Multiple Sinusoid Signal Sub-Nyquist Sampling Method Based on Multi-channel Time Delay Sampling System

Abstract

The disclosure discloses a multiple sinusoid signal sub-Nyquist sampling method based on a multi-channel time delay sampling system. The method includes step 1: initializing; step 2: enabling multiple sinusoid signals x(t) to respectively enter N′ parallel sampling channels after the multiple sinusoid signals are divided, wherein a sampling time delay of adjacent channels is τ, and the number of sampling points of each channel is N; step 3: combining sampled data of each sampling channel to construct an autocorrelation matrix R.sub.xx, and estimating sampling signal parameters c.sub.m of each channel and a set of frequency parameters {circumflex over (f)}.sub.m by utilizing the ESPRIT method; step 4: estimating signal amplitudes α.sub.m and another set of frequency parameters f.sub.m.sup.′ through the estimated parameters c.sub.m and the sampling time delay τ of each channel by utilizing the ESPRIT method; and step 5: reconstructing 2K frequency parameters {circumflex over (f)}.sub.m through the two sets of estimated minimum frequency parameters f.sub.m and f.sub.m.sup.′ by utilizing a closed-form robust Chinese remainder theorem, and screening out K correct frequency parameters {{circumflex over (f)}.sub.k}.sub.k=0.sup.K-1 through sampling rate parameters. The disclosure is configured to solve problems of frequency aliasing and image frequency aliasing occurring in real-valued multiple sinusoid signal sub-Nyquist sampling.

Claims

1. A multiple sinusoid signal sub-Nyquist sampling method based on a multi-channel time delay sampling system, specifically comprising: step 1: initializing; step 2: enabling multiple sinusoid signals x(t) to respectively enter N′ parallel sampling channels after the multiple sinusoid signals are divided, and enabling each sampling channel to evenly sample the signals at a low speed with the same sampling rate, wherein a sampling time delay of adjacent channels is τ, and the number of sampling points of each channel is N; step 3: combining sampled data of each sampling channel to construct an autocorrelation matrix R.sub.xx, and estimating sampling signal parameters c.sub.m of each channel and a set of frequency parameters f.sub.m by utilizing the ESPRIT method; step 4: estimating signal amplitudes α.sub.m and another set of frequency parameters f.sub.m through the estimated parameters c.sub.m and the sampling time delay τ of each channel by utilizing the ESPRIT method; and step 5: reconstructing 2K frequency parameters {circumflex over (f)}.sub.m through the two sets of estimated minimum frequency parameters f.sub.m and f.sub.m.sup.′ by utilizing a closed-form robust Chinese remainder theorem, and screening out K correct frequency parameters {{circumflex over (f)}.sub.k}.sub.k=0.sup.K-1 through sampling rate parameters.

2. The sampling method according to claim 1, wherein step 1 is specifically: it is assumed that the to-be-sampled multiple sinusoid signal x(t) is composed of K frequency components and expressed as: $\begin{array}{cc}x\left(t\right)=\underset{k=0}{\overset{K-1}{.Math.}}.Math.a_k.Math.\cos \left(2.Math.\pi .Math..Math.f_k.Math.t+{\varphi }_k\right),& \left(7\right)\end{array}$ wherein f.sub.k is a k.sup.th to frequency component of the signal, φ.sub.k is a phase of the signal, a.sub.k(a.sub.k≠0,α.sub.k∈) is an amplitude parameter of the signal, and T is a duration of the signal; and an Euler formula e.sup.jt=cos t+j sin t is utilized, and the formula (7) is rewritten as: $\begin{array}{cc}\begin{array}{c}x\left(t\right)=.Math.\underset{k=0}{\overset{K-1}{.Math.}}.Math.a_k.Math.\cos \left(2.Math.\pi .Math..Math.f_k.Math.t+{\varphi }_k\right)\\ =.Math.\frac{1}{2}.Math.\underset{k=0}{\overset{K-1}{.Math.}}.Math.a_k\left(e^{j\left(2.Math.\pi .Math..Math.f_k.Math.t+{\varphi }_k\right)}+e^{-j\left(2.Math..Math.\pi .Math..Math.f_k.Math.t+{\varphi }_k\right)}\right)\\ =.Math.\frac{1}{2}.Math.\underset{m=0}{\overset{2.Math.K-1}{.Math.}}.Math.c_m.Math.e^{j.Math..Math.2.Math..Math.\pi .Math..Math.f_m.Math.t},t\in \left[0,T\right),\end{array}& \left(8\right)\end{array}$ wherein $\begin{array}{cc}{c}_m=\{\begin{array}{cc}{a}_m.Math.e^{j.Math..Math.{\varphi }_m}& 0\le m\le K-1\\ a_{m-k}.Math.e^{-j.Math..Math.{\varphi }_n}& K\le m\le 2.Math.K-1\end{array},& \left(9\right)\\ f_m=\{\begin{array}{cc}{f}_m& 0\le m\le K-1\\ -f_{m-k}& K\le m\le 2.Math.K-1\end{array},& \left(10\right)\end{array}$ in this way, a real signal solving problem of the K frequency components is transformed into a complex exponential signal solving problem of 2K frequency components, and it is assumed that a maximum frequency component f.sub.max of the signal is known a priori, namely 0≤f.sub.k<f.sub.max,∀k∈{0,1, . . . , K−1}; and meanwhile, the signal sampling rate and a time delay meets: $\begin{array}{cc}\frac{{\mathrm{\Gamma \Gamma }}^{\prime }.Math.M}{2}>f_{\max },& \left(11\right)\end{array}$ wherein M is a greatest common divisor of the sampling rate f.sub.s and a time delay reciprocal $\frac{1}{\tau },$ Γ and Γ′ are respectively quotients of f.sub.s, and $\frac{1}{\tau }$ divided by M, and Γ and Γ′ are prime numbers of each other.

3. The sampling method according to claim 1, wherein step 2 is specifically: the sampling channels evenly sample the signals with the sampling rate is f.sub.s≤f.sub.max, and a sampling value of an n′.sup.th channel is expressed as: $\begin{array}{cc}{x}_{n^{\prime }}\left[n\right]=\underset{m=0}{\overset{2.Math.K-1}{.Math.}}.Math.c_m.Math.e^{-j.Math..Math.2.Math..Math.\pi .Math..Math.n^{\prime }.Math.f_k.Math.\tau }.Math.e^{j.Math..Math.2.Math..Math.\pi .Math..Math.{\mathrm{nf}}_k/f_s},0\le n<N,& \left(12\right)\end{array}$ wherein n′=0,1, . . . ,N′−1.

4. The sampling method according to claim 1, wherein step 3 is specifically: x.sub.n′,t′==[x.sub.n′], x.sub.n′[i′+1], x.sub.n, . . . , x.sub.n′[i′+2K]] is made to comprise 2K+1 continuous sampling values to construct the autocorrelation matrix $\begin{array}{cc}{R}_{\mathrm{xx}}=\underset{n^{\prime }=0}{\overset{N^{\prime }-1}{.Math.}}.Math.\underset{I^{\prime }=0}{\overset{I-1}{.Math.}}.Math.{x_{n^{\prime },i^{\prime }}\left(x_{n^{\prime },i^{\prime }}\right)}^H,& \left(13\right)\end{array}$ wherein I=N−K is the number of vectors x.sub.n′,t′ capable of being composed of the sampling values of each channel; if no image frequency aliasing exists, a sampling condition of $N^{\prime }\ge 2,N\ge .Math.2.Math.K+\frac{2.Math.K}{N^{\prime }}.Math.$ ensures that a rank of the autocorrelation matrix R.sub.xx is 2K, and a requirement of an ESPRIT algorithm for an input matrix is met; and if image frequency aliasing exists, N′≥4K,N≥4K is a sufficient and unnecessary condition to ensure that the rank of the autocorrelation matrix R.sub.xx meets the requirement of the ESPRIT algorithm, then the autocorrelation matrix R.sub.xx is solved by utilizing the ESPRIT algorithm, a set of frequency parameter minimum solutions {f.sub.m}.sub.m=0.sup.2K-1 is obtained, the set of solutions meets f.sub.k+f.sub.2K-K-1=f.sub.s, and meanwhile, a series of amplitudes {c.sub.me.sup.−j2πn′f.sup.m.sup.t}.sub.n′=01,1, . . . N′−1.sup.m=0,1, . . . ,2K-1 are further obtained by utilizing the ESPRIT algorithm to solve the matrix R.sub.xx.

5. The sampling method according to claim 1, wherein step 4 is specifically: $\tau >\frac{1}{f_{\max }}$ is set, amplitudes {c.sub.me.sup.−j2πn′f.sup.m.sup.t}.sub.n′=0,1, . . . ,N′−1.sup.m=0,1, . . . ,2K-1 serve as a single sinusoid sampling process, and {f.sub.m.sup.′,c.sub.m}.sub.m=0.sup.2K-1 is obtained through the same method as step 3, wherein f.sub.m.sup.′ is a frequency parameter minimum solution under a virtual sampling rate $\frac{1}{\tau },$ and c.sub.m comprises an amplitude and phase information of a real-valued multiple sinusoid signal as defined in a formula (9).

6. The sampling method according to claim 1, wherein step 5 is specifically: 2K−1 positive frequency estimation values {{circumflex over (f)}.sub.m}.sub.m.sup.2K-1 are solved through two sets of solved frequency parameter minimum solutions {f.sub.m,f.sub.m.sup.′}.sub.m.sup.2K-1 by utilizing the closed-form robust Chinese remainder theorem, and K frequencies ${\left\{{\hat{f}}_k|{\hat{f}}_k\le \frac{{\mathrm{\Gamma \Gamma }}^{\prime }.Math.M}{2}\right\}}_{k=0}^{K-1}$ are selected therefrom as frequency values of K real-valued multiple sinusoid signals.

Description

BRIEF DESCRIPTION OF FIGURES

[0028] FIG. 1 is a block diagram of a multi-channel time delay sampling system of the disclosure.

[0029] FIG. 2 is a result diagram of signal parameter estimation in the absence of image frequency aliasing of the disclosure.

[0030] FIG. 3 is a result diagram of signal parameter estimation in the presence of image frequency aliasing of the disclosure.

[0031] FIG. 4 is a result diagram of parameter estimation with a signal-to-noise ratio being 10 dB of the disclosure.

[0032] FIG. 5A-5B is a result of time domain waveform reconstruction with a signal-to-noise ratio being 10 dB of the disclosure.

DETAILED DESCRIPTION

[0033] The technical solutions in embodiments of the disclosure will be described clearly and completely below in combination with accompanying drawings in the embodiments of the disclosure, and obviously, the embodiments described are only a part of the embodiments of the disclosure, and not all of them. Based on the embodiments in the disclosure, all other embodiments obtained by those of ordinary skill in the art without creative labor are all within the protection scope of the disclosure.

Embodiment 1

[0034] According to a multiple sinusoid signal sub-Nyquist sampling method based on a multi-channel time delay sampling system, the sampling method is specifically: there are N′≥2 adjacent and parallel sampling channels with time delays τ, sampling rates of the N′ channels are the same, and there is a relative delay in sampling time of the N′ channels.

[0035] Step 1: initialization is conducted.

[0036] Step 2: after being divided, multiple sinusoid signals x(t) respectively enter N′ parallel sampling channels, each sampling channel evenly samples the signals at a low speed with the same sampling rate, a sampling time delay of adjacent channels is r, and the number of sampling points of each channel is N.

[0037] Step 3: sampled data of each sampling channel are combined to construct an autocorrelation matrix R.sub.xx, and sampling signal parameters c.sub.m of each channel and a set of frequency parameters f.sub.m are estimated by utilizing the ESPRIT method.

[0038] Step 4: signal amplitudes α.sub.m and another set of frequency parameters f.sub.m.sup.′ are estimated through the estimated parameters c.sub.m and the sampling time delay τ of each channel by utilizing the ESPRIT method.

[0039] Step 5: 2K frequency parameters {circumflex over (f)}.sub.m are reconstructed through the two sets of estimated minimum frequency parameters f.sub.m and f.sub.m.sup.′ by utilizing a closed-form robust Chinese remainder theorem, and K correct frequency parameters {{circumflex over (f)}.sub.k}.sub.k=0.sup.K-1 are screened out through sampling rate parameters.

[0040] Further, step 1 is specifically: it is assumed that the to-be-sampled multiple sinusoid signal x(t) is composed of K frequency components and expressed as:

[00013]$\begin{array}{cc}x\left(t\right)=\underset{k=0}{\overset{K-1}{.Math.}}.Math.a_k.Math.\cos \left(2.Math.\pi .Math..Math.f_k.Math.t+{\varphi }_k\right).& \left(7\right)\end{array}$

[0041] f.sub.k is a k.sup.th frequency component of the signal, φ.sub.k is a phase of the signal, α.sub.k(α.sub.k≠0,α.sub.k∈) is an amplitude parameter of the signal, and T is a duration of the signal. An Euler formula e.sup.jt=cos t+j sin t is utilized, and the formula (7) is rewritten as:

[00014]$\begin{array}{cc}\begin{array}{c}x\left(t\right)=.Math.\underset{k=0}{\overset{K-1}{.Math.}}.Math.a_k.Math.\cos \left(2.Math.\pi .Math.f_k.Math.t+{\varphi }_k\right)\\ =.Math.\frac{1}{2}.Math.\underset{k=0}{\overset{K-1}{.Math.}}.Math.a_k\left(e^{j.Math.〈2.Math.\pi .Math.f_k.Math.t+{\varphi }_k)}+e^{-j.Math.〈2.Math.\pi .Math.f_k.Math.t+{\varphi }_k)}\right)\\ =.Math.\frac{1}{2}.Math.\underset{m=0}{\overset{2.Math.K-1}{.Math.}}.Math.c_m.Math.e^{j.Math..Math.2.Math..Math.\pi .Math..Math.f_m.Math.t},t\in \left[0,T\right),\end{array}& \left(8\right)\end{array}$

[0042] wherein

[00015]$\begin{array}{cc}{c}_m=\{\begin{array}{cc}{a}_m.Math.e^{j.Math..Math.{\varphi }_m}& 0\le m\le K-1\\ a_{m-K}.Math.e^{-j.Math..Math.{\varphi }_m}& K\le m\le 2.Math.K-1\end{array},.Math.\mathrm{and}& \left(9\right)\\ f_m=\{\begin{array}{cc}{f}_m& 0\le m\le K-1\\ -f_{m-K}& K\le m\le 2.Math.K-1\end{array}.& \left(10\right)\end{array}$

[0043] In this way, a real signal solving problem of the K frequency components is transformed into a complex exponential signal solving problem of 2K frequency components, and it is assumed that a maximum frequency component f.sub.max of the signal is known a priori, namely 0≤f.sub.k<f.sub.max,∀k∈{0,1, . . . ,K−1}. Meanwhile, the signal sampling rate and a time delay should meet:

[00016]$\begin{array}{cc}\frac{{\mathrm{\Gamma \Gamma }}^{\prime }.Math.M}{2}>f_{\max }.& \left(11\right)\end{array}$

[0044] M is a greatest common divisor of the sampling rate f.sub.s and a time delay reciprocal

[00017]$\frac{1}{\tau },$

Γ and Γ′ are respectively quotients of f.sub.s and

[00018]$\frac{1}{\tau }$

and divided by M, and Γ and Γ′ are prime numbers of each other.

[0045] Further, step 2 is specifically: the sampling channels evenly sample the signals with the sampling rate f.sub.s≤f.sub.max and a sampling value of an n.sup.th channel is expressed as:

[00019]$\begin{array}{cc}{x}_{n^{\prime }}\left[n\right]=\underset{m=0}{\overset{2.Math.K-1}{.Math.}}.Math.c_m.Math.e^{-j.Math..Math.2.Math..Math.\pi .Math..Math.n^{\prime }.Math.f_k.Math.\tau }.Math.e^{J^{2.Math.m.Math.f_k/f_s}},0\le n<N.& \left(12\right)\end{array}$

[0046] n′=0,1, . . . ,N′−1

[0047] Further, step 3 is specifically:

[0048] x.sub.n′,t′=[x.sub.n′],x.sub.n′[i′+1],x.sub.n′ . . . ,x.sub.n′[i′+2K]] is made to include 2K+1 continuous sampling values to construct the autocorrelation matrix

[00020]$\begin{array}{cc}{R}_{\mathrm{xx}}=\underset{n^{\prime }=0}{\overset{N^{\prime }-1}{.Math.}}.Math.\underset{I^{\prime }=0}{\overset{I-1}{.Math.}}.Math.{x_{n^{\prime }.Math.i^{\prime }}\left(x_{n^{\prime }.Math.i^{\prime }}\right)}^H.& \left(13\right)\end{array}$

[0049] I=N−K is the number of vectors x.sub.n′,i′ capable of being composed of the sampling values of each channel. If no image frequency aliasing exists, a sampling condition of

[00021]$N^{\prime }\ge 2,N\ge .Math.2.Math.K+\frac{2.Math.K}{N^{\prime }}.Math.$

ensures that a rank of the autocorrelation matrix R.sub.xx is 2K, and a requirement of an ESPRIT algorithm for an input matrix is met. If image frequency aliasing exists, N′≥4K,N≥4K is a sufficient and unnecessary condition to ensure that the rank of the autocorrelation matrix R.sup.xx meets the requirement of the ESPRIT algorithm, then the autocorrelation matrix R.sub.xx is solved by utilizing the ESPRIT algorithm, a set of frequency parameter minimum solutions {f.sub.m}.sub.m=0.sup.2K-1 is obtained, the set of solutions meets f.sub.k+f.sub.2K-k-1=f.sub.s, and meanwhile, a series of amplitudes {c.sub.me.sup.−j2πn′f.sup.m.sup.t}.sub.n′=0,1, . . . ,N′-1.sup.m=0,1, . . . ,2K-1 are further obtained by utilizing the ESPRIT algorithm to solve the matrix R.sub.xx.

[0050] Further, step 4 is specifically:

[00022]$\tau >\frac{1}{f_{\max }}$

is set, the amplitudes {c.sub.me.sup.−j2πn′f.sup.m.sup.t}.sub.n′=0,1, . . . ,N′-1.sup.m=0,1, . . . ,2K-1 serve as a single sinusoid sampling process, and {f.sub.m.sup.′,c.sub.m}.sub.m=0.sup.2K-1 is obtained through the same method as step 3. f.sub.m.sup.′ is a frequency parameter minimum solution under a virtual sampling rate

[00023]$\frac{1}{\tau },$

and c.sub.m includes an amplitude and phase information of a real-valued multiple sinusoid signal as defined in the formula (9).

[0051] Further, step 5 is specifically: 2K−1 positive frequency estimation values {{circumflex over (f)}.sub.m}.sub.m.sup.2K-1 are solved through the two sets of solved frequency parameter minimum solutions {f.sub.m,f.sub.m.sup.′}.sub.m.sup.2K-1 by utilizing the closed-form robust Chinese remainder theorem, and K frequencies

[00024]${\left\{{\hat{f}}_k|{\hat{f}}_k\le \frac{{\mathrm{\Gamma \Gamma }}^{\prime }.Math.M}{2}\right\}}_{k=0}^{K-1}$

are selected therefrom as frequency values of K real-valued multiple sinusoid signals.

Embodiment 2

[0052] Noiseless Experiment

[0053] The number of frequency components of a to-be-measured signal is set to be K=4, a maximum frequency of the signals is set to be 60 MHz, a sampling rate of each delay channel is f.sub.s=30 MHz and a sampling time delay of each channel is

[00025]$\tau =\frac{1}{4.Math.0}.Math.\mathrm{.Math.s}.$

In the absence of signal image frequency aliasing, the number of sampling channels is set to be N′=.sup.2, and the number of sampling points of each channel is N=12. FIG. 2 shows a reconstruction effect of signal amplitudes α.sub.k and frequency parameters f.sub.k. In the presence of signal image frequency aliasing, the number of the sampling channels is set to be N′=16, and the number of the sampling points of each channel is N=16. FIG. 3 shows a reconstruction effect of the signal amplitudes α.sub.k and the frequency parameters f.sub.k. It can be seen that in both cases of the absence of signal image frequency aliasing and the presence of signal image frequency aliasing, a sampling structure and method reconstruct the amplitude parameters and frequency parameters of the signals without errors.

Embodiment 3

[0054] Noise Experiment

[0055] The number of frequency components of a to-be-measured signal is set to be K=.sup.4, a maximum frequency of the signals is set to be 100 MHz, no signal image frequency aliasing exists, a sampling rate of each delay channel is f.sub.s=14 MHz, and a sampling time delay of each channel is

[00026]$\tau =\frac{1}{1.Math.7}.Math.\mathrm{.Math.s}.$

The number of sampling channels is set to be N′=.sup.4, the number of sampling points of each channel is N=.sup.50, and a signal-to-noise ratio is 10 dB. FIG. 4 shows a reconstruction effect of signal amplitudes α.sub.k and frequency parameters f.sub.k. FIG. 5 shows a reconstruction effect of a time domain waveform of the signals. Experimental results show that in the case that the signal-to-noise ratio is 10 dB, a sampling structure and method can estimate signal parameters well and recover the time domain waveform of the signals.