# Time Domain Equalizer Design Using Bit Error Rate Minimization for UWB Systems

- 859 Downloads

## Abstract

Ultra-wideband (UWB) communication systems occupy huge bandwidths with very low power spectral densities. This feature makes the UWB channels highly rich in resolvable multipaths. To exploit the temporal diversity, the receiver is commonly implemented through a Rake. The aim to capture enough signal energy to maintain an acceptable output signal-to-noise ratio (SNR) dictates a very complicated Rake structure with a large number of fingers. Channel shortening or time domain equalizer (TEQ) can simplify the Rake receiver design by reducing the number of significant taps in the effective channel. In this paper, we first derive the bit error rate (BER) of a multiuser and multipath UWB system in the presence of a TEQ at the receiver front end. This BER is then written in a form suitable for traditional optimization. We then present a TEQ design which minimizes the BER of the system to perform efficient channel shortening. The performance of the proposed algorithm is compared with some generic TEQ designs and other Rake structures in UWB channels. It is shown that the proposed algorithm maintains a lower BER along with efficiently shortening the channel.

### Keywords

Cyclic Prefix Rake Receiver Effective Channel Asymmetric Digital Subscriber Line Multiuser System## 1. Introduction

Channel shortening is an equalization technique which forces the effective channel impulse response (combined channel and equalizer) to be confined within a desired temporal window. Channel shortening or time domain equalizers (TEQs) have been used in communication systems since the early 1970s [1, 2, 3, 4]. The earlier usage of TEQs was to reduce the number of states in sequence estimation and thus simplify the process. TEQ designs were reinvestigated in the 1990s to mitigate the intersymbol interference (ISI) produced due to inadequate cyclic prefix (CP) in multicarrier modulation (MCM) systems [5, 6, 7, 8, 9, 10]. Each of these designs uses a particular cost function, which may be general or system specific, to perform efficient channel shortening. TEQ has also been proposed to simplify multiuser detection in a large set of users [11]. The TEQ in this case eliminates some users' signals to effectively reduce the size of the user set.

A major problem encountered in UWB systems is to capture enough multipaths through a Rake receiver [12] to maintain a sufficient output signal-to-noise ratio (SNR). An All-Rake (A-Rake) or ideal Rake is not a suitable choice in a dense multipath channel. A Partial-Rake (P-Rake) is easy to implement but provides suboptimum performance. On the other hand, a Selective-Rake (S-Rake) captures a certain number of the strongest multipaths which may not necessarily arrive in successive temporal bins. Therefore, the operational window of the S-Rake may be long enough to cause ISI. Channel shortening can help to mitigate this problem [13, 14, 15, 16]. The presence of the TEQ insures that the channel energy is concentrated into the desired number of multipaths that are available in consecutive bins. As a result, loosely speaking, the Rake receiver enjoys the benefits of S-Rake performance or better in the structure of a P-Rake. Improved SNR is also critical in extending the area of coverage. With a TEQ before the Rake reception, the Rake can be implemented with a smaller number of fingers. This not only simplifies the receiver front end but also the rest of the signal processing and the manufacturing cost involved. Hence, channel shortening in UWB receivers can help in designing a simple and cost effective structure.

UWB communications systems are entirely different from the MCM systems for which a TEQ is commonly proposed. First of all, UWB is a wireless scenario with extremely dense multipath channels. Standard UWB channel models, namely CM1 to CM4 [17], are much more complex than those used in wired line MCM systems, for example, carrier serving area (CSA) loops in asymmetric digital subscriber line (ADSL). Furthermore, to make the UWB receiver design practically simple, a large number of channel taps must be eliminated. This makes the shortened channel window very much smaller than the suppressed channel. Hence, the problem of TEQ design appears in its extreme form. In UWB systems, channel energy capture is crucial to maintain a good output SNR, whereas in most of the existing TEQ designs, except [7, 8, 18], channel delay spread or bit rate is more critical. Also, none of the existing designs considers a multiuser system. The TEQs presented in [13, 14] are very simple to implement but have moderate performance. Whereas the designs presented in [15, 16] perform relatively better but exploit some UWB channel specific parameters. Again, none of them is developed for a multiuser environment. Recently, a TEQ design was proposed which directly minimizes the bit error rate (BER) of cyclic prefixed-based systems [18]. Since traditional UWB systems do not use cyclic prefix and are baseband, we need to derive the BER of a multiuser system in the presence of a TEQ at the receiver front end. To our knowledge, no such system model or analysis is available in the literature for UWB systems. We consider a multiuser system in contrast to most of the existing TEQ designs which assume a single user environment. With some realistic assumptions, we then present an algorithm which performs channel shortening by optimizing the BER of the system.

The remainder of the paper is organized as follows: in Section 2, we briefly discuss the system model used in this paper. The probability of error model and its optimization is derived in Sections 3 and 4, repectively. Performance and complexity analyses are given in Sections 5 and 6, respectively. Section 7 describes the simulation setup followed by the simulation results. The conclusion is given in Section 8.

## 2. System Architecture

where Open image in new window are the multipath gain coefficients, Open image in new window is the delay of the Open image in new window th cluster, Open image in new window is the delay of Open image in new window th multipath component relative to the Open image in new window th cluster arrival time Open image in new window , Open image in new window is the number of clusters, Open image in new window is the number of multipaths within a cluster, and Open image in new window represents the log-normal shadowing associated with multipath amplitudes. Equation (2) is the simplified form of (1) where the multipath gain coefficients Open image in new window and their arrival times Open image in new window are assumed to have absorbed all the statistical properties of Open image in new window , Open image in new window , Open image in new window and Open image in new window , and the channel contains Open image in new window number of multipaths.

We consider an impulse radio (IR) UWB system using pulses Open image in new window of width Open image in new window seconds. In a multiuser environment of Open image in new window simultaneously active users, the unmodulated signalling waveform of the Open image in new window th user is given by

where Open image in new window is the number of pulse repetitions, Open image in new window is the pulse repetition time, Open image in new window is the chip duration such that there are Open image in new window chips within Open image in new window , and Open image in new window is the time hopping (TH) sequence for the Open image in new window th user.

Let Open image in new window be the data sequence available at the Open image in new window th user. We assume that Open image in new window is a wide sense stationary random process with equiprobable symbols. Binary pulse position modulation (BPPM) and binary phase shift keying (BPSK) schemes are considered. Hence, the signal transmitted by the Open image in new window th user can be given as

where Open image in new window for BPPM, Open image in new window for BPSK, Open image in new window is the available power for the Open image in new window th user, and Open image in new window is the modulation index for BPPM and can be chosen to optimize the performance.

It is reasonable to assume that Open image in new window is less than the multipath arrival delay bin and no overlapping between the multipath occurs, that is, only resolvable multipaths are considered. A TEQ Open image in new window is present at the receiver front end before the Rake reception:

where Open image in new window is the Open image in new window th filter coefficient and Open image in new window is the temporal spacing between any two consecutive filter taps.

The received signal from the Open image in new window th user will experience an effective channel of length Open image in new window such that

where Open image in new window is the effective channel, Open image in new window is the channel from the the Open image in new window th user, " Open image in new window '' represents convolution operation, Open image in new window and Open image in new window is the associated delay.

Therefore, the Open image in new window th user signal at the TEQ output is

The additive white Gaussian noise (AWGN) Open image in new window with zero mean and variance Open image in new window will also be processed through the TEQ and can be considered as filtered noise. Hence the signal available for Rake reception is

where Open image in new window is the total noise available at the TEQ output.

## 3. Probability of Error Model

We assume that the receiver knows a typical transmitted waveform and uses it as the correlation template. The template waveform Open image in new window is assumed to be real and synchronized with the TH code of the user of interest and its Open image in new window th multipath arrival time. This means that the TH code Open image in new window for the user of interest is known at the receiver. Each finger of the Rake receiver correlates Open image in new window multipaths along with the noise. The user Open image in new window is the user of interest whose TH code is known at the receiver and the Open image in new window th finger of the Rake is under consideration. In this situation, only the Open image in new window th multipath from the Open image in new window th user contributes to the desired signal energy. All other multipaths from the Open image in new window th user can be accounted for self-interference. Whereas, Open image in new window multipaths from all other users can be regarded as multiple access interference (MAI). The noise, which has now been filtered through the TEQ, is also correlated and contributes through each Rake finger.

Assume that Open image in new window represents the cross-correlation between the template and the received waveform associated with the Open image in new window th multipath from the Open image in new window th user at the Open image in new window th Rake finger for any of the modulation schemes:

where the integral is evaluated over one pulse repetition period, therefore, the index Open image in new window has been dropped. Similarly, Open image in new window is the power of the filtered noise Open image in new window available at the Open image in new window th Rake finger output, such that

Since the actual separation between the Rake fingers is negligible, the channel coefficients from a particular user to any Rake finger can be assumed to be the same. Thus, the contribution of the Open image in new window th user signal power at the Open image in new window th Rake finger output due to multipath channel can be given as

The power available at the Open image in new window th finger output due to the received signal from all users and correlated noise is

Hence, the total received power is the summation of all Rake fingers' output as given below:

As the Open image in new window th user is the user of interest, the TEQ shortens the channel for this user only. In this case, the total received power in (13) can be rewritten in terms of the desired signal Open image in new window , self-interference Open image in new window , multiple access interference (MAI) Open image in new window , and the total noise Open image in new window as follows:

where Open image in new window .

where Open image in new window and Open image in new window .

The instantaneous probability of error for the Open image in new window th user can now be given as

where Open image in new window represents the complementary Gaussian distribution function.

We refer to the term Open image in new window in (16) as shortening signal-to-interference and noise ratio (SSINR) represented by Open image in new window . Optimization of this term will not only shorten the channel but also optimize the BER of the system. It is important to note that a common standard Gaussian approximation (SGA) approach is used when MAI is considered. The proposed method turns out to minimize the instantaneous BER in the low to moderate SNR region where the SGA is accurate. Let each user have unity transmit power available, that is, Open image in new window , then

## 4. BER Optimization Algorithm

The maximization of (17) can be classified into the category of single Rayleigh quotient optimization [20, 21]. Any existing approach can be used to find the optimum solution if the BER is defined in a proper matrix form. Therefore, we first derive the BER in a form which is suitable for optimization. To the knowledge of the authors no such expression is available in the literature for UWB systems. To represent Open image in new window in the matrix form we define the following terms.

Let Open image in new window be the TEQ vector. Open image in new window is the effective channel vector for the Open image in new window th user such that Open image in new window , where Open image in new window is the convolution matrix of the Open image in new window th user channel Open image in new window . Similarly, Open image in new window is a submatrix of Open image in new window containing Open image in new window consecutive rows from the Open image in new window th to Open image in new window row and Open image in new window contains the rest of the rows. The correlation vector for all multipaths from the Open image in new window th user at the Open image in new window th finger is Open image in new window . The vector for the noise entering the TEQ is Open image in new window and Open image in new window is the corresponding convolution matrix. Therefore, Open image in new window is the filtered noise processed through the TEQ. The correlation amplitude of the filtered noise at each finger is Open image in new window .

Hence, each term in (16) can be written in the matrix form as follows:

where Open image in new window is the Open image in new window th row removed version of Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window and Open image in new window is a matrix such that Open image in new window .

where Open image in new window , Open image in new window , Open image in new window and Open image in new window .

From (20) and (25), it is important to note that the contribution of the noise to the SSINR can be reduced by choosing a large value of Open image in new window . Also, if the TH codes of the users are sufficiently orthogonal, we have Open image in new window , which makes MAI significantly small.

where Open image in new window is the eigenvector corresponding to maximum eigenvalue of Open image in new window and Open image in new window is the Cholesky factor of Open image in new window .

The above optimization, as used in many other TEQ designs [7, 8, 9], is performed iteratively to choose the best location of the shortened channel window in the effective channel. The iterative process slides the shortened window from the beginning till the end of the effective channel and chooses the location where the cost function is maximum. It is also possible to define a particular location of the shortened window, but it may not necessarily be an optimum solution.

## 5. Performance Analysis

where Open image in new window is a partition of Open image in new window having any consecutive Open image in new window rows, Open image in new window is the remaining part and the term optimized can be referred to as Open image in new window .

It is evident from (29) and (30) that in an attempt to maximize the cost function given in (29), the MSSNR TEQ also enhances Open image in new window , that is, the self-interference, MAI and the noise available within the window. On the other hand, the proposed TEQ keeps the unwanted power terms to their minimum.

where Open image in new window is the denominator of (23).

This shows that the SSINR of the proposed TEQ will always be greater than MSSNR TEQ in a multiuser and/or AWGN environment. In a single user and noise-free system, both of them will have same performance if the self-interference is neglected. It is also interesting to note that making the MSSNR TEQ more efficient in terms of Open image in new window by increasing the value of Open image in new window will further worsen its performance.

## 6. Complexity Analysis

A very important issue is the relative complexity of the proposed solution. One can think that the simplification in the Rake structure is now transformed into the complexity of the proposed TEQ design. In fact, the proposed solution can be considered as a TEQ followed by a P-Rake. Since the P-Rake does not need a search algorithm for the arriving multipaths, its complexity is negligible as compared to the TEQ complexity. Hence, the overall complexity of the proposed solution, that is, TEQ plus P-Rake, actually lies in the TEQ design. In this section, we briefly analyze the complexity of the proposed TEQ with the S-Rake design. The comparison can be made on different sets of criteria. Here we compare both designs for initial evaluation on the basis of the number of multipaths collected, that is, Open image in new window . The complexity of the proposed TEQ lies in calculating the parameters used in (25) and then performing the optimization. The complexity of the S-Rake lies in searching a subset of Open image in new window strongest multipaths in a channel which is Open image in new window multipaths long.

where Open image in new window .

For the S-Rake, (32) must be computed for Open image in new window multipaths from Open image in new window users at Open image in new window fingers of the Rake. In total, there are Open image in new window values of Open image in new window that must be squared, leading to Open image in new window multiplies. Also, the term Open image in new window must be multiplied by Open image in new window for Open image in new window combinations of Open image in new window and Open image in new window , and the numerator must be divided by the denominator for Open image in new window combinations of Open image in new window and Open image in new window . Everything else in the (32) requires much fewer computations and can be ignored. Thus, (32) requires Open image in new window multiplies and Open image in new window divisions, or Open image in new window operations.

For the proposed design, (25) must be computed. Efficient techniques utilizing reuse of computations [23] can reduce the complexity of evaluating Open image in new window and Open image in new window to Open image in new window . The other terms are mostly summations, and are generally cheaper than Open image in new window . Thus, maximizing the Rayleigh quotient, which is Open image in new window [23], is more complex than computing the Rayleigh quotients in (25), and the overall complexity is Open image in new window .

Another issue is memory use. The S-Rake stores all the values of Open image in new window and the related index Open image in new window . Infact, the memory usage is directly proportional to the duration of the operational window of the S-Rake. This is another disadvantage of S-Rake's long operational window as shown in Figure 2. It is evident that the complexity of the S-Rake increases in dense multipath channels (large Open image in new window ) and with increasing number of users (large Open image in new window ). If both values increase simultaneously, the complexity grows in a quadratic fashion. The complexity of the proposed TEQ is independent of the channel length and the number of users but it grows with cubic power of the TEQ length. Therefore, the TEQ length must be chosen very carefully. For a numerical example, the CM3 profile is roughly Open image in new window taps long at a sampling rate of Open image in new window ns. In a multiuser system with Open image in new window and the TEQ length of Open image in new window , the complexity of the S-Rake is Open image in new window whereas the complexity of the proposed TEQ is roughly Open image in new window . Infact, in dense multipath channels the complexity of both designs is comparable, but when it comes to the memory usage the proposed TEQ outperforms the S-Rake. As depicted in Figure 2, the operational window of S-Rake is Open image in new window to Open image in new window times larger than the operating window of the proposed TEQ. Hence, the S-Rake needs Open image in new window to Open image in new window times more memory from CM1 to CM4 channels.

## 7. Simulation Results

The performance of the proposed BER minimization TEQ is compared with the MSSNR TEQ [7, 8], A-Rake, P-Rake, and S-Rake [22] in CM1, CM2, CM3, and CM4 environments. All users are provided with random semiorthogonal TH codes and employ TH-BPPM and/or TH-BPSK. Channel coefficients are generated at a sampling rate of Open image in new window GHz with Open image in new window for CM1, Open image in new window for CM2, Open image in new window for CM3, and Open image in new window for CM4. P- and S-Rake are capturing the first Open image in new window and the strongest Open image in new window multipaths, respectively. A-Rake is providing a lower bound by capturing all the multipaths and gathering the total available signal energy except the self-interference. First-order Gaussian derivative pulses of Open image in new window ns with center frequency 3 GHz are used. The transmit antenna effects are modeled via random low pass filtering which changes the shape of the transmitted pulse to the second order Gaussian pulse. The modulation index Open image in new window is 2 ns and the chip duration Open image in new window is 5 ns. Other system parameters, for example, Open image in new window , Open image in new window , Open image in new window , and Open image in new window are either kept constant to a certain value or varied in different simulations.

Extensive simulations were performed to test the capabilities of the proposed BER minimization TEQ design. The results are generated by averaging the performance parameter through Monte Carlo simulations. As an SGA approach is used, all the simulations for BER are performed in low to moderate SNR range. Since the performance depends upon many factors, each factor is considered individually.

*x*-axis and close to unity. The small gap to perfection is due to self-interference.

## 8. Concluding Remarks

In this paper, we consider a realistic UWB scenario with all the main factors which may affect the Rake receiver performance. We derive an expression for the BER of the this system in the presence of a TEQ at the receiver front end. Based on the derived formula, we propose a TEQ design which directly attempts to optimize the BER of the system while pushing the effective channel energy within the desired temporal window. We compared the BER performance of the proposed design with P-Rake, S-Rake, and MSSNR TEQ with A-Rake as realistic lower bound. It is shown that the proposed TEQ performs better than the MSSNR TEQ, S-Rake, and P-Rake and is confirmed through simulations. All the major factors which may affect the performance of the proposed TEQ are simulated and discussed. It is shown that the proposed TEQ outperforms the considered MSSNR TEQ and the Rake architectures in any performance aspect. Especially, the proposed TEQ maintains a lower BER while shortening the dense multipath channels to a desired small temporal window. Hence, with the proposed TEQ design, an UWB Rake receiver can be designed with significantly less number of fingers/correlators without compromising the receiver performance in terms of the BER. This will also simplify the receiver architecture and analysis that follow the Rake.

## Notes

### Acknowledgments

The views expressed in this paper are those of the authors, and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U.S. Government. This document has been approved for public release; distribution unlimited.

### References

- 1.Cantoni A, Butler P: Properties of the eigenvectors of persymmetric matrices with applications to communication theory.
*IEEE Transactions on Communications*1976, 24(8):804-809. 10.1109/TCOM.1976.1093391MATHMathSciNetCrossRefGoogle Scholar - 2.Falconer DD, Magee FR Jr.: Adaptive channel memory truncation for maximum likelihood sequence estimation.
*Bell System Technical Journal*1973, 52(9):1541-1562.MATHCrossRefGoogle Scholar - 3.Forney GD Jr.: Maximum-likelihood sequence estimation of digital sequences in the presence of inter symbol interference.
*IEEE Transactions on Information Theory*1972, 18(3):363-378. 10.1109/TIT.1972.1054829MATHMathSciNetCrossRefGoogle Scholar - 4.Magee FR Jr.: A comparison of compromise Viterbi algorithm and standard eqaulization techniques over band limited channels.
*IEEE Transactions on Communications*1975, 23(3):361-367. 10.1109/TCOM.1975.1092802CrossRefGoogle Scholar - 5.Al-Dhahir N, Cioffi JM: Efficiently computed reduced-parameter input-aided MMSE equalizers for ML detection: a unified approach.
*IEEE Transactions on Information Theory*1996, 42(3):903-915. 10.1109/18.490553MATHCrossRefGoogle Scholar - 6.Miyajima T, Ding Z: Second-order statistical approaches to channel shortening in multicarrier systems.
*IEEE Transactions on Signal Processing*2004, 52(11):3253-3264. 10.1109/TSP.2004.836537CrossRefGoogle Scholar - 7.Melsa PJW, Younce RC, Rohrs CE: Impulse response shortening for DMT transceivers.
*IEEE Transactions on Communications*1996, 44(12):1662-1672. 10.1109/26.545896CrossRefGoogle Scholar - 8.Yin C, Yue G: Optimal impulse response shortening for discrete multitone transceivers.
*Electronics Letters*1998, 34(1):35-36. 10.1049/el:19980011CrossRefGoogle Scholar - 9.Balakrishnan J, Martin RK, Johnson CR Jr.: Blind adaptive channel shortening by sum-squared auto-correlation minimization (SAM).
*IEEE Transactions on Signal Processing*2003, 51(12):3086-3093. 10.1109/TSP.2003.818892MathSciNetCrossRefGoogle Scholar - 10.Martin RK, Balakrishnan J, Sethares WA, Johnson CR Jr.: A blind adaptive TEQ for multicarrier systems.
*IEEE Signal Processing Letters*2002, 9(11):341-343. 10.1109/LSP.2002.804423CrossRefGoogle Scholar - 11.Medvedev I, Tarokh V: A channel-shortening multiuser detector for DS-CDMA systems.
*Proceeding of the 53rd IEEE Vehicular Technology Conference (VTC '01), May 2001, Rhodes, Greece*3: 1834-1838.CrossRefGoogle Scholar - 12.Rajeswaran A, Somayazulu VS, Foerster JR: Rake performance for a pulse based UWB system in a realistic UWB indoor channel.
*Proceedings of the IEEE International Conference on Communications (ICC '03), May 2003, Anchorage, Alaska, USA*4: 2879-2883.Google Scholar - 13.Husain SI, Choi J: Single correlator based UWB receiver implementation through channel shortening equalizer.
*Proceedings of the 11th Asia-Pacific Conference on Communications (APCC '05), October 2005, Perth, Wash, USA*610-614.Google Scholar - 14.Husain SI, Choi J: Blind adaptive channel shortening by unconstrained optimization for simplified UWB receiver design.
*Proceedings of the 3rd International Symposium on Wireless Communication Systems (ISWCS '06), September 2006, Valencia, Spain*443-446.Google Scholar - 15.Husain SI, Yuan J, Zhang J: Modified channel shortening receiver based on MSSNR algorithm for UWB channels.
*Electronics Letters*2007, 43(9):535-537. 10.1049/el:20070584CrossRefGoogle Scholar - 16.Husain SI, Yuan J, Zhang J: Rake performance after channel shortening by decay factor optimization in UWB channels.
*Proceeding of the 66th IEEE Vehicular Technology Conference (VTC '07), October 2007, Baltimore, Md, USA*1204-1207.Google Scholar - 17.Foerster JR,
*et al*.:*Channel modelling sub-committee report final.*Working Group for Wireless Personal Area Networks, Monterey, Calif, USA; February 2003.Google Scholar - 18.Martin RK, Ysebaert G, Vanbleu K: Bit error rate minimizing channel shortening equalizers for cyclic prefixed systems.
*IEEE Transactions on Signal Processing*2007, 55(6):2605-2616.MathSciNetCrossRefGoogle Scholar - 19.Saleh AAM, Valenzuela R: A statistical model for indoor multipath propagation.
*IEEE Journal on Selected Areas in Communications*1987, 5(2):128-137.CrossRefGoogle Scholar - 20.Golub GH, Van Loan CF:
*Matrix Computations*. The Johns Hopkins University Press, Baltimore, Md, USA; 1996.MATHGoogle Scholar - 21.Watkins DS:
*Fundamentals of Matrix Computations*. John Wiley & Sons, New York, NY, USA; 1991.MATHGoogle Scholar - 22.Gezici S, Chiang M, Poor HV, Kobayashi H: Optimal and suboptimal finger selection algorithms for MMSE rake receivers in impulse radio ultra-wideband systems.
*EURASIP Journal on Wireless Communications and Networking*2006, 2006:-10.Google Scholar - 23.Martin RK, Vanbleu K, Ding M,
*et al*.: Implementation complexity and communication performance tradeoffs in discrete multitone modulation equalizers.*IEEE Transactions on Signal Processing*2006, 54(8):3216-3230.CrossRefGoogle Scholar

## Copyright information

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.