CN101902417A - Super-exponential iterative blind equalization method based on ant colony optimization - Google Patents

Abstract

The invention discloses an ant colony optimized orthogonal wavelet super-exponential iterative blind equalization method, which uses an ant colony algorithm to find the optimal weight vector as the initial weight vector value of the equalizer input, thereby avoiding local convergence of the algorithm Condition. The method has a positive feedback mechanism to speed up the convergence speed, uses the super-exponential iteration (SEI) method to whiten the data, uses the orthogonal wavelet transform to decorrelate the signal, and makes full use of the global convergence of the ant colony algorithm. The underwater acoustic channel simulation results show that compared with the orthogonal wavelet super-exponential iterative blind equalization method (WT-SEI-CMA), this method has better convergence speed and steady-state error, and the eye diagram after equalization is clearer and more compact . Therefore, this method has certain practical value.

Description

Super-exponential iterative blind equalization method based on ant colony optimization

technical field

The invention relates to an orthogonal wavelet transform super-exponential iterative blind equalization method, in particular to an orthogonal wavelet transform super-exponential iterative blind equalization method based on ant colony optimization.

Background technique

In the underwater communication system, the inter-symbol interference (Inter-Symbol Inter-ferences, ISI) brought by the multipath effect and the limited bandwidth is the main factor affecting the communication quality, which needs to be eliminated by equalization technology (see literature: Guo Yecai , Chao Yang. Design and simulation of super-exponential iterative blind equalizer based on orthogonal wavelet transform [J]. Journal of System Simulation, 2009, 21(20)-: 6556-6559). The blind equalization algorithm that does not require training sequences only uses the statistical characteristics of the received signal itself to equalize the signal, but its convergence speed is slow and the steady-state error is also large. Literature (O Shalvi, E Weinstein. Super-Exponential Methods for Blind Deconvolution [J]. IEEE Trans. Inform. Theory, 1993, vol.39, 504-519.) proposed a super-exponential convergence at a nearly super-exponential speed Blind equalization algorithm (Super-Exponential, SE) and iterative algorithm (Super-Exponential Iterative, SEI). Compared with the CMA algorithm, the SEI algorithm introduces a whitening matrix Q, which whitens the input signal of the equalizer and accelerates the convergence speed, but it still cannot meet the requirements of engineering realizability. Literature (Guo Yecai, Ding Xuejie, Guo Fudong, etc. Cascade Adaptive Blind Equalization Algorithm Based on Normalized Constant Modulus Algorithm [J]. Journal of System Simulation, 2008, 20(17): 4647-4650; Han Yingge, Guo Yecai , Li Baokun et al. Blind equalization algorithm for Orthogonal Wavelet Transforms with momentum term[J]. Journal of System Simulation; 2008, 20(6): 1559-1562; Cooklev T An Efficient Architecture for Orthogonal Wavelet Transforms[J].IEEESignal Processing Letters (S1070-9980), 2006, 13(2): 77~79.) showed that: performing wavelet transform on the input signal of the equalizer and performing energy normalization processing can reduce the correlation between signal and noise, thus effectively Speed up the convergence speed, but the wavelet blind equalization algorithm still finds the optimal weight vector according to the gradient direction, it is sensitive to the initialization of the weight vector, improper initialization will make the algorithm converge to a local minimum, or even diverge. The Ant Colony Algorithm (ACO) was first proposed by Italian scholar Dorigo M on the basis of the research on the collective behavior of real ant colonies in nature. It is a heuristic bionic parallel intelligent evolutionary algorithm based on population. It has the characteristics of intelligent search, global optimization, robustness, positive feedback, distributed computing, and easy combination with other algorithms. It is also very good for harsh situations such as function discontinuity, non-differentiability, and local extreme points. search capability. The optimal solution is sought through the evolution process of the group composed of candidate solutions; through the positive feedback mechanism, the optimization process of the algorithm is accelerated, and the global optimal solution is quickly found, which greatly reduces the possibility of falling into local convergence.

Contents of the invention

The purpose of the present invention is aimed at the shortcomings of the traditional constant modulus blind equalization method (CMA) with slow convergence speed, large steady-state error and local convergence. On the basis of orthogonal wavelet transform theory, a blind equalization method based on ant colony optimization with orthogonal wavelet super-exponential iteration (ACO-WT-SEI) is proposed. This method makes full use of the characteristics of the global random search and positive feedback mechanism of the ant colony algorithm, initializes the weight vector of the equalizer, uses the SEI method to whiten the data, and uses the orthogonal wavelet transform to reduce the autocorrelation of the signal.

In order to achieve the above object, the present invention adopts the following technical solutions:

The present invention is based on the ant colony optimization orthogonal wavelet transform super-exponential iterative blind equalization method, comprising the following steps:

a.) Pass the transmitted signal a(k) through the impulse response channel to obtain the channel output vector x(k), where k is a time series, the same below;

b.) Obtain the input signal of the equalizer by using the channel noise n(k) and the channel output vector x(k) described in step a: y(k)=x(k)+n(k);

c.) The error signal e(k) is introduced into the whitening matrix Q through the SEI algorithm, and the input signal y(k) of the equalizer is whitened by using the whitening matrix Q;

d.) The input signal y(k) after step c.) is subjected to orthogonal wavelet transform to obtain the output vector of orthogonal wavelet: R(k)=y(k)V, wherein V is orthogonal wavelet transform matrix;

Also include the following steps:

e.) Randomly generate the initial population W=[W ₁ , W ₂ ,...,W _M ], where the i-th ant individual W _i corresponds to the i-th weight vector of the equalizer, where 0<i ≤ M, i and M Both are natural numbers, which are used as the initial position of ants;

f.) The reciprocal of the cost function of the equalizer is used as the objective function of the ant colony algorithm optimization:

$\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span>}$

In the formula, J(W _i )=J _CMA is the cost function of the equalizer;

g.) Each ant uses the objective function described in step f.) to optimize one step or after completing the optimization of all M weight vectors, update the residual pheromone:

f _ij (t+1)=(1-ρ)f _ij (t)+Δf _ij ,

表示第k只蚂蚁在本次循环中留在第i个权向量和第j个权向量之间的路径上的信息素，ρ为信息素挥发系数，取值范围为ρ∈[0，1)，Δf_ij表示蚂蚁在本次循环中在第i个权向量和第j个权向量之间的路径上留下的信息素，f_ij(t)为信息素更新第t步所对应的信息素；In the formula,

Indicates the pheromone left by the kth ant on the path between the i-th weight vector and the j-th weight vector in this cycle, ρ is the pheromone volatilization coefficient, and the value range is ρ∈[0, 1) , Δf _ij represents the pheromone left by the ant on the path between the i-th weight vector and the j-th weight vector in this cycle, f _ij (t) is the pheromone corresponding to the t-th step of pheromone update ;

h.) Obtain the corresponding weight vector value when the objective function is optimal, and use this weight vector as the initialization weight vector of the equalizer.

The invention discloses an ant colony optimized orthogonal wavelet super-exponential iterative blind equalization method, which uses an ant colony algorithm to find the optimal weight vector as the initial weight vector value of the equalizer input, thereby avoiding local convergence of the algorithm Condition. The method has a positive feedback mechanism to speed up the convergence speed, uses the super-exponential iteration (SEI) method to whiten the data, uses the orthogonal wavelet transform to decorrelate the signal, and makes full use of the global convergence of the ant colony algorithm. The underwater acoustic channel simulation results show that compared with the orthogonal wavelet super-exponential iterative blind equalization method (WT-SEI-CMA), this method has better convergence speed and steady-state error, and the eye diagram after equalization is clearer and more compact . Therefore, this method has certain practical value.

Description of drawings

Figure 1: Block diagram of SEI method based on orthogonal wavelet transform;

Figure 2: Schematic diagram of blind equalization with orthogonal wavelet transform for ant colony optimization;

Figure 3: (a) mean square error curve, (b) output signal of SEI, (c) WT-SEI output, (d) ACA-WT-SEI output;

Figure 4: (a) mean square error curve, (b) output signal of SEI, (c) WT-SEI output, (d) ACA-WT-SEI output.

Detailed ways

The orthogonal wavelet super-exponential iteration is introduced into the blind equalization method (CMA), and the orthogonal wavelet super-exponential iterative blind equalization method (WT-SEI) is obtained. Its schematic diagram is shown in Figure 1. This method introduces the whitening matrix Q through the SEI algorithm, which whitens the input signal of the equalizer, and then uses the wavelet to perform orthogonal wavelet transform on the signal, and performs energy normalization processing, and then in the transform domain, uses the wavelet transform The final signals adjust the weight coefficients of the equalizer, which reduces the autocorrelation of these signals and speeds up the convergence speed.

In Figure 1, a(k) represents the signal transmitted by the transmitter, which is a white IID sequence with a variance of 1; c(k) is the channel impulse response vector; n(k) is the channel noise, which is generally assumed to be a Gaussian white noise sequence , and is independent of the signal statistics. y(k) is the input sequence of the equalizer, and y(k)=[y(k), y(k-1),..., y(k-L+1)] ^T ; W(k) is the equalizer Weight vector; ψ(·) is a nonlinear function that produces error e(k); z(k) is the output sequence of the equalizer. The equalizer input signal is

y(k)=x(k)+n(k)=c ^T (k)a(k)+n(k) (1)

式中，

为小波变换系数、

为尺度变换系数，

和

为均衡器的权系数，J为最大尺度；k_J＝N/2^j-1(j＝1，2，…，J)为尺度j下小波函数的最大平移。则Let V be the orthogonal wavelet transform matrix,

In the formula,

is the wavelet transform coefficient,

is the scaling coefficient,

and

is the weight coefficient of the equalizer, J is the maximum scale; k _J =N/2 ^j -1 (j=1, 2,..., J) is the maximum translation of the wavelet function at scale j. but

R(k)＝y(k)V (2)

z(k)=W ^H (k)R(k) (3)

In the formula, H is the conjugate transpose. At this time, the iterative formula for calculating the Q matrix is

$\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\frac{{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; R</span>}}^{<span\; class="notranslate">\; *</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; R</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}_{\mathrm{<span\; class="notranslate">\; m</span>}}{\mathrm{<span\; class="notranslate">\; R</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; R</span>}}^{<span\; class="notranslate">\; *</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; ]</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$

In the formula, μ _m represents the iteration step size of Q matrix calculation. The iterative formula of the weight vector is

$\mathrm{<span\; class="notranslate">\; W</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; W</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&mu;</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; e</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; R</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

分别表示对小波变换系数r_j，n(k)、尺度变换系数s_J，n(k)平均功率估计，可由下式递推得到In the formula, e(k)=y(k)(|y(k)| ² -R) is the error function, R=E[|a(k)| ⁴ ]/E[|a(k)| ² ] is the modulus of the emission sequence a(k), and μ is the step size.

represent the average power estimation of wavelet transform coefficient r _{j, n} (k) and scale transform coefficient s _{J, n} (k) respectively, which can be obtained recursively by the following formula

${\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&beta;\&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; r</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

${\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&beta;\&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&beta;</span>}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$

In the formula, β is the smoothing factor, r _j,n (k) is the wavelet transform coefficient of the nth signal decomposed in wavelet space j layer, s _j,n (k) represents the maximum decomposition layer j in the scale space Scaling coefficients for n signals. Equations (2)-(7) constitute a super-exponential iterative blind equalization method based on orthogonal wavelet transform (Super-Exponential Iterative Based on Orthogonal Wavelet Transform, WT-SEI).

The orthogonal wavelet super-exponential iterative blind equalization algorithm of the ant colony optimization of the present invention is as follows:

The traditional orthogonal wavelet super-exponential iterative blind equalization method (WT-SEI-CMA) uses the super-exponential iterative algorithm to calculate the Q matrix at each iteration of the weight vector, and optimizes the step factor through the matrix, and the input signal of the equalizer It plays the role of whitening, and performs orthogonal wavelet transform on it, and the signal autocorrelation matrix approximates a banded distribution (see literature: Guo Yecai, HanYingge. Orthogonal Wavelet Transform Based Sign Decision Dual-mode Blind Equalization Algorithm[C]//2008 9 ^th International Conference on Signal Processing Proceedings, Beijing, China. USA: The Institute of Electrical and Electronics Engineers Inc. 2009: 80-83. Han Yingge, Guo Yecai, Wu Zuolin, etc. Design and Algorithm of Multi-mode Blind Equalizer Based on Orthogonal Wavelet Simulation research [J]. Journal of Instrumentation, 2008, 29(7): 1441-1445), its autocorrelation decreased. The traditional blind equalization algorithm first constructs a cost function, and uses this cost function to calculate the gradient of the equalizer weight vector, so as to determine the iterative equation of the equalizer weight, but this method is essentially a gradient descent search that only considers the local area method, lack of global search ability, easy to converge to local minimum solution.

The invention adopts the ant colony algorithm to find the optimal solution to overcome the defects of WT-SEI-CMA. First use the ant colony optimization blind balance method to balance a very small piece of data, use the positive feedback mechanism of the ant colony algorithm and the pheromone update characteristics to find the weight vector when the objective function is optimal, and use this set of weight vectors as positive The initialization weight vector of the intersection wavelet blind equalization method.

as shown in picture 2. The present invention randomly generates a group of weight vectors, each ant corresponds to each group of weight vectors in turn, uses the weight vector as the decision variable of the ant colony algorithm, uses the equalizer input signal as the input of the ant colony algorithm, and combines the cost function of the CMA algorithm , determine the evolutionary objective function of the ant colony algorithm, use the ant colony algorithm to solve the cost function of the equalizer, and search for the best equalizer weight. In this way, the ant colony algorithm is introduced into the orthogonal wavelet super-exponential iterative blind equalization method, which is called the ant colony optimization-based orthogonal wavelet super-exponential iterative blind equalization method (ACO-WT-SEI). In this method, the characteristics of the ant colony algorithm's global search and positive feedback mechanism are used to find the best equalizer weight, unlike the CMA algorithm, which relies on the guidance of gradient information to adjust the equalizer weight.

The optimization process of ant colony algorithm is as follows:

Whether it is the constant modulus blind equalization method or the orthogonal wavelet blind equalization method, the cost function is:

${\mathrm{<span\; class="notranslate">\; J</span>}}_{\mathrm{<span\; class="notranslate">\; CMA</span>}}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; (</span>{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; CM</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$

是均衡器的模值。现结合式(8)式来说明优化过程。1初始化权向量的产生where z(k) is the output of the equalizer,

is the modulus value of the equalizer. Now combine formula (8) to illustrate the optimization process. 1 Generation of initialization weight vector

The ant colony algorithm operates on the optimization of each ant in the group. Before the optimization operation, the initial data of the initial search point of the ant colony must be initialized, that is, the weight vector corresponding to each ant is initialized, and its initial value is determined to be [-1, 1 ], and use a random method to generate a certain number of weight vectors. Each individual ant corresponds to a weight vector of the equalizer, and the number of weight vectors is the size of the ants. Assume that the randomly generated initial population W=[W ₁ , W ₂ ,...,W _M ], one ant individual W _i (0<i≤M) corresponds to a weight vector of the equalizer, which is used as the initial position of the ant.

2 Generation of objective function

The purpose of the blind equalization method is to iterate the cost function to the minimum value to obtain the best equalizer weight, while the purpose of the ant colony algorithm optimization is to obtain the individual corresponding to the maximum value of the objective function. The reciprocal of the cost function is used as the objective function of the ant colony algorithm optimization, then there is

$\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; \&CenterDot;</span><span\; class="notranslate">\; \&CenterDot;</span><span\; class="notranslate">\; \&CenterDot;</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 9</span>}<span\; class="notranslate">\; )</span>$

In the formula, J(W _i )=J _CMA is the cost function of the equalizer, W _i is the weight vector individual of the equalizer generated by the ant colony algorithm, which is used as the initial pheromone for the optimization of the ant colony algorithm.

3 Update of pheromones

Each optimization of the ant colony algorithm must receive a certain input signal, which is provided by the input signal. After entering the ant colony algorithm, it first decides whether to perform local optimization or global optimization according to the transition probability, and the new The location is limited within the feasible region (see literature: Zhou Jianxin, Yang Weidong, Li Qing. Improved ant colony algorithm and simulation for solving continuous function optimization problems [J]. Journal of System Simulation, 2009, 21(6): 1685-1688). Every time an ant moves to a new location, it will compare the new location, which will strengthen or weaken the pheromone. If it is enhanced, move to a new position, and release the pheromone of the new position to the environment at the same time, otherwise continue to test other positions (see literature: Wang Lei, Wu Qidi. Application of ant colony algorithm in continuous space optimization problem [J ]. Control and Decision Making, 2003, 18(1): 45-48). In order to prevent the residual pheromone from submerging the heuristic information, after each ant optimizes one step or completes the optimization of all M weight vectors, the residual pheromone is updated, as shown in the following formula:

f _ij (t+1)=(1-ρ)f _ij (t)+Δf _ij , (10)

表示第k只蚂蚁在本次循环中留在路径ij上的信息素，ρ为信息素挥发系数，取值范围为ρ∈[0，1)，Δf_ij表示蚂蚁在本次循环中在第i个权向量和第j个权向量之间的路径上留下的信息素，f_ij(t)为信息素更新第t步所对应的信息素；In the formula,

Indicates the pheromone left by the kth ant on the path ij in this cycle, ρ is the pheromone volatilization coefficient, and the value range is ρ∈[0, 1), Δf _ij represents the pheromone of the ant in the ith cycle in this cycle The pheromone left on the path between the weight vector and the jth weight vector, f _ij (t) is the pheromone corresponding to the tth step of pheromone update;

4 Selection of the best weight vector

Local optimization and global optimization are performed through the transition probability of the ant colony algorithm, and the optimization results are limited to the feasible region, the optimal solution is retained, and then the optimal weight vector is solved by updating the pheromone, and the objective function in each generation The individual with the largest weight vector is selected, considering the real-time performance of the algorithm when extracting the best individual and the fact that the blind equalization algorithm must satisfy the zero-forcing condition, and finally obtain the corresponding weight vector value when the objective function is optimal, and use this weight The vector serves as the initialization weight vector of ACO-WT-SEI.

Embodiment 1 Two-path underwater acoustic channel simulation

The unit impulse response of the two-path underwater acoustic channel is c=[-0.35, 0, 0, 1], the transmitted signal is 8PSK, the weight length of the equalizer is 16, and the signal-to-noise ratio is 25dB. In the SEI algorithm, the fourth tap The coefficient is set to 1, the rest are 0, the step size μ _SEI =0.00015, μ _m =0.02; in the WT-SEI algorithm, the 11th tap coefficient is set to 1, the rest are 0, the step size μ _WT-SEI =0.01, μ _m =0.002; step size of ACA-WT-SEI μ _ACA-WT-SEI =0.004, μ _m =0.004. The input signal of each channel is decomposed by DB4 orthogonal wavelet, the decomposition level is 2 layers, the initial value of power is set to 4, and the forgetting factor β=0.9999; 500 Monte Carlo simulation results are shown in Figure 3.

Figure 3(a) shows that ACA-WT-SEI is about 8000 steps faster than SEI and about 4000 steps faster than WT-SEI in terms of convergence speed. In terms of steady-state error, ACA-WT-SEI reduces nearly 7dB compared with SEI, and reduces nearly 6dB compared with WT-SEI. Figure 3(b, c, d) shows that the output constellation diagram of ACA-WT-SEI is clearer and more compact than that of SEIA and WT-SEI.

Embodiment 2 mixed phase channel

The mixed phase channel is c=[0.3132 -0.1040 0.8908 0.3134], the transmitted signal is 16QAM, the weight length of the equalizer is 16, and the signal-to-noise ratio is 25dB. In the SEI algorithm, the third tap coefficient is set to 1, and the rest are set to 0. Long μ _SEI ＝0.00005, μ _m ＝0.02; in the WT-SEI algorithm, the fourth tap coefficient is set to 1, the rest are 0, the step size μ _WT-SEI ＝0.0005, μ _m ＝0.02; ACA-WT-SEI Step size μ _ACA-WT-SEI = 0.0006, μ _m = 0.0006. The input signal of each channel is decomposed by DB4 orthogonal wavelet, the decomposition level is 2 layers, the initial value of power is set to 4, and the forgetting factor β = 0.9999; 500 Monte Carlo simulation results are shown in Figure 4.

Figure 4(a) shows that ACA-WT-SEI is about 4000 steps faster than SEI and WT-SEI in terms of convergence speed. In the steady-state error, compared with SEI, ACA-WT-SEI has reduced by nearly 2.2dB, and compared with WT-SEI, it has reduced by nearly 1.8dB. Figure 4(b, c, d) shows that the output constellation of ACA-WT-SEI is clearer and more compact than that of SEI and WT-SEI.

Claims (1)

1. A kind of orthogonal wavelet transform super-exponential iterative blind equalization method based on ant colony optimization, comprising the steps: a.) Pass the transmitted signal a(k) through the impulse response channel to obtain the channel output vector x(k), where k is a time series, the same below; b.) Obtain the input signal of the equalizer by using the channel noise n(k) and the channel output vector x(k) described in step a: y(k)=x(k)+n(k); c.) The error signal e(k) is introduced into the whitening matrix Q through the SEI method, and the input signal y(k) of the equalizer is whitened by using the whitening matrix Q; d.) The input signal y(k) after step c.) is subjected to orthogonal wavelet transform to obtain the output vector of orthogonal wavelet: R(k)=y(k)V, wherein V is orthogonal wavelet transform matrix; It is characterized in that it also includes the following steps: e.) Randomly generate the initial population W=[W ₁ , W ₂ ,...,W _M ], where the i-th ant individual W _i corresponds to the i-th weight vector of the equalizer, where 0<i≤M, i and M Both are natural numbers, which are used as the initial position of ants; f.) The reciprocal of the cost function of the equalizer is used as the objective function of the ant colony algorithm optimization: $\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span>}$ In the formula, J(W _i )=J _CMA is the cost function of the equalizer; g.) Each ant uses the objective function described in step f.) to optimize one step or after completing the optimization of all M weight vectors, update the residual pheromone: f _ij (t+1)=(1-ρ)f _ij (t)+Δf _ij , In the formula,

Indicates the pheromone left by the kth ant on the path between the i-th weight vector and the j-th weight vector in this cycle, ρ is the pheromone volatilization coefficient, and the value range is ρ∈[0,1 ), Δf _ij represents the pheromone left by the ant on the path between the i-th weight vector and the j-th weight vector in this cycle, f _ij (t) is the information corresponding to the pheromone update step t white; h.) Obtain the corresponding weight vector value when the objective function is optimal, and use this weight vector as the initialization weight vector of the equalizer.

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