JP2014229924A - Compressed signal restoring device, compressed signal restoring method, and program - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a compressed signal restoring device capable of restoring an original signal more accurately than before.SOLUTION: A decoding unit inputs a signal y=Ψxobtained by compressing, with a predetermined observation matrix Ψ, an observation signal xof an i-th sensor, which is represented as x=z+zby a common signal zwhich is a common portion of the signal observed by the i-th sensor, and a signal zwhich is a remaining signal excluding the common signal z, and obtains an observation signal xat which an expression (9) is the smallest under constraint conditions of expressions (10)-(13), and outputs it as a restoration signal of an original signal.

Description

  The present invention relates to encoding and decoding for storing or transmitting time-series signals and other signals, and more particularly to a compressed signal decompression apparatus, a compressed signal decompression method, and a program for decompressing a compressed signal.

  In a sensor network in which a large number of sensors are connected by a wireless network, data acquired by each sensor is frequently transmitted and received. Sensors that have low processing power and cannot be supplied with power from an external power source are often used. Therefore, in order to save power, the amount of information transmitted by the sensor can be reduced. A technique for reducing the amount of calculation performed by itself is regarded as important.

One of them is a technique called compressed sensing. This uses the fact that the results of measurements by sensors that are in the same application and geographically close to each other are highly similar, and represents the information of each sensor as information common to other sensors and other information. Thus, the amount of information is compressed. Non-Patent Document 1 proposes a compression sensing method using a single sensor. Here, the transmission apparatus linearly transforms the signal x of the number of samples N to be measured using the M × N observation matrix Ψ, thereby reducing the information on the dimension M, which is smaller than the dimension of x, y = Compress to Ψx and transmit. The receiving apparatus performs the restoration on the assumption that the non-zero element of the expression s = Φ ⁻¹ x in the base Φ of the original signal x is very small (hereinafter referred to as “sparse”). In this case, a signal exactly equal to the original signal x is to be decoded.

Further, as a method of performing compressed sensing on a wireless network composed of a large number of sensors, in Non-Patent Document 2, the observation signal x _{i of} each i-th sensor is

The compression sensing of the signal represented by is proposed. here,

It is. a _i is the amplitude of the observation signal of the i-th sensor, θ _i is a coefficient representing the phase of the observation signal of the i-th sensor, and z _c is a signal common to all channels. Also,

Where j is an imaginary unit (j ² = −1), DFT is a discrete Fourier transform, and IDFT is an inverse transform of DFT. That is, this model is a model in which the common part z _{c of the} signals observed by the sensors is the signal z _{c, i} having the same shape but having the group delay and the attenuation of the entire amplitude.

In Non-Patent Document 2, (1) to estimate the _{x i,} (2) using the estimated _{x i} and the observed signal to estimate the amplitude _{a i} and phase theta _i, the amplitude _{a i} and phase theta of the (3) Estimation Each signal is obtained in the order of estimating the common signal z _{c, i} using _i and (4) estimating the remaining signal z _{d, i} using the estimated common signal and the observed signal. In the optimizations (1) to (4), optimization is performed according to individual optimization formulas.

E. Hale, W. Yin, and Y. Zhang, “Fixed-point continuation for` 1-minimization: Methodology and convergence, ”SIAM Journal on Optimization, vol. 19, no. 3, pp. 1107-1130, 2008. Yoshifumi Shiraki, Yu Kamamoto, Takehiro Moriya, "Restoration of multi-channel signals with common elements with attenuation and delay", Proceedings of the IEICE Conference, IEICE, August 28, 2012 , Volume 2012, p.62

  Non-Patent Document 1 is a compression sensing method for a single sensor, and thus cannot be directly applied to a sensor network composed of a large number of sensors.

According to Non-Patent Document 2, compression sensing of a large number of sensors can be performed, but there is a problem that a signal x _i that exactly matches the original signal x _i ′ cannot always be decoded. This is because the method of Non-Patent Document 2 optimizes the estimation of each signal using the estimation results of other signals that have already been estimated. Therefore, if the estimation accuracy of the previously estimated signal is low, the estimation accuracy of the remaining signals is low. By also affecting. In addition, since the estimation of the sparse terms z _{d, i} was performed after obtaining the common signal, there was no guarantee that the estimation result obtained by the equation (1) was optimal in terms of least squares to the original observed signal. Therefore, an object of the present invention is to provide a compressed signal restoration apparatus that can restore an original signal with higher accuracy than in the past.

  The compressed signal decompression apparatus of the present invention is a compressed signal decompression apparatus that restores compressed signals from Nc sensors, and includes a decoding unit.

The decoding unit represents an index i as an integer between 1 and Nc, a signal common to all sensors as a signal z _c, and a common part of signals observed by the i-th sensor as a common signal z _{c, i} And the remaining signal excluding the common signal z _{c, i} among the signals observed by the i-th sensor is represented as a signal z _{d, i} ,

The i-th observation signals x _i of the sensors as input compressed signal y _{i =} [psi _i x _i by a predetermined observation matrix Ψ _i, e c, the i-th auxiliary signal common sensor _i represented in And a _{c, i} is a coefficient representing the amplitude of the common signal element of the i-th sensor with respect to the auxiliary common signal e _{c, i} , and θ _{c, i} is the common signal element of the i-th sensor with respect to z _{c, i + 1} Φ is a predetermined base matrix, and the expression s _c = Φ ⁻¹ z _{c of} the common signal z _{c in} the base Φ is not sparse, and the signal z _{d, i in} the base Φ Assume that the expression s _{d, i} = Φ ⁻¹ z _{d, i} is sparse with respect to the index i of all sensors, and ‖ · ‖ ₂ is an L2 norm that is the sum of squares of each element of the vector, and ‖ · ‖ ₁ is L1 norm is the absolute value sum of each element of the vector, the set of the Ω index i That Omega = and {1,2, ···, Nc} and the _{w i,} and weight for the predetermined individual signal _{s d, i,} the ∀ a universal quantifier, imaginary units j, DFT Discrete Fourier transform, IDFT is the inverse transform of DFT, and the symbol Λ

And define

Under the constraints of

There obtains observation signals x _i which minimizes, and outputs a decoded signal of the original signal.

  According to the present invention, the original signal can be restored with higher accuracy than in the prior art.

The figure which shows the outline | summary of the compression sensing by many sensors. 1 is a block diagram illustrating a configuration of a compressed signal decompression apparatus according to a first embodiment of the present invention. The flowchart which shows operation | movement of the compressed signal decompression | restoration apparatus of Example 1 of this invention. The flowchart which shows operation | movement of the compressed signal decompression | restoration apparatus of Example 1 of this invention. The block diagram which shows the structure of the signal estimation part of the compressed signal decompression | restoration apparatus of Example 1 of this invention. The flowchart which shows operation | movement of the signal estimation part of the compressed signal decompression | restoration apparatus of Example 1 of this invention. The block diagram which shows the structure of the compression signal decompression | restoration apparatus of Example 2 of this invention. The flowchart which shows operation | movement of the compressed signal decompression | restoration apparatus of Example 2 of this invention. The block diagram which shows the structure of the signal estimation part of the compressed signal decompression | restoration apparatus of Example 2 of this invention. The flowchart which shows operation | movement of the signal estimation part of the compressed signal decompression | restoration apparatus of Example 2 of this invention. The block diagram which shows the structure of the compression signal decompression | restoration apparatus of Example 3 of this invention. The flowchart which shows operation | movement of the compressed signal decompression | restoration apparatus of Example 3 of this invention. The block diagram which shows the structure of the signal estimation part of the compressed signal decompression | restoration apparatus of Example 3 of this invention. The flowchart which shows operation | movement of the signal estimation part of the compressed signal decompression | restoration apparatus of Example 3 of this invention. The block diagram which shows the structure of the compressed signal decompression | restoration apparatus of Example 4 of this invention. The flowchart which shows operation | movement of the compressed signal decompression | restoration apparatus of Example 4 of this invention. The block diagram which shows the structure of the signal estimation part of the compressed signal decompression | restoration apparatus of Example 4 of this invention. The flowchart which shows operation | movement of the signal estimation part of the compressed signal decompression | restoration apparatus of Example 4 of this invention. The block diagram which shows the structure of the compressed signal decompression | restoration apparatus of Example 5 of this invention. The flowchart which shows operation | movement of the compressed signal decompression | restoration apparatus of Example 5 of this invention. The block diagram which shows the structure of the signal estimation part of the compressed signal decompression | restoration apparatus of Example 5 of this invention. The flowchart which shows operation | movement of the signal estimation part of the compressed signal decompression | restoration apparatus of Example 5 of this invention.

  Hereinafter, embodiments of the present invention will be described in detail. In addition, the same number is attached | subjected to the structure part which has the same function, and duplication description is abbreviate | omitted.

<Outline of compressed sensing>
Hereinafter, compression sensing by a number of sensors will be described with reference to FIG. FIG. 1 is a diagram showing an outline of compression sensing by a large number of sensors. As shown in FIG. 1, Nc sensors (Nc is an integer of 1 or more) (first sensor 9-1, second sensor 9-2,..., Nc sensor 9-Nc) and one central receiver There are eight. The first sensor 9-1, the second sensor 9-2,..., The Nc sensor 9 -Nc measure different signals and transmit them to the central receiver 8. Here, information acquired in each unit time by each sensor is represented by a complex vector, and the length of this vector is N (N is a positive integer). That is, it is assumed that information acquired by Nc sensors in a certain unit time can be generally written as x ′ _i ∈C ^ N, i = 1, 2,..., Nc (index i is an integer between 1 and Nc).

  The observation signal model of the present invention is also used in Non-Patent Document 2.

This model can be expressed as Here, the common signal z _{c, i,} which is a common part of the signals observed by the i-th sensor, is a signal obtained by adding a group delay and an overall amplitude attenuation to the signal z _c common to all sensors.

It is. z _{d, i} is the remaining signal obtained by removing the common signal z _{c, i} from the signals observed by the i-th sensor. Here, Φ is a predetermined basis matrix, and the expression s _c = Φ ⁻¹ z _{c in} the base Φ of the common signal z _c is not sparse, _and the base Φ of the remaining signals z _{d, i} other than the common signal It is assumed that the expression s _{d, i} = Φ ⁻¹ z _{d, i} is sparse with respect to the index i of all sensors. Hereinafter, s _{d, i} is referred to as “individual signal”.

As shown in FIG. 2, each sensor can be regarded as an observation device 9 including a measurement unit 91 and a compression unit 92. The measurement unit 91 transmits a signal x ^ _i of length N observed every unit time to the compression unit 92. The compressing unit 92 converts the signal x ^ _i having a length N into a compressed signal y _i εC ^M having a length M and outputs the signal. However, M is a positive integer smaller than N. It is assumed that this conversion is linear. That is, the compressing unit 92 uses a predetermined observation matrix Ψ _i ∈C ^{M × N.}

The signal y _i compressed by the above is obtained and transmitted.

The central receiver 8 has a compressed signal restoration device 1 for restoring the original signal x ^ _i from the input signal _yi . As shown in FIG. 2, the compressed signal decompression apparatus 1 includes a decoding unit 11. The present invention relates to a technique for estimating each signal x ^ _i in the decoding unit 11.

<Points of the present invention>
In the observation signal model represented by Equation (1), if we focus on the relationship between each sensor

Can be written as Here, e _{c, i} is an auxiliary common signal of the i-th sensor, a _{c, i} is a coefficient representing the amplitude of the common signal element of the i-th sensor with respect to the auxiliary common signal e _{c, i} , θ _{c, i} is a coefficient representing the phase of the common signal element of the i-th sensor with respect to z _{c, i + 1} . Using this relationship, the decoding unit 11 of the present invention uses the input signal y _i to

The

Minimizing under such constraints as (seek smallest observed signal x _i) by, decoding the estimated values x _i of the original signal x ^ _i (S11). This minimization can be performed by applying ADMM (Alternating Direction Method of Multiplier, Reference Non-Patent Document 1). That is,

Is minimized for each of the signal and the coefficients x _i , z _{c, i} , z _{d, i} , e _{c, i} , s _{d, i} , a _{c, i} , θ _{c, i.} it allows decoding the estimated value _{x i} of the original signal x _{^ i.} Here, ‖ · ‖ ₂ is the square sum (L2 norm) of each element of the vector, ‖ · ‖ ₁ is the absolute value sum (L1 norm) of each element of the vector, and Ω is a set of indexes i, Ω = {1, 2,..., Nc}. Ω ⁻ is a set of indexes i excluding Nc, that is, Ω ⁻ = {1, 2,..., Nc−1}. Let Φ be a predetermined basis matrix. Also, λ _i , μ _{c, i} , ν _{c, i} , ξ _{d, i} are variables (undetermined multipliers) for the constraint conditions of the equations (10), (11), (12), and (13), respectively. w _i is a weight for the individual signal s _{d, i} and is a predetermined constant. α, β, γ, and σ are weights for adjusting the convergence speed when solving the minimization problem, and are preset constants (parameters). Such terms of α, β, γ, and σ serve as penalty terms for avoiding overfitting. T represents transposition of a matrix or a vector.
(Reference Non-Patent Document 1) D. Gabay and B. Mercier, “A dual algorithm for the solution of nonlinear variational problems via finite element approximation,” Computers & Mathematics with Applications, vol. 2, no. 1, pp. 17- 40, 1976.

  The compressed signal decompression apparatus 1 according to the first embodiment will be described with reference to FIGS. 2, 3, and 4. FIG. 2 is a block diagram showing the configuration of the compressed signal decompression apparatus 1 of this embodiment. 3 and 4 are flowcharts showing the operation of the compressed signal decompression apparatus 1 of this embodiment. As described above, each sensor can be regarded as the observation device 9 including the measurement unit 91 and the compression unit 92. The compressed signal decompression apparatus 1 includes a decoding unit 11, and the decoding unit 11 includes a signal estimation unit 111, a parameter storage unit 112, a multiplier update unit 113, and an end determination unit 114.

<Parameter storage unit 112>
The parameter storage unit 112 stores a predetermined observation matrix Ψ _i , a base matrix Φ, and the total number Nc of sensors. In addition, predetermined objective function parameters α, β, γ, σ and weights w _i for individual signals are stored. _{Ψ i, Φ, Nc, α} , β, γ, σ, w i is a fixed value. Furthermore, the parameter storage unit 112, and the initial value of the signal and the coefficient _{x i, z c, i,} z d, i, e c, i, s d, i, a c, i, θ c, i, object The initial values of undetermined multipliers λ _i , μ _{c, i} , ν _{c, i} , ξ _{d, i} for the constraints in the function are stored in advance. Random values may be set for these initial values.

<Signal estimation unit 111>
The signal estimation unit 111 converts the objective function H of the above equation (14) into the signal and the coefficients x _i , z _{c, i} , z _{d, i} , e _{c, i} , s _{d, i} , a _{c, i} , θ _{c. , I} to obtain an estimated value of each signal and coefficient, and the signal and coefficient x _i , z _{c, i} , z _{d, i} , e _{c, i} , The values of s _{d, i} , a _{c, i} , θ _{c, i} are updated (S111). Specific processing of the signal estimation unit 111 will be described later.

<Multiplier update unit 113>
The multiplier update unit 113 uses the signals and coefficients estimated by the signal estimation unit 111 to determine undetermined multipliers λ _i , μ _{c, i} , ν _{c, i} , ξ _d, for the constraint condition in the objective function of Equation (14) _{. i} is updated (S113). Specifically, the value of each multiplier is updated by the following formula.

<End determination unit 114>
The end determination unit 114 determines whether or not a predetermined end condition is satisfied. If the predetermined condition is not satisfied, the end determination unit 114 controls the signal estimation unit 111 and the multiplier update unit 113 to estimate each signal and coefficient. Repeat control of the multiplier update process is executed (S114). If satisfies a predetermined condition, and outputs a signal x _i found, the process ends.

As the predetermined end condition, for example, “whether or not the number of repetitions k of the signal estimation unit 111 has reached a predetermined number It _max ” is used. The end determination unit 114 determines that “a predetermined condition has been satisfied” when k exceeds It _max . Alternatively, it is possible to use "whether the update amount of the signal estimator 111 a signal x _i to be updated at is smaller than a predetermined threshold value ε '. In this case, the end determining unit 114, the entire signal after being updated x _i by the signal estimator 111, when the entire signal before the update was ^~ x _i,

Is satisfied, it is determined that “a predetermined condition has been satisfied”.

<Specific Processing of Signal Estimation Unit 111>
Hereinafter, the details of the signal estimation unit 111 will be described with reference to FIGS. 5 and 6. FIG. 5 is a block diagram illustrating a configuration of the signal estimation unit 111 of the compressed signal decompression apparatus 1 according to the present embodiment. FIG. 6 is a flowchart showing the operation of the signal estimation unit 111 of the compressed signal decompression apparatus 1 of this embodiment. The signal estimation unit 111 includes a whole signal estimation unit 1111, an element signal estimation unit 1112, a common signal estimation unit 1113, an individual signal estimation unit 1114, and an amplitude / phase estimation unit 1115. These components may execute the processes in any order. For example, the overall signal estimation unit 1111, the element signal estimation unit 1112, the common signal estimation unit 1113, the individual signal estimation unit 1114, and the amplitude phase estimation unit It is preferable to execute the processes in the order of 1115.

<Overall signal estimation unit 1111>
The overall signal estimation unit 1111 receives the input signal y _i and z _{c, i} , z _{d, i} , λ _i , α, Ψ _i , Nc stored in the parameter storage unit 112 as input, and corresponds to y _i An estimated value x _i of the decoded signal to be obtained is obtained (S1111). Specifically, the overall signal estimation unit 1111

The search of _{x i} by minimizing the _{x i,} stored in the parameter storage unit 112 (S1111). This minimization is for example

It is executed as follows. The overall signal estimation unit 1111 performs minimization for each of i = 1, 2,..., Nc (S1111). I is a unit matrix of size N. Equation (20) is obtained by extracting only the term related to the variable x _i to be estimated from Equation (14).

<Element signal estimation unit 1112>
The element signal estimation unit 1112 stores x _i , z _{c, i} , z _{d, i} , λ _i , α, μ _{c, i} , a _{c, i} , e _{c, i} , β stored in the parameter storage unit 112. The estimated values z _{c, i} and z _{d, i} of the element signals for each sensor are obtained and stored in the parameter storage unit 112 (S1112).

  Specifically, the element signal estimation unit 1112 determines that i = 1.

Z _{c, i is} obtained by minimizing z _{c, i} (S1112). This minimization is

It is done as follows. In addition, the element signal estimation unit 1112 determines that i = 2, 3,..., Nc−1.

Z _{c, i is} obtained by minimizing z _{c, i} (S1112). This minimization is

It is done as follows. Here, * represents a Hermitian transpose matrix.
In addition, the element signal estimation unit 1112 determines that i = Nc.

Z _{c, i is} obtained by minimizing z with respect to z _{c, i} (S1112). This minimization is

It is done as follows. Expressions (22), (24), and (26) are obtained by extracting only terms related to the variables z _{c and i} to be estimated from Expression (14).

  In addition, the element signal estimation unit 1112 has i = 1, 2,.

Is minimized with respect to z _{d, i} and z _{d, i} is output (S1112). This minimization is for example

It is done as follows. Equation (28) is obtained by extracting only the term related to the variable z _{d, i} to be estimated from Equation (14).

Element signal estimation unit 1112, _{z c} stored in the parameter storage unit _{112, i} and _{z d,} the value of _i, and updates each _{z c} was determined by minimizing the _{above, i} and _{z d,} the value of _i (S1112).

<Common signal estimation unit 1113>
The common signal estimation unit 1113 is configured to store z _{c, i} , θ _{c, i} , μ _{c, i} , a _{c, i} , e _{c, i} , β, ν _{c, i} , Λ, z stored in the parameter storage unit 112. _The auxiliary common signal ec _{, i} is estimated using _{c, i + 1} , γ (S1113). Specifically, the common signal estimation unit 1113 is

E _{c, i is} obtained by minimizing e _{c, i} (S1113). This minimization is

It is done by seeking. The common signal estimation unit 1113 obtains this for each of i = 1, 2,..., Nc−1 (S1113). Equation (30) is obtained by extracting only terms related to _{ec, i} to be estimated from Equation (14).

The common signal estimation unit 1113 updates the value of ec _{, i} stored in the parameter storage unit 112 to the value of ec _{, i} obtained by the above-described minimization (S1113).

<Individual signal estimation unit 1114>
Individual signal estimation unit 1114, ξ _d, i stored in the parameter storage unit _{112, z d, i, Φ} , s d, i, σ, with _{w i,} estimates the individual signal _{s d, i} ( S1114). Specifically, the individual signal estimation unit 1114

S _{d, i is} obtained by minimizing s _{d, i} (S1114). This minimization is

It is done by seeking. Here, sgn is a function that outputs a code for each element, and ◯ is a symbol that represents multiplication for each element. It is assumed that max {| · |, 0} is calculated for each element. This calculation can be performed by applying the soft thresholding method of Non-Patent Document 1.

The individual signal estimation unit 1114 obtains the equation (33) for each of i = 1, 2,..., Nc (S1114). Expression (32) is obtained by extracting only the term related to the signal s _{d, i} to be estimated from Expression (14).

Individual signal estimation unit 1114 updates _{s d} stored in the parameter storage unit _112, the value of _{i, s d} was determined by minimizing the _above, the value of _i (S1114).

<Amplitude phase estimation unit 1115>
The amplitude / phase estimation unit 1115 stores z _{c, i} , z _{c, i + 1} , μ _{c, i} , a _{c, i} , ec _{, i} , β, ν _{c, i} , Λ, γ stored in the parameter storage unit 112. Is used to estimate the amplitude coefficient a _{c, i} and the phase coefficient θ _{c, i} (S1115). Specifically, the amplitude phase estimation unit 1115

The seek _{a c, i} by minimizing _{a c,} the _i (S1115). This minimization is

It is done by seeking. The amplitude / phase estimation unit 1115 performs this for each of i = 1, 2,..., Nc−1 (S1115). Expression (34) is obtained by extracting only the term related to the amplitude coefficient _{ac, i} to be estimated from Expression (14).
In addition, the amplitude / phase estimation unit 1115

The by minimizing the θ _{c, i,} determining the phase coefficients θ _{c, i} (S1115). This minimization is performed by a numerical calculation method or the like. The amplitude / phase estimation unit 1115 performs minimization for each of i = 1, 2,..., Nc−1 (S1115). Equation (36) is obtained by extracting only the term related to the phase coefficient θ _{c, i} to be estimated from Equation (14).

Amplitude phase estimation unit 1115, _{a c} stored in the parameter storage unit _112, the values of _i and theta _{c, i,} and updates the respective values of _{a c, i} and theta _{c, i} obtained by minimizing the above (S1115).

  As described above, according to the compressed signal restoration apparatus 1 of the present embodiment, unlike the conventional method, it is possible to perform restoration so as to avoid overfitting by sequentially and estimating the parameter estimation. The original signal can be restored with higher accuracy than before.

Next, the compressed signal decompression apparatus 2 according to the second embodiment will be described with reference to FIGS. FIG. 7 is a block diagram showing the configuration of the compressed signal decompression apparatus 2 of this embodiment. FIG. 8 is a flowchart showing the operation of the compressed signal decompression apparatus 2 of this embodiment. FIG. 9 is a block diagram illustrating the configuration of the signal estimation unit 211 of the compressed signal decompression apparatus 2 of the present embodiment. FIG. 10 is a flowchart showing the operation of the signal estimation unit 211 of the compressed signal decompression apparatus 2 of this embodiment. As shown in FIG. 9, the signal estimation unit 211 of the compressed signal restoration device 2 of the present embodiment has a configuration in which the amplitude phase estimation unit 1115 included in the signal estimation unit 111 of the compressed signal restoration device 1 of the first embodiment is omitted. Therefore, as shown in FIG. 10, the operation (S211) of the signal estimation unit 211 of the present embodiment is the same as that of the first embodiment except that step S1115 is omitted, and thus the description thereof is omitted. In this embodiment, step S1115 can be omitted by setting predetermined values for a _{c, i} and θ _{c, i} .

  As described above, according to the compressed signal restoration apparatus 2 of the present embodiment, the amplitude phase estimation process (step S1115) is omitted when it is not necessary to consider the amplitude and phase in advance or when the amplitude and phase are known. Thus, the same effect as in the first embodiment can be obtained.

  Next, the compressed signal decompression apparatus 3 according to the third embodiment will be described with reference to FIGS. 11 to 14. FIG. 11 is a block diagram showing the configuration of the compressed signal decompressing apparatus 3 of this embodiment. FIG. 12 is a flowchart showing the operation of the compressed signal decompressing apparatus 3 of this embodiment. FIG. 13 is a block diagram illustrating a configuration of the signal estimation unit 311 of the compressed signal decompression apparatus 3 according to the present embodiment. FIG. 14 is a flowchart showing the operation of the signal estimation unit 311 of the compressed signal decompression apparatus 3 of this embodiment. The signal estimation unit 311 of the compressed signal decompression apparatus 3 according to the present exemplary embodiment has a configuration in which the common signal estimation unit 1113 and the amplitude / phase estimation unit 1115 are omitted from the signal estimation unit 111 according to the first exemplary embodiment. Furthermore, the signal estimation unit 311 of this embodiment includes an element signal estimation unit 3112 instead of the element signal estimation unit 1112 of the first embodiment. Further, the parameter storage unit 112 of the first embodiment is changed to the parameter storage unit 312 in the present embodiment. Further, the multiplier update unit 113 of the first embodiment is changed to a multiplier update unit 313 in the present embodiment.

When the amplitude and phase are known as in the case of the second embodiment, it is not necessary to introduce _{ec and i} . Therefore, in this embodiment, in the signal estimation unit 311, instead of (Equation 14),

An example will be described in which the value of the signal is obtained by minimizing the objective function J expressed by the following equation for each of the signals x _i , z _{c, i} , z _{d, i} , s _{d, i} .

  In order to execute the above processing, in the present embodiment, each component of the first embodiment is changed as follows.

<Parameter storage unit 312>
The only difference from the parameter storage unit 112 of the first embodiment is that parameters γ, ec _{, i} , a _{c, i} , θ _{c, i} , ν _{c, i that} are not used in the equation (14 ′) are not stored. .

<Signal estimation unit 311>
The signal estimation unit 311 minimizes the objective function J represented by the above equation (14 ′) for each of the signals x _i , z _{c, i} , z _{d, i} , s _{d, i} , A value is obtained (S311). Then, the values of the signals x _i , z _{c, i} , z _{d, i} , s _{d, i} stored in the parameter storage unit 312 are updated to the values of the respective signals obtained by minimization (S311).

<Overall signal estimation unit 1111>
The configuration, input / output, and function of the overall signal estimation unit 1111 are the same as those in the first embodiment.

<Element signal estimation unit 3112>
The element signal estimation unit 3112 uses x _i , z _{c, i} , z _{c, i + 1} , z _{d, i} , λ _i , α, μ _{c, i} , β stored in the parameter storage unit 312 for each sensor. The estimated values z _{c, i} and z _{d, i} of the element signal are obtained and stored in the parameter storage unit 312 (S3112).

  Specifically, the element signal estimation unit 3112 determines that i = 1.

Is minimized with respect to z _{c, i} and z _{c, i} is output (S3112). This minimization is

It is done as follows.
In addition, the element signal estimation unit 3112 calculates i = 2, 3,..., Nc−1.

Is minimized with respect to z _{c and i} , and z _{c and i} are output (S3112). This minimization is

It is done as follows.
In addition, the element signal estimation unit 3112 determines that i = Nc.

Is minimized with respect to z _{c and i} , and z _{c and i} are output (S3112). This minimization is

It is done as follows.
Also elements signal estimation unit 3112 i = 1, 2, ..., for Nc, seeking _{z d, i} by minimizing the equation (28) _{z d,} for _i, stored in the parameter storage unit 312 z _d, updates the value of _{i z d} obtained by _minimizing, to the value of _i (S3112). This minimization is performed, for example, as shown in Equation (29).

<Individual signal estimation unit 1114>
The configuration, input / output, and function of the individual signal estimation unit 1114 are the same as those in the first embodiment.

<Multiplier update unit 313>
The multiplier update unit 313 updates the value of each multiplier according to the following equation instead of the equations (15) to (18) in the first embodiment (S313).

That is, the multiplier update unit 313 of the present embodiment is the same as the multiplier update unit 113 of the first embodiment, except that Expression (17) is not used and Expression (16) is replaced with Expression (16 ′). .

<End determination unit 114>
The configuration, input / output, and function of the end determination unit 114 are the same as those in the first embodiment.

  As described above, according to the compressed signal restoration apparatus 3 of the present embodiment, the common signal estimation process (step S1113), the amplitude, when it is not necessary to consider the amplitude and the phase in advance or when the amplitude and the phase are known. The same effect as that of the first embodiment can be obtained by omitting the phase estimation process (step S1115) and using the equation (14 ′) instead of the equation (14).

  Next, the compressed signal decompression apparatus 4 according to the fourth embodiment will be described with reference to FIGS. 15 to 18. FIG. 15 is a block diagram showing the configuration of the compressed signal decompressing apparatus 4 of this embodiment. FIG. 16 is a flowchart showing the operation of the compressed signal decompressing apparatus 4 of this embodiment. FIG. 17 is a block diagram illustrating a configuration of the signal estimation unit 411 of the compressed signal decompression apparatus 4 according to the present embodiment. FIG. 18 is a flowchart showing the operation of the signal estimation unit 411 of the compressed signal decompression apparatus 4 of this embodiment. The signal estimation unit 111, the parameter storage unit 112, and the multiplier update unit 113 in the first embodiment are changed to a signal estimation unit 411, a parameter storage unit 412, and a multiplier update unit 413 in the present embodiment, respectively. The overall signal estimator 1111, the element signal estimator 1112, the common signal estimator 1113, and the amplitude / phase estimator 1115 according to the first embodiment are the overall signal estimator 4111, the element signal estimator 4112, and the common signal estimator in the present embodiment, respectively. 4113 and the amplitude / phase estimation unit 4115 are changed.

In the first to third embodiments, it is assumed that there is one common signal (z _c ). However, the first embodiment can be extended even when there are a plurality of common signals. The case where there are a plurality of common signals is considered, for example, when each sensor observes a plurality of observation targets (sources) and acquires signals.
That, j = 1, 2, ..., as Ns (Ns total number of common signal), the signal _{x i} of the i-th sensor,

It can be described as follows. Ωs is a set of indexes of common signals Ωs = {1, 2,..., Ns}. Here, z _{c, i, k} are k-th common signals in the i-th sensor. Each common signal is

Suppose you can write in a relationship like Here, a _{i, k} is a coefficient representing the amplitude of the k-th common signal element in the i-th sensor for the common signal z _{c, k} , and θ _{c, k} is _k in the i-th sensor for z _c, k. The coefficient represents the phase of the th common signal element. a _{i, k and} θ _{c, k} are real scalar quantities, respectively.

  In order to execute the above processing, in the present embodiment, each component of the first embodiment is changed as follows.

<Parameter storage unit 412>
The parameter storage unit 412 stores a predetermined compression matrix A _i , observation matrix ψ _i , and base matrix Φ. Further, in the process of the decoding unit 41, parameters α, β, γ, σ, w necessary for obtaining an estimated value of the signal, and signals and coefficients x _i , z _{c, i, k} , z _{d, i , e c, i, k,} s d, i, a c, i, k, θ c, i, k, λ i, μ c, i, k, ν c, i, k, ξ d, i early Store the value in advance. Note that a random value may be set in advance as the initial value. Note that a _{c, i, k} is a coefficient representing the amplitude of the k-th common signal element in the i-th sensor with respect to the auxiliary common signal e _{c, i, k} , and θ _{c, i, k} is z _{c, i + 1,} a coefficient representing the phase of the k-th common signal component in the i-th sensor for _k.

<Signal estimation unit 411>
The signal estimation unit 411 replaces the equation (14) with the following equation (40), and converts the signal and coefficients x _i , z _{c, i, k} , z _{d, i} , ec _{, i, k} , s _{d, i} , A _{c, i, k} , θ _{c, i, k} are minimized to obtain signal and coefficient values (S411). Then, the signal stored in the parameter storage unit 412 and the coefficient _{x i, z c, i,} k, z d, i, e c, i, k, s d, i, a c, i, k, θ c, The values of _{i and k} are updated to the values of the respective signals and coefficients obtained by minimization.

<Overall signal estimation unit 4111>
The overall signal estimation unit 4111 receives the input signal y _i and z _{c, i, k} , z _{d, i} , λ _i , α, Ψ _i , Nc, Ns stored in the parameter storage unit 412 as inputs, The estimated value x _i of the decoded signal corresponding to y _i is obtained (S4111). Specifically, the overall signal estimation unit 4111

The search of _{x i} by minimizing the _{x i,} stored in the parameter storage unit 412 (S4111). This minimization is for example

It is executed as follows. The overall signal estimation unit 4111 obtains x _i by calculating Equation (42) for each of i = 1, 2,..., Nc (S4111).

<Element signal estimation unit 4112>
The element signal estimation unit 4112 stores x _i , z _{c, i, k} , z _{d, i} , λ _i , α, μ _{c, i, k} , a _{c, i, k} , e stored in the parameter storage unit 412. _{Using c, i, k} , β, the estimated values z _{c, i, k} and z _{d, i} of the element signals for each sensor are obtained and stored in the parameter storage unit 412 (S4112).
Specifically, the element signal estimation unit 4112 determines that i = 1.

Z _{c, i, k is} obtained by minimizing z with respect to z _{c, i, k} (S4112). This minimization is

It is done as follows. Here, Ωs \ k represents a set obtained by removing k from Ωs.
In addition, the element signal estimation unit 4112 determines that i = 2, 3,..., Nc−1.

Request _{z c, i, k} by minimizing the _{z c, i,} for _k (S4112). This minimization is

It is done as follows.
In addition, the element signal estimation unit 4112 determines that i = Nc.

Z _{c, i, k is} obtained by minimizing z with respect to z _{c, i, k} (S4112). This minimization is

It is done as follows.

  In addition, the element signal estimation unit 4112 has i = 1, 2,.

Is minimized with respect to z _{d, i} and z _{d, i} is output (S4112). This minimization is for example

It is done as follows.

The element signal estimation unit 4112 obtains the values of z _{c, i, k} and z _{d, i} obtained by the above-described minimization of the values of z _{c, i, k} and z _{d, i} stored in the parameter storage unit 412. (S4112).

<Common signal estimation unit 4113>
The common signal estimation unit 4113 is configured to store z _{c, i, k} , θ _{c, i, k} , μ _{c, i, k} , a _{c, i, k} , e _{c, i, k} , stored in the parameter storage unit 412. The auxiliary common signals ec _{, i, k} are estimated using β, ν _{c, i, k} , Λ, z _{c, i + 1, k} , γ (S4113). Specifically, the common signal estimation unit 4113 is

_{Ec, i, k is} obtained by minimizing ec _{, i, k} (S4113). This minimization is

It is done by seeking. The common signal estimation unit 4113 obtains this for each of i = 1, 2,..., Nc−1 (S4113). Common signal estimation unit 4113 updates _{e c} stored in the parameter storage unit _{412, i,} the value of _{k, e c} obtained by minimization of the _{above, i,} the value of _k (S4113).

<Individual signal estimation unit 1114>
The configuration, input / output, and function of the individual signal estimation unit 1114 are the same as those in the first embodiment.

<Amplitude phase estimation unit 4115>
The amplitude and phase estimation unit 4115 stores z _{c, i, k} , θ _{c, i, k} , μ _{c, i, k} , a _{c, i, k} , e _{c, i, k} , stored in the parameter storage unit 412. Using β, νc _{, i, k} , Λ, z _{c, i + 1, k} , γ, the amplitudes a _{c, i, k} and the phase coefficients θ _{c, i, k} are estimated (S4115). Specifically, the amplitude phase estimation unit 4115

A, _{c, i, k is} obtained by minimizing a with respect to a, _{c, i, k} (S4115). This minimization is

It is done by seeking. The amplitude / phase estimation unit 4115 performs this for each of i = 1, 2,..., Nc−1 (S4115).
In addition, the amplitude / phase estimation unit 4115

The θ _{c, i,} by minimizing the _k, the phase coefficients θ _{c, i,} obtains the _k (S4115). This minimization is performed by a numerical calculation method or the like. The amplitude phase estimation unit 4115 performs minimization for each of i = 1, 2,..., Nc−1 (S4115).

The amplitude / phase estimation unit 4115 obtains the values of a _{c, i, k} and θ _{c, i, k} stored in the parameter storage unit 412 by the above-described minimization, a _{c, i, k} and θ _{c, i. , K} are updated respectively (S4115).

<Multiplier update unit 413>
The multiplier updating unit 413 is configured such that the signal estimator 411 estimates λ _i , α, x _i , z _{c, i, k} , z _{d, i} , μ _{c, i, k} , β, ν _{c, i, k} , γ. , Ec _{, i, k} , ec _{, i-1, k} , _{sd, i} , Λ, _{ac, i, k} , θc _{, i-1, k} , ξd _{, i} , σ, Φ Then, the undetermined multipliers λ _i , μ _{c, i, k} , ν _{c, i, k} , ξ _{d, i} corresponding to each constraint condition of the equation (40) are updated (S413). Specifically, the multiplier update unit 413 updates the value of each multiplier according to the following equation (S413).

<End determination unit 114>
The configuration, input / output, and function of the end determination unit 114 are the same as those in the first embodiment.

  Thus, according to the compressed signal decompression apparatus 4 of the present embodiment, the same effects as those of the first embodiment can be obtained even when there are a plurality of common signals.

  Next, the compressed signal decompression apparatus 5 according to the fifth embodiment will be described with reference to FIGS. 19 to 22. FIG. 19 is a block diagram showing the configuration of the compressed signal decompression apparatus 5 of this embodiment. FIG. 20 is a flowchart showing the operation of the compressed signal decompression apparatus 5 of this embodiment. FIG. 21 is a block diagram illustrating a configuration of the signal estimation unit 511 of the compressed signal decompression apparatus 5 according to the present embodiment. FIG. 22 is a flowchart showing the operation of the signal estimation unit 511 of the compressed signal decompression apparatus 5 of this embodiment. The signal estimation unit 111 and the multiplier update unit 113 of the first embodiment are changed to a signal estimation unit 511 and a multiplier update unit 513 in the present embodiment, respectively. Further, the compressed signal decompression apparatus 5 of the present embodiment includes an index exchanging unit 515 and a correspondence table storage unit 516 that the compressed signal decompression apparatus 1 of the first embodiment does not have. The element signal estimation unit 1112, the common signal estimation unit 1113, and the amplitude / phase estimation unit 1115 of the first embodiment are changed to an element signal estimation unit 5112, a common signal estimation unit 5113, and an amplitude / phase estimation unit 5115, respectively, in this embodiment. .

  In the fourth embodiment, it is assumed that there are a plurality of common signals. However, in some cases, it is difficult to distinguish between the common signals (there is ambiguity), so that optimization may not be successful. is there. In the fifth embodiment, in order to avoid this, the common signal index is changed so that the power of the common signal is increased.

  In order to execute the above processing, in the present embodiment, each component of the first embodiment is changed as follows.

<Correspondence table storage unit 516>
The correspondence table storage unit 516 stores a correspondence table representing a correspondence relationship between the index of the common signal before the conversion and the index after the conversion (after rearranging in the order of power). The initial value of the correspondence table is such that the index before conversion and the index after conversion are the same (no change).

<Index switching unit 515>
The index exchanging unit 515 receives z _{c, i, k} from the element signal estimation unit 5112 and pays attention to any predetermined i-th sensor, z _c for each k = 1, 2,..., Ns. _{, I, k} are calculated (S515). The index switching unit 515 then converts the index k before rearrangement so that the post-conversion index k ′ = 1, 2,..., Ns is in the descending order of the L2 norm of z _{c, i, k ′.} The correspondence relationship with the converted index k ′ is obtained, and the content of the correspondence table in the correspondence table storage unit 516 is updated (S515).

  The element signal estimator 5112, the common signal estimator 5113, the amplitude phase estimator 5115, and the multiplier updater 513 basically perform the same operations as those in the fourth embodiment, but for the index k of the common signal, the correspondence table storage unit Referring to the correspondence table stored in 516, the calculation is performed after replacing each variable in the order of the converted index. The configuration, input / output, and functions of other parts are the same as those in the first embodiment.

  Thus, according to the compressed signal decompression apparatus 5 of the present embodiment, even when it is difficult to distinguish between the common signals, the common signal indexes are switched so that the powers of the common signals are in descending order. Thus, optimization can be performed.

  In Examples 1 to 4, the sparsity of individual signals was assumed. In this embodiment, this restriction is eliminated. Specifically, as the signal estimation unit 611,

Or

Can be minimized as an objective function. In this case, the individual signal estimation unit 1114 may be removed from each embodiment, and each variable corresponding to each of the equations (60) to (62) may be minimized.

In the first to fourth embodiments, the weight w _i is a fixed value, but the w _i may be updated by a method such as FPC (Fixed-Point Continuation, Non-Patent Document 1). In the multiplier updating unit 713, the weight coefficient w _i is further updated.

<Multiplier update unit 713>
The multiplier update unit 713 updates the weight w _i stored in the parameter storage unit based on the following update formula in addition to the processing of the multiplier update units 113 to 413 described in any of the first to fourth embodiments ( S713).

Here, η is a predetermined constant. However, when w _i is smaller than a predetermined lower limit value w _min, w _min may be output.

  Whether or not to perform the above-described weight update process

You may judge by. That is, it is possible to perform a weight update process according to the expression (63) if the expression (64) is true, and not perform a weight update process if the expression (64) is false. Here, ε _{d, i} is a predetermined constant.

<End determination unit 714>
In addition to the equation (19), the end determination unit 714 uses the following equation as the “predetermined end condition” in the end determination unit 114 of the first embodiment.

Is used. That is, when the condition that “the number k of repetitions of the signal estimation unit reaches a predetermined number It _max ” and “the updated weight w _i is smaller than w _min ” is satisfied, “the predetermined condition is satisfied Is determined.

  In addition, the various processes described above are not only executed in time series according to the description, but may be executed in parallel or individually according to the processing capability of the apparatus that executes the processes or as necessary. Needless to say, other modifications are possible without departing from the spirit of the present invention.

  Further, when the above-described configuration is realized by a computer, processing contents of functions that each device should have are described by a program. The processing functions are realized on the computer by executing the program on the computer.

  The program describing the processing contents can be recorded on a computer-readable recording medium. As the computer-readable recording medium, for example, any recording medium such as a magnetic recording device, an optical disk, a magneto-optical recording medium, and a semiconductor memory may be used.

  The program is distributed by selling, transferring, or lending a portable recording medium such as a DVD or CD-ROM in which the program is recorded. Furthermore, the program may be distributed by storing the program in a storage device of the server computer and transferring the program from the server computer to another computer via a network.

  A computer that executes such a program first stores, for example, a program recorded on a portable recording medium or a program transferred from a server computer in its own storage device. When executing the process, the computer reads a program stored in its own recording medium and executes a process according to the read program. As another execution form of the program, the computer may directly read the program from a portable recording medium and execute processing according to the program, and the program is transferred from the server computer to the computer. Each time, the processing according to the received program may be executed sequentially. Also, the program is not transferred from the server computer to the computer, and the above-described processing is executed by a so-called ASP (Application Service Provider) type service that realizes the processing function only by the execution instruction and result acquisition. It is good.

  Note that the program in this embodiment includes information that is used for processing by an electronic computer and that conforms to the program (data that is not a direct command to the computer but has a property that defines the processing of the computer). In this embodiment, the present apparatus is configured by executing a predetermined program on a computer. However, at least a part of these processing contents may be realized by hardware.

Claims (8)

A compressed signal decompression device for decompressing compressed signals from Nc sensors,
An index i is an integer of 1 to Nc, a signal common to all sensors is a signal z _c , a common part of signals observed by the i-th sensor is a common signal z _{c, i} , The remaining signal excluding the common signal z _{c, i} among the signals observed by the th sensor is represented as a signal z _{d, i} ,

The i-th signal observation signals x _i of the sensor is compressed by a predetermined observation matrix [psi _i y _{i =} [psi _i x _i, represented in an input,
Let e _{c, i be} the auxiliary common signal of the i-th sensor, a _{c, i be} a coefficient representing the amplitude of the common signal element of the i-th sensor with respect to the auxiliary common signal e _{c, i} , and θ _{c, i} be The coefficient representing the phase of the common signal element of the i-th sensor with respect to z _{c, i + 1} , Φ is a predetermined base matrix, and the expression s _c = Φ ⁻¹ z _c of the common signal z _{c in} the base Φ is not sparse, the signal _{z d,} representation _{s d} in basal [Phi of _{i, i} = [Phi ^-1 z _{d, i} is assumed to be sparse with respect to the index i of all sensor, ‖ - ‖ ₂ vectors and L2 norm is a sum of squares of the elements, ‖ - ‖ ₁ is L1 norm is the absolute value sum of each element of the vector, the set of indices i and Omega, ie Ω = {1,2, ···, Nc }, W _i is a weight for a predetermined individual signal s _{d, i} , and ∀ is a generic symbol J is an imaginary unit, DFT is a discrete Fourier transform, IDFT is an inverse transform of DFT, and the symbol Λ is

And define

Under the constraints of

A compressed signal decompression apparatus that includes a decoding unit that obtains an observation signal x _i that minimizes and outputs it as a decoded signal of the original signal. The compressed signal decompression device according to claim 1,
λ _i , μ _{c, i} , ν _{c, i} , ξ _{d, i} are assumed to be undetermined multipliers for the above constraints, and α, β, γ, σ are adjusted for convergence speed when solving the minimization problem And T represents the transpose of the matrix or vector,
The decoding unit

Is minimized for each of the signal and the coefficients x _i , z _{c, i} , z _{d, i} , e _{c, i} , s _{d, i} , a _{c, i} , θ _{c, i.} A signal estimator for estimating the signals and coefficients x _i , z _{c, i} , z _{d, i} , e _{c, i} , s _{d, i} , a _{c, i} , θ _{c, i} ,
A multiplier updating unit for updating the undetermined multipliers λ _i , μ _{c, i} , ν _{c, i} , ξ _{d, i} based on the estimated signals and coefficients;
Until the predetermined condition is satisfied, the signal estimation unit, and an end determination unit that repeatedly controls the processing of the multiplier update unit,
A compressed signal decompression apparatus including: The compressed signal decompression device according to claim 2,
* Is a symbol representing a Hermitian transpose matrix, I is a unit matrix of size N, and the signal estimation unit is

An overall signal estimator for obtaining the signal x _i by
For i = 1,

For i = 2, 3,..., Nc-1,

For i = Nc,

To obtain the signal z _{c, i} ,
For i = 1, 2,..., Nc,

An element signal estimator for obtaining signals z _{d, i} by
For i = 1, 2,..., Nc−1,

A common signal estimator for obtaining signals _{ec, i} by
sgn is a function that outputs a code for each element, ◯ is a symbol that represents multiplication for each element, and max {| · |, 0} is calculated for each element,
For i = 1, 2,..., Nc,

An individual signal estimation unit for obtaining the signal s _{d, i} by
For i = 1, 2,..., Nc−1,

To obtain the coefficients a _{c, i} ,

An amplitude phase estimation unit for obtaining the coefficients theta _{c, i} by minimizing On θ _{c, i,}
A compressed signal decompression apparatus including: The compressed signal decompression device according to claim 1,
λ _i , μ _{c, i} , ξ _{d, i} are undetermined multipliers for the constraint conditions, α, β, σ are weights for adjusting the convergence speed when solving the minimization problem, and T Denote the transpose of a matrix or vector,
The decoding unit

Is minimized for each of the signal and the coefficients x _i , z _{c, i} , z _{d, i} , s _{d, i} , so that the signal and the coefficients x _i , z _{c, i} , z a signal estimator for estimating _{d, i} , s _{d, i} ;
A multiplier updating unit for updating the undetermined multipliers λ _i , μ _{c, i} , ξ _{d, i} based on the estimated signals and coefficients;
Until the predetermined condition is satisfied, the signal estimation unit, and an end determination unit that repeatedly controls the processing of the multiplier update unit,
A compressed signal decompression apparatus including: A compressed signal decompression device for decompressing compressed signals from Nc sensors,
An index i is an integer of 1 to Nc, a signal common to all sensors is a signal z _c , a common part of signals observed by the i-th sensor is a common signal z _{c, i} , The k-th common signal in the i-th sensor is expressed as z _{c, i, k,} and the remaining signals excluding the common signal z _{c, i} among the signals observed by the i-th sensor are the signals z _{d, i.} Ns is the total number of common signals, Ωs is a set of common signal indexes, ie, Ωs = {1, 2,..., Ns}, j = 1, 2,. Is the imaginary unit, DFT is the discrete Fourier transform, IDFT is the inverse transform of DFT, and the symbol Λ is

And define

The observed signal _{x i} of the i th sensor as input compressed signal _{y i} = [psi _i x _i by a predetermined observation matrix [psi _i represented _{in, a i,} common _k signal _{z c,} the i for _k real number representing the th and k-th common signal is a scalar real number representing the amplitude component coefficient in the sensors, theta _{c, k} and z _c, of the k-th common signal component in the i-th sensor for _{the k} phase And λ _i , μ _{c, i, k} , ν _{c, i, k} , ξ _{d, i} are undetermined multipliers for the constraints, and α, β, γ, σ are , And a weight for adjusting the convergence speed when solving the minimization problem, and T represents a matrix or vector transpose,
e _{c, i,} and _k and k-th auxiliary common signal at the i th _{sensor, a c, i, k} the auxiliary common signal _{e c, i,} k-th common signal component in the i-th sensor for _{the k} , _{C, i, k} is a coefficient representing the phase of the k-th common signal element in the i-th sensor with respect to z _{c, i + 1, k} , Φ is a predetermined base matrix, common signal _z expressed in basal [Phi of _{c s c} = Φ ^-1 z _c is not sparse, the signal _{z d,} representation _{s d} in basal [Phi of ^{_{i, i = Φ -1 z d}} , i all sensors ‖ · ‖ ₂ is the L2 norm that is the sum of squares of the elements of the vector, ‖ · ‖ ₁ is the L1 norm that is the sum of the absolute values of the elements of the vector, and Ω Is a set of sensor indexes i, that is, Ω = {1, 2,..., Nc}. And, Omega ^- the set of indices i, except for Nc, i.e. ^{Ω - = {1,2, ···,} Nc-1} and the individual signals predetermining _{w i s d,} the weight for _i,

Output in an objective function _{H 2} represented, signal and coefficient _{x i, z c, i,} k, z d, i, s d, by minimizing for each _i, as a decoded signal of the original signal A compressed signal decompression apparatus including a decoding unit. A compressed signal decompression method for decompressing compressed signals from Nc sensors,
An index i is an integer of 1 to Nc, a signal common to all sensors is a signal z _c , a common part of signals observed by the i-th sensor is a common signal z _{c, i} , The remaining signal excluding the common signal z _{c, i} among the signals observed by the th sensor is represented as a signal z _{d, i} ,

The i-th signal observation signals x _i of the sensor is compressed by a predetermined observation matrix [psi _i y _{i =} [psi _i x _i, represented in an input,
Let e _{c, i be} the auxiliary common signal of the i-th sensor, a _{c, i be} a coefficient representing the amplitude of the common signal element of the i-th sensor with respect to the auxiliary common signal e _{c, i} , and θ _{c, i} be The coefficient representing the phase of the common signal element of the i-th sensor with respect to z _{c, i + 1} , Φ is a predetermined base matrix, and the expression s _c = Φ ⁻¹ z _c of the common signal z _{c in} the base Φ is not sparse, the signal _{z d,} representation _{s d} in basal [Phi of _{i, i} = [Phi ^-1 z _{d, i} is assumed to be sparse with respect to the index i of all sensor, ‖ - ‖ ₂ vectors and L2 norm is a sum of squares of the elements, ‖ - ‖ ₁ is L1 norm is the absolute value sum of each element of the vector, the set of indices i and Omega, ie Ω = {1,2, ···, Nc }, W _i is a weight for a predetermined individual signal s _{d, i} , and ∀ is a generic symbol J is an imaginary unit, DFT is a discrete Fourier transform, IDFT is an inverse transform of DFT, and the symbol Λ is

And define

Under the constraints of

A compressed signal restoration method including a decoding step of obtaining an observation signal x _i that minimizes and outputting as a decoded signal of the original signal. The compressed signal decompression method according to claim 6,
λ _i , μ _{c, i} , ν _{c, i} , ξ _{d, i} are assumed to be undetermined multipliers for the above constraints, and α, β, γ, σ are adjusted for convergence speed when solving the minimization problem And T represents the transpose of the matrix or vector,
The decoding step includes

Is minimized for each of the signal and the coefficients x _i , z _{c, i} , z _{d, i} , e _{c, i} , s _{d, i} , a _{c, i} , θ _{c, i.} A signal estimation sub-step for estimating the signals and coefficients x _i , z _{c, i} , z _{d, i} , e _{c, i} , s _{d, i} , a _{c, i} , θ _{c, i} ,
A multiplier update sub-step for updating the undetermined multipliers λ _i , μ _{c, i} , ν _{c, i} , ξ _{d, i} based on the estimated signals and coefficients;
An end determination substep for repeatedly controlling the signal estimation substep and the multiplier update substep until a predetermined condition is satisfied;
A compressed signal decompression method comprising:   The program which gives the instruction | command which should perform the compressed signal decompression | restoration method of Claim 6 or 7 with respect to a computer.

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