An MMSE Channel Estimation Algorithm Based on AR Model

Abstract

The present invention belongs to the technical field of OFDM communication, and specifically relates to an MMSE channel estimation algorithm based on the AR model. First, the present invention performs circular extrapolation on the channel in the frequency domain, so that its frequency-domain autocorrelation matrix changes from a Toeplitz matrix to a circulant matrix. Then, the circulant AR model is used to model the channel, and the autocorrelation matrix of the extrapolated channel is solved. Since extrapolation makes the autocorrelation matrix a circulant matrix, the complexity of the matrix inversion operation in the AR-MMSE algorithm is greatly reduced; moreover, the AR model can well fit peak spectra of different shapes; the simulation results show that for various channel types, the method of the present invention does not require additional parameter tuning, and compared with the classical MMSE estimation algorithm, it can achieve good working performance and demonstrates excellent robustness.

Description

Technical Field

[0001] The present invention belongs to the technical field of OFDM communication, and particularly relates to an MMSE channel estimation algorithm based on an AR model. Background Art

[0002] In the problem of channel estimation in wireless communication, traditional MMSE channel estimation needs to assume the statistical characteristics of the channel. For example, it is usually assumed that the channel delay power spectrum (PDP) is exponentially distributed or uniformly distributed to calculate the autocorrelation of the channel, and then the channel estimator is obtained according to the MMSE criterion. Therefore, the performance of traditional MMSE channel estimation depends to a large extent on whether the assumed model matches the actual channel. When the real channel does not conform to the assumed distribution, a large estimation error will occur.

[0003] Given a segment of signal, the autoregressive model (AR model) can automatically fit the distribution of the signal spectrum and can calculate all the autocorrelation coefficients of the signal based on a small number of samples. Therefore, the present invention uses the AR model to fit the power delay spectrum of the frequency-domain channel to be estimated, and then calculates the frequency-domain autocorrelation of the channel. Summary of the Invention

[0004] The purpose of the present invention is to provide an MMSE channel estimation algorithm based on an AR model with strong robustness and low complexity.

[0005] The MMSE channel estimation algorithm based on the AR model provided by the present invention is abbreviated as the AR-MMSE channel estimation method. First, the channel to be estimated is extrapolated into a cyclic structure, and the cyclic AR model is used to calculate the autocorrelation of the circular channel; then, MMSE estimation is performed. When calculating MMSE (the minimum mean square error of the data carrier channel), the inversion of the autocorrelation matrix is involved. Since the channel becomes circular after extrapolation, the autocorrelation of the channel changes from the traditional Toeplitz matrix to a circulant matrix at this time. Compared with the traditional Toeplitz matrix, the inversion complexity of the circulant matrix is greatly reduced. The simulation results show that for various channel types, the AR-MMSE method can achieve good working performance without additional parameter tuning, demonstrating excellent robustness.

[0006] The following are the symbol descriptions in this article: Bold uppercase letters are matrices, and bold lowercase letters are column vectors. I is the identity matrix. (·) T , (·) * and (·) H are transpose, conjugate, and conjugate transpose respectively. (·) -1 denotes inversion, denotes the pseudoinverse. diag(A) denotes the column vector composed of the diagonal elements of A, and diag(a) denotes the diagonal matrix with the elements in a as the diagonal elements. vec(·) denotes column vectorization. ‖·‖2 denotes the Euclidean norm, and ‖·‖ F denotes the Frobenius norm. E{·} denotes taking the expectation. fft(a) denotes performing the fast Fourier transform on the vector a, and ifft(a) denotes performing the inverse fast Fourier transform on the vector a.

[0007] The present invention provides an AR-MMSE channel estimation algorithm, and the specific steps are as follows:

[0008] (1) Reduction of the problem

[0009] (1.1) For an M r ×M t , with N f subcarriers in the MIMO-OFDM system, the signal x (i) is transmitted on the i-th subcarrier, and the signal received at the receiving end is as follows:

[0010] y (i) =H (i) x (i) +z (i) , i = 1, …, N f , (1)

[0011] where is the channel matrix with rank K. Assume E{‖x‖ 2} = P t , that is, the transmission power is P t , is the zero-mean circularly symmetric complex Gaussian noise, is the noise variance.

[0012] The channel of the i-th carrier can be expressed as

[0013]

[0014] where f i represents the frequency of the -th subcarrier. Each element H(f) is as follows, obtained by the Fourier transform of h(t):

[0015]

[0016] h(t) is the channel time-domain impulse response:

[0017]

[0018] where τ p represents the delay of the p-th path, γp Denotes the amplitude of the p-th path. δ(.) is the δ function;

[0019] The autocorrelation function of H(f) is:

[0020]

[0021] Where, is the power delay profile. Δf is the frequency interval.

[0022] The channel estimation for the overall MIMO channel can be equivalently considered as separately estimating each of the M r M t elements of the MIMO individually. Next, for simplicity of representation, the problem will be described in a single-stream form.

[0023] First, obtain the least-squares channel estimate at the pilot:

[0024]

[0025] is the set composed of pilot position indices, and N p is the number of pilots;

[0026] Then, the minimum mean square error (MMSE) estimate of all data carrier channels is expressed as:

[0027]

[0028] Where, N d represents the number of data. The MMSE estimation matrix W MMSE is:

[0029]

[0030] is the autocorrelation matrix, and the element in its i-th row and j-th column is R H (Δf), where Δf represents the frequency interval between the i-th pilot and the j-th pilot carrier. Similarly, the element in the i-th row and j-th column of R pd is also R H (Δf), and at this time, Δf represents the frequency interval between the i-th pilot and the j-th data carrier. is the noise power, is the signal power

[0031] Therefore, the key problem solved by the present invention is: to give new methods for calculating R pp and R pd to improve the performance and robustness of channel estimation, and at the same time significantly reduce the computational complexity of (8).

[0032] (2) Problem Solving

[0033] (2.1) Circular Extrapolation of the Channel

[0034] Since the autocorrelation matrix R pp is a Toeplitz matrix, the computational complexity of calculating (8) using the well-known Levinson-Durbin algorithm is If the structure of the autocorrelation matrix can be transformed into a circulant matrix, then its inversion complexity can be reduced to To make the autocorrelation matrix a circulant matrix, the channel needs to be transformed, that is, the channel is extrapolated to be in a circular form that can be connected end to end.

[0035] To extrapolate the channel into a circular form that can be connected end to end, the optimization goal is equivalent to making the extrapolated channel as smooth as possible in the frequency domain. The smoothness in the frequency domain is quantified by the IDFT of the sequence.

[0036] Let M = N p + N e , and F is an M-point IDFT matrix:

[0037]

[0038] where

[0039] As an example, a P×P diagonal matrix is used:

[0040]

[0041] (but not limited to the above form) to construct:

[0042]

[0043] The objective equation is as follows:

[0044]

[0045] where N e is the total extrapolation length. The minimization in equation (12) is equivalent to making the power delay profile (PDP) of the extrapolated channel more concentrated at the zero time instant. The more concentrated the channel PDP is, the smoother the channel will be in the frequency domain, thus achieving a periodic cyclic effect of connecting end to end.

[0046] Partition F L by columns, and write the objective function in (11) as:

[0047]

[0048] where Then, the closed-form solutions of x and y can be written by least squares:

[0049]

[0050] (2.2) Calculate the channel autocorrelation using the cyclic AR model

[0051] Different from the traditional AR model, the cyclic AR model needs to impose boundary conditions on the signal sequence. A cyclic AR model with a period of N (N is even) is expressed as follows:

[0052]

[0053] where h is the signal sequence, are the AR model coefficients, n is the order of the AR model, e is the noise, and the noise variance is σ 2 .

[0054] The above equation can be represented in matrix form as

[0055]

[0056] Note that the following symbol convention will be used next: For the vector b = [b1,..., b N T , define B = circ(b1,..., b N ), which means taking b as the first column and cyclically obtaining the circulant matrix B by columns.

[0057] Let A = circ(a0,..., a n , 0,..., 0), and the above equation can be further simplified as:

[0058] Ah = e (17)

[0059] Let C = A -1 , and write (16) as

[0060]

[0061] Equivalent to:

[0062]

[0063] Multiply the left side of equation (15) by h * [f - k] and take the expectation to obtain the autocorrelation of h:

[0064]

[0065] Write (20) in matrix form:

[0066] ​

[0067] By sampling the channel, \(r(0),\ldots,r(n)\) can be obtained. Both \(c\) and \(\sigma\) 2 can be considered as functions of \(a\), denoted as \(c(a\) (m) ) and \(\sigma\) 2 (a (m) ) respectively to represent their values in the \(m\)-th iteration. Next, an iterative method [1] can be used to solve the coefficients of the cyclic AR model, that is, to solve (21). The specific solution method is as follows:

[0068] Let:

[0069]

[0070] Next, iterative calculations are performed for \(m = 0, 1,\ldots\):

[0071]

[0072]

[0073] Until convergence. The proof of the convergence of (23) and (24) is given in [1].

[0074] After obtaining \(a\) by solving, construct \(A=\text{circ}(a_0,\ldots,a\) n ,0,\ldots,0)\), and obtain the expression of the complete autocorrelation through Equation (17):

[0075]

[0076] That is:

[0077] \(R = \sigma\) 2 \cdot(AA\) H ) -1 , (26)

[0078] \(AA\) H is a circulant matrix. Therefore, the inverse of \(AA\) can be quickly solved using the fast inversion algorithm for circulant matrices

[0079] Before the start of iteration, the Levinson - Durbin algorithm can be used to solve the coefficients \(a\) (0) and \(\sigma\) 2 (a (0) ) of the non - periodic AR model as the values of the 0 - th iteration. Note that the complexity of the Levinson - Durbin algorithm here is \(O(n\) 2 ), where \(n\ll N\) and \(n\) is the order of the AR model.

[0080] (3) Initialization algorithm for cyclic AR model calculation

[0081] First, the Yule - Walker equations are given, which are the basis for calculating the coefficients of the AR model [2]: r(0), …, r(n) are the known autocorrelations of the sequence:

[0082]

[0083] Let

[0084] θ = [a1, …, a n T (28)

[0085] Note that both θ and σ 2 are related to the order n. To emphasize this, rewrite Equation (27) as:

[0086]

[0087] where the definitions are:

[0088]

[0089] Next, the steps of the Levinson - Durbin algorithm for solving the Yule - Walker equations are given. LDA is an algorithm for recursive solution. It is specifically summarized as Algorithm 1.

[0090] Note that we use the following notation: for the vector x = [x1, …, x n T , define:

[0091]

[0092] Levinson - Durbin algorithm:

[0093] Input: r(0), …, r(n)

[0094] Output: θ n ,

[0095] Initialization:

[0096] For i = 0, …, n - 1, calculate:

[0097]

[0098] (4) Summary of the cyclic AR model solution algorithm (please rewrite the following)

[0099] Cyclic AR model autocorrelation solution algorithm:

[0100] Input: r(0), …, r(n) obtained by sampling the channel; ​​

[0101] Output: Complete channel autocorrelation matrix

[0102] Initialization: Use the Levinson-Durbin algorithm to solve the coefficients of the non-periodic AR model as the initial values, that is:

[0103] {a (0) ,σ 2 (a (0) )}=Levinson-Durbin algorithm(r(0),…,r(n))

[0104] Perform the following operations until convergence:

[0105]

[0106] Calculation: R = σ 2 ·(AA H ) -1 .

[0107] (5) Fast Circular Matrix Inversion Algorithm

[0108] Typically, the complexity of inverting an N×N matrix is If the matrix is circulant, the inversion complexity can be further reduced. A fast algorithm for inverting circulant matrices is given here.

[0109] Define F as the N-point discrete Fourier transformation (DFT) matrix

[0110]

[0111] in,

[0112] So is the N-point inverse Fourier transform matrix. And there is

[0113] The circulant matrix C has the following properties:

[0114]

[0115] make Then we have D = diag(d). This means D is a diagonal matrix whose diagonal elements are the Fourier transform of c. In other words, we have C = circ(ifft(d)).

[0116] According to (32), we can further obtain:

[0117]

[0118] Invert C, we get:

[0119]

[0120] Therefore, we only need to take the reciprocal of each element of d and then perform the inverse Fourier transform to obtain the first column of the inverse matrix of C. That is,

[0121] This is the fast inversion algorithm for circulant matrices. Its computational complexity is the complexity of the fast Fourier transform (FFT)

[0122] (6) Summary of the AR-MMSE channel estimation process

[0123] The first step: First, perform the least squares (LS) estimation on the received signal y to obtain the estimated channel at the pilots

[0124] The second step: According to (14), perform circumferential extrapolation of the channel to obtain

[0125] The third step: Perform DFT interpolation on to complete the channels at all real subcarriers and extrapolated subcarriers The interpolated is still in a circular form with the head and tail connected;

[0126] The fourth step: The AR order is n. Sample to obtain r(0), …, r(n), and use the circulant AR model solution algorithm to solve for the complete circulant autocorrelation matrix to obtain the MMSE estimator

[0127] The fifth step: W MMSE Filter to obtain the filtered channel

[0128] The sixth step: Remove the extrapolated virtual part to obtain the final estimated channel

[0129] The present invention combines the AR model with the MMSE channel estimation, has a low algorithm complexity, and utilizes the characteristic of the AR model to automatically capture the signal spectral peak, has excellent robustness and good channel estimation performance, and can be applied to various communication scenarios. Brief description of the drawings

[0130] Figure 1 It is the flowchart of the method of the present invention.

[0131] Figure 2 Schematic diagram of channel circumference extrapolation.

[0132] Figure 3 The following is a comparison curve of the channel estimation performance between AR-MMSE and traditional MMSE under RA channel.

[0133] Figure 4 The following is a comparison curve of the channel estimation performance of AR-MMSE and traditional MMSE under ETU channel. DETAILED DESCRIPTION

[0134] First, we will demonstrate the effect of channel circular extrapolation. We intercept a 48-subcarrier RA channel and set the number of extrapolation points to N. e =16. Figure 2 In the figure, the black curve is a section of the original channel, and the red virtual carrier channels at both ends are obtained by extrapolation. The entire extrapolated channel is extended to form the blue part on the right, which verifies that the channel has indeed become a circular form connected end to end through extrapolation.

[0135] Next in Figure 3 and Figure 4 In

[15] , the performance of traditional MMSE channel estimation and AR-MMSE channel estimation is compared under RA channel and ETU channel respectively. The pilot interval is set to 4. For traditional MMSE channel estimation, the power delay profile (PDP) of the channel is assumed to follow an exponential distribution. Calculating the frequency domain autocorrelation of the channel requires knowing τ rms Parameters, but in actual channel estimation, it is impossible to know the exact τ rms , so the choice often relies on experience. For the traditional Wiener filtering method, we traverse several different τ rms Settings. From Figure 3 and Figure 4 As can be seen from the figure, for traditional MMSE channel estimation methods, parameter selection has a significant impact on the results. However, AR-MMSE channel estimation does not require excessive assumptions about the channel distribution and can automatically capture and fit the channel's spectral peaks in the time domain. This allows for excellent estimation performance regardless of channel type.

Claims

1. An MMSE channel estimation algorithm based on AR model, characterized in that: The specific steps are as follows: (1) Summary of the Problem (1.1) For an M r ×M t , there are N f subcarriers MIMO-OFDM system, sending signal x on the i-th subcarrier (i) , the signal received by the receiving end is as follows: y (i) =H (i) x (i) +z (i) ,i=1,…,N f , (1) in, is a channel matrix of rank K; let E{‖x‖ 2 =P t , that is, the transmission power is P t , is zero-mean cyclically symmetric complex Gaussian noise, is the noise variance; The channel of the i-th carrier is expressed as Among them, f i represents the frequency of the th subcarrier; each element H(f) is as follows, obtained by Fourier transform of h(t): h(t) is the channel time domain impulse response: where τ p represents the delay of the pth path, γ p represents the amplitude of the pth path, δ(.) is the δ function; The autocorrelation function of H(f) is: in, is the power delay spectrum, Δf is the frequency interval; The channel estimation of the MIMO channel as a whole is equivalent to the M r M t Each element is estimated separately; in order to simplify the representation, the problem will be described in a single-stream form; First, obtain the least squares channel estimate at the pilot: is a set of pilot position indices, N p is the number of pilot frequencies; Then, the minimum mean square error (MMSE) estimate of all data carrier channels is expressed as: Among them, N d Represents the number of data, MMSE estimation matrix W MMSE for: is the autocorrelation matrix, whose element in row i and column j is R H (Δf), Δf represents the frequency interval between the i-th pilot and the j-th pilot carrier; similarly, R pd The element in row i and column j of is also R H (Δf), where Δf represents the frequency interval between the i-th pilot and the j-th data carrier; is the noise power, is the signal power; So the problem is: given the calculation R pp and R pd method to improve the performance and robustness of channel estimation while significantly reducing the computational complexity of (8); (2) Problem Solving (2.1) Circular extrapolation of the channel Since the autocorrelation matrix R pp To solve the problem (8) using the Levinson-Durbin algorithm, the structure of the autocorrelation matrix needs to be transformed into a circulant matrix to reduce the computational complexity. In order to transform the autocorrelation matrix into a circulant matrix, the channel needs to be transformed, that is, the channel needs to be extrapolated and transformed into a circular form with end-to-end connections. To this end, the optimization goal is equivalent to making the extrapolated channel as smooth as possible in the frequency domain; the IDFT of the sequence is used to quantify its smoothness in the frequency domain; Let M = N p +N e , F is the M-point IDFT matrix: in, Use a P×P diagonal matrix: To construct: The objective equation is as follows: Among them, N e is the total extrapolation length; the minimization in Equation (12) is equivalent to making the power delay profile (PDP) of the extrapolated channel more concentrated at time zero; the more concentrated the channel PDP is, the smoother the channel will be in the frequency domain, thus achieving a periodic cycle effect with end-to-end connection; F L Divide the columns into blocks and write the objective function (11) as: in, Then, the closed-form solutions for x and y can be written using least squares: (2.2) Calculating channel autocorrelation using the cyclic AR model The cyclic AR model requires boundary conditions to be imposed on the signal sequence; a cyclic AR model with an even number N as a period is expressed as follows: Where h is the signal sequence, is the AR model coefficient, n is the AR model order, e is the noise, and the noise variance is σ 2 ; The above formula is expressed in matrix form as: The following notational conventions are used below: For a vector b = [b1,…,b N ] T , define B=circ(b1,…,b N ), which means taking b as the first column and looping through the columns to get the circulant matrix B; Let A=circ(a0,…,a n ,0,…,0), the above formula is further simplified as follows: Ah=e (17) Let C = A -1 , (16) can be written as: is equivalent to: Multiply equation (15) by h on the left * [fk] and find the expectation to get the autocorrelation of h: Write (20) in matrix form: By sampling the channel, we can obtain r(0),…,r(n); c and σ 2 Are all functions of a, respectively using c(a (m) ), σ 2 (a (m) ) represents their values in the mth iteration; Next, the iterative method is used to solve the coefficients of the cyclic AR model, that is, to solve (21). The specific solution method is as follows: make: Next, for m=0,1,…, perform iterative calculation: Until convergence; After solving a, construct A=circ(a0,…,a n ,0,…,0), the complete autocorrelation expression is obtained through formula (17): Right now: R=σ 2 ·(AA H ) -1 , (26) AA H It is a circulant matrix, so the circulant matrix fast inversion algorithm can be used to quickly solve it. Before the iteration begins, the Levinson-Durbin algorithm is used to solve the coefficient a of the non-periodic AR model. (0) and σ 2 (a (0) ), as the value of iteration 0; The initialization method for cyclic AR model calculation is as follows: First, the Yule-Walker equation is given, which is the basis for calculating the AR model coefficients; r(0),…,r(n) are known serial autocorrelations: make: θ=[a1,…,a n ] T (28) θ and σ 2 All are related to the order n, and formula (27) can be written as: Which defines: The steps of the Levinson-Durbin algorithm to solve the Yule-Walker equation are given below; The following symbols are used: for a vector x = [x1,…,x n ] T ,definition: The Levinson-Durbin algorithm is: Input: r(0),…,r(n); Output: initialization: For i=0,…,n-1, calculate: The algorithm for solving the autocorrelation of the cyclic AR model is: Input: r(0),…,r(n) obtained by sampling the channel; Output: Complete channel autocorrelation matrix Initialization: Use the Levinson-Durbin algorithm to solve the coefficients of the non-periodic AR model as the initial values, that is: {a (0) ,σ 2 (a (0) )}=Levinson-Durbin algorithm(r(0),…,r(n)); Perform the following operations until convergence: Calculation: R = σ 2 ·(AA H ) -1 .

2. The MMSE channel estimation algorithm based on the AR model according to claim 1, characterized in that The fast inversion algorithm for circulant matrices is as follows: Define F as the N-point discrete Fourier transform (DFT) matrix: in, So is the N-point inverse Fourier transform matrix; and The circulant matrix C has the following properties: make Then we have D = diag(d), that is, D is a diagonal matrix whose diagonal elements are the Fourier transform of c; in other words, we have C = circ(ifft(d)); According to (32), we can get: To invert C, we have: Therefore, take the reciprocal of d element by element, and then perform inverse Fourier transform to obtain the first column of the inverse matrix of C, that is, 3. The MMSE channel estimation algorithm based on the AR model according to claim 2, characterized in that The AR-MMSE channel estimation process is as follows: Step 1: First, perform least squares (LS) estimation on the received signal y to obtain the estimated channel at the pilot Step 2: According to (14) Perform circular extrapolation of the channel and obtain Step 3: Perform DFT interpolation to complete the channels at all real subcarriers and extrapolated subcarriers The interpolated It is still in the form of a circle connected end to end; Step 4: AR order is n, The sampling is obtained by r(0),…,r(n), and the cyclic AR model solving algorithm is used to solve the complete cyclic autocorrelation matrix to obtain the MMSE estimator. Step 5: W MMSE right Filter and get the filtered channel Step 6: Remove the extrapolated virtual part to get the final estimated channel

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