CN101867387A - Signal reconstruction technical scheme for sampling with rate lower than Nyquist rate - Google Patents

Abstract

The invention relates to a signal reconstruction technical scheme for sampling with rate lower than nyquist rate. The signal reconstruction scheme comprises the following steps of: firstly, solving a transformation basic matrix (signals x to be reconstructed thereon are sparse on the transformation basic matrix), modulating the transformation basic matrix with a cutting sequence, filtering by means of a digital filter, sampling the filtered signals, then generating compressive sensing arithmetic operators, and finally obtaining reconstructed signals by solving an optimized problem through the sparse inverse transformation. The invention aims to provide the signal reconstruction technical scheme for sampling with rate lower than the Nyquist rate, satisfy the needs for people in solving the contradictions between the increasingly widening signal bandwidth as well as the increasingly quickening sampling frequency and the requirements for reducing processing time and various costs reduce the signal sampling frequency and data transmission and storage cost, further remarkably shorten the signal processing time and reduce the computational expense, and accurately reconstruct signals.

Description

Be lower than Nyquist rate sampling signal reconstruction technical scheme down

Technical field

The present invention is a kind of signal reconstruction technical method, belongs to signals collecting and reconstruction field.

Background technology

Nyquist sampling theorem is to instruct the most important theories basis of how to sample.It points out that sampling rate must reach the above accurately reconstruction signal of twice of signal highest frequency.For broadband signal, will bring immense pressure to signals collecting, storage and processing with the Nyquist rate sampling, and nyquist sampling theorem is not used to the sparse property of this type of signal.Compressed sensing (Compressive Sensing, CS) theory is pointed out, as long as signal is compressible or is sparse at certain transform domain, so just can with one with the incoherent observing matrix of transform-based with on conversion gained higher-dimension signal projection to a lower dimensional space, just can from these a spot of projections, reconstruct original signal by finding the solution an optimization problem then, can prove that such projection has comprised the enough information of reconstruction signal with high probability.There are a large amount of sparse signals in the reality,, need to solve wherein crucial signal reconstruction technical problem for reaching to be lower than the purpose of Nyquist rate to its sampling and accurate reconstruct.

Summary of the invention

Technical problem: the purpose of this invention is to provide the signal reconstruction technical scheme that is lower than under the Nyquist rate sampling, satisfy that people solve signal bandwidth broadening day by day, sample frequency day by day speeds and require to reduce the processing time and various cost between the needs of contradiction.Reduce signal sampling frequency and transfer of data and storage cost, and then reduce signal processing time significantly and assess the cost and can accurate reconstruction signal be target of the present invention.

Technical scheme: the signal reconstruction scheme that is lower than under the Nyquist rate sampling of the present invention, at first obtain transform-based matrix (treating that reconstruction signal x should be sparse thereon), with the cutting sequence it is modulated then, then by digital filter filtering, again to filtered signal down-sampling, produce the compressed sensing operator thus, at last by finding the solution an optimization problem and obtaining reconstruction signal through sparse inverse transformation.

Figure 1 shows that direct information sample reconstruction system block diagram, wherein signal reconstruct partly is the signal reconstruction technical scheme that is lower than under the Nyquist rate sampling of the present invention.With the frame is unit, and the value of transform-based matrix Ψ and linear feedback shift register output is the cutting sequence p of ± 1 alternate _c(t) multiply each other, deliver to digital filter and carry out low-pass filtering, obtain compressed sensing operator A after can proving down-sampling ^Cs, in conjunction with input observation sequence y[m] and find the solution a l ₁Optimization problem obtains the coefficient of reconstruct

$\widehat{\mathrm{\α}}=\arg \min {\left|\right|\mathrm{\α}\left|\right|}_{1}s.t.y=A^{\mathrm{cs}}\mathrm{\α}---\left(1\right)$

Below three parts are specifically described.

1, the formation of transform-based matrix Ψ

Might as well suppose that transform domain Ψ is a Fourier transform domain, whole system is carried out data processing based on Frame, according to the character of the corresponding frequency domain frequency displacement of the time domain time delay of Fourier transform, can think Ψ=FFT (E _{N * N}), E wherein _{N * N}Be the unit matrix of N * N, and N is the length of a frame signal.The present invention in other transform domain, only need correspondingly obtain alternative the getting final product of basic matrix Ψ with this transform-based matrix as input.

2, compressed sensing operator A ^CsFormation

If w is the frequency dividing ratio of down-sampling, p _c(t) speed is P, then w and P and

Relation satisfy (here Sampling rate for low rate ADC in the AIC part)

Consider that certain moment t begins in (corresponding this moment s sampled point) frame signal duration T.Formula (2) two ends then obtain a frame signal length N and measured value length M and satisfy with the duration T that multiply by a frame

Utilize basic matrix Ψ modulation cutting sequence p among the present invention _c(t) back low-pass filtering again down-sampling obtain compressed sensing operator A ^CsThat is,

$(\mathrm{\Ψ}\·p_c)*h\left(r\right)=\underset{k=s+1}{\overset{s+N}{\mathrm{\Σ}}}{\mathrm{\ψ}}_n\left(k\right)p_c\left(k\right)h(r-k){|}_{r=w\·m}---\left(3\right)$

Relation by analog-and digital-filter impulse responses

In the formula (3)

So formula (3) is converted into

Wherein m, n are integer and m ∈ [1, M], n ∈ [1, N], and that obtains enough big up-to-date style (3) as frame length N can think the approximate of following formula with formula

And AIC is output as

The information rate that contains in the simulation-intelligence sample hypothetical simulation signal is limited, can think and in the unit interval, can represent original signal, think that promptly analog signal x (t) is made of discrete continuous base of limited weighting or basic dictionary element with the continuous basic function of limited quantity:

$x\left(t\right)=\underset{n=1}{\overset{N}{\mathrm{\Σ}}}{\mathrm{\α}}_n{\mathrm{\ψ}}_n\left(t\right)---\left(6\right)$

So formula (5) can turn to

Write as matrix form and be y=A ^Csα.Therefore, down-sampling output promptly is approximately compressed sensing operator A as the formula (3) among the present invention ^Cs

3, signal reconstruction

The signal reconstruction demand is separated a l among the present invention ₁Optimization problem, as the formula (1).The compressed sensing operator A that obtains according to reconstruct ^CsWith the observation signal y[m that receives] find the solution l ₁Optimize the coefficient that obtains reconstruct If basic matrix is chosen as the Fourier transform basic matrix, then to coefficient

Make IFFT and can recover original signal

Beneficial effect: have a large amount of sparse signals in the reality, but nyquist sampling theorem is not used to the sparse property of this type of signal.The signal reconstruction technical scheme is descended in the Nyquist rate sampling that is lower than that the present invention provides, can sample to original signal with the frequency that is lower than Nyquist rate, and with high probability reconstruct original signal, improve the validity of analog signal digital, and can reduce transfer of data and storage cost, and then reduce signal processing time significantly and assess the cost.Can satisfy that people solve signal bandwidth broadening day by day, sample frequency day by day speeds and require to reduce the processing time and various cost between the needs of contradiction.

With the input analog signal be 1MHz, 2MHz and 4MHz sinusoidal signal and be example, it is sparse on Fourier transform domain.Parameter value is as shown in table 1.

The table 1 parameter table of winning the confidence

With aforementioned sinusoidal signal and be analog input (as Fig. 2 (a)), through AIC output digital information (as Fig. 2 (c)), use therein sample frequency is 25% of Nyquist frequency, i.e. 2MHz.Know that by comparison diagram 2 (c) and Fig. 2 (a) AIC output information measured value speed ratio original signal speed is also low, use that the present invention provides simulation-intelligence sample reconfiguration scheme come the reconstruct original signal, effect is as shown in Figure 3.Signal is by accurately reconstruct (as Fig. 3 (a)) as seen from the figure, and spectrum component is clear, nothing assorted (as Fig. 3 (b)) frequently.Through calculating reconstruction signal

With the norm of the relative error vector of original signal x be $\left|\right|r\left|\right|=\left|\right|x-\hat{x}/x\left|\right|=7.4063\×{10}^{-14}.$

The relative error value of analytical sampling frequency next frame restoring signal and original signal

The relative error r here is a vector that length is N, for ease of observing the norm of getting the relative error vector r of 600 Monte Carlo simulation reconstruction signals || and r|| compares, as table 2.

Reconstruct relative error under different AIC sampling rates during table 2N=256

When finding out frame length N=256 by table 2, even if the AIC sampling rate is low to moderate 14.45% of Nyquist frequency, reconfiguration system also can be complied with big probability reconstruct prime information, and reconstruction signal relative error vector norm is 10 ^-14The order of magnitude has reached to be lower than the accurate reconstruct purpose of signal under the Nyquist rate sampling.

Certainly; the present invention also can have other various embodiments; under the situation that does not deviate from spirit of the present invention and essence thereof; those of ordinary skill in the art work as can make various corresponding changes according to the present invention, but these corresponding changes and distortion all should belong to the protection range of the appended claim of the present invention.

Description of drawings

Fig. 1 is an intelligence sample reconfiguration system block diagram

Each form of a frame signal among Fig. 2 AIC, wherein (a) is that original signal (b) is digital information (d) the original signal spectrogram that signal (c) the low speed sampling after pseudorandom modulation and the low-pass filtering obtains

Fig. 3 be the reconstruction signal design sketch wherein (a) be the α of reconstruct for the x of reconstruct (b)

Embodiment

The invention provides the signal reconstruction technical scheme that is lower than under the Nyquist rate sampling, satisfy and reduce signal sampling frequency and transfer of data and storage cost, and then reduce the signal processing time and the accurately needs of reconstruction signal that assess the cost also significantly.The signal reconstruction technical scheme that is lower than under the Nyquist rate sampling of the present invention, at first obtain transform-based matrix (treating that reconstruction signal x should be sparse thereon), with the cutting sequence it is modulated then, then by digital filter filtering, again to filtered signal down-sampling, produce the compressed sensing operator thus, at last by finding the solution an optimization problem and obtaining reconstruction signal through sparse inverse transformation.Here the transform-based matrix can be Fourier's basic matrix, can also choose the Gabor basic matrix of total variation norm matrix, the oscillator signal of wavelet basis matrix, the functions of bounded variation, the Curvelet basic matrix of picture signal with discontinuous edge and the orthogonal basis dictionary that is made of a plurality of orthogonal basiss or the like.The above transform-based is not that the present invention is imposed any restrictions, and every technical scheme essence according to the present invention changes any simple modification, change and the equivalent structure that above embodiment did, and all still belongs in the protection range of technical solution of the present invention.

Concrete mode is as follows:

1, determines system parameters: before signal reconstruction begins, need system parameterss such as specification signal reconstruct frame length, signal degree of rarefication, cutting sequence generating rate.

2, calculate compressed sensing operator A ^Cs: the flow process that provides according to scheme calculates compressed sensing operator A ^CsWait for receiving observation vector, prepare the reconstruct original signal.

3, reconstruction signal: receive observation vector, in conjunction with calculate compressed sensing operator A ^CsFind the solution l ₁Optimize the coefficient that obtains reconstruct

Try to achieve the recovery value of original signal again according to corresponding conversion.

Claims (2)

1. be lower than the signal reconstruction technical scheme under the Nyquist rate sampling, it is characterized in that at first obtaining transform-based matrix (treating that reconstruction signal x should be sparse thereon), with the cutting sequence it is modulated then, then by digital filter filtering, to filtered signal down-sampling, obtain reconstruction signal by finding the solution an optimization problem at last again.

2. be lower than Nyquist rate sampling signal reconstruction technical scheme down according to right 1 is described, it is characterized in that obtaining of compressed sensing operator, adopt with cut sequence modulation transform-based matrix and by behind the digital filtering again the method for down-sampling obtain.

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