THE NEW BLIND EQUALIZATION TECHNIQUES TO IMPROVE BANDWIDTH EFFICIENCY IN WIRELESS DATA COMMUNICATION SYSTEM FOR THE INTERNET ACCESS
K. SKOWRATANANONT AND D. RATANAPANICH
Graduate School, Assumption University
Bangkok, Thailand.
The cost function of the Constant Modulus Algorithm (CMA) when used to equalize a fractionally spaced channel, which satisfies certain length and nondegeneracy conditions, has been shown to possess only global minima in the absence of channel noise. However, in the presence of channel noise, the performance of CMA can suffer because of the existence of undesired local minima which correspond to various transmission delays. This paper introduces two new techniques based upon the Gram-Schmidt method and the Recursive Least Squared (RLS) algorithm to obtain convergence to an optimal delay where the Mean Squared Error (MSE) is minimum. Simulations are presented to support the excellence of these techniques
INTRODUCTION
Existing wiring, whether cabling for television or twisted pairs for telephony, is being enlisted to access the Internet far faster than is possible with conventional modems [1]. But how can access be made if the existing wiring is not up to the job or if, as happens in many parts of the world, there is no wiring. In spite of some apparent limitations, fixed wireless technology is a strong competitor because it can be designed quickly and is much cheaper than wired technologies to install and operate. However, just like other high speed wireless communication systems, the fixed wireless technology for the Internet access suffers with the problem so called multipath propagation which causes a severe bit error rate to the reconstruction of the transmitted signals. Many methods such as space diversity, frequency diversity, or Viterbi algorithm have been used in the existing systems to obtain the better signal retrieval. However, these methods need to reserve about 20% of every packet to transmit the training sequence and require the knowledge of the training sequence to store at the receiver. Hence, the bandwidth usage is not efficient.
There have been increasing interest in “Blind Equalization” during the last decade due to the potential increase in bandwidth efficiency [2]. This is because the training sequence known at the receiver is not necessary. Hence, the bit rate can be improved. Among the several classes of methods, the Fractionally Spaced Constant Modulus Algorithm (FS-CMA) has proven to be very successful due to its simplicity and robust convergence properties [3] as shown in Fig.1a. However, in the presence of channel noise, FS-CMA may have local minima with poor MSE (Fig.1b). Local minima may also appear for a baud spaced constant modulus algorithm (BS-CMA) even in the absence of channel noise. Therefore, the major problem is the choice of the equalizer initialization so that the equalizer converges towards a desired global minimum.
THE CONSTANT MODULUS EQUALIZER
For simplification the transmitted sequence is assumed to be binary phase shift keying (BPSK) of alphabet (1). At the receiver, the received signals with the effect of the multipath propagation phenomenon are passed to equalizer to reconstruct the transmitted signals. The standard CMA cost function can be expressed as
JCMA(W)=E[(y2(k)-1)2](1)
where the y(k) and W are respectively the equalizer output and the equalizer weight vector. With a stochastic gradient update, the equalizer weight vector W is adaptively adjusted according to
W(k+1)=W(k)-(y2(k)-1)X(k)y(k)(2)
where the is the step size and X(k) is the equalizer input vector (or the received signal vector).
This paper describes two additional techniques for the CMA such that the equalizer is more likely to converge to a global minima with a correspondingly minimum MSE. The first technique [4] relies on recent work [5] in which demonstrates the adaptive blind simultaneous reconstruction of multiple sources and relies on the assumption that the input symbol is independent and identically distributed (i.i.d.). Therefore, the orthogonality constraint can be applied to find different transmission delay fractionally spaced equalizers. The second technique introduced in this paper relies on the proof in Zeng’s work [6] that the CMA equalizer is approximately a scaled version of a Minimum Mean Squared Error (MMSE) equalizer. By using the link between CMA and MMSE equalizers, the different transmission delays can be approximated by the CMA equalizer output. The new initializations which correspond to different delays, can then be found by the RLS algorithm
THE GRAM-SCHMIDT BASED METHOD FOR CMA (GS-CMA)
As mentioned, this technique exploits the fact that the transmitted signals in digital communication systems can be assumed to be zero mean and i.i.d., and thus E(sisj)=0; ij where si is the transmitted symbol at delay i. Therefore, the ideal equalizer outputs that correspond to different transmission delays will also be orthogonal. This orthogonality is written as
E(yi(k)yj(k))=0 ij(3)
where the equalizer output at delay i, yi(k)=XT(k)Wi , (*)T is a transpose operation, and Wi is the equalizer weight vector providing the output at delay i. Therefore, eq.(3) can be rewritten as
E(WiTX(k)XT(k)Wj)=WiTRXXWj=0(4)
where RXX is the autocorrelation matrix of the observation vector. Hence, we require Wj to be orthogonal to RXXWi. By exploiting the Gram-Schmidt procedure, the equalizer weight vectors, which correspond to the different transmission delays, are given by
Wj(k+1)=Wj(k)+[j-1I=1(WiT(k+1)RXXWj(k))/(WiT(k+1)RXXWi(k+1))]Wi(k+1) (5)
where m and n are, respectively, the order of the channel and the length of the equalizer. To Implement the above technique, we exploit the standard CMA in eq.(2) to adapt the equalizer taps. At each iteration, the equalizer weight vector from eq(2) is used in eq(5) to adapt the m+n-1 equalizer weight vectors. Once the equalizer weight vector from eq(2) converges to a solution for a particular delay, eq(5) will provide solutions for other equalization delays. Since each equalizer weight vector from eq(5) is the desired equalizer setting up to a multiplicative constant, the equalizer parameters are scaled such that the equalizer output power is unity.
As mentioned earlier, the different minima have various depths which are due to the noise amplification factor. As the noise amplification factor is equal to the squared l2 norm of the equalizer weight vector, comparison of the norm of each solution is appropriate. The equalizer weight vector which has the minimum squared norm will then substitute those of the CMA, whereas any other delay equalizers are switched off. Hence, the complexity is decreased.
RECURSIVE CROSS CORRELATION BASED METHOD FOR CMA (RCC-CMA)
Similar to the GS-CMA, this method is proposed to select the best solution of the equalizer
weight vector corresponding to the minimum MSE. This method exploits the relationship between the CMA and MMSE equalizers where the MMSE equalizer can be approximated by
Wim=RXX-1RXsiWi(6)
where RXX=E(X(k)XT(k)) and RXsi=E(X(k)s(k-i)) are respectively the autocorrelation matrix of the equalizer input vector and the cross correlation vector between the equalizer input vector and the transmitted signal at delay i. Wim and Wi respectively denote the MMSE and CMA equalizers at delay i. Assuming that the CMA equalizer retrieves the transmission signal with delay i, we form another equalizer such that its output does not have any contribution from s(k-i).
Based upon the above relation in eq.(6), we suggest the RLS algorithm for finding the equalizer corresponding to a different delay. The estimated autocorrelation and cross correlation matrices can be written as [7]
RXX(k)=kn=1k-iX(n)XT(n)(7)
RiXs(k)=kn=1k-iX(n)s(n-i) (8)
where is a forgetting factor. Isolating the term corresponding to n=k from the rest of the summation on the right hand side of eq(7) and eq(8), we may write
RXX(k)=RXX(k-1)+X(k)XT(k)(9)
RXsi(k)= RXsi(k-1)+X(k)s(k-i) (10)
Let P(k)=RXX(k)-1 and using the matrix inversion lemma[iv] with eq(9) and eq(10), we can write
P(k)=-1P(k-1)- -1K(k)XT(k)P(k-1)(11)
where
K(k)=(-1P(k-1)X(k))/(1+-1XT(k)P(k-1)X(k))(12)
Substituting eq(10) to eq(12) in eq(6), we obtain
Wim(k)=Wim(k-1)+K(k)(s(k-i)- WimT(k-1)X(k))(13)
where Wim(k) denotes the new equalizer corresponding to delay i. Notice that eq(11) and eq(12) do not require an explicit matrix inverse and thereby provide a recursive method for computing P(k), and hence a lower complexity approach to calculate eq(6). Eq(13) provides an iterative way to estimate Wim(k) provided s(k-i) is known. We assume with high probability that by simply using the CMA algorithm, the equalizer has converged such that its output is approximately equal to s(k-i), i.e. y(k)s(k-i). To find Wi+dm(k) in eq(13) corresponding to a new delay (i+d), we replace s(k-i) by y(k-d)/|y(k-d)|. Then, we can write the reinitialized equalizer vector at time k as
Wi+d(k)= Wi+d(k-1)+K(k)(y(k-d)/|y(k-d)|)- Wi+dT(k-1)X(k))(14)
The range of d should be –(M+N-1)d(M+N-1) in order to consider all possible delays where M and N-1 denote respectively a known channel order and subequaliser order. The searching procedure is repeated until the equalizer parameter with the smallest mean squared error is obtained.
SIMULATION RESULTS
To demonstrate that in the noisy there are local minima in the CMA error surface for the FS-CMA, two subchannels in the fixed wireless system were chosen as 1+0.5z-1 and 1-0.5z-1. The inverted error surface represents the performance of the CMA algorithm with and without channel noise in Fig.1a and b. As mentioned earlier, the CMA algorithm is capable of reconstructing the transmitted signal without mistakes in the absence of channel noise. Fig.1a displays the standard CMA error surface has four identical minima (equal zero) where the parameters of the equalizers are [0.5,0.5] and [1,-1]. The first pair correspond to an equalizer output of s(k) and the second to s(k-1) as shown in Fig.1a. However, in the presence of channel noise, CMA algorithm still provides the fair solutions which are global minima together with the unacceptable solutions which are the local minima. As seen in Fig.1b, these four minima have different depths. By implementing the update equation in eq(2), simulations of 150 symbols and one tap in each sub channel were used. Fig.2 depicts the FS-CMA error surface and the trajectories of the FS-CMA and the technique based upon eq.(1) with 20dB SNR. When initialized to W=[-1.4,1.4], the FS-CMA equalizer (solid line) converges to the solution for s(k-1) with JCMA=0.0196. To ensure that the equalizer weight vector in eq(5) does not converge to zero vector, small offsets are required between the initialization of the equalizers over the different delays. Thus, the second equalizer (dashed line) is initialized at [-1.41,1.39]. With the first equalizer at one of the local minima, the second equalizer converge to the global minima of JCMA=0.005. However, the existence of the overshoot in the Gram Schmidt trajectory is due to the necessity to scale the equalizer output power.
It was shown in Fig.2 that the error surface associated with the standard CMA algorithm is multi-modal. Therefore, the convergence to a particular minimum depends on the initialization. Fig.3 depicts the convergence trajectory of the standard CMA algorithm and the GS-CMA for 16 different initializations on a circle of radius 2. The algorithms were run over 120 iterations for each initialization. It is shown in Fig.3a that the standard CMA, Initialized from positions1, 5-9 and 13-16, converges to local minima. By using the GS-CMA, the minimum norm of the equalizer weight vectors between the standard CMA from eq.(2) (dashed line) and the Gram-Schmidt from eq(5) (dash-dot line) is selected at symbol no. 120. Fig.3b represents “x” for the convergence of the GS-CMA initialized at 1, 5-9, 13-16. Note that the GS-CMA converges only to one of the symmetrical global minima, whereas the standard CMA may converge (depending on its initialization) to one of the two undesired local minima.
Next we consider the RCC-CMA technique with the same channel and initialization used in GS-CMA. With 20dB SNR, a simulation of 150 symbols was implemented. As shown in Fig.4, the trajectory of the CMA (solid line) converges to the solution for s(k-1) with JCMA =0.0196 whereas the trajectory of the RCC-CMA (dashed line) converges to global minimum of JCMA=0.005.
CONCLUSION
The GS-CMA technique provides a way to find simultaneously the different delay equalizers, whereas the RCC-CMA technique finds the different delays iteratively. However, both methods have been shown to succeed the minimum mean squared error. Hence, it is more likely to reconstruct the transmitted signal properly. Obviously, these two proposed techniques are also capable of reconstructing the transmitted signals without the knowledge of the training sequence. Hence, the bit rate can be improved. Since the high speed wireless data transmission such as the fixed wireless technology is becoming the modus operandi of the Internet access, these two techniques will be appropriated solutions to enhance the performance of the Internet.
(a)(b)
FIG.1 THE INVERTED ERROR SURFACE FOR FS-CMA (A) NOISELESS CASE AND (B) 20DB OF SNR
FIG.2 CONTOUR PLOT FOR FS-CMA, “O”: CMA AND GS-CMA INITIALIZATION POINTS, “*”: THE GLOBAL MINIMA, “X”: THE LOCAL MINIMA, THE SOLID LINE IS THE TRAJECTORY OF FS-CMA, AND THE DASHED LINE IS THE TRAJECTORY OF GS-CMA.
(a)(b)
FIG.3 CONVERGENCE PARAMETER TRAJECTORIES OF STANDARD CMA AND GS-CMA, “O”: INITIAL POINT, “X”: GS-CMA EQUALIZER WEIGHT VECTOR AT SAMPLE NO. 120 INITIALIZED AT 1, 5-9 AND 13-16, THE DASHED LINE IS THE TRAJECTORY OF FS-CMA, AND THE DASH-DOT LINE IS THE TRAJECTORY OF GS-CMA
FIG.4 CONTOUR PLOT FOR FS-CMA, “+”: CMA AND RCC-CMA INITIALIZATION POINTS, “*”: THE GLOBAL MINIMA, “X”: THE LOCAL MINIMA, THE SOLID LINE IS THE TRAJECTORY OF FS-CMA, AND THE DASHED LINE IS THE TRAJECTORY OF RCC-CMA.
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[iv]