KR101401388B1 - Method for Signal Processing with Toeplitz Jacket Matrices - Google Patents

Abstract

Provided is a signal processing method using a Toeplitz Jacket (TJ) matrix. The communications method according to an embodiment of the present invention converts an input signal by using the TJ matrix. The TJ matrix is generated by pre-multiplying a left side of a circulant Jacket matrix by a reverse matrix of a diagonal matrix and post-multiplying a right side of the circulant Jacket matrix by the diagonal matrix. Accordingly, the present invention can process various signals more effectively, and enhance the system performance ultimately.

Description

[0001] The present invention relates to a signal processing method using a Toeplitz Jacket matrix,

The present invention relates to signal processing, and more particularly, to a method and system for processing signals using a Toeplitz matrix.

The Toeplitz matrix is the following n × n matrix T:

에 대해

이다.here,

About

to be.

This Toeplitz matrix can be applied to various fields. That is to say, when solving differential and integral equations, finding answers in physics, mathematics, statistics, and complex information theory, finding the channel capacity of a Gaussian channel, solving the distortion function of an autoregressive source In addition, it has been applied variously to derive the results related to the eigenvalue distribution theorem, encoding region, spectrum estimation, watermarking, combination analysis, voice enhancement, interference cancellation, image restoration, sensor network, adaptive filtering, graphic model and noise cancellation.

Here, the mentioned Toeplitz matrix is classified into two types as follows.

에 대해

이다.The first classification is formed by a circulation matrix, where each row vector is rotated from left to right in relation to the previous column vector. In particular, the rotation as in equation (1)

About

to be.

에 대해

이다.
The second Toeplitz matrix is a negacycle matrix. Here,

About

to be.

SUMMARY OF THE INVENTION It is an object of the present invention to provide various signal processing methods and systems using a TJ (Toeplitz Jacket) matrix.

According to an aspect of the present invention, there is provided a communication method including: transforming an input signal using a Toeplitz Jacket (TJ) matrix; And modulating and transmitting the transformed signal in the transforming step, wherein the TJ matrix multiplies the left side of the Jacket circulation matrix by an inverse matrix of the diagonal matrix, and a matrix obtained by multiplying the right side of the circulation matrix by the diagonal matrix to be.

In the diagonal matrix, the element of (n, n) may be k ^{n -1} .

In addition, the diagonal matrix may have 0 elements other than the elements of (n, n).

The TJ matrix may be the following 4x4 matrix.

Also, the TJ matrix may be the following 8x8 matrix.

Receiving the converted signal in the converting step; And inversely transforming the received signal using an inverse matrix of the TJ matrix.

According to another aspect of the present invention, there is provided a communication method comprising: transforming an input signal using an inverse matrix of a TJ (Toeplitz Jacket) matrix; And modulating and transmitting the transformed signal in the transforming step, wherein the TJ matrix multiplies the left side of the Jacket circulation matrix by an inverse matrix of the diagonal matrix, and a matrix obtained by multiplying the right side of the circulation matrix by the diagonal matrix to be.

In the diagonal matrix, the element of (n, n) may be k ^{n -1} .

In addition, the diagonal matrix may have 0 elements other than the elements of (n, n).

Receiving the converted signal in the converting step; And inversely transforming the received signal using the TJ matrix.

As described above, according to the embodiments of the present invention, it is possible to perform various signal processing more effectively using a TJ (Toeplitz Jacket) matrix, and ultimately improve the performance of the system.

의 고속 알고리즘, 그리고,
도 3은 본 발명이 적용가능한 통신 시스템을 도시한 도면이다.1 shows a high-speed algorithm of [TJ] ₄ ,
2 is a cross-

Speed algorithm,
3 is a diagram showing a communication system to which the present invention is applicable.

Hereinafter, the present invention will be described in detail with reference to the drawings.

을 만족하면 행렬 M을 Jacket 행렬이라 한다. 즉, 역행렬은 원소단위의 역과 transpose로 얻어진다. 이에 대한 동치(equivalent)형태는 Jacket 행렬이 다음 관계를 만족할 때이다.If the n-th order matrix M is the inverse of its inverse matrix

The matrix M is called a Jacket matrix. In other words, the inverse matrix is obtained by the inverse of the element unit and the transpose. The equivalent form for this is when the Jacket matrix satisfies the following relation:

The Toeplitz Circulant matrix is applied in several places. For example, x in the following equation represents the input as a column vector, and when i> 0, t _i is zero.

Therefore, the vector can be expressed as follows.

The matrix elements are as follows.

.

.

The vector y represents the response of a discrete-time causal time-invariant filter with impulse response t _k .

에 대해

되도록 한 경우가 있다. 이러한 경우를 행렬로 표시해보면 다음과 같이 나타낼 수 있다.A special case of the Toeplitz matrix is that each row of the matrix moves to the right one cycle from the top row

About

. If this case is represented by a matrix, it can be expressed as follows.

This type of matrix is called a circulant matrix. This matrix is used for cyclic code study for error correction including DFT (Discrete Fourier Transform).

의 시퀀스에 대한 Szego의 정리이다. 다음식과 같은 영(zero)이 아닌 벡터 x가 존재한다면 복소수 스칼라 α는 행렬 A의 고유값이다.The Toeplitz matrix is a Toeplitz matrix that deals with the role of eigenvalues as n increases infinitely

Szego's theorem on the sequence of The complex scalar α is the eigenvalue of matrix A if there exists a nonzero vector x such as:

In this case, x is an eigenvector of A.

가

와 같이 감소하지 않는 형태의 순서로 되어 있는 경우, 이는 적분을 이용하여 근사화될 수 있으며, Hermitian Toeplitz 행렬

의 시퀀스에 대한 고유값

의 점근적 역할을 다루는 Szego의 정리에 일반성이 손실되지 않음을 의미한다. 이 정리는 몇 가지 만족해야할 기술적인 요구조건이 있다. 예를 들면, 다음 식에 의해 서로 관련되어 있는 계수가 t_k인 Fourier series가 존재해야 한다는 조건이다.Eigenvalue

end

, It can be approximated using integration, and the Hermitian Toeplitz matrix

Eigenvalues for the sequence of

The generalizability is not lost in Szego's theorem which deals with the asymptotic role of. This theorem has some technical requirements to be satisfied. For example, the condition that a Fourier series with a coefficient t _k , which are related to each other by the following equation, must exist.

는 함수 f를 결정하고, 그 역도 마찬가지로 가능하다. 따라서, 행렬의 시퀀스는 가끔 T_n(f)로 정의된다. 만약 T_n(f)가 Hermitian이라면 즉, T_n(f)^*=T_n(f)이라면, t_-k = t^* _k이고 f는 실수 값이다.So,

Determines the function f, and vice versa. Thus, the sequence of the matrix is sometimes defined as T _n (f). If T _n (f) is Hermitian, ie, T _n (f) ^* = T _n (f) then t _-k = t ^* _k and f is a real number.

Assuming a reasonable assumption, Szego's theorem for the continuous function F in the range of f can be expressed as:

On the other hand, the Jacket matrix does not contain any zero elements. A Jacket matrix with all elements of modulo 1 is called a complex Hadamard matrix. If K is a Jacket matrix, the hybrid matrix H = PDKD ^T P ^{T is} also a Jacket matrix for all permutation matrices P, P ^T and all invertible diagonal matrices D, D ^T.

The Jacket matrix expressed in this form is called equivalent. Finding all the Jacket matrices up to the equiva- lence is a very difficult problem and so far we have found it until the order n ≤ 5.

Jacket matrices can be applied theoretically from harmonic analysis to quantum information theory and also applied to signal processing. In particular, it is used to realize spatial multiplexing of a wireless communication network and to analyze unitary beamforming in a Multiple-Input Multiple-Output (MIMO) system.

The case of the Toeplitz Jacket matrix is of interest as a way to obtain a new example of a Complex Hadamard matrix. Because the general Toeplitz matrix has n-1 degrees of freedom compared to the degree of freedom of the circulating matrix, it seems plausible at a glance. However, it has been found that the nature of Jacket imposes a serious limitation on the structure of the Toeplitz matrix, and the following results must be proved.

Theorem 1. All Toeplitz Jacket matrices correspond to a cyclic Jacket matrix.

Cycle and Toeplitz Jacket Matrix: An example of a cyclic Jacket matrix is as follows.

을 링(ring)

의 invertible 원소라 놓으면,

에 대해 다음과 같은 원소를 갖는 길이가 n인 행벡터

를 상정한다.Example 1. Let n≥2 be an integral number,

Ring,

If you put invertible circle of,

A row vector having a length n having the following elements

.

인 순환 complex Hadamard 행렬은 DFT행렬과 일치한다.Here, i means a complex imaginary number. The first row

The cyclic complex Hadamard matrix corresponds to the DFT matrix.

Prove another example of a cyclic Jacket matrix as follows.

은 일종의 순환 Jacket 행렬이 된다. 단, n > 4 인 경우 치환행렬 P로부터 발생하는 행렬은 unimodular 원소를 갖지 않기 때문에 complex Hadamard 행렬이 아님을 주의하여야 한다.Example 2. Let I _n and J _n as each unit matrix with every element of 1, n-order matrix, the roots of the quadratic equation α _n ^{α 2 + (n-2)} α + 1 = 0. Then,

Becomes a kind of cyclic Jacket matrix. Note that the matrices generated from the permutation matrix P are not complex Hadamard matrices since they do not have unimodular elements.

Finally, if n = 4, we prove the following example.

Example 3. When a, b, and c are non-zero complex numbers, all of the following matrices are 4-dimensional Toeplitz Jacket matrices.

이

의 원소단위의 역행렬(element- wise inverse) 이라면

임을 쉽게 알 수 있다.if,

this

Wise inverse of the element-wise inverse of

.

이 대각 원소가

인

대각 행렬이라 한다.In order to propose a new Toeplitz matrix from the old Toeplitz matrix,

This diagonal element

sign

Diagonal matrix.

Proposition 1. If T _n is an n-dimensional Toeplitz matrix, a is an arbitrary number, and b is a non-zero complex number, then the following equation is also the Toeplitz matrix.

임을 쉽게 알 수 있다.Proof: The (i, j) element of T ' _n

.

Proposal 1. Assuming that T _n is a Toeplitz Jacket matrix [TJ] _n , we can express as follows by Equation (1).

일 때 만족함을 쉽게 알 수 있고,

일 때 또한 만족함을 보임으로서 증명을 시작한다. T의 첫 번째 두 개의 행 내에서 식 (2)의 조건을 고려하면, 즉,

인 경우 다음 식을 얻는다.Proof: The equation (9)

When you are satisfied you can easily find out,

And also demonstrate satisfaction when it starts. Considering the condition of equation (2) in the first two rows of T, that is,

The following equation is obtained.

의 쌍에 대해 식 (2) 를 고려하면 다음 식을 유도 할 수 있다.Next row

(2), the following equation can be derived.

을 더하고 식 (10) 를 이용하면 결론적으로 식 (9)가

일 때 역시 만족함을 알 수 있다. 특히,

일 때 다음식의 결과를 얻을 수 있다.On both sides

(10), and consequently, equation (9)

It can be seen that it is also satisfactory. Especially,

The result of the following can be obtained.

의 경우에 식 (12)가 만족됨을 증명한다. 이 식이 어떤 l에 대해 만족한다고 가정하면 행

의 쌍에 대해 식 (2)의 조건을 다음과 같이 생각해볼 수 있다.Now through mathematical induction

(12) is satisfied. Assuming that this expression is satisfactory for any l,

The condition of equation (2) can be considered as follows.

을 양측에 더하고 식 (10)를 통해 간소화하면 l+1 에 대해 결론적으로 식 (12)의 검증에 이른다. 결국 식 (12)가 연속되는 t_-1과 t_-l-1항을 연결시켜 주는 것을 확인함으로써 증명이 완료된다.Missing Port

Is added to both sides and simplified by Eq. (10), the conclusion of Eq. (12) is finally obtained for 1 + 1. Finally, the proof is completed by confirming that (12) connects the consecutive t _-1 and t _-l-1 terms.

As a result of Proposition 1 and Proposition 1, the number of free parameters in the Toeplitz Jacket matrix is at least n-1, exactly the same as in the circulation matrix. This is not normal, but can be seen as a factor that assures the effectiveness of theorem 1. I intend to prove this again.

이라 놓는다. 여기서

은 n 번째 근을 의미한다. 명제 1에 의해 다음 행렬은 C 는 Toeplitz 행렬이다.Proof of theorem 1: Let T _n be a Toeplitz matrix as in equation (1)

. here

Means the nth root. The following matrix, by Proposition 1, is the Toeplitz matrix.

에 대해

임을 보임으로서 충분하다.

인 것은 확실하다. 반면에 제안 1 에 의하면 다음과 같은 결론을 얻을 수 있다.In addition, this matrix is a circular matrix. To confirm this,

About

It is enough to show that.

It is sure to be. On the other hand, according to Proposal 1, the following conclusions can be obtained.

Theorem 1 gives a direct way to obtain the Toeplitz Jacket matrix from the circulating Jacket matrix as follows.

은 정수, a 와 b는 영이 아닌 복소수라 하고 C_n은 n 차 순환 Jacket 행렬이라 하자, 그러면 다음 행렬Inference 1.

Let a and b be a complex number, not a zero, and C _{n be} an nth order cyclic Jacket matrix, then the next matrix

Becomes an n-order Toeplitz Jacket matrix with two free parameters.

We will now briefly discuss the Toeplitz real Hadamard matrix.

Inference 2. If H is the Toeplitz real Hadamard matrix, H is either cyclic or negacyclic.

일 때

이 되어 행렬은 순환한다. 반면에

이면

이 되어 negacycle이 된다.Proof: From the proposal 1, the following conclusion can be obtained. If the share If (9)

when

And the matrix is circulated. On the other hand

If

Becomes a negative cycle.

인 경우에는 real 순환 Hadamard 행렬이 존재하지 않는다.

, There is no real circulating Hadamard matrix.

Example 4. The following matrix is called the Toeplitz real Hadamard matrix and is equivalent.

In order to construct a Toeplitz Jacket matrix, a Jacket circulation matrix is required first, and any Jacket circulation matrix can be used to construct this matrix.

3.1. 4 × 4 Toeplitz Jacket matrix

To construct a 4 × 4 Toeplitz Jacket matrix, we first need a 4 × 4 circulant matrix. The following 4 × 4 circulation matrix C ₄ is selected.

The following diagonal matrix D4 is multiplied by C4 to the left and right to construct a 4 × 4 Toeplitz Jacket matrix [TJ] ₄ .

를 곱하면 다음과 같이 단위행렬이 됨을 알 수 있다.Therefore, [TJ] ₄ and

, We can see that it becomes a unit matrix as follows.

의 고속 알고리즘이 도 2에 도시되어 있다.
A fast algorithm of [TJ] ₄ is shown in Figure 1,

2 is shown in FIG.

3.2. 8 × 8 Toeplitz Jacket matrix

를 다음과 같이 구성한다. To construct an 8 × 8 Toeplitz Jacket matrix, an 8 × 8 Jacket circulation matrix is required. For this, we set n = 8, a = 1, b = 0,

Is constructed as follows.

This vector can be rewritten as:

The matrix consisting of the first row of the circulant matrix x (1,0) is called a cyclic jacket matrix C ₈ , and can be expressed as follows.

Here, C ₈ does not include free parameters. Thus, we can construct an 8-dimensional, 2-parameter Toeplitz Jacket matrix using C ₈ in a similar way to 4 × 4.

과

를 곱하면 단위행렬이 됨을 알 수 있다.In the case of 8 × 8, as shown in equation (22)

and

It can be seen that the matrix is a unit matrix.

3 is a communication system to which the present invention is applicable. 3, a communication system to which the present invention can be applied includes a transmitting end 100 including a coding unit 110 and a transmission unit 120, a receiving end including a receiving unit 210 and a decoding unit 220 200 are connected to each other so that they can communicate with each other.

The encoding unit 110 of the transmitting terminal 100 encodes the input signal by converting the input signal by multiplying the input signal by the TJ matrix and the transmitting unit 120 modulates and transmits the encoded signal by the encoding unit 110. [

The receiving unit 210 of the receiving end 200 receives and demodulates the signal transmitted from the transmitting unit 120. The decoding unit 220 multiplies the receiving signal received from the receiving end 200 by the inverse matrix of the TJ matrix, And decodes it.

The decoding unit 220 of the receiving end 200 multiplies the received signal by the TJ matrix and outputs the resultant signal by multiplying the received signal by the TJ matrix, It is also possible to implement decoding by converting the signal.

Also, the TJ matrix can be applied to signal processing other than coding and decoding of signals.

While the present invention has been particularly shown and described with reference to exemplary embodiments thereof, it is to be understood that the invention is not limited to the disclosed exemplary embodiments, but, on the contrary, It will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the present invention.

100: Transmitting terminal 110:
120: transmission unit 200:
210; Receiving unit 220:

Claims (10)

Transforming an input signal using a TJ (Toeplitz Jacket) matrix; And
And modulating and transmitting the converted signal in the converting step,
Wherein the TJ matrix comprises:
Wherein the left side of the Jacket circulation matrix is multiplied by the inverse of the diagonal matrix, and the right side of the Jacket circulation matrix is multiplied by the diagonal matrix, and the 4x4 matrix is the following matrix.

The method according to claim 1,
Wherein the diagonal matrix comprises:
(n, n) is k ^{n -1} .
3. The method of claim 2,
Wherein the diagonal matrix comprises:
(n, n) is zero.
delete delete The method according to claim 1,
Receiving the converted signal in the conversion step; And
And inversely transforming the received signal using an inverse matrix of the TJ matrix.
Transforming an input signal using an inverse matrix of a TJ (Toeplitz Jacket) matrix; And
And modulating and transmitting the converted signal in the converting step,
Wherein the TJ matrix comprises:
Wherein the left side of the Jacket circulation matrix is multiplied by the inverse of the diagonal matrix, and the right side of the Jacket circulation matrix is multiplied by the diagonal matrix, and the 4x4 matrix is the following matrix.

8. The method of claim 7,
Wherein the diagonal matrix comprises:
(n, n) is k ^{n -1} .
9. The method of claim 8,
Wherein the diagonal matrix comprises:
(n, n) is zero.
8. The method of claim 7,
Receiving the converted signal in the conversion step; And
And inversely transforming the received signal using the TJ matrix.

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