# Fast Multi-Symbol Based Iterative Detectors for UWB Communications

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## Abstract

Ultra-wideband (UWB) impulse radios have shown great potential in wireless local area networks for localization, coexistence with other services, and low probability of interception and detection. However, low transmission power and high multipath effect make the detection of UWB signals challenging. Recently, multi-symbol based detection has caught attention for UWB communications because it provides good performance and does not require explicit channel estimation. Most of the existing multi-symbol based methods incur a higher computational cost than can be afforded in the envisioned UWB systems. In this paper, we propose an iterative multi-symbol based method that has low complexity and provides near optimal performance. Our method uses only one initial symbol to start and applies a decision directed approach to iteratively update a filter template and information symbols. Simulations show that our method converges in only a few iterations (less than 5), and that when the number of symbols increases, the performance of our method approaches that of the ideal Rake receiver.

## Keywords

Channel Impulse Response Information Symbol Rake Receiver Multiple Access Interference Generalize Likelihood Ratio Test## 1. Introduction

Ultra-wideband (UWB) impulse radio (IR) transmits ultra-short pulses at low power spectral density where the information is encoded via pulse-amplitude modulation (PAM) or via pulse-position modulation (PPM). The IR-UWB systems show some important merits including: coexistence with current narrowband signals, high multiple-access capacity and fine timing resolution [1, 2, 3]. Fine timing resolution property helps the receiver to resolve distinct dense multipath components and provides high degrees of diversity whilst the low power spectral density enables sharing of the RF spectrum with limited mutual interference.

One of the major challenges in UWB system is to deal with the dense multipath channel. Indeed, each transmitted UWB pulse arrives at the receiver as hundreds of replicas with different delays, amplitudes and phases [4, 5, 6]. To collect the available diversity, Rake receivers [7, 8] employ a large number of fingers to capture the multipath energy [9]. However, channel estimation error can degrade the Rake's performance and the accurate estimation of the gains and delays of channel paths incurs considerable computational cost [10].

As an alternative to the Rake receiver, the transmitted reference (TR) method [8, 11, 12, 13, 14] sends a reference signal along with the data-modulated signal. The receiver can simply be an autocorrelation receiver which demodulates the data by correlating the delayed reference signal and the data-modulated signal. The advantage of the TR method compared to the Rake method is that it is easier to implement because it does not require explicit channel estimation. However, the main drawback of TR-based methods is that the noise induced in the reference signal severely degrades the error performance.

In [15], decision-directed autocorrelation (DDA) receivers are proposed to detect the current symbol by correlating the current information waveform with a waveform template generated by all previously decoded symbols. However, the DDA receivers detect the information symbols successively and the current detected symbol has no contribution to the preceding symbol detection. To relieve the noise effect of the reference signal in TR system, further enhancement techniques exploit the multi-symbol differential detection [16, 17] to jointly detect Open image in new window consecutive symbols. The generalized likelihood ratio test (GLRT) approach for the multi-symbol case is derived and exhaustive search is performed on all Open image in new window symbol possibilities to find the optimal one [16]. The practical implementation of the method suffers from the exponential computational complexity in terms of block size Open image in new window . A reduced complexity algorithm is devised in [17] by introducing the sphere decoding algorithm (SDA). An approximate algorithm based on the Viterbi algorithm (VA) is also presented in [17]. Although SDA and VA reduce the complexity relative to exhaustive search, and are effective for small Open image in new window , they require considerable computational effort when Open image in new window is large.

In this paper, we propose a fast multi-symbol iterative detection method. The method harvests the benefits from the concept of the multiple symbols detection and outputs a better bit error rate (BER) performance than the single symbol TR system whilst exhibiting a low computational complexity ( Open image in new window , where Open image in new window is block size and Open image in new window is the maximum number of iterations). Following the description of general iterative method, two particular low-complexity detectors are designed and evaluated in the simulation experiments. Although the proposed method cannot guarantee to achieve the same performance as the GLRT-based detector in the general case, experimental results show that the BER performance of the method is very close to that of the GLRT when Open image in new window (the signal-to-noise ratio (SNR) gap is less than Open image in new window dB). Further experiments demonstrate that a few iterations ( Open image in new window iterations) are sufficient for the detectors to converge.

The rest of the paper is organized as follows. Section 2 introduces the UWB signal model. Section 3 describes the multi-symbol transmitted reference system with GLRT detection. Section 4 develops two fast multi-symbol transmitted reference-based detectors. Section 5 shows the numerical results for a constant channel and random channels, respectively. Section 6 concludes the paper.

## 2. Signal Model

where Open image in new window is a transmitted monocycle waveform with support set Open image in new window , the Open image in new window 's are the modulated symbols, the Open image in new window 's are the user-specific pseudorandom time-hopping (TH) codes and Open image in new window is its frame duration. Because the energy of one single pulse is limited in UWB communication systems, each symbol is transmitted using Open image in new window frames so that the receiver can collect enough energy to recover the signal. Thus, the symbol duration is Open image in new window . The TH codes Open image in new window are integers chosen from Open image in new window so that multiple users can access the channel concurrently and the transmission time of Open image in new window th monocycle waveform is delayed with Open image in new window seconds. Due to the highly-frequency selective feature of UWB channel, the frame duration is chosen such that Open image in new window , where Open image in new window is the maximum excess delay of the channel. This condition eliminates intersymbol interference (ISI). The energy of one pulse is Open image in new window .

where Open image in new window is the channel template, Open image in new window denotes the convolution operation and Open image in new window denotes the noise including multiple access interference (MAI) and an additive white Gaussian noise (AWGN) with zero mean and two-sided power spectral density Open image in new window . The noiseless received signal energy in each frame is defined as Open image in new window and is proportional to the pulse energy Open image in new window .

- (i)
Transmitted Reference (TR) [12] with Open image in new window = 1 if Open image in new window is even, otherwise Open image in new window = Open image in new window .

- (ii)
Multi-Symbol Differential Encoder (MSDE) [17] with Open image in new window and Open image in new window where Open image in new window is a multi-symbol block index and Open image in new window .

- (iii)
Multi-Symbol Transmitted Reference (MSTR) with Open image in new window and Open image in new window where Open image in new window is a multi-symbol block index and Open image in new window .

In this paper, our focus is on the MSTR encoder in this paper. All these encoders employ the first modulated symbol as the reference signal in each block and the TR scheme [12] can be viewed as a special case of MSTR scheme where Open image in new window . For MSDE case, the current transmitted symbols are encoded differentially with the previous encoded symbols and the first symbol is used as an initial symbol, while in MSTR case, the current transmitted symbol is the same as the information symbol except the first one, which is used to generate the reference template.

## 3. GLRT-Based Multi-Symbol Detection

by assuming that the channel is quasi-static over the interval Open image in new window .

where Open image in new window is the integration interval of the correlator, and Open image in new window .

Some remarks are now of interest.

where Open image in new window is the decision variable for ATR.

where Open image in new window is the Open image in new window -function Open image in new window

Open image in new window Unlike the ideal Rake receiver, which correlates the receive signal with noiseless template, the TR scheme uses the noisy reference signal as a template in one symbol case and the best estimated reference signal using (11) in the multi-symbol case. However, the TR system does not explicitly estimate the channel parameters and only requires the correlation coefficients Open image in new window , Open image in new window , Open image in new window , Open image in new window evaluated in (14).

Open image in new window As seen in (11), the variance of the reference signal Open image in new window decays as Open image in new window increases when Open image in new window . In turn, the accuracy of the multi-symbol detection is improved and converges to the performance of ideal Rake receiver as Open image in new window

Open image in new window The global optimal value of Open image in new window can be obtained by using exhaustive search [16] or sphere decoding [17]. However, the computational cost of the exhaustive search method grows exponentially with the number of symbols Open image in new window . Sphere decoding method searches all the lattice points inside a given radius and reduces the complexity of the exhaustive search method on average. However, the expected complexity of SDA is still exponential for fixed SNR and increases significantly when SNR is low [18].

## 4. A Fast MSTR Detection Method

In this section, we develop an iterative MSTR detection algorithm by avoiding the high computational complexity of GLRT-based detectors (e.g., exhaustive search [16] and SDA [17]). Similar to the TR detection scheme, the proposed method first generates a reference template by using the initial symbol only, and then estimates the information symbols by correlating the reference template with the symbol waveforms. Furthermore, with the help of the information from multiple transmitted symbols, our method manages to suppress the reference template noise. However, our method also generates additional noise-cross-signal and noise-cross-noise terms which do not appear in the case of an ideal Rake receiver with perfect channel knowledge.

where Open image in new window can be found in (12).

This means that at the first step the estimated symbols are obtained by correlating the waveform corresponding to Open image in new window with the Open image in new window th symbol waveform. Hence, the BER performance is the same as that of the ATR in (19).

### 4.1. Weight Selections

- (i)
Hard Decision for MSTR (HD-MSTR)

which indicates that Open image in new window in (23). Also note that, the template is a scaled version of the GLRT template estimate given the detected symbols Open image in new window as shown in (11).

where Open image in new window is the standard deviation of the random variable. In general, the larger the mean-standard deviation ratio, the better the BER performance. Thus, in the case of HD-MSTR, if more correct symbols are detected for the current iteration, during the next iteration, the reference template is improved and then the method potentially results in better BER performance. The iterative procedure runs back and forth until no symbol is changed or the maximum number of iterations is reached.

(ii) Soft Decision for MSTR (SD-MSTR)

An intuitive idea of the SD-MSTR detector is that the decision variable Open image in new window obtained in each iteration reflects the reliability of the detected symbol Open image in new window . The larger the value of Open image in new window , the more we can trust the accuracy of the detected symbol Open image in new window . Hence, the corresponding symbol deserves higher weight in the representation of the reference template for next iteration.

where the two terms in (31) are the posterior probabilities of correct and erroneous detection of the symbol Open image in new window conditioned on the decision variable Open image in new window . If these probabilities are the same, that means it does not matter which decision we make. This represents the most unreliable case and the weight should be set to zero. The larger the probability of correct detection, the higher weight we should put on this decision. Note that the weight Open image in new window of the known reference symbol Open image in new window is set to 1, Open image in new window should be Open image in new window and Open image in new window ranges from Open image in new window indicating how much the averaged signal Open image in new window contributes to the final template depending on accuracy of the estimated symbol Open image in new window .

where Open image in new window is the probability density function (pdf) of the standard normal distribution.

where Open image in new window and Open image in new window can be found in (17) and (18) which require the frame energy Open image in new window and the noise power Open image in new window to evaluate Open image in new window and Open image in new window , but they are easy to estimate and store at the receiver side.

Now, we can summarize our method in the following steps for one block symbol detection.

*Input*:

Correlation matrix Open image in new window defined in (14), where Open image in new window , Open image in new window , the maximum number of iterations Open image in new window , channel statistics Open image in new window and Open image in new window for the SD-MSTR case.

Step 1.

Initialize Open image in new window , Open image in new window , Open image in new window .

Step 2.

Step 3.

Obtain the decision variables by (24).

Step 4.

Obtain the detected symbols by (26).

Step 5.

Set Open image in new window for the HD-MSTR case or update the weights for Open image in new window based on (31), (35), (36) in the SD-MSTR case.

Step 6.

If Open image in new window and Open image in new window goto Step 2, otherwise output Open image in new window and exit.

### 4.2. Convergence and Discussions

Open image in new window The convergence rate also affects the practical value of the method (e.g., a system with a tight constraint on decoding delay) and the number of iterations affects the performance. These will be verified by the numerical simulation that the proposed method converges to the stable performance curve within a few iterations (usually Open image in new window iterations).

Open image in new window Comparing with MSDD, we choose MSTR as the encoding scheme which allows the algorithm to detect symbol Open image in new window directly without any further processing.

Open image in new window Instead of evaluating each iteration's reference template Open image in new window explicitly, the method computes the decision variables by linear combination of the correlation coefficients Open image in new window which can be computed in the first iteration and reused later.

Open image in new window The HD-MSTR only requires the coefficients Open image in new window which is the same as the GLRT approach meanwhile the SD-MSTR requires some additional channel statistical information to update the weights for each iteration.

Open image in new window For each iteration, Step 3 requires Open image in new window multiplications and Open image in new window additions to attain the decision variables for all Open image in new window symbols. In Step 4, Open image in new window sign operations are performed to obtain detected symbols. No arithmetic is required for HD-MSTR in Step 5, while the SD-MSTR performs Open image in new window times Gaussian pdf evaluation and needs Open image in new window additions and Open image in new window divisions to normalize weights. We can treat the computational costs of sign operation and Gaussian pdf evaluation as being constant, and then the computational complexity of the both detectors for each iteration is Open image in new window where Open image in new window is the block size. Note that the complexity of the proposed method is independent of the channel realizations whilst the computational complexity of SDA relies on the specific realization of channels and SNR.

## 5. Numerical Results

This section compares the BER performance of the proposed methods (HD-MSTR and SD-MSTR) and the MSTR based on exhaustive search (ES-MSTR) as benchmark. Two kinds of channel schemes are evaluated: one is a constant channel with fixed CIR parameters, and the other is a random channel based on Saleh and Valenzuela (SV) channel model.

### 5.1. Constant Channel

At the transmitter side, the pulse Open image in new window is the second derivative of a Gaussian function with normalized unit energy and pulse width Open image in new window . The number of frames per symbols is Open image in new window . For the UWB channel model, we employ the resolvable multipath assumption such that Open image in new window as studied in [12, 13, 19] and then Open image in new window in (18) can be approximated with the number of paths Open image in new window . In this simulation, Open image in new window is Open image in new window and the energy of impulse channel response (CIR) Open image in new window which means Open image in new window in this scheme. As we have shown in Section 3, if the number of symbols in one block Open image in new window is equal to Open image in new window or the maximum number of iterations Open image in new window is equal to Open image in new window , then the system outputs the same performance as ATR scheme in [12]. Note that there is a Open image in new window gap between the ATR curve in the following figures and the one in [12]. This is because the definition of frame energy in [12] is twice as the one of ours. In this subsection, we only consider single user case with Open image in new window for all Open image in new window . Multiuser case will be shown in next subsection.

#### 5.1.1. BER with Different Block Size

Comparing the performance of HD-MSTR and SD-MSTR detectors in Figures 1 and 2, respectively, the difference is obvious when Open image in new window is small. The SD-MSTR outperforms the HD-MSTR, with about Open image in new window of SNR gain when Open image in new window and around Open image in new window gain when Open image in new window . The difference becomes trivial when Open image in new window is Open image in new window or larger. This indicates that the SD-MSTR method can offer additional advantages for low complexity UWB systems with small Open image in new window and but its advantage decreases with increasing Open image in new window . Bearing in mind that the SD-MSTR requires some statistical channel information ( Open image in new window , Open image in new window in (17) and (18)) and the Gaussian pdf calculation of the system, it is more likely that the simpler HD-MSTR algorithm would be implemented if Open image in new window is large.

Compared with HD-MSTR and SD-MSTR, the ES-MSTR has an advantage when Open image in new window is small (if Open image in new window , about Open image in new window gain for HD-MSTR and Open image in new window for SD-MSTR) and the performance gap becomes smaller when Open image in new window is larger. When Open image in new window , the gap reduces to around Open image in new window for HD-MSTR case and about Open image in new window for the SD-MSTR case. This shows that with the increasing value of Open image in new window the difference between the optimal ES-MSTR method and our proposed methods decreases rapidly and that the gap can be ignored for a sufficient large Open image in new window . Furthermore, the ES-MSTR incurs much higher computational cost than our MSTR method.

#### 5.1.2. BER with Different Iterations

### 5.2. SV Channel Model

where Open image in new window models the double-sided Rayleigh distributed amplitudes with exponentially decaying profile.

where Open image in new window is the average received energy per frame and the frame repetition factor is Open image in new window (to compare with [17]) while the integral interval is Open image in new window ns and the frame duration is Open image in new window ns to preclude the IFI.

#### 5.2.1. Single User Scenario

Furthermore, when Open image in new window and BER = Open image in new window , there is about Open image in new window gap between the ideal Rake receiver and our algorithm. As we expected, the performance over random channels is worse than the one with constant channels.

#### 5.2.2. Multiuser Scenario

## 6. Conclusion

In this paper, we propose fast detection methods for MSTR transmissions. The proposed MSTR detectors obtain the decision variables by summing up the correlation of different symbol waveforms, each properly weighted by the reliability of detected symbols and iteratively updating the weights and detected symbols. With different updating methods for the weights, two detectors are proposed: Hard Decision for MSTR (HD-MSTR) detector and Soft Decision for MSTR (SD-MSTR) detector, where HD-MSTR obtains the template based only on the previous detected symbols, while SD-MSTR constructs the template with additional information from the decision variables. Enhanced BER performance relative to the ATR scheme and the fast convergence property of these detectors are shown by the simulation results. Due to its simplicity, low computational complexity and near optimal performance for Open image in new window , the method is promising for realistic UWB applications.

## Notes

### Acknowledgments

Part of this work is supported by the Georgia Tech Ultrawideband Center of Excellence (http://www.uwbtech.gatech.edu/). The authors would like to thank the anonymous reviewers and the guest editor for their helpful comments which improved the quality of this paper.

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