent data sequence.

[0013] Further, step two specifically includes:

[0014] The m sequence of the coherent data packet is taken as a pilot sequence, the division step of the coarse estimation frequency grid is designed according to the length and symbol rate of the m sequence; after the received signal is compensated for frequency offset at different frequency grid points, the correlation operation is performed between the received signal and the local pilot sequence, and the frequency grid point satisfying the maximum ambiguity function is regarded as the coarse estimated main Doppler angular frequency.

[0015] Further, the division step is 1 / 4 of the main lobe width of the autocorrelation function.

[0016] Further, in step four, the Newton orthogonal matching pursuit method is used to realize secondary convergence and fine correction of the Doppler coefficient, specifically:

[0017] The Newton second-order partial derivative optimization is performed on the cost function of the coarse estimated Doppler coefficient obtained in step two by using the Newton orthogonal matching pursuit method.

[0018] Further, the transmitting signal frame structure comprises an LFM frame header, a guard interval and a coherent data packet arranged in sequence.

[0019] Further, the preset frequency range is determined by the relative speed between the signal transmitting end and the receiving end.

[0020] A Newton orthogonal matching pursuit-based underwater acoustic double spread channel parameter estimation device comprises one or more processors for implementing a Newton orthogonal matching pursuit-based underwater acoustic double spread channel parameter estimation method.

[0021] An electronic device comprises:

[0022] One or more processors;

[0023] A storage device for storing one or more programs, which, when executed by the electronic device, cause the electronic device to implement the Newton orthogonal matching pursuit-based underwater acoustic double spread channel parameter estimation method.

[0024] A computer-readable storage medium having a program stored thereon, which, when executed by a processor, implements the Newton orthogonal matching pursuit-based underwater acoustic double spread channel parameter estimation method.

[0025] The present application has the following advantages:

[0026] (1) The present application uses an m sequence as a pilot signal, and utilizes the autocorrelation peak reduction of the m sequence when there is a Doppler frequency offset to achieve coarse step estimation of the Doppler coefficient.

[0027] (2) The present application uses a Newton orthogonal matching pursuit method to off-grid estimate the channel Doppler coefficient, breaking through the resolution limit of the grid method.

[0028] (3) The present application uses a block iterative estimation method, which can realize real-time tracking estimation of the Doppler coefficient in a time-varying Doppler channel. BRIEF DESCRIPTION OF DRAWINGS

[0029] Figure 1 is a flowchart of the Newton orthogonal matching pursuit-based underwater acoustic double spread channel parameter estimation method of one embodiment of the present application.

[0030] Figure 2 is a structure diagram of a signal frame in one embodiment of the present application.

[0031] Figure 3 is a structure diagram of block iterative estimation of one embodiment of the present application.

[0032] Figure 4is the estimation result of time delay-doppler of one embodiment of the present application under the condition of 10dB signal to noise ratio.

[0033] Figure 5 is the estimation result of time delay-doppler of one embodiment of the present application under the condition of 10dB signal to noise ratio. DETAILED DESCRIPTION

[0034] The present application will be described in detail below with reference to the accompanying drawings and preferred embodiments, and the purposes and effects of the present application will become more apparent. It should be understood that the specific embodiments described herein are merely intended to explain the present application and are not intended to limit the present application.

[0035] Technical terms are explained as follows:

[0036] QPSK: Quadrature Phase Shift Keying, quadrature phase shift keying;

[0037] LFM: Linear Frequency Modulation, linear frequency modulation.

[0038] Reference Figure 1 The method for estimating parameters of underwater acoustic double-spread channel based on Newton orthogonal matching pursuit proposed by one embodiment of the present application comprises the following steps:

[0039] (1) Signal synchronization

[0040] The purpose of signal synchronization is to find when the signal reaches the receiving end. Considering that the LFM signal is not sensitive to the Doppler effect, as shown in Figure 2 , the transmission signal frame structure is composed of an LFM frame header, a guard interval and a coherent data packet. The correlation method is used to perform correlation operation on the received signal by using the same local LFM frame header copy as the transmission signal. The starting time of the signal is determined according to the correlation peak, and signal synchronization is realized. Then, the synchronized signal is sampled, down-mixed to the baseband, and processed by a low-pass filter, so as to facilitate subsequent estimation of the time delay-doppler of the signal.

[0041] The coherent data packet is composed of an m sequence (as a pilot sequence) with sharp autocorrelation characteristics of a near impulse function and a subsequent data sequence. The transmission signal is QPSK modulated as:

[0042]

[0043] wherein, represents the transmission signal after QPSK modulation, Re{} represents taking the real part, represents the sequence of the nth transmission symbol, N is the length of the transmission symbol sequence, is a pulse shaping filter, T is a symbol period, is the carrier frequency.

[0044] QPSK modulated transmitted signal After passing through the dual extended channel with multipath number q, the received signal is obtained :

[0045]

[0046]

[0047] in, represents the impulse response of the dual-extended channel, represents the impulse function, represents the Doppler coefficient of the kth path, represents the amplitude attenuation coefficient of the kth path, represents the channel propagation delay of the kth path, For noise.

[0048] To receive the signal Sampling, down-mixing and low-pass filtering yields:

[0049]

[0050]

[0051]

[0052] in, Indicates the received signal The signal is sampled, down-mixed and low-pass filtered. is the sampling rate, represents the amplitude delay attenuation coefficient of the kth path, Indicates identity; represents the Doppler angular frequency of the kth path, represents the noise of the nth transmitted symbol.

[0053] Will The calculation formula is expressed in matrix form as follows:

[0054]

[0055]

[0056]

[0057]

[0058]

[0059] wherein, is in the form of a matrix, is in the form of a matrix, is in the form of a matrix, is a discrete time delay, is a discrete time delay amplitude matrix, denotes the horizontal and vertical coordinates in the discrete time delay amplitude matrix; is in the form of a matrix.

[0060] (2) Coarse estimation of the Doppler coefficient of the pilot sequence

[0061] In an underwater acoustic communication system, the Doppler coefficients of different paths can be considered to fluctuate in a small range around a main Doppler coefficient , which reflects the dominant Doppler frequency shift of signal transmission, and the corresponding main Doppler angular frequency is , that is, , wherein the fluctuation range is . After signal synchronization, the main Doppler coefficient is coarsely estimated from the signal processed by the low-pass filter, specifically as follows:

[0062] The signal processed by the low-pass filter is divided into blocks in the time domain, and the signal is divided into multiple data blocks, as shown in FIG. 1, wherein the first data block is a pilot sequence data block. Then, the Doppler coefficients of different data blocks are estimated. Figure 3

[0063] First, the Doppler coefficient of the pilot sequence data block is coarsely estimated, specifically as follows: within a preset frequency range , the grid is divided by a fixed step, and the ambiguity function method is used to coarsely estimate the Doppler coefficient of the received signal by using the sharp autocorrelation characteristics of the m sequence of the coherent data packet. wherein,

[0064] The preset frequency range is determined by the relative speed between the signal sending end and the receiving end.

[0065] Specifically, according to the length of the m sequence and the symbol rate, the division step of the coarse estimation frequency grid is designed. The m sequence has the sharp autocorrelation characteristics of approaching the impulse function. When there is a Doppler frequency offset (carrier frequency offset), a persistent phase rotation will be introduced between the pilot sequence in the received signal and the local pilot sequence, which will destroy the coherent accumulation process. As a result, the correlation peak height will decrease significantly. Based on the above characteristics, the ambiguity function method is used to coarsely estimate the main Doppler angular frequency in a relatively wide frequency range ​​​​​​between the two The grid is divided, and the received signal is compensated for frequency offset after being divided at different frequency grid points, and then correlated with the local pilot sequence. The frequency grid point that satisfies is considered to be the coarsely estimated main Doppler angle frequency.

[0066] For the NOMP-based estimation method, its essence is a local optimization method based on Newton's method, whose convergence highly depends on the initial point being located in the effective convex region near the global optimal solution (main lobe peak value), and the target function (cross-correlation value) being approximately twice differentiable near the initial point. The purpose of coarsely estimating the main Doppler angle frequency is to provide an initial point close enough to the true Doppler coefficient, to ensure that Newton's method converges to the global optimal solution (main lobe peak value), rather than falling into a local extremum (side lobe). This requires the maximum possible error of coarse estimation to be strictly less than the effective width of the main lobe of the autocorrelation function. This "effective width" usually refers to the frequency offset range corresponding to the main lobe peak value falling to a certain significant level (such as -3 dB). In order to ensure that the initial point of the NOMP-based estimation method falls within the effective convex region of the main lobe of the m sequence autocorrelation function, the fixed step size of the coarse estimation is set to 1 / 4 of the main lobe width, that is:

[0067]

[0068] where L is the length of the m sequence, is the symbol rate, and the main lobe width of the autocorrelation function (ACF) is .

[0069] This ensures that even if the true frequency offset is located in the middle of two grid points, it still ensures that the initial point of Newton's method is located in the range where the target function can be well approximated by a quadratic function, satisfying the necessary condition for the convergence of Newton's method.

[0070] (3) Time delay estimation of pilot sequence

[0071] According to the coarse estimation result of the Doppler coefficient, the signal processed by the low-pass filter is resampled, thereby realizing preliminary compensation of the signal processed by the low-pass filter to eliminate the main Doppler offset; then, matching filtering is performed between the signal and the local m sequence, and different correlation peaks are generated to estimate different multipath time delays, specifically:

[0072] The data packet of the received signal after preliminary compensation is correlated with a copy of the local m sequence, and different path pilot m sequences will form different correlation peaks with the copy of the local m sequence, and different multipath time delays are estimated according to different correlation peaks:

[0073]

[0074] wherein, is the sampling rate, corr is the phasor correlation operation, B is the matrix form of the local m-sequence copy,

[0075] (4) NOMP off-grid refinement estimation of pilot sequence

[0076] Based on the estimated time delay and the coarse estimation result of the Doppler coefficient, the method of Newton orthogonal matching pursuit is used to realize the secondary convergence and refinement correction of the Doppler coefficient, realize the off-grid estimation of the parameter, and obtain the estimated frequency offset of the first data block.

[0077] The secondary convergence here is to optimize the first-order partial derivative of the Newton second-order partial derivative of the coarse estimation result of the Doppler coefficient, update the Doppler coefficient, and realize the refinement estimation.

[0078] The estimation problem of refinement estimation can be written as:

[0079]

[0080] Expanding it obtains

[0081]

[0082] In the above formula, only and are related, so the optimization problem is transformed into:

[0083]

[0084] The second-order derivative is used to realize the secondary convergence of the estimated parameter

[0085]

[0086]

[0087]

[0088]

[0089]

[0090] Indicates the Hadamard product operator.

[0091] The updated estimated Doppler coefficient is

[0092]

[0093] wherein, indicates the coarse estimation result of the frequency offset of the lth iteration.

[0094] ​​Figure 4 The figure shows the delay-Doppler estimation result of the present invention under the condition of 10dB signal-to-noise ratio. The figure also shows the estimation result of the traditional grid-based method. The comparison results show that the estimation accuracy of the method of the present invention is better than that of the traditional grid-based method.

[0095] (5) Block-by-block iterative estimation to achieve time-varying Doppler tracking

[0096] The estimated frequency offset of the first data block is used as the frequency offset of the second data block of the block-divided processed signal, and frequency offset compensation and channel equalization are performed on the second data block to obtain the self-signal of the equalized second data block, and the estimated frequency offset of the second data block is calculated; and so on, the estimated frequency offset of the εth data block is used as the frequency offset of the ε+1th data block, and frequency offset compensation and channel equalization are performed on the ε+1th data block. Then, the estimated frequency offset of the ε+1th data block is recalculated based on the equalization result of the ε+1th data block, thereby realizing tracking and estimation of the time-varying Doppler coefficient.

[0097] Among them, according to the real time-varying Doppler coefficient of the εth data block As the initial value of the Doppler coefficient of the ε+1th data block ,Right now , perform frequency offset compensation for the ε+1th data block, and obtain ,right Perform channel equalization and set the symbol after equalization to ; Using the symbols before and after equalization to perform feedback correction on the real time-varying Doppler coefficient of the ε+1 data block, the estimation problem can be written as:

[0098]

[0099] in

[0100]

[0101] Therefore, for the The Doppler coefficient angular frequency of each data block is corrected as follows:

[0102]

[0103] in, is the cost function of the true time-varying Doppler coefficient of the ε+1th data block. After equalization, the symbol The matrix form of the offset.

[0104] Figure 5 The figure shows the simulation results of time-varying Doppler channel tracking estimation under 10dB signal-to-noise ratio. Figure 5 It can be seen from the figure that the method of the present invention can track the time-varying Doppler coefficient well.

[0105] Corresponding to the foregoing underwater acoustic double spread channel parameter estimation method based on Newton orthogonal matching pursuit, the application also provides an embodiment of an underwater acoustic double spread channel parameter estimation device based on Newton orthogonal matching pursuit.

[0106] The underwater acoustic double spread channel parameter estimation device based on Newton orthogonal matching pursuit provided by the embodiment of the application comprises one or more processors for implementing the underwater acoustic double spread channel parameter estimation method based on Newton orthogonal matching pursuit in the foregoing embodiments.

[0107] The embodiment of the underwater acoustic double spread channel parameter estimation device based on Newton orthogonal matching pursuit of the application can be applied to any device with data processing capability, which can be a device or apparatus such as a computer. The device embodiment can be implemented by software, or by hardware or a combination of software and hardware. Taking software implementation as an example, as a logically meaningful device, it is formed by reading the corresponding computer program instructions in the non-volatile memory into the memory for execution by the processor of the device with data processing capability in which the device is located. From the hardware level, the device with data processing capability in which the device based on the self-attention mechanism of the application is located, in addition to the processor, the memory, the network interface, and the non-volatile memory, the device with data processing capability in which the device is located in the embodiment usually includes other hardware according to the actual functions of the device with data processing capability, and details are not described here.

[0108] The implementation process of the functions and roles of each unit in the above device is specifically described in the implementation process of the corresponding steps in the above method, and is not described here.

[0109] For the device embodiment, since it basically corresponds to the method embodiment, the related parts are described in the part of the method embodiment. The device embodiments described above are only illustrative, and the units described as separate components can or can not be physically separated, and the components displayed as units can or can not be physical units, that is, they can be located in one place, or they can be distributed on multiple network units. According to actual needs, part or all of the modules can be selected to achieve the purpose of the application scheme. Those skilled in the art can understand and implement it without creative labor.

[0110] The embodiment of the application also provides a computer readable storage medium having a program stored thereon, and the program is executed by a processor to implement the underwater acoustic double spread channel parameter estimation method based on Newton orthogonal matching pursuit in the foregoing embodiments.

[0111] The computer readable storage medium can be an internal storage unit of any of the aforementioned devices with data processing capability, such as a hard disk or a memory. The computer readable storage medium can also be an external storage device, such as a plug-in hard disk, a SmartMedia card (SMC), an SD card, a Flash card, etc. Further, the computer readable storage medium can include both an internal storage unit of any of the aforementioned devices with data processing capability and an external storage device. The computer readable storage medium is used to store the computer program and other programs and data required by the aforementioned devices with data processing capability, and can also be used to temporarily store data that has been output or will be output.

[0112] Those skilled in the art can understand that the above description is only preferred examples of the application and is not used to limit the application. Although the application is described in detail with reference to the foregoing examples, those skilled in the art can still modify the technical solutions recorded in the foregoing examples or make equivalent replacements for part of the technical features. Any modification, equivalent replacement, etc. made within the spirit and principle of the application shall be included in the protection scope of the application.

Claims

1. A method for underwater acoustic double-extended channel parameter estimation based on Newton orthogonal matching pursuit, characterized in that: The method comprises the following steps: Step 1: Using the cross-correlation method, a copy of the local LFM frame header, identical to the transmitted signal, is correlated with the received signal. The signal start time is determined based on the correlation peak to achieve signal synchronization. The synchronized signal is then sampled, mixed down to baseband, and processed through a low-pass filter to obtain the processed signal. Step 2: Divide the processed signal into blocks in the time domain, use the m-sequence of the coherent data packet as a pilot sequence, divide the grid with a fixed division step size within a preset frequency range, and use the sharp autocorrelation characteristics of the m-sequence to roughly estimate the Doppler coefficient of the processed signal obtained in step 1 using an ambiguity function method; Step 3: resampling the processed signal obtained in step 1 according to the rough estimation result of the Doppler coefficient obtained in step 2, thereby achieving preliminary compensation for the processed signal obtained in step 1 to eliminate the main Doppler shift; Then, matched filtering is performed between the local m-sequence and different multipath delays are estimated based on the generated different correlation peaks. Step 4: Based on the time delay estimated in step 3 and the rough estimation result of the Doppler coefficient obtained in step 2, a quadratic convergence refinement correction of the Doppler coefficient is performed using the Newton orthogonal matching pursuit method to achieve off-grid estimation of the Doppler coefficient and obtain a corrected Doppler coefficient; Step 5: Using the corrected Doppler coefficient as the initial value of the Doppler coefficient of the second data block of the processed signal divided into blocks, frequency offset compensation and channel equalization are performed on the second data block to obtain the self-signal of the equalized second data block, and then the true time-varying Doppler coefficient of the second data block is estimated; and similarly, using the true time-varying Doppler coefficient of the εth data block as the initial value of the Doppler coefficient of the ε+1th data block, frequency offset compensation and channel equalization are performed on the ε+1th data block, and then the true time-varying Doppler coefficient of the ε+1th data block is re-estimated based on the equalization result of the ε+1th data block, thereby achieving tracking estimation of the time-varying Doppler coefficient.

2. The underwater acoustic dual-extended channel parameter estimation method based on Newton orthogonal matching pursuit according to claim 1 is characterized in that: The coherent data packet includes an m-sequence having a sharp autocorrelation characteristic close to an impulse function and a subsequent data sequence.

3. The underwater acoustic dual-extended channel parameter estimation method based on Newton orthogonal matching pursuit according to claim 1 is characterized in that: The second step specifically includes: The m-sequence of the coherent data packet is used as a pilot sequence, and the division step size of the coarsely estimated frequency grid is designed according to the length of the m-sequence and the symbol rate. The received signal is correlated with the local pilot sequence after frequency offset compensation is performed at different divided frequency grid points, and the frequency grid point that satisfies the maximum ambiguity function is regarded as the coarsely estimated main Doppler angular frequency.

4. The underwater acoustic dual-extended channel parameter estimation method based on Newton orthogonal matching pursuit according to claim 3 is characterized in that: The division step is 1 / 4 of the main lobe width of the autocorrelation function.

5. The underwater acoustic dual-extended channel parameter estimation method based on Newton orthogonal matching pursuit according to claim 3 is characterized in that: In step 4, the Newton orthogonal matching pursuit method is used to implement quadratic convergence and refinement correction of the Doppler coefficient, specifically: The Newton orthogonal matching pursuit method is used to optimize the cost function of the roughly estimated Doppler coefficient obtained in step 2 using the Newton second-order partial derivative.

6. The underwater acoustic dual-extended channel parameter estimation method based on Newton orthogonal matching pursuit according to claim 1 is characterized in that: The transmission signal frame structure includes an LFM frame header, a guard interval and a coherent data packet arranged in sequence.

7. The underwater acoustic dual-extended channel parameter estimation method based on Newton orthogonal matching pursuit according to claim 1 is characterized in that: The preset frequency range is determined by the relative speed between the signal transmitting end and the receiving end.

8. An underwater acoustic dual-extended channel parameter estimation device based on Newton orthogonal matching pursuit, characterized in that: The method comprises one or more processors for implementing the underwater acoustic dual-extended channel parameter estimation method based on Newton orthogonal matching pursuit according to any one of claims 1 to 7.

9. An electronic device, characterized in that: include: one or more processors; A storage device for storing one or more programs, which, when executed by the electronic device, enables the electronic device to implement the underwater acoustic dual-extended channel parameter estimation method based on Newton orthogonal matching pursuit as described in any one of claims 1 to 7.

10. A computer-readable storage medium, characterized in that A program is stored thereon, and when the program is executed by a processor, the underwater acoustic dual-extended channel parameter estimation method based on Newton orthogonal matching pursuit according to any one of claims 1 to 7 is implemented.

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