CN106649198B - A kind of method of higher-dimension signal reconstruction quality in detection wireless sensor network - Google Patents
A kind of method of higher-dimension signal reconstruction quality in detection wireless sensor network Download PDFInfo
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Abstract
The invention discloses a kind of methods of higher-dimension signal reconstruction quality in detection wireless sensor network, are a kind of computational methods using force zero estimation and the performance indicator for detecting reconstructed signal after Linear Minimum Mean-Square Error Estimation.During signal reconstruction in wireless sensor network, wireless sensor collected signal can be indicated with the form of Fourier expansion.In view of the influence of noise in acquisition signal, need to rebuild signals and associated noises using estimator.When network size is larger, when signal dimension and larger wireless sensor quantity i.e. to be treated, the present invention provides a kind of methods for capableing of the easy probability density function for accurately obtaining generalized circular matrix characteristic value again, to the accurate detection performance index that can easily calculate system.
Description
Technical Field
The invention relates to a method for detecting reconstruction quality of a high-dimensional signal in a wireless sensor network, in particular to a computing method for detecting a mean square error index of a reconstructed signal after applying zero forcing estimation (ZF) and linear minimum mean square error estimation (LMMSE), and belongs to the technical field of signal reconstruction in the wireless sensor network.
Background
In the signal reconstruction process in the wireless sensor network, the random vandermonde matrix plays an important role. The signals collected by the wireless sensor can be expressed in the form of Fourier series expansion. The acquired signals may be represented as a product of a vandermonde matrix and a vector representing the spectrum of the signal, taking into account the different geographical locations at which each sensor is located. The noisy signal needs to be reconstructed by an estimator taking into account the noise effects in the acquired signal.
The classical algorithm has zero forcing estimation and minimum mean square error estimation, and after the original signal is reconstructed by using the two estimators, the mean square error of the reconstructed signal needs to be calculated, so that the accuracy of signal reconstruction is determined. The calculation of the mean square error requires solving the probability density function of the eigenvalues of the previously mentioned vandermonde matrix.
When the dimension of the acquired signal, the number of sensors, and the bandwidth of the acquired signal are low, the probability density function of the eigenvalues of the vandermonde matrix can be estimated by a monte carlo numerical calculation method. However, when the above parameter values are large, the calculation amount required for the computation of the monte carlo value increases exponentially, and therefore, a method capable of easily and accurately obtaining the probability density function of the matrix eigenvalue is required.
Disclosure of Invention
The invention provides a method for simply, conveniently and accurately obtaining a probability density function of a characteristic value of a vandermonde matrix, thereby accurately and conveniently calculating a detection performance index of a system.
The invention adopts the following technical scheme for solving the technical problems:
the invention provides a method for detecting reconstruction quality of a high-dimensional signal in a wireless sensor network, wherein the wireless sensor network is composed of m wireless sensors, each wireless sensor respectively collects a d-dimensional signal, the bandwidth of the signal is represented as n, and the collected signal is represented as y ═ VHa+nwWhere y denotes the measured value of the signal, a denotes the signal to be reconstructed, nwRepresenting a white noise signal, V representing a dimension ndX m van der waals matrix.
When the signal is reconstructed by adopting a zero-forcing estimation method or a minimum mean square error estimation method, the mean square errors of the reconstruction results are respectivelyAndwherein,is the variance of a white noise signal, β ═ nd/m，Eλ{. -HThe characteristic value of (2).
Solving the probability density function f of lambda byλ(d, β, λ), specifically:
1) constructing an optimization equation:and satisfies the conditional equation:wherein, mupP-order moment of λ;
2) fixed point position f for integral operation in optimization equation and condition equation by Gaussian product-solving ruleλthe weighted sum of the (d, β, λ) values is expressed as:
wherein λ isjIs the coordinate of the jth sampling point defined in the Gaussian product-solving rule, N is the number of sampling points, wjTo correspond to lambdajThe weight of (2);
3) solving the optimization equation and the conditional equation to obtain the equation corresponding to lambdajF of (a)λ(d,β,λj) Optimal values to fit fλ(d,β,λ)。
According to f obtainedλthe (d, beta, lambda) can calculate the mean square error of the reconstruction result, so that the quality of signal reconstruction by adopting a zero forcing estimation method or a minimum mean square error estimation method is measured according to the mean square error.
As a further technical scheme of the invention, the signal reconstructed by adopting the zero forcing estimation method is as follows:
as a further technical scheme of the invention, the signal reconstructed by adopting the minimum mean square error estimation method is as follows:wherein I represents an identity matrix.
As a further technical solution of the present invention, the elements of the vandermonde matrix V are:wherein, Vs,tThe element of row s, column t, x representing VtIndicating the geographic location of the t-th wireless sensor.
As a further aspect of the present invention, a is an ndA dimension vector.
As a further technical solution of the present invention, y is an m-dimensional vector.
Compared with the prior art, the invention adopting the technical scheme has the following technical effects: when the dimension of the collected signals, the number of the sensors and the bandwidth value of the collected signals are large, the probability density function of the characteristic value of the vandermonde matrix can be simply, conveniently and accurately obtained by adopting the technical scheme of the invention.
Drawings
Figure 1 is a numerical plot of the system mean square error calculated in an embodiment of the present invention.
Detailed Description
The technical scheme of the invention is further described in detail by combining the drawings and the specific embodiments:
in a wireless sensor network consisting of m wireless sensors, each sensor is responsible for acquiring a d-dimensional signal, the bandwidth of which is denoted by n. The acquired signal can be represented by the signal that needs to be reconstructed as: y is VHa+nwWherein y represents a measured value of the signal and is an m-dimensional vector; a tableThe signal to be reconstructed is denoted by ndA dimension vector; n iswRepresents a white noise signal; v represents a dimension ndX m, the elements of the s-th row and t-th column of the matrix can be expressed as:wherein x istIndicating the geographic location of the t-th wireless sensor.
When applying zero forcing estimation (ZF), the reconstructed signal is expressed as:the mean square error of the reconstruction result can therefore be written as:
when minimum mean square error estimation (LMMSE) is applied, the reconstructed signal is expressed as:the mean square error of the reconstruction result can therefore be written as:wherein I represents an identity matrix.
When the values of m, n, d are large, n is assumed to bedthe ratio of m is represented as β ═ ndM, then the mean square error of the reconstructed signal using zero forcing estimation or minimum mean square error estimation can be further accurately estimated as:
wherein λ is the matrix β VVHThe characteristic value of (2). Thus, the Probability Density Function (PDF) f of λλ(d, β, λ) requires a method to be given, where d and β are two of their parameters.
Before calculating the Probability Density Function (PDF) of λ, the p-moment of λ is known and is noted μp. The p-moment of λ is described in detail in A.Nordio, C.C. -F.Chiasserini, E.Viterbi, "Performance of linear field reconstruction techniques with noise and uncertainly sensor locations," IEEETrans. on Signal Processing, Vol.56, No.8, pp.3535-3547, Aug.2008.
Constructing an optimization equation:and satisfies the conditional equation:
fixed point position f for integral operation in optimization equation and condition equation by Gaussian product-solving ruleλthe weighted sum of the (d, β, λ) values is expressed as:
wherein λ isjIs the coordinate of the jth sampling point defined in the Gaussian product-solving rule, N is the number of sampling points, wjTo correspond to lambdajThe weight of (2) is also obtained by a gaussian product-finding rule. Thus, by solving the optimization equation and the conditional equation, corresponding to each λjF of the sampling pointλ(d,β,λj) The optimum can be found and fitted to give fλ(d,β,λ)。
In a simulation test environment, we test when d is 1 or 4,And m is ndassuming that the system respectively adopts zero forcing estimation and minimum mean square error estimation methods to reconstruct signals, the system mean square error obtained by the Monte Carlo method is respectively compared with the system mean square error calculated by the method provided by the invention.
As can be seen from an observation of FIG. 1, the minimum mean square error estimation method performs better than the zero forcing estimation method because the objective of the minimum mean square error is to minimize the mean square error, more importantly, as the beta value increases, the mean square error calculated by the present invention is very similar to the mean square error result obtained by the Monte Carlo method, but the present invention greatly reduces the amount of calculation of the mean square error, and in FIG. 1, the points "□" and "x" represent the simulation results obtained by the Monte Carlo method, and the dotted line and solid line represent the simulation results obtained by the method of the present invention.
The above description is only an embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art can understand that the modifications or substitutions within the technical scope of the present invention are included in the scope of the present invention, and therefore, the scope of the present invention should be subject to the protection scope of the claims.
Claims (6)
1. The method for detecting reconstruction quality of high-dimensional signals in a wireless sensor network comprises m wireless sensors, wherein each wireless sensor acquires a d-dimensional signal, the bandwidth of the signal is represented as n, and the acquired signal is represented as y-VHa+nwWhere y denotes the measured value of the signal, a denotes the signal to be reconstructed, nwRepresenting a white noise signal, V representing a dimension ndX m van der waals matrix, characterized in that when signal is reconstructed using zero-forcing estimation or minimum mean square error estimation, the reconstruction is performedThe mean square error of the result is respectivelyAndwherein,is the variance of a white noise signal, β ═ nd/m，Eλ{. -HA characteristic value of (d);
solving the probability density function f of lambda byλ(d, β, λ), specifically:
1) constructing an optimization equation:and satisfies the conditional equation:wherein, mupP-order moment of λ;
2) fixed point position f for integral operation in optimization equation and condition equation by Gaussian product-solving ruleλthe weighted sum of the (d, β, λ) values is expressed as:
wherein λ isjIs the coordinate of the jth sampling point defined in the Gaussian product-solving rule, N is the number of sampling points, wjTo correspond to lambdajThe weight of (2);
3) solving the optimization equation and the conditional equation to obtain the equation corresponding to lambdajF of (a)λ(d,β,λj) Optimal values to fit fλ(d,β,λ)；
According to f obtainedλthe (d, beta, lambda) can calculate the mean square error of the reconstruction result, so that the quality of signal reconstruction by adopting a zero forcing estimation method or a minimum mean square error estimation method is measured according to the mean square error.
2. The method for detecting reconstruction quality of high-dimensional signals in wireless sensor networks according to claim 1, wherein the signals reconstructed by the zero-forcing estimation method are:
3. the method for detecting reconstruction quality of high-dimensional signals in a wireless sensor network according to claim 1, wherein the signals reconstructed by the minimum mean square error estimation method are:wherein I represents an identity matrix.
4. The method for detecting reconstruction quality of high-dimensional signals in wireless sensor networks according to claim 1, wherein the elements of the vandermonde matrix V are:wherein, Vs,tThe element of row s, column t, x representing VtIndicating the geographic location of the t-th wireless sensor.
5. The method of claim 1, wherein a is ndA dimension vector.
6. The method of claim 1, wherein y is an m-dimensional vector.
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| CN103107815A (en) * | 2013-01-09 | 2013-05-15 | 华为技术有限公司 | Method and device for signal reconstruction |
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