AU2012200795A1 - Analog compressed sensing sampling method and system based on random cyclic matrices - Google Patents

Abstract

The present invention relates to an analog compressed sensing sampling method and system based on random cyclic matrices. The analog compressed sensing sampling method based on random cyclic matrices, including: Step 1, determining the processing delay r of a multiplier, a low pass filter with a cut-off frequency of 1/2T and an analog-to-digital converter with a sampling frequency of I/T, in which I/T > B, B being the maximum bandwidth of the sub-bands in input signal; Step 2, delaying the input signal at time point I = to+(i-1)T, in which i =1, 2, ..., m, by (i-1)r time period and inputting it into the multiplier, and multiplying the delayed input signal by the sequence p,(t), in which i =1, 2, ..., m, wherein the sequence p;(t) is the pseudo random sequence generated by an m-sequence generator and circularly shifted in accordance with an integer generated by a random integer generator, and wherein the input signal at time point t after being delayed by (i-l)T time period is the same as that at time to, to being the starting time of sampling, and wherein m ? 3.5Nlog(M)log(NlogM)log 2N, N being the number of the sub-bands in the input signal and M being the length of the pseudo random sequence; and Step 3, inputting the output signal of the multiplier into the low pass filter with the cut-off frequency of 1/2T, and inputting the output signal of the low pass filter with the cut off frequency of 112T into the analog-to-digital converter with the sampling frequency of 1/ T. (Figure 2 for abstract) receiver Figure 1 z,(ti=12,..., m Input x(t) y(n) signal delay LPF Timing switch p(t),i = 1,2,.Rad cyclic ; delayiner Timing switch Pt m-sequence generator Figure 2

Description

AUSTRALIA Patents Act COMPLETE SPECIFICATION (ORIGINAL) Class Int. Class Application Number: Lodged: Complete Specification Lodged: Accepted: Published: Priority Related Art: Name of Applicant: Beijing University of Posts and Telecommunications Actual Inventor(s): Xiaofeng Tao, Qimei Cui, Xianjun Yang, Xiaodong Xu, Ping Zhang Address for Service and Correspondence: PHILLIPS ORMONDE FITZPATRICK Patent and Trade Mark Attorneys 367 Collins Street Melbourne 3000 AUSTRALIA Invention Title: ANALOG COMPRESSED SENSING SAMPLING METHOD AND SYSTEM BASED ON RANDOM CYCLIC MATRICES Our Ref: 935115 POF Code: 507511/515932 The following statement is a full description of this invention, including the best method of performing it known to applicant(s): 6006q Analog Compressed Sensing Sampling Method and System Based on Random Cyclic Matrices This application claims priority from Chinese Application No. 201110457171.3 filed on 30 December 2011, the contents of which are to be taken as incorporated herein by this reference. FIELD OF THE INVENTION This invention relates to analog compressed sensing sampling, especially the analog compressed sensing sampling method based on random cyclic matrices and the system thereof. DESCRIPTION OF THE RELATED ART The compressed sensing technology was first brought forward in 2006 with the fundamental ideology as: if an unknown signal is sparse or compressible on a known orthogonal basis or overcomplete orthogonal basis (such as Fourier transform basis and wavelet basis), it only requires a small amount of linear and non-adaptive measurements to accurately recover the original signal. To facilitate analyzing on problems and make full use of the accumulations of the discrete signal processing domain, compressed sensing was initially designed for discrete signal. Later on, it was generalized to analog domain. In Random Filters for Compressive Sampling and Reconstruction issued in 2006 on the literature for IEEE International Conference on Acoustics, Speech and Signal Processing, the signal for random filter was still supposed to be discrete. In A Nyquist folding analog-to-information receiver issued in 2008 on literature for the 42th Asilomar Coference on Signal, Systems and Computers, the Nyquist folding analogy-to-information receiver applied pre-modulated signal. However, the scope of application is relatively restricted. In the Analog-to-Information Conversion via Random Demodulation issued in 2006 on IEEE DCAS, the proposed analog-to-information converter was for narrowband signal and was not applicable for wideband signal. In From Theory to Practice: Sub-Nyquist sampling of Sparse Wideband Analog Signals issued in 2010 on IEEE Journal of Selected Topics in Signal Processing, the wideband modulation converter was proposed for multiband signal with multiple of parallel branches at a speed lower than Nyquist rate. However, when the signals are very sparse, the number of parallel branches will be increased rapidly, which greatly increases the complexity of the hardware. In conclusion, researches on analog compressed sensing are still in initial stages.

Therefore, it is in urgent need of an analog compressed sensing method on sparse wideband signal with low hardware complexity. SUMMARY OF THE INVENTION Based on the abovementioned problems existing in prior art, this invention provides an analog compressed sensing sampling method based on random cyclic matrices and the system thereof. The present invention provides an analog compressed sensing sampling method based on random cyclic matrices, including: Step 1, determining the processing delay r of a multiplier, a low pass filter with a cut-off frequency of 1/2T and an analog-to-digital converter with a sampling frequency of 1/T, in which I/T B, B being the maximum bandwidth of the sub-bands in input signal; Step 2, delaying the input signal at time point t = to+(i-1)r, in which i =1, 2, ... , m, by (i 1)r time period and inputting it into the multiplier, and multiplying the delayed input signal by the sequence p,(t), in which i =1, 2, ... , m, wherein the sequence p,(t) is the pseudo random sequence generated by an m-sequence generator and circularly shifted in accordance with an integer generated by a random integer generator, and wherein the input signal at time point t after being delayed by (i-1)r time period is the same as that at time to, to being the starting time of sampling, and wherein m > 3.5Nlog(M)log(NlogA4)log 2 N, N being the number of the sub bands in the input signal and M being the length of the pseudo random sequence; and Step 3, inputting the output signal of the multiplier into the low pass filter with the cut off frequency of 1/2T, and inputting the output signal of the low pass filter with the cut-off frequency of 1/2T into the analog-to-digital converter with the sampling frequency of I/T. In an example, wherein in Step 2, the input signal is inputted into L branches in parallel; at time point t = to+(i-1)r, in which i =1, 2, ... , [m/L], the input signal in the le branch, after a delay of (i-1)r time period, is inputted into the multiplier and multiplied by the sequence p, 1 (t) which is the pseudo random sequence generated by the m-sequence generator and circularly shifted in accordance with an integer generated by the random integer generator, in which 1=1, ... , L and i =1, 2, ... , [m/L]; and the cyclic shift number Cl; of the pseudo random sequence in each branch is obtained by serial-parallel converting the integer generated by the same random integer generator, in which 1=1, ... , L and i=1, 2, ..., [m/L]. In an example, wherein in Step 2 the input signal is inputted to L branches in parallel; at time point t = to+(i-l)r, in which i =1, 2, ... , [m/L], the input signal in the th branch, after a delay of (i-l)z time period, is inputted into the multiplier and multiplied by the sequence p,,(t) which is the pseudo random sequence generated by the m-sequence generator and circularly shifted in accordance with an integer generated by the random integer generator, in which 1=1, ... , L and i=1, 2, ..., [m/L]; and the cyclic shift number Cy of the pseudo random sequence in each branch is generated by mutually independent random integer generators, in which 1=1, ..., L and i =1, 2, ..., mI/L], the L m-sequence generators are independent with each other. In an example, wherein the m-sequence generator is replaced by a ZC sequence generator, the length of which is a prime number. In an example, wherein the m-sequence generator is replaced by a sequence generator, whose Fourier transform has unit magnitude. The present invention provides an analog compressed sensing sampling system based on random cyclic matrices, including a first delayer, a multiplier, a low pass filter with a cut-off frequency of 1/2T, an analog-to-digital converter with a sampling frequency of IT, a random integer generator, an m-sequence generator and a second delayer, in which I/T > B, B being the maximum bandwidth of the sub-bands in input signal; the first delayer is provided for delaying the input signal at time point t = to+(i-1)r, in which i =1, 2, ..., m, by (i-l)T time period and inputting it into the multiplier, wherein the input signal at time point t after being delayed by (i-l)T time period is the same as that at time to, to being the starting time of sampling, and wherein m > 3.5Nlog(M)log(NlogM4)log 2 N, N being the number of the sub-bands in the input signal and M being the length of the pseudo random sequence, and wherein T is the processing delay of the multiplier, the low pass filter with the cut-off frequency of 1/2T and the analog-to-digital converter with the sampling frequency of lIT; the second delayer is provided for circularly shifting the pseudo random sequence generated by the m-sequence generator in accordance with an integer generated by the random integer generator; the multiplier is provided for multiplying the output signals of the first delayer and the second delayer; the low pass filter with the cut-off frequency of 112T is provided for filtering the output signal of the multiplier; and the analog-to-digital converter with the sampling frequency of 1/T is provided for sampling the output signal of the low pass filter with the cut-off frequency of 1/2T. In an example, wherein the input signal is inputted to L branches in parallel, each branch including a first delayer, a second delayer, a multiplier, a low pass filter with the cut-off frequency of 112T and an analog-digital converter with the sampling frequency of IT; and the analog compressed sensing sampling system further includes a serial-parallel converter module, for carrying out serial-parallel conversion for the integer generated by the random integer generator and inputting the converted integer into the second delayer, wherein the cyclic shift number Cy of the pseudo random sequence in each branch is generated by the same random integer generators, in which 1=1, ... , L and i =1, 2, ... , [m/L]. In an example, wherein the input signal is inputted to L branches in parallel, each branch including a first delayer, a second delayer, a multiplier, a random integer generator, an m-sequence generator, a low pass filter with the cut-off frequency of 1/2T and an analog-digital converter with the sampling frequency of 1/T; the cyclic shift number Cy of the pseudo random sequence in each branch is generated by mutually independent random integer generators, in which 1, ... , L and i =1, 2, ... , [m/L] and the pseudo random sequence generated by the m-sequence generator in each branch is independent from each other. In an example, wherein the m-sequence generator is replaced by a ZC sequence generator, the length of which is a prime number. In an example, wherein the m-sequence generator is replaced by a sequence generator, whose Fourier transform has unit magnitude. This invention is able to take samples on the sparse analog signals at a speed lower than the Nyquist rate. It is suitable for wideband signals and has lowered the hardware realization complexity and improved the recovery performance of the analog compressed sensing. BRIEF DESCRIPTION OF THE DRAWINGS Further detailed explanations will be made hereinafter in conjunction with the drawings. Figure 1 is the schematic of multiband signal; Figure 2 is the first analog compressed sensing sampling system based on random cyclic matrices; Figure 3 is the second analog compressed sensing sampling system based on random cyclic matrices; Figure 4 is the third analog compressed sensing sampling system based on random cyclic matrices. DETAIL DESCRIPTION OF THE EMBODIMENTS This invention provides a sampling method based on compressed sensing technology for sparse wideband signal, especially for multiband signal x(t) , which is a real-valued continuous signal bandlimited to F = 1/ 2T,,,1/2T,,) F = 1/2T,,,1/2T,,). The multiband signal model M is defined as follows: M contains all signals x(t) , whose frequency support x(t) is contained within a union of N disjoint bands in F . In addition, the bandwidth of each sub-band is not greater than B, and the positions of the sub-band f are random. Schematic drawing of multiband signal as shown in Figure 1, of which N=6 and signals are located at fl, f2, f3, -fl, -f2, -f3 respectively. For example, the multiband signal under ideal circumstance may be expressed as: x(t)= a sinc(B(t -t )) cos(2pf (t-t )) ,wherein, I, is the offset of time. This invention offers three analog compressed sensing sampling methods based on random cyclic matrices, which will be explained respectively hereinaftero The specific steps for the first analog compressed sensing sampling method based on random cyclic matrices include: (1) initialization stage: Suppose the current time is to, input signal x(t) multiplies the pseudo random sequence p(t) generated by m-sequence generator to obtain signal %Kt), signal Vf(t) then passes through the low pass filter (LPF) with the cut-off frequency of 1/2T and low speed analog-to-digital converter (ADC) with a sampling frequency of 1/T (I / T 3

B)

to generate discrete sequence y, (n); (2) Cyclic delay stage: Suppose processing time of the initialization stage is t , then at time t = to + (i- 1)t,i= 2,K ,m, input signal was delayed by(i- 1)t time period,i= 2,K ,m, then the random cyclic shift of the pseudo random sequence p,(t),i= 2,K ,m is multiplied by the signal of x(t) from the time of to + (i- 1)t -(i- l)t =to to obtain a signal Y(t),i= 2,K ,m, signal Y(t) then passes through the low pass filter with the cut-off frequency of 1/2T and low speed ADC with a sampling frequency of I/T( /T 3 B) to generate discrete sequence y,(n),i= 2,K ,m, of which the number of random cyclic shift is C,i= l,...,m which is generated by random integer generator. The value of m should satisfy m 3 3.5N log(M)log(Nlog M)log 2 N, of which N is the number of sub-band in input signal x(t), M is the length of pseudo random sequence. The specific steps for the second analog compressed sensing sampling method based on random cyclic matrices include: (1) Initialization stage: Suppose the current time is to , L parallel branches are applied, at the th branch, input signal x(t) is multiplied by the pseudo random sequence p(t) generated by m-sequence generator to obtain signal Yp(t),l= 1,K L , signal ,(t),l= 1,K L passes through the low pass filter with the cut-off frequency of 1/2T and low speed ADC with a sampling frequency of I/T to generate discrete sequence y, (n),l= 1,K L; (2) Cyclic delay stage: Suppose processing time of the initialization stage is I , at time t= t o+ (i- 1)t ,i= l,2,K ,[m/L], at the th branch, input signal was delayed by(i- 1)t time period, i= 2,K ,[m / L] , then multiplied by the random cyclic shift of pseudo random sequence p 1 (t),l= 1,K L; i= 2,K ,[m/L] to obtain signal Vp(t),l= 1,K L;i= 2,K ,(m/L], signal V(t) passes through low pass filter with the cut-off frequency of 1/2T and low speed ADC with sampling frequency of I/T to obtain discrete sequence y.,(n), / = 1,K L;i = 2,K ,[m/ L], of which the number of cyclic shift of pseudo random sequence for each branch is C, I= 1,...L;i= 1,2,K ,[m / L] which was obtained after serial parallel conversion for the integer generated by the same random integer generator. The specific steps for the third analog compressed sensing sampling method based on random cyclic matrices include: (1) Initialization stage: Suppose the current time is t , a total number of L parallel branches are applied. At the Ith branch, input signal x(t) is multiplied by the pseudo random sequence generated by m-sequence generator p,(t),l = 1,K L to obtain signal Vp,(t),l = 1,K L, signal Vp (t),l = 1,K L passes through low pass filter with the cut-off frequency of 1/2T and low speed ADC with sampling frequency of l/T to obtain discrete sequence y,,(n),l = 1,K L , of which each pseudo random sequence p,(t) was generated by m-sequence generators that are mutually independent, i.e. p,(1) is mutually independent; (2) Cyclic delay stage: Suppose processing time of the initialization stage is t , at t = to+ (i- 1)t ,i= 1,2,K ,[m / L], at the e* branch, input signal was delayed by(i- 1)t time period,i= 1,2,K ,[m /L], then was multiplied by the random cyclic shift of pseudo random sequence p,,(t),1= 1,K L; i= 2,K ,[m / L] to obtain signal Y(t),l= 1,K L; i= 2,K ,[m / L], signal p (t) passes through low pass filter with the cut-off frequency of 1/2T and low speed ADC with the sampling frequency of I/T to generate discrete sequence y,,(n), 1= 1,K L;i= 2,K ,[m/L] , of which the digit of cyclic shift of pseudo random sequence for each branch is C,l= 1,...L;i= 1,2,K ,[m/L] which is generated by random integer generators that are mutually independently. In the following text, more comprehensive descriptions are made on the three analog compressed sensing sampling methods based on random cyclic matrices proposed in this invention and demonstrative examples are given. Application example 1: Based on the sampling system shown in Figure 2, this example gives detailed sampling and signal recovery steps of the single-branch realization method of analog compressed sensing sampling based on random cyclic matrices: D Sampling stage (1) initialization stage: Suppose the current time is to, input signal x(t) multiply pseudo random sequence p(t) generated by m-sequence generator to obtain signal Yi(t). The signal rp(t) then passes through low pass filter with the cut-off frequency of 1/2T and low speed ADC with the sampling frequency of I /T to generate discrete sequence y,(n); (2) Cyclic delay stage: Suppose processing time of the initialization stage is t , at time t= to+ (i- 1)t ,i= 2,K ,m , input signal was delayed by (i- 1)t time period, i= 2,K ,m , then is multiplied by the random cyclic shift of pseudo random sequence p, (t), i = 2,K , m to obtained V$t), i = 2,K , m , signal V(t) passes through low pass filter with the cut-off frequency of 1/2T and low speed ADC with sampling frequency of 1/T to obtain discrete sequence y,(n),i= 2,K ,m, of which the number of random cyclic shift is generated by random integer generator. Time switch in Figure 2 closes once in a time interval of t . E Signal recovery stage (1) Expression of measurement: The sampled data generated by each cyclic shift is y,(n),i= 1,K ,m, and n= 1,K ,N, rewrite the measurement into the mxN matrix Y. Set Q = yyT, all columns of each matrix V meeting Q = VVH may be spanned as span(Y). In addition, as Q is positive semidefinite, therefore, there's always such as matrix V that enables Q = VV". The arithmetic product of the eigenvector of matrix Q and the square root of the eigenvalue may be used as the column vector of matrix V. (2) Expression of measurement matrix: Measurement matrix is made by three steps of multiplication with pseudo random sequence, passing low pass filter and low speed ADC. Specifically, measurement matrix A = SFD, of which S is the m' M random sequence p, (t) with the value of ± 1, m' M of which m is the times of cyclic shift, M is the length of sequence p, (t). Suppose M is an odd number. F is the reordered discrete Fourier matrix of M' M , F=IF(M+I)/ 2 ,K ,- (M+I)/2] , of which F,=[q**,K q(M-')*] , q= e- 2,lM D = diag(d + 2 K d (M +1)/ 2 ) w herein d = MM 0 D(1- q )/2jpm, m 0 (3) Signal recovery algorithm based on compressed sensing: recover the sampled signal by utilizing compressed sensing, which is made by two steps. The first step is to get the nonzero support S of the signal by V = AU , and the second step is to get the specific expression of the signal according to S. Step 1: Use OMP algorithm based on MMV (M-OMP algorithm) to find the nonzero support S under V = AU; Step 2: Based on S, compute Zs [n]= AsY[n], z,[n]= 0, i f S , interpolate signal into z,[n], W= nM (44= , of which n is integer, the discrete sequence of the input signal x(t) 0 ,otherwise is x[n] = * (A{n] * h, [n])e 2 p'" , where h, [n] is the impulse response of low pass filter. is Application example 2: Based on the sampling system shown in Figure 3, this example gives detailed sampling and signal recovery steps of the multi-branch integrated random sequence generator and random integer generator of analog compressed sensing sampling based on random cyclic matrices DSampling stage (1) Initialization stage: Suppose the current time is to , a total number of L parallel branches are applied. At the l1h branch, input signal x(t) is multiplied by the pseudo random sequence p(t) generated by m-sequence generator to obtain signal Y (t), / = 1,K L, signal S(t), I= 1,K L passes through low pass filter with the cut-off frequency of 1/2T and low speed ADC with sampling frequency of l/T to obtain discrete sequence y,, (n),1 = 1,K L; (2) Cyclic delay stage: Suppose processing time of the initialization stage is I , at time t = to+ (i- 1)t ,i= 1,2,K ,[m / L], at the th branch, input signal was delayed by (i- 1)t time period, i = 1, 2,K ,[m / L], then is multiplied by the random cyclic shift of pseudo random sequence p,, (t),l= 1,K L; i= 2,K ,[m / L] to obtain signal p (t), l= 1,K L; i= 2,K ,[m / L], signal Y$(t) passes through low pass filter with the cut-off frequency of 1/2T and low speed ADC with sampling frequency of l/T to obtain discrete sequence y,,(n), / = 1,K L;i = 2,K ,[m / L], of which the number of cyclic shift of pseudo random sequence for each branch is obtained by serial-parallel converting the integer generated by the same random integer generator . Time switch in Figure 3 closes once in a time interval of t . D Signal recovery stage (1) Expression of measurement: sampled data generated at the (h branch and the it' cyclic shift is y, 1 (n), of which 1= 1,K L, i= 1,K ,m/L, i= 1,K ,[m/L], and n= 1,K ,N, rewrite the measuremnt into the mxN matrix Y= [y 11 ,K y 1 ,y 1 ,K ,yL 2 ,K YLIL]T , of which y 1 = [y[1],K yP,[N]]. Set Q = YY , all columns of each matrix V satisfying Q = VV" may be spanned as span(Y). In addition, as Q is positive semidefinite, therefore, there's always such as matrix V that enables Q = VVH . The arithmetic product of the eigenvector of matrix Q and the square root of the eigenvalue may be used as the column vector of matrix V. (2) Expression of measurement matrix: Measurement matrix is made by three steps of multiplication with random sequence, passing low pass filter and low speed ADC. Specifically, measurement matrix A= SFD, of which S corresponds to the T' M random sequence p,(t) with the value of ± 1, m' M m is the multiplication of the number of parallel braches L and the times of cyclic shift for each branch m / L , M is the length of sequence p , ( 1). Suppose M is an odd number. F is the rearranged discrete Fourier matrix of M' M, F =I[F(,+ >,2,K ,. .(M+1)/2] , of which [qo",K q(A- 1)] , q= e- j2pl tl/M ,m=0 D= diag(dt+,)/ 2 ,K d (M+ 1)/2), wherein d,, = 0 t(1- q m )/2jpm, m 0 (3) Signal recovery algorithm based on compressed sensing: recover the sampled signal by utilizing compressed sensing, which is made by two steps. The first step is to get the nonzero support of the signal by V=AU, and the second step is to get the specific expression of the signal according to S. Step 1: Use OMP algorithm based on MMV (M-OMP algorithm) to find the nonzero support Sunder V = AU; Step 2: Based on S, compute Zs[n]= AtsY[n], z,[n]= 0,il S, interpolate the signal into Y1P/= i , of which the discrete sequence of the input signal x(t) is t 0 ,otherwise x[n] 0 ((n]* h,[n])e 2 pf, T , and h, [n] is the impulse response of low pass filter. As Application example 3: Based on the sampling system shown in Figure 4, this example gives detailed sampling and signal recovery steps of the multi-branch integrated independent random sequence generator and random integer generator of analog compressed sensing sampling based on random cyclic matrices 0 Sampling stage (1) Initialization stage: Suppose the current time is to , a total number of L parallel branches are applied. At the ph branch, input signal x(t) is multiplied by the pseudo random sequence generated by m-sequence generator p,(t),1= 1,K L to obtain signal .Vp(t),l= 1,K L, signal Yp(t),l= 1,K L passes low pass filter with the cut-off frequency of 1/2T and low speed ADC with sampling frequency of l/T to obtain discrete sequence y,, (n), I = 1,K L , of which the pseudo random sequence p, (t) is mutually independent; (2) Cyclic delay stage: Suppose processing time of the initialization stage is t , at time t = tD+ (i- 1)t ,i= 1,2,K ,[m / L], at the l branch, input signal was delayed by (i- l)t time period, i = 1, 2,K ,[m / L], then is multiplied by pseudo random sequence after random cyclic shift p,,(t),l = 1,K L; i = 2,K ,[m / L] to obtain signal Vp (), / = 1,K L; i = 2,K ,[m / L] signal .$rt) passes low pass filter with the cut-off frequency of 1/2T and low speed ADC with the sampling frequency of I/T to obtain discrete sequence y,,(n), 1 = 1,K L;i= 2,K ,[m / L], of which the digit of cyclic shift of pseudo random sequence for each branch is generated by random integer generators that are mutually independently. The timing switch in Figure 4 closes once in a time interval o.f t . E Signal recovery stage (1) Expression of measurement: The sampled data generated at the 1 0 branch and i'* cyclic shift is y,,(n) , of which 1= 1,K L , i= 1,K ,[m/L], and n= 1,K ,N, rearite the measurement into the mxN matrix Y= [y 1 1 ,K y, 1 ,y 1 ,K ,y, 2 ,K YL,mIL]' , of which 1; = [yF,[1],K y 1 [N]]. Set Q = yyT, all columns of each matrix V meeting Q= VV"H may be spanned as span(Y). In addition, as Q is positive semidefinite, therefore, there's always such as matrix V that enables Q = VVH . The arithmetic product of the eigenvectors of matrix Q and the square root of the eigenvalues may be used as the columns of matrix V. (2) Expression of measurement matrix: Measurement matrix is made by three steps of multiplication with random sequence p,(t), passing low pass filter and low speed ADC. Specifically, measurement matrix A= SFD, of which S corresponds to the m' M random sequence p,(t) with the value of ± 1, m' M m is the multiplication of the number of parallel braches L and the times of cyclic shift for each branch m / L , M is the length of sequence p,(t). Suppose M is an odd number. F is the reordered discrete Fourier matrix of M' M, F =[F(M+I)/ 2 ,K 3F (A+ >/2] of which P, = [qo*,K ,q(M- IT)"]' q= e- j2pM -(P1412] of w il/M ,Fm= 0 D = diag(d(M+ )/ 2 ,K d (M+) 2 ), wherein d,,,= 1M M0 t(l- qm)/2jpm, m' 0 (3) Signal recovery algorithm based on compressed sensing: recover the sampled signal by utilizing compressed sensing, which is made by two steps. The first step is to get the nonzero support of the signal by V = AU , the second step is to get the specific expression of the signal according to S.

Step 1: Use OMP algorithm based on MMV (M-OMP algorithm) to find the nonzero support Sunder V = AU ; Step 2: Based on S, compute Zs[n] = A'Y[n], z, [n] = 0, i f S, interpolate the signal into ( P z,[n],W nMnI , the discrete sequence of input signal x(t) is tO ,otherwise x[n] ' (Yf(n]* h,[n])e 2 pfn T , of which h,[n] is the impulse response of low pass filter. In the abovementioned examples, ZC(zadoff-chu) sequence generator may be used to replace m-sequence generator, of which the expression of ZC sequence is: x, (n) = e- jpun(n+ 1)/Nzc OE n Nz - 1, of which u is the root for generating ZC sequence, when the length of the ZC sequence is a prime number, its cyclic shift and sequence are mutual orthogonal. Replace m-sequence generator by ZC sequence generator, the measurement matrix meeting the RIP feature in the compressed sensing principle may be obtained by random cyclic shift. Then analog compressed sensing realized by hardware with low complexity may be obtained. In the abovementioned case, the sequence generator whose Fourier transform has unit magnitude may also be used to replace m-sequence generator, because the circular matrix generated by this kind of sequence can be diagonalized by discrete Fourier transform in other words, if matrix B is a cyclic matrix: B = FZ diag(Feb)F" = FZ AFM Where, b is the first column of the cyclic matrix, F, is the discrete Fourier 1 - a oi- 1Xk- 1) transformation matrix of M' M , and F.(i,k)- e . If A A= I B'B= FA'FF AFm = FLA'AF,= 1. Matrix generated by this sequence by random cyclic shift can meet the RIP feature in compressed sensing. The abovementioned contents are just optimal implementation modes of this invention, and the scope of protection under this invention is not restricted accordingly. Any person skilled in the art may make appropriate modifications or changes within the technical scope disclosed by this invent. However, such modifications or changes must be covered within the scope of protection under this invention.

Claims (10)

1. An analog compressed sensing sampling method based on random cyclic matrices, including: Step 1, determining the processing delay r of a multiplier, a low pass filter with a cut-off frequency of 1/2T and an analog-to-digital converter with a sampling frequency of l/T, in which I/T> B, B being the maximum bandwidth of the sub-bands in input signal; Step 2, delaying the input signal at time point I = to+(i-1)r, in which i =1, 2, ... , m, by (i l)r time period and inputting it into the multiplier, and multiplying the delayed input signal by the sequence p,(t), in which i =1, 2, ... , m, wherein the sequence pi(t) is the pseudo random sequence generated by an m-sequence generator and circularly shifted in accordance with an integer generated by a random integer generator, and wherein the input signal at time point t after being delayed by (i-i )r time period is the same as that at time to, to being the starting time of sampling, and wherein m >_ 3.5Nlog(M)log(NlogM)log 2 N, N being the number of the sub bands in the input signal and M being the length of the pseudo random sequence; and Step 3, inputting the output signal of the multiplier into the low pass filter with the cut-off frequency of 1/2T, and inputting the output signal of the low pass filter with the cut-off frequency of 1/2T into the analog-to-digital converter with the sampling frequency of I/T.

2. The analog compressed sensing sampling method according to claim I, wherein in Step 2, the input signal is inputted into L branches in parallel; at time point t = to+(i-l)r, in which i =1, 2, ... , [m/L], the input signal in the le branch, after a delay of(i-1)r time period, is inputted into the multiplier and multiplied by the sequence pl.i(t) which is the pseudo random sequence generated by the m-sequence generator and circularly shifted in accordance with an integer generated by the random integer generator, in which 1=1, ... , L and i=1, 2, ... , [m/L]; and the cyclic shift number C; of the pseudo random sequence in each branch is obtained by serial-parallel converting the integer generated by the same random integer generator, in which 1=1, ... , L and i =1, 2, ... , [m/L].

3. The analog compressed sensing sampling method according to claim 1, wherein in Step 2 the input signal is inputted to L branches in parallel; at time point t = to+(i-l)r, in which i =1, 2, ... , [m/L], the input signal in the th branch, 14 after a delay of (i-1)r time period, is inputted into the multiplier and multiplied by the sequence pli(t) which is the pseudo random sequence generated by the m-sequence generator and circularly shifted in accordance with an integer generated by the random integer generator, in which l=1, ... , L and i=1, 2, ... , [m/L]; and the cyclic shift number C,; of the pseudo random sequence in each branch is generated by mutually independent random integer generators, in which =1, ..., L and i =1, 2, ..., [m/L], the L m-sequence generators are independent with each other.

4. The analog compressed sensing sampling method according to claim 1, 2 or 3, wherein the m-sequence generator is replaced by a ZC sequence generator, the length of which is a prime number.

5. The analog compressed sensing sampling method according to claim 1, 2 or 3, wherein the m-sequence generator is replaced by a sequence generator, whose Fourier transform has unit magnitude.

6. An analog compressed sensing sampling system based on random cyclic matrices, including a first delayer, a multiplier, a low pass filter with a cut-off frequency of 1/2T, an analog-to-digital converter with a sampling frequency of 1/T, a random integer generator, an m-sequence generator and a second delayer, in which l/T > B, B being the maximum bandwidth of the sub-bands in input signal; the first delayer is provided for delaying the input signal at time point t = to+(i-l)r, in which i =1, 2, ..., m, by (i-l)T time period and inputting it into the multiplier, wherein the input signal at time point I after being delayed by (i-I)T time period is the same as that at time to, to being the starting time of sampling, and wherein m > 3.5Mog(M)log(NlogAM)log 2 N, N being the number of the sub-bands in the input signal and M being the length of the pseudo random sequence, and wherein r is the processing delay of the multiplier, the low pass filter with the cut-off frequency of 1/2T and the analog-to-digital converter with the sampling frequency of 1/T; the second delayer is provided for circularly shifting the pseudo random sequence generated by the m-sequence generator in accordance with an integer generated by the random integer generator; the multiplier is provided for multiplying the output signals of the first delayer and the second delayer; 15 the low pass filter with the cut-off frequency of 112T is provided for filtering the output signal of the multiplier; and the analog-to-digital converter with the sampling frequency of l/T is provided for sampling the output signal of the low pass filter with the cut-off frequency of 1/2T.

7. The analog compressed sensing sampling system according to claim 6, wherein the input signal is inputted to L branches in parallel, each branch including a first delayer, a second delayer, a multiplier, a low pass filter with the cut-off frequency of 1/2T and an analog-digital converter with the sampling frequency of 1/T; and the analog compressed sensing sampling system further includes a serial-parallel converter module, for carrying out serial-parallel conversion for the integer generated by the random integer generator and inputting the converted integer into the second delayer, wherein the cyclic shift number Cy of the pseudo random sequence in each branch is generated by the same random integer generators, in which 1=1, ... , L and i =1, 2, ... , [m/L].

8. The analog compressed sensing sampling system according to claim 6, wherein the input signal is inputted to L branches in parallel, each branch including a first delayer, a second delayer, a multiplier, a random integer generator, an m-sequence generator, a low pass filter with the cut-off frequency of 1/2T and an analog-digital converter with the sampling frequency of 1/T; the cyclic shift number Cy of the pseudo random sequence in each branch is generated by mutually independent random integer generators, in which 1=1, ... , L and i =1, 2, ... , [m/L] and the pseudo random sequence generated by the m-sequence generator in each branch is independent from each other.

9. The analog compressed sensing sampling system according to claim 6, wherein the m sequence generator is replaced by a ZC sequence generator, the length of which is a prime number.

10. The analog compressed sensing sampling system according to claim 6, wherein the m sequence generator is replaced by a sequence generator, whose Fourier transform has unit magnitude. 16