Blind channel equalization with colored source based on constrained optimization methods

The constrained minimum output variance method (CMOV), which is also known as the constrained minimum output energy method (CMOE), has been applied to directly blind equalize a linear channel and proven effective with white inputs (Tsaisanis et al. (1999)). It is believed that the method introduced in Tsaisanis can work for a linear system with colored source. In this paper, we correct this misunderstanding, and prove that a colored input will cause the equalizer to incorrectly converge. Consequently, we introduce a new blind channel equalizer algorithm based on the CMOV, but with different constraints. Our proposed algorithm works for channels with either white or colored input, and performs equivalently to the trained minimum mean-square-error (MMSE) equalizer under high SNR. Our proposed algorithm may be regarded as a generalized version of the CMOV in Tsaisanis. However, unlike CMOV, the proposed method exploits the input sequence statistics. As a result, the proposed method performs better than the original CMOV even when the input is white.


I. INTRODUCTION
In digital communication, due to the multi-path effect in channels, the signal is subject to inter-symbol-interference (ISI).The ISI will increase the symbol error rate (SER) at the receiver, sometimes making a correct estimation of the sent signal impossible.As a result, equalizers are required to remove the channel distortion.Roughly speaking, two kinds of equalizers in digital communication systems exist: data aided (trained) equalizers and blind equalizers.For data aided equalizers, a reference signal is required, increasing the bandwidth.As a result, a blind equalizer is preferred in highspeed communication systems due to its potential of reducing the ISI without increasing the overhead costs.
Blind channel equalization relies solely on the channel output and some a priori statistical knowledge of the input of the channel.Blind system equalization for a single input can be divided into two categories: single input signal output (SISO) configurations, and single input multiple output (SIMO) configurations.Note that all SISO blind identification and equalization algorithms explicitly or implicitly exploit the high-order statistics of the input and output signals.As a result they all suffer from local minima or slow convergence [2,3].
The SIMO configuration can be obtained from the exploitation of time (over-sampling) or spatial (multiantenna) diversity of the received signal.The TXK algorithm developed by Tong et al. [4] first proved the channel information could be blindly estimated using only secondorder statistics by exploiting the diversity.Different SIMO blind channel estimation and equalization algorithms have been proposed such as the sub-channel matching algorithm [5], subspace algorithm [6], linear prediction algorithm [7,8] and outer product decomposition algorithm [9,10] etc. (details about these and other algorithms can be find in [2] and its references).The popularity of SIMO, rather than SISO, blind channel identification and equalization comes from their fast convergence and efficient computation.
Most of the SIMO based blind channel estimation and equalization methods assume the channel input is white; as a result, the designed equalizers are sensitive to the color of the input.Several researchers have attempted to extend the TXK method to solve colored input problem [11,12], but the algorithms either entail many restrictions or heavy computation.The subspace based method introduce by Moulines et al. [6] could work for colored input.However the equalizer design requires a two-step procedure, first to estimate the channel coefficients, then to invert the channel effect using either zero forcing or MMSE.Moreover, the method does not exploit the input statistic, which could improve equalizer performance.
Tsatsanis and Xu [1] proposed a direct blind equalization method by incorporating the CMOV, which is widely used in array signal processing.Based on their algorithm, the blind equalizer achieves a performance close to the trained MMSE equalizer for channels with white input at high SNR.However, the algorithm does not work for colored input, though it is believed to [1,2].
The objective here is to design a direct and efficient blind equalizer that works for channels with colored input.The main contribution of this paper is the introduction of a new, constrained minimum output variance criterion that is used to develop a new blind equalizer that performs equivalently to the trained MMSE equalizer.Moreover, by exploring the input statistics, our new algorithm improves the performance of CMOV method [1] for a white input.n k i represent white noise.The SIMO channel in vector form is

II. PROBLEM DEFINITION
.
Here we assume that ( ) is the finite impulse response (FIR) of channel i with order q.We can represent the summation in (1): x( ) (2) with the following definitions: , where M is the number of taps in the FIR equalizer ( ) block Toeplitz matrix.We can denote the i th column of ( ) and we define the channel coefficient vector as 3) and (4) will be useful in the following.Our problem is to find ( ) g k i based on the following assumptions: AS1) the input s( ) k is unknown, but the second order statistic is known and has full-rank;

AS2) ( )
T h has full column rank and the Z-transforms ( ) mean noise with variance 2 n σ .
These are common assumptions in multi-channel blind identification and equalization problems [5][6][7][8][9][10].AS1 has been used in [12] to extend the TXK method for colored input.Similar to the vector structure defined in (2), we define the equalizer g vector as: We want the output of the equalizer equal to the delayed input, i.e.
where x R is the channel output covariance matrix, However, for our problem, the input s(n-a) and channel ( ) are not available.We desire to find a blind equalizer g with performance close to g mse that works for both white and colored input.We examine the CMOV based method introduced by Tsatsanis and Xu [1] next.Using the method of Lagrange, the equalizer g yielded is Note that for a white input s(n), there is only a scale difference from the optimum MMSE equalizer, and the obtained minimum output variance is: However in blind channel equalization, the channel information 1 h d + is unknown.Tsatsanis and Xu resorted to Capon's max/min approach [14] to estimate C d Pd q Pq H I P q P q P M d P q and channel coefficients vector h as in (4) so that The estimated is thus obtained by maximizing the minimum output variance in (12), i.e.
The solution is the eigenvector corresponding to the minimum eigenvalue of . Proof in [1] shows The blind equalizer in [1] is summarized by the following steps: 1. Estimate the correlation matrix x R and compute Estimate ĥcapon as the eigenvector of A corresponding to the minimum eigenvalue; 3. Using the estimated ĥcapon , compute the CMOV equalizer via (11).Note that algorithm developed by Tsatsanis and Xu requires that the order of the equalizer must be bigger than 3(q+1), and "d is not allowed to take any of the first or last 2q allowable lag[s]".These restrictions ensure that (16) holds.Also, it is believed that above algorithms "are not sensitive to the color of the input" [1].However, our simulations (refer to section V) show the bias of the algorithm for blindly equalizing a channel with colored input.Since the estimation of ĥcapon and the proof of ( 16) do not require white input, the estimation of channel will not be affected by colored inputs.However, the very basis of the constrained optimization (10) will generate a biased equalizer in this case, which means the equalizer calculated by (11) cannot eliminate the channel effect for non-white inputs.
The reason for the failure of the CMOV method is insufficient constraints.The overall response of the p channels and p equalizers can be regard as a SISO FIR filter f(n) with order M q + , as shown in Fig. 2. We want this FIR filter f(n) to be a pure delay, which means only one coefficient of f(n) can be nonzero, which can be expressed as: Fig. 2. The overall response of the multi-channel and equalizer as shown in Fig. 1.
The variance of the output of the FIR filter is where ( ) . For white input, we have Using the constraint ( ) g , the minimum output variance is achieved when all other coefficients in f are zeros, except f(d), i.e. the filter f is a pure delay of d samples.
However, for colored input, the constraint g cannot guarantee f will converge to a pure delay.Actually, the overall response f will force the frequency component of y(n) to be the one's complement of s(n), which is easily obtained by analysis of g TX in (11) [15].

IV. NEW ALGORITHM
Our new algorithm is still based on the constrained minimum output variance method.As we discussed earlier, the failure of the TX CMOV method is caused by inadequate constraints.In order to design a new equalizer with performance close to the trained MMSE equalizer, we first give a proposition.(correlation between delayed input and input vector), then g differs from the MMSE equalizer only by a scalar factor gain.
Proof: This is a constrained optimization problem, which can be solved using Lagrange multipliers.First define the cost function where λ is a Lagrange multiplier.Minimizing the cost function yields Comparing ( 22) with ( 8), we see only a scalar factor difference between H g and the MMSE equalizer.Proposition 1 provides the theoretical background of our new algorithm.However, as with the method in TX [1], this constrained optimization requires the channel information that is not available.In order to estimate the channel information, we need to resort to the Capon max/min method.
The minimum output variance is obtained when Based on the Capon method, we can find the channel information by maximizing the minimum output variance ( ) min . However, it is not easy to directly apply the max/min method.We define an extended vector of r [0, ,0, ( ),0, ,0, ( 1), (1),0, ,0] This extended vector is constructed by reversing the vector r and placing p-1 zeros in front of, between, and after it.Then we construct the The following relation holds: This lemma is easily proven by direct substitution.A method based on the Z-transform can also be applied here [6].
Using (24) and lemma 1, we rewrite (23) as ( ) At this point, we can apply The Capon max/min principle to estimate h ( ) We see that ĥcapon is equal to the eigenvector corresponding to the minimum eigenvalue of . Thus, our algorithm for blind channel equalizer design can be summarized in the following steps:

V. PERFORMANCE ANALYSIS
In this section, we will see that the performance of our proposed blind equalizer is close to that of the trained MMSE equalizer for high SNR.As shown in (22) and step 4 of our algorithm, if the estimated ĥcapon equal to the correct channel coefficient vector h , then the blind equalizer differs from the MMSE equalizer only by a scale factor, which can be corrected by comparing the power of the equalizer output and system input.Consequently, our developed equalizer will have equivalent performance to the trained MMSE equalizer.As a result, how well the estimated channel coefficients vector relates to the true vector will determine the performance of the proposed equalizer.To see this, we need to prove the following lemma.Lemma 2. Under the condition of Proof: We follow the same steps as in [1], i.e. we first prove ; and then prove this solution is unique under the condition given by lemma 2.
. Lemma 1 of [1] easily yields: as 2 0 − , the number of equations is greater or equal to the number of unknowns, so we can only have one solution, i.e. ' h h = , which contradicts our assumption.So h is the only solution under our assumption.Due to the Toeplitz structure, this argument also works for colored input but d will take a wider range of values that depend on the value of r.As a result, the given condition is a necessary condition of colored inputs.
Lemma 2 also shows that our proposed method provides a method of blind channel identification and the identifiable condition.It is interesting to see that, although based on different constraints, both our method and the Tsatsanis method can be used to estimation channel coefficients.This is because the underlying estimation implicitly makes use of the subspace method, which is insensitive to colored inputs.
Here, we prove the performance of our proposed blind channel equalizer will be close to the trained MMSE equalizer as 2 0 v σ → .However, as proved by Tsatsanis and Xu, there will be a small penalty in the CMOE method.This penalty can be completely removed by using the Power of R (POR) method introduced in [16] to improve the performance of our proposed equalizer, that is, we can calculate the matrix . It has been proven that the performance of the CMOE equalizer will asymptotically converge to that of the MMSE equalizer.Since this only affects the estimation of h , our proposed method will still work for colored input.

VI. SIMULATIONS
We simulate a SIMO communication system as shown in Fig. 1.The channel is modeled by ( ) where f(t) is a raised-cosine pulse limited in 6T with roll-off factor 0.10 and with an oversampling factor q=3.
In the first simulation, the input s(n) is generated by filtering an i.i.d.4-PAM signal with an FIR filter whose impulse response is [1 -.3 0.14 0.12]; and SNR=11 dB.Fig. 3 shows the channel identification result.We find that both algorithms identify the channel coefficients, but our method identifies the channel better than does the CMOV method in [1].We notice that as the SNR becomes small, the improvement is more pronounced.Fig. 4 shows the overall response f(n) defined in (16).Our algorithm successfully equalizes the channel distortion while the CMOV method fails.
Next we simulate both algorithms for white input at same SNR (15 dB) and compare the performance of our proposed method and CMOV.In this situation, we find that our proposed algorithm still outperforms the CMOV algorithm by approximately 8 dB; and as the SNR increases, this becomes more pronounced.This is because our method uses the input signal statistics.Due to limited space, a detailed analysis of this issue and performance comparison with other available techniques will appear in a journal paper.

VII. CONCLUSIONS
A new direct blind linear system equalization method has been developed based on the constrained optimum method.Unlike the previous CMOV, our proposed algorithm is guaranteed to work for either white or colored inputs with performance close to the MMSE equalizer under high SNR.One byproduct of this algorithm is the effective estimation of the channel information.The new algorithm can be regarded as a generalization of the Tsatsanis and Xu CMOV.However, our method makes use of the input statistics, and so it performs better than that algorithm.

Fig. 1
Fig. 1 shows the base-band representation of a SIMO data communication system with input s(k).The left part of the figure represents the multi-channels ( ) h k i with multiple

Fig. 1 .
Fig. 1.SIMO blind channel estimation and equalizationIII.ANALYSIS OF EXISTING CMOV METHODTsatsanis and Xu, borrowing from array signal processing, proposed a CMOV method which successfully solves the above problem when the input s(n) is white and the SNR of the measured x(n) is high.The equalizer is developed using the constrained optimization: }

1 . 5 .
Estimate the input statistics r , Remove the phrase and scale factor gain of H g .

−
, as the SNR → ∞ , the ĥcapon estimated in (26) differs from the true channel coefficients by a phase and scale factor, i.e.