CN103684690A

CN103684690A - Compressed sensing based channel shortening filter design method

Info

Publication number: CN103684690A
Application number: CN201310635165.1A
Authority: CN
Inventors: 李有明; 刘小青; 雷鹏; 季彪; 朱星; 陈斌
Original assignee: Ningbo University
Current assignee: Ningbo University
Priority date: 2013-12-02
Filing date: 2013-12-02
Publication date: 2014-03-26
Anticipated expiration: 2033-12-02
Also published as: CN103684690B

Abstract

The invention discloses a compressed sensing based channel shortening filter design method. The method includes: firstly, dividing received signals received by a receiving end of a single input single output system into multiple modules; secondly, selecting any one of the modules, computing a cross correlation matrix of the selected module and a module formed by all corresponding symbols in sending signals, and computing an autocorrelation matrix of the selected module; thirdly, according to the acquired cross correlation matrix and autocorrelation matrix as well as set maximum signal-to-noise ratio loss, acquiring an upper bound of mean square error appreciation after the selected module passes a channel shortening filter; finally, determining the channel shortening filter by adopting a reconstruction method in compressed sensing and according to the upper bound of the mean square error appreciation. The method has the advantages that the filter designed by the method is low in computation complexity and high in computation accuracy, computation complexity of an application system can be lowered, and degree of freedom of the application system is increased.

Description

Channel shortening filter design method based on compressed sensing

Technical Field

The present invention relates to a channel shortening technique in a communication system, and in particular, to a channel shortening filter design method based on compressed sensing.

Background

With the rapid growth of communication services, the demands on communication quality and rate are also increasing. The impulse response of a wideband communication channel is long, extends over tens of symbol periods, and causes severe intersymbol interference. In a single-input single-output system, in order to reduce intersymbol interference, a cyclic prefix needs to be added before each symbol, but the length of the cyclic prefix should not be smaller than the length of the impulse response of the wideband communication channel. However, a long cyclic prefix is required under a broadband communication channel, which may seriously reduce the transmission rate of the single-input single-output system. In response to these problems, a channel shortening filter may be used to shorten the impulse response length of a wideband communication channel, thereby achieving the purposes of suppressing intersymbol interference and reducing the transmission rate loss caused by the cyclic prefix. However, the complexity of the system increases proportionally with the number of non-zero taps in the channel shortening filter, which results in a very high complexity of the system using the conventional non-sparse filter, and thus the sparse filter is receiving more and more attention.

While a reasonable and effective sparse filter is an important guarantee for the superior performance of a communication system, in the existing research of the sparse filter, the calculation complexity of the design method of the sparse filter is always poor, such as an Orthogonal Matching Pursuit (OMP) method, even if the expected design complexity is achieved, the accuracy of the sparse filter is sacrificed. Therefore, it is of practical significance to research a sparse filter with low design complexity at high accuracy.

Disclosure of Invention

The technical problem to be solved by the invention is to provide a channel shortening filter design method based on compressed sensing, and the filter obtained by design not only can increase the degree of freedom of a single-input single-output system, but also has low calculation complexity and high calculation accuracy.

The technical scheme adopted by the invention for solving the technical problems is as follows: a channel shortening filter design method based on compressed sensing is characterized by comprising the following steps:

firstly, at a sending end of a single-input single-output system, the sending end transmits and sends signals to a receiving end through a broadband communication channel;

② at the receiving end of the single input single output system, starting from the 1 st symbol, with continuous N_fDividing the received signal received by the receiving end into one period of symbols

A module, wherein N represents the total number of symbols contained in the received signal, and 1 is less than or equal to N_fN and assuming N_fCan be divided exactly by N;

then, one module is randomly selected from all modules for receiving signals, Nyquist sampling is carried out on each symbol in the selected module, the sampling frequency is l, l sampling values of each symbol in the selected module are obtained, and the kth sampling value of the ith symbol in the selected module is recorded as y_i,kWherein l is ∈ [1, N ∈ ]_f]，1≤i≤N_f，1≤k≤l；

Then obtaining the symbol corresponding to each symbol position in the selected module from the sending signal, and then calculating the cross-correlation matrix between the module formed by all the corresponding symbols obtained from the sending signal and the selected module, and recording the cross-correlation matrix as R_yxAnd calculating the autocorrelation matrix of the selected module, denoted as R_yy；

Let omega denote channel shortening filter, let gamma_maxIndicating a set maximum signal-to-noise ratio loss, then according to gamma_maxDetermining the upper bound of mean square error increment of the selected module after omega, marking as epsilon,

wherein,

ε_x=E[x_k-Δ ²]，ε_x=E[x_k-Δ ²]the middle symbol, "| |" is a modulo symbol, E [ x_k-Δ ²?]Expression to x_k-Δ ²Statistical average of (a), x_k-ΔRepresenting a symbol corresponding to the ith symbol position in the selected module in the transmitted signal, and obtaining a kth sampling value x in l sampling values after l Nyquist sampling_kA signal value obtained after delaying delta, wherein delta is an integer and is more than or equal to 0 and less than or equal to N_f，

Is r_ΔConjugate transpose of (r)_Δ=R_yxI_Δ，I_ΔRepresents (N)_f+ν)×(N_fIn the unitary matrix of the + v) dimensionV represents the maximum memory length of the wideband communication channel, L^-1Is the inverse matrix of L, L^HIs a conjugate transpose of L, L being to R_yyOne obtained after Cholesky decomposition (l.times.N)_f)×(l×N_f) Lower triangular matrix of dimensions, R_yy=LL^H；

Fourthly, determining omega according to epsilon by adopting a reconstruction method in a compressed sensing theory, wherein the concrete process is as follows: fourthly-1, assuming that the number of the finally determined nonzero tap coefficients in the omega is K, and making I₀The initial set of indices is represented as,

let r be₀Denotes the initial residual error, r₀=L^-1r_ΔLet N denote the number of iterations, the initial value of N is 1, where K is greater than or equal to 1 and less than or equal to N_fV, n is less than or equal to 2K; fourthly-2, calculating the residual error r after the last iteration when the nth iteration is carried out_n-1The residual error r after the last iteration is related to the correlation coefficient of each column in L_n-1The correlation coefficient with the j-th column in L is denoted as σ_j，Wherein j is more than or equal to 1 and less than or equal to l multiplied by Nf,

is represented by r_n-1By conjugate transposition of L^H(j) Denotes the conjugate transpose of L (j), L (j) denotes the jth column in L, where the symbol "|" is a modulo symbol; then, the largest K correlation coefficients are selected from all the correlation coefficients, and the subscripts of the selected K correlation coefficients form a set, which is marked as c_n(ii) a Fourthly-3, order omega_sRepresents one (N)_fA + v) x 1-dimensional intermediate column vector, and let the initial ω be_sIs 0; then calculating the index set I after the last iteration_n-1And c_nIs marked as b_n，b_n=I_n-1∪c_n(ii) a Then the initial ω_sThe middle subscript belongs to b_nAll elements of (a) constitute the initial ω in sequence_sThe subvector of (2), denoted as ω_s(：,b_n) According to

To obtain omega_s(：,b_n) A value of each element in (a); finally according to omega_s(：,b_n) Updates the initial ω by the value of each element in (1)_sWhere the symbol "U" is a union operation symbol, L^H(：,b_n) Represents L (: b is b_n) Conjugate transpose of, L (: b is b_n) Indicates that the column number in L belongs to b_nAll of the columns of (a) make up a sub-matrix of L,

is represented by (L)^H(：,b_n) A pseudo-inverse of); 4, from the updated omega_sSelecting K elements with the maximum absolute value, and forming the subscripts of the selected K elements into an index set of the nth iteration, which is marked as I_nThen the residual error of the nth iteration is calculated and recorded as r_n，r_n=L^-1r_Δ-L^H(：,I_n)ω_s(：,I_n) Wherein L is^H(：,I_n) Represents L (: i, I_n) Conjugate transpose of, L (: i, I_n) Indicates that the column number in L belongs to I_nAll columns of (a) constitute a sub-matrix of L, ω_s(：,I_n) Represents updated ω_sThe middle subscript being of formula I_nAll elements of (a) in sequence constitute an updated ω_sThe subvectors of (1); fourthly-5, reserving updated omega_sMiddle and omega_s(：,I_n) And the value of the element corresponding to each element in (b), and ω to be updated_sThe values of the other elements are set to 0 and are recorded again as omega_s'; fourthly-6, judging whether n is less than or equal to 2K, if n is less than or equal to 2K, comparing r_nIf r is the magnitude of the square of the modulus of (2) and ∈_nIf the square value of the modulus is larger than epsilon, let n = n +1, then return to the step of (r) -2 to continue execution, proceed the next iteration, if r is larger than epsilon_nIs less than or equal to epsilon, the iterative process is ended and let ω = ω_s'; such asIf n is not more than 2K, ending the iteration process and making omega = omega_s'; wherein n = n +1 and ω = ω_s"=" in' is an assigned symbol.

In the step I, the sending signal is a baseband complex signal, and the broadband communication channel is a discrete time invariant noisy channel.

Compared with the prior art, the invention has the advantages that:

1) the channel shortening filter designed by the method is a sparse filter, and the number of nonzero tap coefficients is less than that of the traditional non-sparse filter, so that the calculation complexity of a system using the sparse filter designed by the method is effectively reduced, and the degree of freedom of a single-input single-output system is increased.

2) Because the method of the invention takes the maximum signal-to-noise ratio loss as the constraint condition to design the channel shortening filter, the user can set different maximum signal-to-noise ratio loss values according to different actual conditions, thus the compromise between the calculation complexity and the calculation precision can be simply realized, namely if the user has low requirement on the precision, the maximum signal-to-noise ratio loss can be properly increased, thereby reducing the precision requirement, but the calculation complexity can also be reduced, namely the calculation speed is accelerated; if the requirement of the user on the precision is higher, the maximum signal-to-noise ratio loss can be properly reduced, and the precision is improved by improving the computational complexity.

3) Compared with the existing sparse filter designed based on the orthogonal matching tracking method, the method provided by the invention has the advantages that the index set of the sparse filter generated after the last iteration is detected and corrected in the process of each iteration, so that the calculation precision is improved, and meanwhile, the iteration times of the method provided by the invention are greatly reduced compared with the existing orthogonal matching tracking method due to the inherent advantages of the reconstruction method, so that the calculation complexity is reduced.

4) Under the same constraint condition (the same number of tap coefficients), the channel shortening filter designed by the method has lower calculation complexity and higher calculation precision compared with the sparse filter designed by the prior art.

Drawings

FIG. 1 is a block diagram of the application of the method of the present invention;

FIG. 2a is a block diagram of an overall implementation of the method of the present invention;

FIG. 2b is a block diagram of a specific process of determining a channel shortening filter according to an upper bound of a mean-square error increment after a selected module passes through the channel shortening filter by using a reconstruction method in a compressive sensing theory in FIG. 2 a;

FIG. 3 is a schematic diagram showing the comparison of simulation times of a channel-shortening filter designed by the method of the present invention and a channel-shortening filter designed by the conventional Orthogonal Matching Pursuit (OMP) method at different cyclic prefix lengths;

fig. 4 is a schematic diagram showing the comparison of the residual errors of the channel-shortening filter designed by the method of the present invention and the channel-shortening filter designed by the conventional Orthogonal Matching Pursuit (OMP) method at different cyclic prefix lengths.

Detailed Description

The invention is described in further detail below with reference to the accompanying examples.

The invention provides a channel shortening filter design method based on compressed sensing, the general implementation block diagram of which is shown in fig. 2a, and the method comprises the following steps:

firstly, at a sending end of a single-input single-output system, the sending end transmits and sends signals to a receiving end through a broadband communication channel. In the implementation process, the transmission signal can adopt a baseband complex signal, and the broadband communication channel can adopt a discrete time invariant noisy channel.

② at the receiving end of the single input single output system, starting from the 1 st symbol, with continuous N_fDividing the received signal received by the receiving end into one period of symbolsA module, wherein N represents the total number of symbols contained in the received signal, i.e. the total number of symbols contained in the transmitted signal, and 1 ≦ N_fN and assuming N_fCan be divided evenly by N.

Then, one module is randomly selected from all modules for receiving signals, Nyquist sampling is carried out on each symbol in the selected module, the sampling frequency is l, l sampling values of each symbol in the selected module are obtained, and the kth sampling value of the ith symbol in the selected module is recorded as y_i,kWherein l is ∈ [1, N ∈ ]_f]，1≤i≤N_f，1≤k≤l。

Then obtaining the symbol corresponding to each symbol position in the selected module from the sending signal, and then calculating the cross-correlation matrix between the module formed by all the corresponding symbols obtained from the sending signal and the selected module, and recording the cross-correlation matrix as R_yxAnd calculating the autocorrelation matrix of the selected module, denoted as R_yy。

wherein,

εx₌E[|x_k-Δ|²]，ε_x=E[|x_k-Δ|²]the middle symbol, "| |" is the modulo symbol, E [ | x [ ]_k-Δ|²]Is it expressed as finding x_k-Δ ²Statistical average of (a), x_k-ΔRepresenting a symbol corresponding to the ith symbol position in the selected module in the transmitted signal, and obtaining a kth sampling value x in l sampling values after l Nyquist sampling_kA signal value obtained after delaying delta, wherein delta is an integer and is more than or equal to 0 and less than or equal to N_f，

Is r_ΔConjugate transpose of (r)_Δ=R_yxI_Δ，I_ΔRepresents (N)_f+ν)×(N_fColumn Δ +1 in the unitary matrix of dimension + v), v representing the maximum memory length of the wideband communication channel, L^-1Is the inverse matrix of L, L^HIs a conjugate transpose of L, L being to R_yyOne obtained after Cholesky decomposition (l.times.N)_f)×(l×N_f) Lower triangular matrix of dimensions, R_yy=LL^H。

Here, γ_maxThe specific value of (2) is determined by the comprehensive calculation complexity and calculation precision of the user.

And determining omega according to epsilon by adopting a reconstruction method in a compressed sensing theory, wherein the specific process is as shown in figure 2 b: fourthly-1, assuming that the number of the finally determined nonzero tap coefficients in the omega is K, and making I₀The initial set of indices is represented as,

let r be₀Denotes the initial residual error, r₀=L^-1r_ΔLet N denote the number of iterations, the initial value of N is 1, where K is greater than or equal to 1 and less than or equal to N_fV, n is less than or equal to 2K; fourthly-2, calculating the residual error r after the last iteration when the nth iteration is carried out_n-1The residual error r after the last iteration is related to the correlation coefficient of each column in L_n-1The correlation coefficient with the j-th column in L is denoted as σ_j，

Wherein j is more than or equal to 1 and less than or equal to l multiplied by Nf_，

Is represented by r_n-1By conjugate transposition of L^H(j) Denotes the conjugate transpose of L (j), L (j) denotes the jth column in L, where the symbol "|" is a modulo symbol; then, the largest K correlation coefficients are selected from all the correlation coefficients, and the subscripts of the selected K correlation coefficients form a set, which is marked as c_n(ii) a Fourthly-3, order omega_sRepresents one (N)_fA + v) x 1-dimensional intermediate column vector, and let the initial ω be_sIs 0; then calculating the index set I after the last iteration_n-1And c_nIs marked as b_n，b_n=I_n-1∪c_n(ii) a Then the initial ω_sMiddle subscript (where subscript refers to the element at the initial ω)_sPosition in (1), e.g. initial ω_sThe subscript of the 1 st element in (1) belongs to b_nAll elements of (a) constitute the initial ω in sequence_sThe subvector of (2), denoted as ω_s(：,b_n) According to

is represented by (L)^H(：,b_n) A pseudo-inverse of); 4, from the updated omega_sSelecting K elements with the maximum absolute value, and forming the subscripts of the selected K elements into an index set of the nth iteration, which is marked as I_nFor example, if updated ω is given when K =3_sThe 1 st, 3 rd and 6 th elements are the maximum three elements in the sequence, the updated omega_sSubscript group of the largest three elementsThe set of indices is { 136 }; then, the residual error of the nth iteration is calculated and recorded as r_n，r_n=L^-1r_Δ-L^H(：,I_n)ω_s(：,I_n) Wherein L is^H(：,I_n) Represents L (: i, I_n) Conjugate transpose of, L (: i, I_n) Indicates that the column number in L belongs to I_nAll columns of (a) constitute a sub-matrix of L, ω_s(：,I_n) Represents updated ω_sThe middle subscript being of formula I_nAll elements of (a) in sequence constitute an updated ω_sThe subvectors of (1); fourthly-5, reserving updated omega_sMiddle and omega_s(：,I_n) And the value of the element corresponding to each element in (b), and ω to be updated_sThe values of the other elements are set to 0 and are recorded again as omega_s'; fourthly-6, judging whether n is less than or equal to 2K, if n is less than or equal to 2K, comparing the square value of the rn module with the size of epsilon, and if r is less than or equal to 2K_nIf the square value of the norm of rn is less than or equal to epsilon, the iteration process is ended, and omega = omega is made_s'; if n ≦ 2K does not hold, the iterative process ends and let ω = ω_s'; wherein n = n +1 and ω = ω_s"=" in' is an assigned symbol.

Fig. 1 shows a schematic diagram of a channel shortening filter designed by applying the method of the present invention.

The feasibility and effectiveness of the method of the invention are further illustrated by computer simulation.

The simulation environment is ADSL8 standard Carrier Service Area (CSA) loop 7. Set allowed maximum signal-to-noise ratio loss gamma_max1dB, and an IFFT (inverse fourier transform) length of 1024. The cyclic prefix length is from 50 to 120. To ensure the accuracy of the results, the mean value was taken after 1000 simulations.

Fig. 3 shows a comparison of simulation times of a channel-shortening filter designed by the method of the present invention and a channel-shortening filter designed by an existing orthogonal matching pursuit method (OMP) under different cyclic prefix lengths, fig. 4 shows a comparison of residuals of the channel-shortening filter designed by the method of the present invention and the channel-shortening filter designed by the existing orthogonal matching pursuit method (OMP) under different cyclic prefix lengths, fig. 3 and fig. 4 show that the present invention-16 and the present invention-24 in fig. 3 correspond to channel-shortening filters having numbers K of

non-zero tap coefficients

16 and 24 designed by the method of the present invention, and the OMP-16 and the OMP-24 correspond to channel-shortening filters having numbers K of

non-zero tap coefficients

16 and 24 designed by the existing orthogonal matching pursuit method (OMP). As can be seen from fig. 3, when the CP length is not changed, the channel-shortening filter designed by the method of the present invention has approximately half less operation time than the channel-shortening filter designed by the conventional Orthogonal Matching Pursuit (OMP) method under the same number of nonzero tap coefficients. As can be seen from fig. 4, under the cyclic prefix CP with the same length, the residual errors of the channel-shortened filters designed by the method of the present invention are all smaller than those of the channel-shortened filters designed by the existing Orthogonal Matching Pursuit (OMP) method, which indicates that the channel-shortened filters designed by the method of the present invention have better performance, and the single-input single-output system using the channel-shortened filters estimates the transmission signals more accurately. In conclusion, the comprehensive performance of the channel shortening filter designed by the method is well improved in terms of time complexity and system accuracy.

Claims

1. A channel shortening filter design method based on compressed sensing is characterized by comprising the following steps:

then, one module is randomly selected from all modules for receiving signals, Nyquist sampling is carried out on each symbol in the selected module, the sampling frequency is l, l sampling values of each symbol in the selected module are obtained, and the kth sampling value of the ith symbol in the selected module is recorded as y_i,kWherein l is ∈ [1, N ∈ ]_f]，1≤i≤Nf_，1≤k≤l；

wherein,

ε_x=E[x_k-Δ²]，ε_x=E[x_k-Δ²]the middle symbol, "| |" is a modulo symbol, E [ x_k-Δ²]Expression to x_k-Δ²Statistical average of (a), x_k-ΔRepresenting a symbol corresponding to the ith symbol position in the selected module in the transmitted signal, and obtaining a kth sampling value x in l sampling values after l Nyquist sampling_kA signal value obtained after delaying delta, wherein delta is an integer and is more than or equal to 0 and less than or equal to N_f，

Is r_ΔConjugate transpose of (r)_Δ=R_yxI_Δ，I_ΔRepresents (N)_f+ν)×(N_fColumn Δ +1 in the unitary matrix of dimension + v), v representing the maximum memory length of the wideband communication channel, L^-1Is the inverse matrix of L, L^HIs a conjugate transpose of L, L being to R_yyOne obtained after Cholesky decomposition (l.times.N)_f)×(l×N_f) Lower triangular matrix of dimensions, R_yy=LL^H；

Fourthly, determining omega according to epsilon by adopting a reconstruction method in a compressed sensing theory, wherein the concrete process is as follows: fourthly-1, assuming that the number of the finally determined nonzero tap coefficients in the omega is K, and making I₀The initial set of indices is represented as,let r be₀Denotes the initial residual error, r₀=L^-1r_ΔLet N denote the number of iterations, the initial value of N is 1, where K is greater than or equal to 1 and less than or equal to N_fV, n is less than or equal to 2K; fourthly-2, calculating the residual error r after the last iteration when the nth iteration is carried out_n-1The residual error r after the last iteration is related to the correlation coefficient of each column in L_n-1The correlation coefficient with the j-th column in L is denoted as σ_j，

Wherein j is more than or equal to 1 and less than or equal to lxN_f，

Is represented by r_n-1By conjugate transposition of L^H(j) Denotes the conjugate transpose of L (j), L (j) denotes the jth column in L, where the symbol "|" is a modulo symbol; then, the largest K correlation coefficients are selected from all the correlation coefficients, and the subscripts of the selected K correlation coefficients form a set, which is marked as c_n(ii) a Fourthly-3, order omega_sRepresents one (N)_fA + v) x 1-dimensional intermediate column vector, and let the initial ω be_sIs 0; then calculating the index set I after the last iteration_n-1And c_nUnion ofIs denoted by b_n，b_n=I_n-1∪c_n(ii) a Then the initial ω_sThe middle subscript belongs to b_nAll elements of (a) constitute the initial ω in sequence_sThe subvector of (2), denoted as ω_s(：,b_n) According toTo obtain omega_s(：,b_n) A value of each element in (a); finally according to omega_s(：,b_n) Updates the initial ω by the value of each element in (1)_sWhere the symbol "U" is a union operation symbol, L^H(：,b_n) Represents L (: b is b_n) Conjugate transpose of, L (: b is b_n) Indicates that the column number in L belongs to b_nAll of the columns of (a) make up a sub-matrix of L,is represented by (L)^H(：,b_n) A pseudo-inverse of); 4, from the updated omega_sSelecting K elements with the maximum absolute value, and forming the subscripts of the selected K elements into an index set of the nth iteration, which is marked as I_nThen the residual error of the nth iteration is calculated and recorded as r_n，r_n=L^-1r_Δ-L^H(：,I_n)ω_s(：,I_n) Wherein L is^H(：,I_n) Represents L (: i, I_n) Conjugate transpose of, L (: i, I_n) Indicates that the column number in L belongs to I_nAll columns of (a) constitute a sub-matrix of L, ω_s(：,I_n) Represents updated ω_sThe middle subscript being of formula I_nAll elements of (a) in sequence constitute an updated ω_sThe subvectors of (1); fourthly-5, reserving updated omega_sMiddle and omega_s(：,I_n) And the value of the element corresponding to each element in (b), and ω to be updated_sThe values of the other elements are set to 0 and are recorded again as omega_s'; fourthly-6, judging whether n is less than or equal to 2K, if n is less than or equal to 2K, comparing r_nIf r is the magnitude of the square of the modulus of (2) and ∈_nOf the dieIf the square value is larger than epsilon, let n = n +1, then return to the step (r-2) to continue execution, carry out the next iteration, if r is larger than epsilon_nIs less than or equal to epsilon, the iterative process is ended and let ω = ω_s'; if n ≦ 2K does not hold, the iterative process ends and let ω = ω_s'; wherein n = n +1 and ω = ω_s"=" in' is an assigned symbol.

2. The method as claimed in claim 1, wherein the transmission signal in step (i) is a baseband complex signal, and the wideband communication channel is a discrete time invariant noisy channel.