Two Blind Adaptive Mobile Receivers in Time-Hopping PAM UWB Impulse Radio System

This paper deals with the design of blind mobile station (MS) receiver for time-hopping (TH) ultrawide-band (UWB) impulse radio (IR) system employing antipodal pulse amplitude modulation (PAM). In the presence of multipath fading channel, batch-mode minimum output energy (MOE) receiver, as well as a blind channel estimator, is ﬁrst developed. To reduce the computational complexity, we propose two blind adaptive algorithms to determine the weight vector of the MS receiver. The rational of the algorithms premises on iteratively maximizing the minimum possible receiver’s output energy. The ﬁrst (indirect) approach is derived by ﬁrst developing an adaptive blind channel estimator, then the updated channel impulse response (IR) is used to calculate the MS receiver’s weight vector. Meanwhile the second (direct) method jointly and iteratively optimizes the weight vector and channel IR to improve system performance. Simulation results demonstrate convergence of both algorithms. Moreover, the algorithms are shown to be robust to multiuser interference (MUI) and near-far problems.


INTRODUCTION
Ultrawide-band (UWB) impulse radio (IR) system has spurred great attention for the promised applications in high-speed short-range indoor wireless communication system.The attractive features of UWB IR technology include low power, low complexity, carrierless modulation, and ample multipath diversity [1][2][3][4][5].The basic structure of UWB impulse radio stems from transmitting a stream of pulses of very short duration (on the order of nanosecond or less) and with very low duty cycle.When such transmissions are applied in multiple access system, the transmission model encompasses time-hopping (TH) M-ary pulse position modulation (MPPM) [5][6][7][8], TH antipodal pulse amplitude modulation (PAM) [4], and direct sequence (DS) binary phase shift keying (BPSK) [9][10][11] schemes.
Among the various modulation and multiple access schemes, we focus on the design of blind mobile station (MS) receiver in UWB IR multiple access communication system employing TH-PAM modulation.In TH-PAM multiple access system, separation of different users is accomplished by assigning user-specific pattern of time shifting of pulses.Though orthogonal TH codes can be chosen in downlink communication such that the pulses for each user are located in nonoverlapping time slots, however, multipath fading induced signature waveform distortion severely degrades system performance.Thereby, the above benefits vanish without accurate channel estimation.A batch-mode blind (nondata aided and only the desired user's TH sequence is known) signal reception scheme is proposed in [12].To make feasible application and reduce the computation load, this paper attempts to develop a blind, adaptive, joint signal extraction and channel impulse response (IR) estimation algorithm in the MS receiver.
We first propose a batch-mode MS receiver that is designed to meet the minimum output energy (MOE) criterion.Such method originated in the area of array signal processing, which is referred to as minimum power distortionless response (MPDR) [13] beamforming.It has been applied in the context of CDMA system for blind interference suppression [14].In order to extract and estimate the UWB channel impulse response blindly, we make use of the attractive feature that the channel parameters can be determined to maximize the minimum possible mean output energy of the MOE receiver.Unfortunately, the proposed batchmode MOE receiver and blind channel estimation algorithm require performing an inverse of the data correlation matrix with a prohibitively large size.It is worthwhile, therefore, to explore adaptive implementation algorithms to reduce the computational complexity.
Both the gradient search and recursive least squares (RLS) [15] based methods are exploited to develop the adaptive algorithms.The indirect approach is derived by first developing an RLS-based channel IR adaptation rule.The updated channel parameters' vector is then used to calculate the MOE receiver's weight vector.The direct method is motivated by the concept of direct blind multichannel equalizers [16].In this approach, the weight vector and channel IR are jointly and iteratively optimized to maximize the minimum possible output energy.Performance of the two adaptive algorithms is comprehensively evaluated.
The remainder of this paper is organized as follows.In Section 2, we formulate the transmitting and receiving signal models of the time-hopping UWB multiple access communication system using binary antipodal PAM modulation.Section 3 highlights the batch processed MOE receiver.The rationale of blind channel parameters' estimation algorithm is also described.Section 4 describes two types of adaptive blind MOE receivers.Simulation results are presented in Section 5. Concluding remarks are finally made in Section 6.

SIGNAL MODEL
In TH UWB IR system, every information symbol (bit) is conveyed by N f data modulated ultrashort pulses over N f frames.There is only one pulse in each frame and the frame duration is T f .The pulse waveform, p(t), is referred to as a monocycle [1] with ultrashort duration T c at the nanosecond scale.We assume that p(t) is normalized within T c , so that Tc 0 |p(t)| 2 dt = 1.Note that T f is usually a hundred to a thousand times of chip duration, T c , which accounts for very low duty cycle.When multiple users are simultaneously transmitted and received, signal separation can be accomplished with user-specific pseudorandom TH codes, which shift the pulse position in every frame.In downlink side, all users are synchronously transmitted; we may establish the data model of the transmitted signal as where t is the clock time of the transmitter.K is the number of active users.a k is the amplitude of the kth user.The ith bit transmitted by kth user is given by d k (i), which takes on ±1 with equal probability.The operation is equivalent to repeating the bit N f times, a repetition coding with code rate 1/N f .Denoting T b as the duration of the N f repeated bits, then T b = N f T f .Suppose each frame is composed of N c time slots each with duration T c , thus, T f = N c T c .User separation is accomplished by user-specific pseudorandom TH code.{c k j } j=0,1,...,N f −1 accounts for the kth user's TH code with period N f .Thereby, c k j T c is the timeshift of the pulse position imposed by the TH sequence employed for multiple access.
In this paper, we adopt the channel models of CM4 (indoor office) and CM9 (the industrial) as described in [17], where "dense" arrivals of multipath components (MPC) were observed and each resolvable delay bin contains significant energy.In these cases, a realization of the impulse response based on a tapped delay line model with regular tap spacings is to be used.Thereby, the time-invariant channel impulse response with (L + 1) resolvable paths can be written as where α l denotes the attenuation coefficient of the lth resolvable path.In writing (2), we have implicitly assumed that the maximum time dispersion is LT c .To simplify the analysis, we assume that the channel parameters vary so slowly that they are essentially constant over observation interval, and In downlink channel, signals are subject to the same fading.Consequently, the received composite waveform at the desired mobile is made up of a weighted sum of attenuated and delayed replicas of the transmitted signal x(t), that is, where n(t) is assumed to be a zero-mean, wide-sense stationary (WSS) Gaussian random process with a power density spectrum that is constant (white), σ 2 , in a finite frequency interval.Evidently, by selecting c k N f −1 = 0, for all k, and T f > LT c (or, equivalently, N c > L), we can guarantee no intersymbol interference (ISI).Note that under multipath fading, the original pulse, p(t), has been distorted and lengthened to p(t) with support of [0, LT c ]: Hence, we may re-express (3) as ( The problem addressed in this paper is the design of MS receiver based on the observation process r(t) with unknown channel parameters and without undesired users' TH sequences.

Minimum output energy (MOE) receiver
As depicted in (4), multipath fading leads to time dispersion of p(t), which adversely affects the orthogonality between the TH codes.In what follows, MUI occurs even in downlink side employing orthogonal (nonoverlapping) TH codes.
At the receiver, chip-matched filtering followed by chip-rate sampling yields a sequence of N c N f -vectors.The samples of chip-matched filter (CMF) output during the ith symbol interval can be expressed as where the N c N f -vector c k represents the chip-rate sampled version (during T b ) of the composite waveform and d Denoting q k,l as the chip-rate samples at the output of CMF (within a bit) of the received waveform coming from the lth path, then l represents a delayed version of the original signature vector, c k .It is evident that c k can be expressed as where From ( 6), we can obtain the correlation matrix of r(i): Without loss of generality, we assume user 1 is the desired user hereafter.In order to recover the desired signal, d 1 (i), an N c N f -by-1 weight vector should be designed to effectively suppress MUI and a sign test is proceeded to determine the information bit: Figure 1 depicts the block diagram of the MOE receiver.The choice of weight vector, w, for the MOE receiver [12,14] aims to minimize the output energy, E{|w H r(i)| 2 } = w H R rr w, while distortionlessly passes the desired signal, yielding arg min Applying Lagrange multiplier method [18] to solve the above-constrained optimization problem, we arrive at the optimal solution of (10): where η = 1/ c T 1 R −1 rr c 1 stands for the minimum output energy at w MOE .In what follows, the output of the MOE receiver and the corresponding signal-to-interference-plusnoise power ratio (SINR) can be obtained as

Blind channel estimation algorithm
Intuitively, η is mainly determined by the desired signal's energy, a 2 1 , since any signature vector, that is not exactly matched to c 1 , is suppressed to minimize the output energy.We may obtain from (7), c 1 = C 1 α, that c 1 is uniquely determined by α.Or equivalently, the estimation error of channel parameter may lead to the output energy degradation.Motivated by the output energy degradation that arises from estimation error, we propose to estimate α such that η is maximized: Upon defining the (L + 1)-by-(L + 1) square matrix Since the correlation matrix R rr is positive definite and C 1 is full column rank, thus, Q 1 is also positive definite.From the theorem related to Rayleigh quotient or the method of Lagrange multipliers [18], the optimal solution of ( 15) can be derived as where e min represents the normalized eigenvector of Q 1 corresponding to its minimum eigenvalue.Note that, in practice, R rr is unknown and needs to be estimated.Under ergodic (with respect to the autocorrelation function) assumption, we may perform time average on the measurements.Hence, for an observation length (window size) of k bits, the estimate of R rr is given by

RECURSIVE IMPLEMENTATION OF THE MOE RECEIVER
As we can observe from ( 11) and ( 14), the batch-mode MOE receiver and the blind channel estimator involve the inversion of R rr .Unfortunately, since the size of R rr , N c N f -by-N c N f , is generally prohibitively large, hence from the computational load point of view, it is crucial to develop iterative algorithms to implement the MOE receiver.In this section, we propose two constrained adaptive optimization techniques.The indirect approach premises on recursively updating the channel parameters and then exploiting the result to derive the weight vector.While the second scheme jointly updates the channel parameters and weight vector which is referred to as direct approach.

Indirect approach
To accommodate the time-varying characteristics and compute Q 1 without performing R −1 rr , we attempt to develop an adaptive algorithm instead of the batch constrained optimization method as described in Section 3.2.Using the method of Lagrange multipliers to solve the constraint optimization problem of (15), we can establish the cost function where λ is the corresponding Lagrange multiplier.The gradient of J with respect to α is given by Setting the result of (19) to zero and multiplying both sides by α H yield Compare (20) with ( 11), then we can obtain λ = 1/η, the inverse of the MOE receiver's output energy.It follows from ( 16) that λ is equivalent to the minimum eigenvalue of Q 1 .
We first deduce the adaptation rule for R rr to accommodate the time-varying characteristics of data vector: From the well-known matrix inversion lemma [11], We arrive at an update equation of R −1 rr (k), Subsequently, Q 1 (k) can be obtained as Substituting ( 24) into (19), we have Employing steepest decent gradient search, we get the update equation for α: where the step size μ is a positive number.To make the algorithm consistent with the unit-norm constraint of (15), we further normalize α(k + 1) at each iteration.Then, the effective signature vector at (k + 1)th iteration can be obtained as It follows from (20) to update λ by Substituting ( 23), ( 27), and (28) into (11), we obtain w(k+1).
In summary, we may generalize the above RLS-gradientbased adaptation algorithm as follows.
Step 2. Accept new data vector r(k) and use (23) to update R −1 rr (k).
Step 9. Go back to Step 2 until converge.
The above steps are displayed in Figure 2(a).Towards the above adaptation rule, the following remarks are in order.
(i) In the absence of signal, the data is full of white noise with correlation matrix R rr (0) = σ 2 I NcN f , thus, we set R −1 rr (0) = (1/σ 2 )I NcN f in Step 1. Intuitively, λ = α H Q 1 α stands for the inverse of the output energy of the MOE receiver.Hence, we set λ(0) = 1/a 2  1 , since the MOE receiver's output energy is mainly determined by the desired signal's power.To meet the constraint α H α = 1 and ensure nontrivial solution, we set α(0) = e 1 .

Delay
Update (ii) To determine the step size μ, we first assume that Q 1 (k), λ(k) converge asymptotically to their respective optimum values lim k→∞ Q 1 (k) = Q 1 and lim k→∞ λ(k) = λ.Let the error vector at time k be defined as c(k) := α(k) − α o , where α o denotes the optimum channel impulse response vector.Exploiting the fact that Q 1 α 0 = λ min α 0 , we may rewrite (26) in terms of the error vector Using eigenvalue decomposition (EVD), we may express Q 1 as where P is a unitary matrix (PP H = P H P = I L+1 ) with column vectors associated with the eigenvectors of with its diagonal elements being the eigenvalues of Q 1 .Substituting (31) into (30), we get Premultiplying both sides of (32) by P H and defining v(k) = P H c(k), we may reformulate (32) as We thus can obtain the nth element of v(k + 1): where v n (0) is the initial value of the nth mode.The stability or convergence of the steepest decent algorithm is assured provided that Since λ − λ n < 0, it follows that a necessary and sufficient condition for convergence is that the μ satisfies where λ max , λ min are the maximum and minimum eigenvalues of Q 1 , respectively.(iii) It is worthy to note that if L N c N f , then finding the minimum eigenvector of Q 1 does not result in computational burden.In what follows, we may develop the RLS-EVD-based adaptation rule in steps.
Step 2. Accept new data vector r(k) and use (23) to update R −1  rr (k).

EURASIP Journal on Wireless Communications and Networking
Step 4. Find α(k) = e min (k), where e min (k) is the minimum eigenvector (normalized) of Q 1 (k).
Step 6. Calculate w(k) where λ min (k) is the minimum eigenvalue of Q 1 (k).
Step 7. Go back to Step 2 until converge.
The above steps are displayed in Figure 2(b).In order to reduce the computation load induced by EVD in Step 4, we may apply the power method [18] that recursively update the eigenvector according to where and c is an arbitrary positive number to make all the eigenvalues of Q 1 (k) positive.In what follows, the eigenvector corresponds to the maximum eigenvalue of Q 1 (k) that is identical to the eigenvector associated with the minimum eigenvalue of Q 1 (k).It is evident that the vector in (38) will asymptotically converge to the maximum eigenvector of Q 1 (k).

Direct approach
Incorporating c 1 = C 1 α, α H α = 1 into the distortionless constraint of (10), we arrive at an alternative constrained optimization problem: The cost function towards (39) can be explicitly written as where vector β comprises the corresponding Lagrange multipliers.The problem addressed in (40) is similar to the linearly constraint minimum power (LCMP) beamforming problem in [13] and the solution is where Substituting the solution of α as deduced in (16), we have The goal for the direct approach is to minimize J(w, α) with respect to w, while maximizing J(w, α) with respect to α.The gradient of J(w, α) with respect to w and α is given by Based on the gradient decent adaptation rule, we obtain two update equations for w and α, respectively as where μ 1 and μ 2 are the step sizes setting for w and α, respectively.To solve β(k) of the above update equations, we exploit the constraint of (39) and enforce C H 1 w(k+1) = α(k) in (44), which yields After some manipulations, we can obtain Substituting ( 47) into ( 44) and (45), we arrive at the update equations where P ⊥ C1 denotes the orthogonal projection matrix with respect to the constraint space C 1 From (17), we can obtain a simple update equation for the correlation matrix The above adaptation algorithm can be summarized as follows.
Step 6. Go back to Step 2 until converge.
The above steps are displayed in Figure 3.To determine the step size μ 1 in (48), we first assume that R rr (k), β(k) converge asymptotically to their respective optimum values lim k→∞ R rr (k) = R rr , lim k→∞ β(k) = β.Let the error vector at time k be defined as d(k) := w(k) − w o , where w o denotes the optimum weight vector.It follows from (41) that C 1 β = R rr w o .Thus, we may rewrite (44) in terms of the error vector Following the same procedures as we derive the step size in indirect approach, we can obtain the necessary and sufficient condition for the convergence where λ max is the maximum eigenvalue of R rr .

PERFORMANCE EVALUATION
In this section, we conduct simulations to evaluate and compare the performance of the proposed two types of adaptive blind receiving schemes.Moreover, the performance of the batch-mode MOE receiver with perfect channel information is also provided as an optimum bound.The channel model applied for simulation is based on (2).Unless otherwise mentioned, we set the parameters N c = 20, N f = 20, L = 8, and K = 10, and signal-to-noise-ratio (SNR) for each user is set to be 20 dB throughout all the simulation examples.Figure 4 measures the SINR performance with respect to the number of iterations (k), where the performance of the ideal batch-mode MOE receiver is provided as an upper bound.Notice that the output SINR, after kth iterations, can be obtained from ( 13) as follows: The convergence characteristics of both direct and indirect adaptation rules are verified in Figure 4.The RLS-gradient and RLS-EVD algorithms of the indirect approach converge to almost the same level, while the direct approach is approximately 7 dB better than the indirect one.However, there is still an 8 dB loss when comparing the direct approach with the ideal case.This is mainly due to the inevitable estimation error.We then examine the convergent behavior of the proposed adaptive channel estimators.A plausible criterion to measure the estimation accuracy is root mean-squared-error (RMSE) that is defined as where N s is the Monte-Carlo trial number that is set to be N s = 50.Figure 5 measures the RMSE performance with respect to k.As depicted in Figure 5, RMSE decreases as k increases for both recursive channel estimators.In the next example, we examine the convergent rate of the indirect scheme under different weight vector sizes.Figure 6 compares the SINR performance with respect to k for N c = N f = 20, 15, 10, respectively, in the presence of 10 equal power (15 dB) users.As we can observe from Figure 6, though the convergence rate is approximately the same, smaller N c × N f achieves essentially better performance.This is mainly due to the fact that larger N c × N f is more sensitive to estimation error.
In the following simulation examples, we fix the number of iterations, k = 5000, and examine the performance under  near-far and MUI environments.Figure 7 presents SINR performance with respect to the number of active users (K) in the system.As verified by the simulation results, both algorithms are not affected by (or insensitive to) MUI.To measure the near-far resistance characteristics, we first set the interferers' amplitudes to be the same, a 2 = a 3 = • • • = a K = η, and define the near-far ratio (NFR) as the interferer-todesired user's power ratio, (η/a 1 ) 2 (in dB).In the presence of 10 equal-power interferers, the SINR performance with respect to NFR is obtained in Figure 8.As we vary NFR from 0 to 10 dB, the SINR performance is essentially invariant, which demonstrates the near-far resistant characteristics.

CONCLUSION
Without exploiting the undesired users' TH sequences, we developed two blind adaptive MS receivers for UWB IR system employing antipodal PAM modulation.Both algorithms are deduced to minimize the MOE receiver's output energy subject to appropriate constraints, which are chosen to maximize the minimal possible output energy.We can infer from the simulation results that the system performance for both receivers is sequentially optimized.Specifically, the direct scheme generally outperforms the indirect one under various scenarios.Compared to the batch-mode MOE receiver, the computation load of both algorithms is extensively reduced.Moreover, simulations confirm reliable convergent rate and the performance is comparable to the ideal case that the channel parameters are well known.

NOTATION
The boldface lower-case and upper-case letters: Represent vectors and matrices, respectively.
[•] T , [•] H : Transpose and complex transpose of a matrix or vector, respectively.E{•}: Expectation (ensemble average).:=: " i s d e fi n e d a s ." " * ": The convolution operation.I M : An identity matrix with size M.

1:
A vector with all elements being 1.

x:
The estimate of parameter x. e i : A vector with all entries zero except for the ith entry, which is one.

Figure 2 :
Figure 2: Implementation of the indirect blind adaptive MOE receiver.

Figure 3 :
Figure 3: Implementation of the direct blind adaptive MOE receiver.

Figure 4 :
Figure 4: SINR performance with respect to the number of iterations.

Figure 5 :
Figure 5: RMSE performance with respect to the number of iterations.

Figure 7 :
Figure 7: SINR performance with respect to the number of active users (K).

Figure 8 :
Figure 8: The SINR performance with respect to NFR.