- Research Article
- Open Access
Approximate Minimum Bit Error Rate Equalization for Fading Channels
© Lorant Kovacs et al. 2010
- Received: 1 October 2009
- Accepted: 8 July 2010
- Published: 26 July 2010
A novel channel equalizer algorithm is introduced for wireless communication systems to combat channel distortions resulting from multipath propagation. The novel algorithm is based on minimizing the bit error rate (BER) using a fast approximation of its gradient with respect to the equalizer coefficients. This approximation is obtained by estimating the exponential summation in the gradient with only some carefully chosen dominant terms. The paper derives an algorithm to calculate these dominant terms in real-time. Summing only these dominant terms provides a highly accurate approximation of the true gradient. Combined with a fast adaptive channel state estimator, the new equalization algorithm yields better performance than the traditional zero forcing (ZF) or minimum mean square error (MMSE) equalizers. The performance of the new method is tested by simulations performed on standard wireless channels. From the performance analysis one can infer that the new equalizer is capable of efficient channel equalization and maintaining a relatively low bit error probability in the case of channels corrupted by frequency selectivity. Hence, the new algorithm can contribute to ensuring QoS communication over highly distorted channels.
- Minimum Mean Square Error
- Radial Basis Function Neural Network
- Dominant Term
- Equalization Algorithm
- Equalizer Coefficient
Broadband radio channels are susceptible to selective fading due to multipath propagation. In this case, the differences among the propagation delays on the paths may amount to a significant fraction of a symbol interval. Hence, frequency selective fading may yield severe performance degradation. As a result, efficient channel equalization techniques prove to be instrumental to combat intersymbol interference (ISI) in order to avoid large scale drops in system performance.
Since the effect of interferences are especially crucial in wireless communication systems, fast channel equalizer algorithms have to be developed which are simple enough to run on the currently available hardware architectures. This paper aims at developing a low complexity channel equalizer algorithm by directly minimizing the BER instead of minimizing the mean-square error or the peak distortion [1, 2]. Unfortunately, the direct minimization of BER with respect to the equalizer coefficients is of exponential complexity due to the large summation when expressing BER as a function of the equalizer coefficients. Thus, we develop a new bound on BER on which basis the equalizer coefficients can be optimized by a fast algorithms. It is also pointed out that the new bound on BER is sharper than the previously known ones listed in .
The first attempts to derive an equalizer based on the minimum BER strategy can be found in the work of Shimbo and Celebiler  and Shamas and Yao . The optimal equalizer coefficients were only sought by exhaustive search, thus real-time adaptivity was not guaranteed. In recent years, some new results have been developed for minimum BER equalization. In , a low-complexity adaptive algorithm is proposed for 2- or 4-state modulation systems but the convergence is rather slow, while in [7, 8] near minimum BER equalization is carried out by radial basis function neural networks which considerably increases the equalizer complexity. On the other hand, very complex equalizer schemes have been proposed for DS-CDMA systems in [9–11]. Paper  investigates the minimization of BER in MIMO systems with linear equalizers based on QPSK modulation. In this model the information sequence is corrupted by multiuser interference and additive noise. BER is derived for the case of two transmitters and approximated by a simple formula. The optimum then is calculated by analytical tools. When the number of transmitters is larger than two then the authors use a sequence of cascade filters. It is important to note that this article does not address the problem of ISI. In paper , BER is minimized subject to some constraints. The authors prove that the constrained BER cost function has only one global minimum and equalization can be achieved by quadratic programming. However, due to the constraints this detector is only optimal in the case of minimum phase channels. In the paper [14, 15], the equalization is carried out by using the Bayes risk criterion. This is a more general approach than BER minimization, but the choice of loss function is left open, realizing that not every loss function will yield fast convergence or yield low error rate. Furthermore, the loss function is very complex and does not lend itself to simple minimization. Therefore, deriving bound on BER for fast minimization with respect to the equalizer coefficients can still yield powerful and real-time channel equalization. The novel algorithm presented by the paper is demonstrated by BPSK modulation scheme.
in Section 2, the communication model is outlined;
in Section 3, BER is expressed as a function of the equalizer coefficients and a gradient based algorithm is discussed for minimizing BER;
in Section 4, a novel method is derived to approximate BER by using the dominant terms;
in Section 5, the new equalizer algorithm will be introduced based on the approximation treated in Section 4;
in Section 6, the performance and convergence properties of the new equalizer algorithms are analyzed numerically.
To describe single-user communication over a fading channel, we use the so-called equivalent discrete time white noise filter model (for further details see ).
The corresponding quantities are defined as follows:
denotes the transmitted information bit at time instant being a sequence of identically distributed independent Bernoulli random variables with ;
the discrete impulse response of the channel is denoted by where denotes the span of ISI;
the noise is denoted by and is assumed to be a stationary zero mean white Gaussian random sequence with constant spectral density ;
where denotes the free parameters of the equalizer which are subject to further optimization;
where denotes the delay of the channel. (For the sake of brevity, here we assume , while the more general treatment for will be given in Section 4.3);
where denotes the support of the overall impulse response.
Both approaches involve the use of linear stochastic approximation schemes [1, 2, 16], but they fell short of providing efficient equalization as the goal functions did not have any direct relationship with BER.
Here, is the value of the weight vector at the th iteration. One must note, that the gradient search with fixed step size in general will not guarantee the convergence to the global minimum. However, as our simulations have demonstrated, over standard wireless channels the algorithm in most cases reached the global optimum.
In the forthcoming discussion, the procedure given by formula (11) is termed as true gradient search (TGS). Unfortunately performing TGS is computationally prohibitive, because of the summation over an exponentially large number of vectors in expression (10). Furthermore, this summation in TGS must be calculated in each step of algorithm (11). Thus, TGS can only be applied in practice if the support of the overall impulse response defined in (4) is very limited. Otherwise, near-optimal algorithms must be sought which lend themselves to real-time implementations. To ease this complexity a new bound is derived on BER.
In this section, we derive a new approximation on BER. The purpose of developing this approximation is to estimate BER with an expression which is a computationally simple function of the equalizer weights. This paves the way towards real-time channel equalization.
In order to derive a bound on BER, one can note that function tends rapidly to zero for negative arguments. As a result, the terms in the summation can differ in several magnitudes. This gives rise to the idea of collecting only the dominant terms to provide a lower bound on BER. This lower bound has been commonly used in other domains, such as reliability analysis and referred to as Li-Silvester bound , where the tiresome calculation of an expected value over a large state space is approximated by only using the dominant terms in the summation.
where denotes the cardinality of .
In this case, is subject to uniform distribution, thus the left hand side of (15) can be very tight if is chosen reasonably high. At the same time, the upper bound tends to be loose as the terms in have rather inaccurate upperbounds. In Section 4.2, we will evaluate the tightness of the bound based on the th dominant sample.
The most "harmful" sequence, denoted by in (i.e., the sequences beginning with ) is since , which indicates to be the absolute dominant term in the summation equation (12).
Note that points to the th smallest element of in absolute value. The extension of index array for will be given in Section 4.3. The second dominant term can be deduced from by changing the sign of the component , because in this case , where is the smallest possible value for decreasing the PD.
and change the sign of the component ;
and change the sign of the component of ; ;
FOR TO find the index set for which is minimal, but and ;
Form the set to be used in the lower bound given in (16).
It is easy to see that the case of (when ) results in a cost function which has minimum value over the same coefficient vector as has the peak distortion. Increasing the value of , the lower bound in (16) tends to the exact and finally the case of and results in the exact minimum of BER.
and change the sign of the component ; ;
and change the sign of the component ; ;
If then and change the sign of the components and ; ; ELSE and change the sign of the component ; ;
Form the set to be used in the lower bound (16).
Unfortunately, calculating the set to find the largest term is of exponential complexity, as must be calculated and arranged in monotone order for all possible .
4.1. Optimization of the Bound
Of course, one can use variable step size in algorithm (19) to improve the speed of convergence. For example, the Armijo rule  can be applied to speed up the convergence. However, simulations showed no improvement by applying this rule. Another problem with this method is its high complexity (the gradient has to be evaluated several times). On the other hand, we may introduce a heuristically chosen step size, such as . The convergence of (19) by using this step size this algorithm is guaranteed by the Kushner-Clark theorem (for more details, see ). In the simulation section, the improvement of convergence achieved by variable step size method is also illustrated.
4.2. A Numerical Example for Calculating the New Bound
Figure 2 shows two curves belonging to the 10 and 20 dB, respectively. From this figure, it can be seen that if and dB then the necessary number of samples . This necessary sample number will increase with respect to the decrease of SNR (in the case of dB the number of samples is seven). This is in line with the reasoning detailed above.
4.3. Handling the Channel Delay
In order to run the proposed algorithm, channel state information is needed (the channel impulse function appears in expression (9)). There are plenty of real time adaptive channel identification algorithms  which provide fast and simple channel state information by using a training sequence where is a transmitted binary sequence known at the receiver and is the corresponding received sequence.
This algorithm minimizes the mean square error between the unknown channel impulse response function and the FIR filter coefficients . Here denotes the received sequence at the output of the channel estimator. Parameters in algorithm (35) converge to the true channel impulse response function in mean square (and in probability) if the degree of the FIR filter is larger than the channel impulse response (overmodeling).
It is noteworthy that the adaptive channel identifier (35) converges rather fast to the true cannel-state because of the narrow eigenvalue-spectrum of the underlying matrices (for further details see ). Hence, the combination of identification and equalization can provide real time solutions for low BER communication.
In this section, a detailed performance analysis is given in which the bit error probability achieved by the different equalization methods are compared with each other.
6.1. Channel Characteristics and Channel-State Information
The channel distortion can be modeled by a tapped delay-line model (see Section 2). If the WSSUS (Wide Sense Stationary Uncorrelated Scattering) assumption is made, then the channel coefficient are uncorrelated, and Gaussian distributed. In the project COST 207 , several wideband propagation models were proposed for the practical realization of both hardware and software simulators in the context of GSM systems for different classes of environments (an other set of models is ITU-R models for third-generation cellular systems). These models are generally described by power delay profiles from which the discrete time equivalent can be derived.
The simulations were performed on three different discrete channels representing multipath propagation derived from the power delay profiles of the above mentioned models. The corresponding channel characteristics are given by their impulse response as follows: , .
Note that channel has the minimum-phase, while and have the nonminimum-phase property. The equalization of nonminimum-phase channels is difficult, because these channels have zeros outside the unit circle, and hence, the inverse of the channel has poles outside the unit circle.
6.2. Performance Analysis
TGS—True Gradient Search (for details see (11));
LISIx—our algorithm with dominant terms in the approximation;
LMS—Least Mean Square algorithm;
MMSE—Off-line calculated Minimum Mean Square Error solution
NOEQ—BER without any equalizer;
AMBER—Adaptive Minimum Bit Error Rate algorithm .
As far as the channel-state information is concerned, we assumed no channel-state information to be available at the receiver side, thus channel equalization was preceded by an adaptive channel identifier algorithm given in (35). In all simulations the delay parameter used in the decision rule (3) was set by exhaustive search. The step size of the gradient-descent-type algorithms was set empirically. The experiments show that the attained BER is not too sensitive to the value of the step size, while the convergence speed is highly dependent on this value as described below. Furthermore the value of the step size depends on the SNR as well, since the error surface tends to be more complicated as SNR increases .
6.2.1. Convergence Analysis
Figure 3 also demonstrates that algorithm LISI4 with fix step size will yield slower convergence than TGS. However, the convergence speed can be increased by modifying the step size in each step according to the rule , which indicates that investigation on the step size can also improve the convergence.
6.2.2. BER versus SNR
In this paper, a novel channel equalizer algorithm has been developed based on approximating the BER by dominant terms. Due to the simplicity of this approximation, a fast equalization algorithm can be obtained, the performance of which falls close to optimum. Since this approximation needs channel state information, the equalizer is preceded by an adaptive channel identifier. The combined convergence of channel identification and the new bound-based equalization is still much faster than the convergence of other algorithms (e.g., LMS, AMBER ). The operational complexity of the new algorithm is also smaller than TGS (for details see (11)). The new method yielded better performance than the traditional ZF and MMSE equalizer algorithms on standard wireless channels. These benefits make the new algorithm suitable for real time applications.
- Proakis J: Digital Communications. 4th edition. McGraw–Hill, New York, NY, USA; 2001.MATHGoogle Scholar
- Biglieri E, Proakis J, Shamai S: Fading channels: information-theoretic and communications aspects. IEEE Transactions on Information Theory 1998, 44(6):2619-2692. 10.1109/18.720551MathSciNetView ArticleMATHGoogle Scholar
- Glave FE: An upper bound on the probability of error due to intersymbol interference for correlated digital signals. IEEE Transactions on Information Theory 1972, 18(3):356-363. 10.1109/TIT.1972.1054828MathSciNetView ArticleMATHGoogle Scholar
- Shimbo SO, Celebiler M: The probability of error due to intersymbol interference and gaussian noise in digital communication systems. IEEE Transactions on Communications 1971, 19(2):113-119. 10.1109/TCOM.1971.1090619View ArticleGoogle Scholar
- Shamash E, Yao K: On the structure and performance of a linear decision feedback equalizer based on the minimum error probability criterion. Proceedings of the 10th International Conference on Communications (ICC '74), 1974, Minneapolis, Minn, USA 1-5.Google Scholar
- Yeh C-C, Barry JR: Adaptive minimum bit-error rate equalization for binary signaling. IEEE Transactions on Communications 2000, 48(7):1226-1235. 10.1109/26.855530View ArticleGoogle Scholar
- Chen S, Mulgrew B, Hanzo L: Least bit error rate adaptive nonlinear equalisers for binary signalling. IEE Proceedings: Communications 2003, 150(1):29-36. 10.1049/ip-com:20030284View ArticleGoogle Scholar
- ChandraKumar P, Saratchandran P, Sundararajan N: Minimal radial basis function neural networks for nonlinear channel equalisation. IEE Vision Image and Signal Processing 2000, 147(5):428-435. 10.1049/ip-vis:20000459View ArticleGoogle Scholar
- Chen S, Samingan AK, Mulgrew B, Hanzo L: Adaptive minimum-BER linear multiuser detection. IEEE Transactions on Signal Processing 2001, 49(6):1240-1247. 10.1109/78.923306View ArticleGoogle Scholar
- Dua A, Desai UB, Mallik RK: Minimum probability of error-based methods for adaptive multiuser detection in multipath DS-CDMA channels. IEEE Transactions on Wireless Communications 2004, 3(3):939-948. 10.1109/TWC.2004.827759View ArticleGoogle Scholar
- Hjorungnes A, Debbah M: Minimum BER FIR receiver filters for DS-CDMA systems. Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM '05), December 2005, St. Louis, Mo, USA 4: 2287-2291.Google Scholar
- Gesbert D: Robust linear MIMO receivers: a minimum error-rate approach. IEEE Transactions on Signal Processing 2003, 51(11):2863-2871. 10.1109/TSP.2003.818160MathSciNetView ArticleGoogle Scholar
- Wang XF, Lu W-S, Antoniou A: Constrained minimum-BER multiuser detection. IEEE Transactions on Signal Processing 2000, 48(10):2903-2909. 10.1109/78.869045View ArticleGoogle Scholar
- Gunther J, Moon T: Minimum bayes risk adaptive linear equalizers. IEEE Transactions on Signal Processing 2009, 57(12):4788-4799.MathSciNetView ArticleGoogle Scholar
- Bertsekas DP: Nonlinear Programming. 2nd edition. Athena Scientific, Belmont, Mass, USA; 1999.MATHGoogle Scholar
- Haykin S: Adaptive Filter Theory. 4th edition. Prentice Hall, New York, NY, USA; 2001.MATHGoogle Scholar
- Li VOK, Silvester JA: Performance analysis of networks with unreliable components. IEEE Transactions on Communications 1984, 32(10):1105-1110.View ArticleGoogle Scholar
- Steele R, Hanzo L: Mobile Radio Communications. 2nd edition. Wiley-IEEE Press, New York, NY, USA; 1999.View ArticleGoogle Scholar
- Failli M: Digital land mobile radio communications COST 207. European Commission; 1989.Google Scholar
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