CN111901058B - Multi-antenna auxiliary broadband spectrum sensing method based on sub-nyquist sampling - Google Patents

Multi-antenna auxiliary broadband spectrum sensing method based on sub-nyquist sampling Download PDF

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CN111901058B
CN111901058B CN202010756759.8A CN202010756759A CN111901058B CN 111901058 B CN111901058 B CN 111901058B CN 202010756759 A CN202010756759 A CN 202010756759A CN 111901058 B CN111901058 B CN 111901058B
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matrix
antenna
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CN111901058A (en
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方俊
王集林
段惠萍
李晴
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University of Electronic Science and Technology of China
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    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04B—TRANSMISSION
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Abstract

The invention belongs to the technical field of wireless communication, and particularly relates to a multi-antenna auxiliary broadband spectrum sensing method based on sub-nyquist sampling. The invention realizes broadband spectrum sensing based on the sub-Nyquist sampling theory, and the multi-antenna auxiliary spectrum sensing method provided by the invention can overcome the sampling bottleneck problem existing in Nyquist sampling of the ultra-wideband spectrum, thereby reconstructing the power spectrum of a signal sent by a master user (PU) in a multi-antenna Secondary User (SU), and enabling the power spectrum to identify the frequency positions of a plurality of single-antenna master users (PU) distributed on a broadband. Experiments show that the invention enables the proposed architecture to have obvious performance advantages by multi-antenna assisted spectrum sensing, and can realize ultra-wideband spectrum sensing with lower hardware complexity in practice.

Description

Multi-antenna auxiliary broadband spectrum sensing method based on sub-nyquist sampling
Technical Field
The invention belongs to the technical field of wireless communication, and particularly relates to a multi-antenna auxiliary broadband spectrum sensing method based on sub-nyquist sampling.
Background
Cognitive radio is an effective means for improving the utilization rate of frequency spectrum and relieving the exhaustion of frequency spectrum resources. Since the present, the study of the present inventors has been conducted extensively and intensively. In cognitive radio, authorized users of a frequency band, called Primary Users (PUs), have priority communication rights. An unauthorized User, called a Secondary User (SU), establishes a communication link that does not interfere with normal communications to the primary User. In the implementation of the cognitive radio system, a radio spectrum needs to be continuously sensed, a current communication environment is analyzed by learning various parameters (each channel occupation condition, interference intensity, noise level, master user power intensity) of a current radio channel, and then system related parameters (transmission frequency, transmission power and the like) are adjusted to establish a wireless communication link in a current optimal mode.
The method is an important research topic in the field of cognitive radio, and is used for efficiently, quickly and reliably monitoring the frequency spectrum in a broadband range. The analysis of the spectrum of the Signal by the spectrum monitoring equipment is based on Digital Signal processing, the monitoring equipment receives a broadband analog Signal, converts the broadband analog Signal into a Digital Signal by an analog-to-Digital converter (ADC), and then sends the Digital Signal to a spectrum detection algorithm of a Digital Signal Processing (DSP) module for analysis, and finally obtains the spectrum distribution. Due to nyquist sampling theorem, a sampled signal can be completely restored to the original signal when the sampling frequency is greater than 2 times the highest frequency in the signal. In a wide frequency band, if nyquist sampling is directly performed on a wide frequency spectrum, an ADC device needs a sampling rate of several GHz or even more than ten GHz, which is a great challenge for the existing ADC device. Therefore, ultra-wideband signal sampling is an important challenge that needs to be overcome for wideband spectrum sensing.
Disclosure of Invention
The invention aims to realize broadband spectrum sensing by utilizing a sub-Nyquist sampling theory, reconstruct the power spectrum of a signal sent by a Primary User (PU) in a multi-antenna Secondary User (SU) and enable the power spectrum to identify the frequency position of a plurality of single-antenna Primary User (PU) transmissions distributed on a broadband. Specifically, the invention overcomes the sampling rate bottleneck through a new sub-nyquist sampling architecture and a compression power spectrum estimation method, reconstructs the power spectrum of the signal transmitted from the PU and obviously reduces the required ultra-wideband sampling rate.
The technical scheme of the invention is as follows:
the design of a sub-Nyquist sampling architecture and a multi-antenna auxiliary broadband spectrum sensing method in a cognitive radio system, wherein the number of antennas configured by a Secondary User (SU) in the system is M, the number of Primary Users (PU) to be identified is K, and M > K is met, the method comprises the following steps:
s1: the received signal X of the secondary user antenna is represented as
X＝HS+W
Wherein the content of the first and second substances,
Figure BDA0002611829570000021
representing the transmission channel matrix from K primary users to M secondary users,
Figure BDA0002611829570000022
indicating that the matrix elements belong to the complex field,
Figure BDA0002611829570000023
representing K modulated complex signal vectors,
Figure BDA0002611829570000024
representing an additive white Gaussian noise vector, and describing a channel matrix H by using a Kronecker model, then
Figure BDA0002611829570000025
Wherein the content of the first and second substances,
Figure BDA0002611829570000026
respectively representing the correlation matrix of the transmission and the reception,
Figure BDA0002611829570000027
is composed of independent elements which are subjected to complex Gaussian distribution with the mean value of 0 and the variance of 1, and transmits a correlation matrix R due to the independence among different main userstIs a diagonal matrix, the diagonal term of which corresponds to the transmission capability of the primary user, assuming that both transmit and receive correlation matrices are known a priori;
s2: sampling a received signal based on a sub-nyquist sampling method, comprising:
adding a delay tau after the receiving antenna of the mth secondary usermComprises the following steps:
τm＝cmT
wherein, cmIs not less than 0 and is integer, T represents Nyquist sampling interval, delayed signal is sampled by synchronous ADC, its sampling interval is set to TsNT, wherein N>0 and is an integer, representing a sub-sampling factor，cm<N；
The m antenna receives the signal as xm(t) let xm[n]Representing x by Nyquist sampling interval Tm(t) sampling, i.e. xm[n]＝xm(nT), N sample point vector of m-th antenna receiving signal at time l
Figure BDA0002611829570000028
Expressed as:
Figure BDA0002611829570000029
total sampling vector of L time points of m-th antenna receiving signal
Figure BDA0002611829570000031
Expressed as:
Figure BDA0002611829570000032
wherein
Figure BDA0002611829570000033
L is a parameter chosen to satisfy the desired resolution of the restored power spectrum, and, therefore,
Figure BDA0002611829570000034
Figure BDA0002611829570000035
denotes the transpose of the mth row of the channel matrix H, S [ l ]]Indicating N-point sampling of the transmitted signal at time L, and a sampling vector at time L
Figure BDA0002611829570000036
Is shown as
Figure BDA0002611829570000037
Wherein the content of the first and second substances,
Figure BDA0002611829570000038
let m receiving antenna at time l delay N point sampling data y of signalm[l]Expressed as:
Figure BDA0002611829570000039
wherein
Figure BDA00026118295700000310
Figure BDA00026118295700000311
Indicating that its constituent elements belong to the real number domain, except for (c)m+1) positions are all 0 except 1, and emExpanded as a matrix
Figure BDA00026118295700000312
Make it divided on the diagonal (c)m+1) position is other than 1, and the rest positions are all 0, and then y ism[l]Is shown as
Figure BDA00026118295700000313
Total sampling vector of signals delayed by m-th receiving antenna at L moments
Figure BDA00026118295700000314
Is shown as
Figure BDA00026118295700000315
Wherein
Figure BDA00026118295700000316
Order to
Figure BDA00026118295700000317
Is divided by the diagonal (c)m+1+ pN) position is other than 1, and the remaining positions are all 0, p is 0, …, L-1;
s3: first, the m-th1And m2Root antenna straightReceiving samples of a received signal
Figure BDA00026118295700000318
And
Figure BDA00026118295700000319
cross correlation of
Figure BDA00026118295700000320
Expressed as:
Figure BDA00026118295700000321
then, the m-th1And m2Sampling of root antenna delayed output signals
Figure BDA00026118295700000322
And
Figure BDA00026118295700000323
is expressed as:
Figure BDA00026118295700000324
in the above formula, E [, ]]The representation is defined by mathematical expectation
Figure BDA00026118295700000325
Definition of
Figure BDA00026118295700000326
In the mth row of the matrix Q, then
Figure BDA00026118295700000327
Calculate out
Figure BDA00026118295700000328
Definition of
Figure BDA00026118295700000329
Figure BDA00026118295700000330
Represents the (n) th thereof1,n2) Element of (1), define gnIs a matrix GTThe (c) th column of (a),
Figure BDA00026118295700000423
expressed as:
Figure BDA0002611829570000041
wherein the content of the first and second substances,
Figure BDA0002611829570000042
thus it can calculate
Figure BDA0002611829570000043
Calculate out
Figure BDA0002611829570000044
Order to
Figure BDA0002611829570000045
Thus, it is possible to provide
Figure BDA0002611829570000046
Expressed as:
Figure BDA0002611829570000047
wherein the content of the first and second substances,
Figure BDA0002611829570000048
is a matrix
Figure BDA0002611829570000049
Represents the sample vector for L instants of the kth signal,
Figure BDA00026118295700000410
a correlation matrix representing a kth signal vector;
definition of
Figure BDA00026118295700000411
Then the process of the first step is carried out,
Figure BDA00026118295700000412
Figure BDA00026118295700000413
Figure BDA00026118295700000414
is that
Figure BDA00026118295700000415
Is determined by the partial observation matrix of (a),
Figure BDA00026118295700000416
wherein the (i, j) th element is
Figure BDA00026118295700000417
When i ═ cm1+1+p1N，j＝cm2+1+p2Element of N, p1,p 20, …, L-1, since the transmission signal is broadly smooth, the transmission signal is not limited to the above-mentioned signals
Figure BDA00026118295700000418
Having the structure Toeplitz, option cmCan satisfy suitable conditions from
Figure BDA00026118295700000419
In and out
Figure BDA00026118295700000420
S4: in order to reduce the computational complexity, based on
Figure BDA00026118295700000421
The sub-matrix T, the power spectra of the K transmission signals are recovered, thereby identifying the frequency location,
Figure BDA00026118295700000422
the invention has the beneficial effects that: the broadband spectrum sensing method has low calculation complexity, and the frequency required by ultra-wideband sampling is greatly reduced through the design of multi-antenna assistance and a sub-Nyquist sampling architecture.
Drawings
FIG. 1 is a multi-coset sampling architecture based on multiple antennas according to the present invention, where the m-th receiving antenna is followed by a value τmPerforming Nyquist sampling on the delayed signals;
fig. 2 shows the power spectrum of a 30ms noisy signal reconstructed by using the sub-nyquist sampling architecture and the wideband spectrum sensing method of the present invention, compared with the power spectrum reconstructed by using the noise-free nyquist sampling method, where SNR is-5 dB, the wideband frequency range is [0,100] MHz, the bandwidths of 5 transmission signals are all 1MHz, and the normalized frequencies of the carrier frequencies are 0.13, 0.31, 0.47, 0.67, and 0.78, respectively.
FIG. 3 is a ROC curve of the method of the present invention at different SNR, with pseudo-accuracy (FPR) on the abscissa and real-accuracy (TPR) on the ordinate.
Fig. 4 is a comparison of the power spectrum of a 30ms noisy signal reconstructed using the sub-nyquist sampling architecture and the wideband spectrum sensing method of the present invention versus the power spectrum reconstructed using the noise-free nyquist sampling method. The SNR is 5dB, the broadband frequency range is [0,100] MHz, the bandwidths of 10 transmission narrow-band signals are all 5MHz, the spectrum occupancy rate is 50%, and the center frequencies of the narrow-band signals are uniformly distributed and do not overlap with each other.
FIG. 5 is a ROC curve based on FIG. 4 at different SNR.
Detailed Description
The invention is described in detail below with reference to the drawings and simulation examples to prove the applicability of the invention.
The invention considers the problem of realizing broadband spectrum sensing in a cognitive radio system by utilizing the sub-Nyquist sampling theory, the number of antennas configured by a Secondary User (SU) in the system is M, the number of Primary Users (PU) to be identified is K, and M & gt K is satisfied. The channel matrix H is described using the Kronecker model, and assuming that the transmit and receive correlation matrices of the channel are known, the received signal X ═ HS + W can be expressed as
Figure BDA0002611829570000051
Wherein the content of the first and second substances,
Figure BDA0002611829570000052
represents the transmission channel matrix from K Primary Users (PU) to Secondary Users (SU),
Figure BDA0002611829570000053
representing K modulated complex signal vectors,
Figure BDA0002611829570000054
representing an additive white gaussian noise vector. Wherein the content of the first and second substances,
Figure BDA0002611829570000055
representing the correlation matrix for transmission and reception, respectively.
Figure BDA0002611829570000061
Consisting of elements that are independent and obey a complex gaussian distribution with a mean of 0 and a variance of 1. Transmission correlation matrix R due to independence between different Primary Users (PUs)tIs a diagonal matrix whose diagonal entries correspond to the transmission capabilities of the Primary User (PU).
In order to solve the sampling bottleneck brought by directly carrying out Nyquist sampling on an ultra-wideband signal, the invention provides a sub-Nyquist sampling architecture, and the delay set after an mth receiving antenna of a Secondary User (SU) is as follows:
τm＝cmT
wherein, cmIs more than or equal to 0 and is an integer, and T represents the Nyquist sampling interval. The delayed signal is sampled by a synchronous ADC with a sampling interval set to TsNT, wherein N>0 and is an integer, represents a sub-sampling factor, and is such that cm<N。
To facilitate the representation of the delayed sample signal, let xm[m]＝xmVector of N sample points at time (nT, l)
Figure BDA0002611829570000062
Expressed as:
Figure BDA0002611829570000063
sampling vector of L time instants
Figure BDA0002611829570000064
Expressed as:
Figure BDA0002611829570000065
wherein
Figure BDA0002611829570000066
L is a parameter selected to meet the desired resolution of the restored power spectrum. Therefore, the temperature of the molten metal is controlled,
Figure BDA0002611829570000067
Figure BDA0002611829570000068
representing the transpose of the mth row of the channel matrix H. S [ l ]]Indicating that N points of the transmitted signal are sampled at time l. Then the sampling vector of the L-time-total
Figure BDA0002611829570000069
Can be expressed as
Figure BDA00026118295700000610
Wherein the content of the first and second substances,
Figure BDA00026118295700000611
the delayed sampling signal is described next, and the N-point sampling data y of the delayed signal of the mth receiving antenna at the time l is madem[l]Expressed as:
Figure BDA00026118295700000612
wherein
Figure BDA00026118295700000613
Except that (c)mAnd +1) the positions are all 0 except 1. E is to bemExpanded as a matrix
Figure BDA00026118295700000614
Make it divided on the diagonal (c)m+1) position is 1, and the rest positions are 0. Will ym[l]Is shown as
Figure BDA00026118295700000615
Will be provided with
Figure BDA00026118295700000616
Is shown as
Figure BDA0002611829570000071
Wherein
Figure BDA0002611829570000072
Order to
Figure BDA0002611829570000073
Is divided by the diagonal (c)m+1+ pN) position is other than 1, and the remaining positions are all 0, p ═ 0, …, L-1.
To facilitate the representation of the delayed signal correlation matrix, a calculation is first made
Figure BDA0002611829570000074
Expressed as:
Figure BDA0002611829570000075
can be derived from the properties of the Kronecker product
Figure BDA0002611829570000076
Then calculating two sensor outputs
Figure BDA0002611829570000077
And
Figure BDA0002611829570000078
is expressed as:
Figure BDA0002611829570000079
next, define
Figure BDA00026118295700000710
Definition of
Figure BDA00026118295700000711
In the mth row of the matrix Q, then
Figure BDA00026118295700000712
Calculate out
Figure BDA00026118295700000713
For convenience of presentation, definitions
Figure BDA00026118295700000714
Figure BDA00026118295700000715
Represents the (n) th thereof1,n2) Element of (1), define gnIs a matrix GTThe (c) th column of (a),
Figure BDA00026118295700000716
expressed as:
Figure BDA00026118295700000717
wherein
Figure BDA00026118295700000718
Thus it can calculate
Figure BDA00026118295700000719
Due to the matrix
Figure BDA00026118295700000720
Is composed of independent elements with a complex Gaussian distribution with a mean value of 0 and a variance of 1, and can be calculated
Figure BDA00026118295700000721
Due to the fact that
Figure BDA00026118295700000722
Thus, it is possible to provide
Figure BDA00026118295700000723
Can be expressed as:
Figure BDA0002611829570000081
wherein the content of the first and second substances,
Figure BDA0002611829570000082
is a matrix
Figure BDA0002611829570000083
Represents the sample vector for L instants of the kth signal,
Figure BDA0002611829570000084
representing the kth signal vectorA correlation matrix.
For convenience of presentation, definitions
Figure BDA0002611829570000085
Then the process of the first step is carried out,
Figure BDA0002611829570000086
Figure BDA0002611829570000087
so far, the signal correlation matrix after the time delay is known
Figure BDA0002611829570000088
Is that
Figure BDA0002611829570000089
Is determined by the partial observation matrix of (a),
Figure BDA00026118295700000810
wherein the (i, j) th element is
Figure BDA00026118295700000811
When i ═ cm1+1+p1N，j＝cm2+1+p2Element of N, p1,p 20, …, L-1. Obviously, it is possible to select from a matrix
Figure BDA00026118295700000812
The power spectrums of K transmission signals are recovered, and the frequency point positions of the K transmission signals are identified according to the power spectrums.
Since the transmission signal is broadly stationary, it is possible to reduce the noise
Figure BDA00026118295700000813
Having a Toeplitz structure, selected
Figure BDA00026118295700000814
Satisfying the following conditions C1 and C2
Figure BDA00026118295700000815
In and out
Figure BDA00026118295700000816
C1：
Figure BDA00026118295700000817
C2：
Figure BDA00026118295700000818
Finding the minimum number of M so that the condition C2 can be satisfied is a minimum sparse scale problem, which has been well studied.
In order to reduce the complexity of the hardware, based on
Figure BDA00026118295700000819
Recovers the power spectra of the K transmission signals and thereby identifies the frequency location.
Figure BDA00026118295700000820
Representation matrix
Figure BDA00026118295700000821
The first delta row and the first delta column. To obtain the T matrix, the conditions C3 and C4 are satisfied.
C3：
Figure BDA00026118295700000822
C4：
Figure BDA0002611829570000091
Wherein
Figure BDA0002611829570000092
Indicating rounding down on x.
Obviously, for a given sub-sampling factor N, the constraint of C4 is that a smaller number of M's are required to recover the T matrix. The power spectrum of the K signals can then be determined by the autocorrelation sequence
Figure BDA0002611829570000093
And DFT is carried out to obtain. Wherein the content of the first and second substances,
Figure BDA0002611829570000094
the first row of the matrix T.
In the simulation, the number of Secondary User (SU) configuration antennas in the system is set to be M-6, and the delay factor { c }mIs set to {0,1,2,6,10,13 }. The number of Primary Users (PU) to be identified is K-5, the bandwidths of the transmission signals are all set to be 1MHz, the carrier frequencies are 13, 31, 47, 67 and 78MHz respectively, and the broadband spectrum range is [0,100]]And (4) MHz. Setting the sampling frequency of each delayed signal of each receiving antenna to be 8MHz, so that the sub-sampling factor N is T/Ts200/8-25. The spectral resolution is set to 25kHz and therefore the required sample length is 8000. The number of sampling instants L is 8000/N is 320. Acquiring 30ms signals to calculate a correlation matrix
Figure BDA0002611829570000095
The signal-to-noise ratio SNR is defined as
Figure BDA0002611829570000096
Wherein, { s [ n ]]Denotes the Nyquist sampling of s (t), NtThe number of sampling points. Sigma2Is the variance of gaussian noise.
FIG. 1 depicts the inventive multi-antenna based multi-coset sampling architecture with a size τ set after the mth receiving antennamIs delayed by T on the delayed signalsNyquist sampling for the sampling interval, thereby obtaining a digital signal sequence.
Fig. 2 is a comparison of the power spectrum of a 30ms noisy signal reconstructed using the inventive sub-nyquist sampling architecture and broadband spectral sensing method versus the power spectrum reconstructed using the noise-free nyquist sampling method. The SNR is-5 dB, the wideband frequency range is [0,100] MHz, the bandwidths of the 5 transmission signals are all 1MHz, and the normalized frequencies of the carrier frequencies are 0.13, 0.31, 0.47, 0.67, and 0.78, respectively. It is clear that the power spectrum of the transmitted signal is reconstructed almost completely correctly with the architecture of the invention and the proposed method.
FIG. 3 is a ROC curve for the invented method at different SNR, with pseudo-accuracy (FPR) on the abscissa and real-accuracy (TPR) on the ordinate. On the basis of the reconstructed power spectrum, signal frequency points on a broadband frequency domain are identified by setting a threshold, and the positions of the energy level which is obviously higher than that of other grid points are considered to be correct, so that a group of (FPR, TPR) can be calculated, the power spectrum is reconstructed under different signal-to-noise ratios, and different ROC curves are drawn. It can be seen that the inventive architecture and method can still achieve reliable frequency point detection when the SNR is-5 dB.
Fig. 4 is a comparison of the power spectrum of a 30ms noisy signal reconstructed using the inventive sub-nyquist sampling architecture and broadband spectral sensing method versus the power spectrum reconstructed using the noise-free nyquist sampling method. The SNR is 5dB, the wideband frequency range is [0,100] MHz, the bandwidths of 10 transmission narrowband signals are all 5MHz, the spectrum occupancy rate is 50% to represent non-sparse spectrum, and the narrowband signal center frequencies are uniformly distributed and thus do not overlap with each other. It can be found that for non-sparse spectrum, the architecture and the proposed method of the present invention can still achieve reliable spectrum sensing.
FIG. 5 is a ROC curve based on FIG. 4 at different SNR.
In summary, the sub-nyquist sampling architecture and the multi-antenna auxiliary spectrum sensing method provided by the invention can overcome the sampling bottleneck problem existing in nyquist sampling of an ultra-wide band spectrum, and realize reliable power spectrum reconstruction and frequency point detection successfully under the condition that the SNR (signal-to-noise ratio) is-5 dB. In addition, the framework provided by the invention does not depend on the sparsity of a broadband spectrum, and experiments show that under the condition of 50% of spectrum occupancy rate, the framework provided by the invention can still realize reliable power spectrum reconstruction and frequency point detection under the condition that the acceptable signal-to-noise ratio (SNR) is 5dB, so that the framework and the method provided by the invention have obvious performance advantages, and can realize ultra-wideband spectrum sensing with lower hardware complexity in practice.

Claims (1)

1. A multi-antenna auxiliary broadband spectrum sensing method based on sub-Nyquist sampling is characterized in that the number of antennas configured by a secondary user in a system is M, the number of main users to be identified is K, and M > K is met, and the method comprises the following steps:
s1: the received signal X of the secondary user antenna is represented as
X＝HS+W
Wherein the content of the first and second substances,
Figure FDA0002996552580000011
representing the transmission channel matrix from K primary users to M secondary users,
Figure FDA0002996552580000012
indicating that the matrix elements belong to the complex field,
Figure FDA0002996552580000013
representing K modulated complex signal vectors,
Figure FDA0002996552580000014
representing an additive white Gaussian noise vector, and describing a channel matrix H by using a Kronecker model, then
Figure FDA0002996552580000015
Wherein the content of the first and second substances,
Figure FDA0002996552580000016
respectively representing the correlation matrix of the transmission and the reception,
Figure FDA0002996552580000017
is composed of independent elements which are subjected to complex Gaussian distribution with the mean value of 0 and the variance of 1, and transmits a correlation matrix R due to the independence among different main userstIs a diagonal matrixThe diagonal term of which corresponds to the transmission capability of the primary user, assuming that both the transmit and receive correlation matrices are known a priori;
s2: sampling a received signal based on a sub-nyquist sampling method, comprising:
adding a delay tau after the receiving antenna of the mth secondary usermComprises the following steps:
τm＝cmT
wherein, cmIs not less than 0 and is integer, T represents Nyquist sampling interval, delayed signal is sampled by synchronous ADC, its sampling interval is set to TsNT, wherein N>0 and is an integer, representing a sub-sampling factor, cm<N；
The m antenna receives the signal as xm(t) let xm[n]Representing x by Nyquist sampling interval Tm(t) sampling, i.e. xm[n]＝xm(nT), N sample point vector of m-th antenna receiving signal at time l
Figure FDA0002996552580000018
Expressed as:
Figure FDA0002996552580000019
total sampling vector of L time points of m-th antenna receiving signal
Figure FDA00029965525800000110
Expressed as:
Figure FDA00029965525800000111
wherein
Figure FDA0002996552580000021
L is a parameter chosen to satisfy the desired resolution of the restored power spectrum, and, therefore,
Figure FDA0002996552580000022
Figure FDA0002996552580000023
denotes the transpose of the mth row of the channel matrix H, S [ l ]]Indicating N-point sampling of the transmitted signal at time L, and a sampling vector at time L
Figure FDA0002996552580000024
Is shown as
Figure FDA0002996552580000025
Wherein the content of the first and second substances,
Figure FDA0002996552580000026
sampling data y of N points of the signals after delaying the m-th receiving antenna at the moment lm[l]Expressed as:
Figure FDA0002996552580000027
wherein
Figure FDA0002996552580000028
Figure FDA0002996552580000029
Meaning that its constituent elements belong to the real number domain, divided by (c)m+1) positions are all 0 except 1, and emExpanded as a matrix
Figure FDA00029965525800000210
Make it divided on the diagonal (c)m+1) position is other than 1, and the rest positions are all 0, and then y ism[l]Is shown as
Figure FDA00029965525800000211
Total sampling vector of signals delayed by m-th receiving antenna at L moments
Figure FDA00029965525800000212
Is shown as
Figure FDA00029965525800000213
Wherein
Figure FDA00029965525800000214
Order to
Figure FDA00029965525800000215
Is divided by the diagonal (c)m+1+ pN) position is other than 1, and the remaining positions are all 0, p is 0, …, L-1;
s3: first, the m-th1And m2Sampling of signals received directly by root antennas
Figure FDA00029965525800000216
And
Figure FDA00029965525800000217
cross correlation of
Figure FDA00029965525800000218
Expressed as:
Figure FDA00029965525800000219
then, the m-th1And m2Sampling of root antenna delayed output signals
Figure FDA00029965525800000220
And
Figure FDA00029965525800000221
is expressed as:
Figure FDA00029965525800000222
in the above formula, E [, ]]The representation is defined by mathematical expectation
Figure FDA00029965525800000223
Definition of
Figure FDA00029965525800000224
In the mth row of the matrix Q, then
Figure FDA00029965525800000225
Calculate out
Figure FDA00029965525800000226
Definition of
Figure FDA00029965525800000227
Figure FDA00029965525800000228
Denotes the (n) th1,n2) Element of (1), define gnIs a matrix GTThe (c) th column of (a),
Figure FDA00029965525800000229
expressed as:
Figure FDA0002996552580000031
wherein the content of the first and second substances,
Figure FDA0002996552580000032
calculate out
Figure FDA0002996552580000033
Calculate out
Figure FDA0002996552580000034
Order to
Figure FDA0002996552580000035
Thus, it is possible to provide
Figure FDA0002996552580000036
Expressed as:
Figure FDA0002996552580000037
wherein the content of the first and second substances,
Figure FDA0002996552580000038
is a matrix
Figure FDA0002996552580000039
Represents the sampling vector of L time instants of the kth transmitted signal,
Figure FDA00029965525800000310
a correlation matrix representing a kth signal vector;
definition of
Figure FDA00029965525800000311
Then the process of the first step is carried out,
Figure FDA00029965525800000312
Figure FDA00029965525800000313
Figure FDA00029965525800000314
is that
Figure FDA00029965525800000315
Is determined by the partial observation matrix of (a),
Figure FDA00029965525800000316
wherein the (i, j) th element is
Figure FDA00029965525800000317
Zhongdang (Chinese character)
Figure FDA00029965525800000318
Figure FDA00029965525800000319
Element of (i) p1,p20, …, L-1, since the transmission signal is broadly smooth, the transmission signal is not limited to the above-mentioned signals
Figure FDA00029965525800000320
Having the structure Toeplitz, according to cmValue is taken from
Figure FDA00029965525800000321
In and out
Figure FDA00029965525800000322
S4: based on
Figure FDA00029965525800000323
Recovers the power spectra of the K transmission signals, thereby identifying the frequency locations,
Figure FDA00029965525800000324
representation matrix
Figure FDA00029965525800000325
The first delta row and the first delta column.
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