Optimal and suboptimal finger selection algorithms for MMSE RAKE receivers in impulse radio ultra-wideband systems

Convex relaxations of the optimal finger selection algorithm are proposed for a minimum mean square error (MMSE) RAKE receiver in an impulse radio ultra-wideband system. First, the optimal finger selection problem is formulated as an integer programming problem with a non-convex objective function. Then, the objective function is approximated by a convex function and the integer programming problem is solved by means of constraint relaxation techniques. The proposed algorithms are suboptimal due to the approximate objective function and the constraint relaxation steps. However, they can be used in conjunction with the conventional finger selection algorithm, which is suboptimal on its own since it ignores the correlation between multipath components, to obtain performances reasonably close to that of the optimal scheme that cannot be implemented in practice due to its complexity. The proposed algorithms leverage convexity of the optimization problem formulations, which is the watershed between 'easy' and 'difficult' optimization problems.


I. INTRODUCTION
Since the US Federal Communications Commission (FCC) approved the limited use of ultra-wideband (UWB) technology [1], communications systems that employ UWB signals have drawn considerable attention. A UWB signal is defined to be one that possesses an absolute bandwidth larger than 500MHz or a relative bandwidth larger than 20% and can coexist with incumbent systems in the same frequency range due to its large spreading factor and low power spectral density. UWB technology holds great promise for a variety of applications such as short-range high-speed data transmission and precise location estimation.
Commonly, impulse radio (IR) systems, which transmit very short pulses with a low duty cycle, are employed to implement UWB systems ([2]- [6]). In an IR system, a train of pulses is sent and information is usually conveyed by the position or the polarity of the pulses, which correspond to Pulse Position Modulation (PPM) and Binary Phase Shift Keying (BPSK), respectively. In order to prevent catastrophic collisions among different users and thus provide robustness against multipleaccess interference, each information symbol is represented by a sequence of pulses; the positions of the pulses within that sequence are determined by a pseudo-random time-hopping (TH) sequence specific to each user [2]. The number N f of pulses representing one information symbol can also be interpreted as pulse combining gain.
Commonly, users in an IR-UWB system employ Rake receivers to collect energy from different multipath components. A Rake receiver combining all the paths of the incoming signal is called an all-Rake (ARake) receiver. Since a UWB signal has a very wide bandwidth, the number of resolvable multipath components is usually very large. Hence, an ARake receiver is not implemented in practice due to its complexity. However, it serves as a benchmark for the performance of more practical Rake receivers. A feasible implementation of multipath diversity combining can be obtained by a selective-Rake (SRake) receiver, which combines the M best, out of L, multipath components [7]. Those M best components are determined by a finger selection algorithm. For a maximal ratio combining (MRC) Rake receiver, the paths with highest signal-to-noise ratios (SNRs) are selected, which is an optimal scheme in the absence of interfering users and inter-symbol interference (ISI). For a minimum mean square error (MMSE) Rake receiver, the "conventional" finger selection algorithm is to choose the paths with highest signal-to-interferenceplus-noise ratios (SINRs). This conventional scheme is not necessarily optimal since it ignores the correlation of the noise terms at different multipath components. In other words, choosing the paths with highest SINRs does not necessarily maximizes the overall SINR of the system.
In this paper, we formulate the optimal MMSE SRake as a nonconvex, integer-constrained optimization, where the aim is to choose the finger locations of the receiver so as to maximize the overall SINR. While computing the optimal finger selection is NP-hard, we present several relaxation methods to turn the (approximate) problem into convex optimization problems that can be very efficiently solved by interior-point methods, which are polynomial time in the worst case, and very fast in practice. These optimal finger selection relaxations produce significantly higher average SINR than the conventional one that ignores the correlations, and represent a numerically efficient way to strike a balance between SINR optimality and computational tractability.
The remainder of the paper is organized as follows. Sec- tion II describes the transmitted and received signal models in a multiuser frequency-selective environment. The finger selection problem is formulated and the optimal algorithm is described in Section III, which is followed by a brief description of the conventional algorithm in Section IV. In Section V, two convex relaxations of the optimal finger selection algorithm, based on an approximate SINR expression and integer constraint relaxation techniques, are proposed. The simulation results are presented in Section VI, and the concluding remarks are made in the last section.
II. SIGNAL MODEL We consider a synchronous, binary phase shift keyed TH-IR system with K users, in which the transmitted signal from user k is represented by: where p tx (t) is the transmitted UWB pulse, E k is the bit energy of user k, T f is the "frame" time, N f is the number of pulses representing one information symbol, and b (k) j/N f ∈ {+1, −1} is the binary information symbol transmitted by user k. In order to allow the channel to be shared by many users and avoid catastrophic collisions, a time-hopping (TH) sequence {c Consider the discrete presentation of the channel, L ] for user k, where L is assumed to be the number of multipath components for each user, and T c is the multipath resolution. Then, the received signal can be expressed as where p rx (t) is the received unit-energy UWB pulse, which is usually modelled as the derivative of p tx (t) due to the effects of the antenna, and n(t) is zero mean white Gaussian noise with unit spectral density. We assume that the time-hopping sequence is constrained Due to the high resolution of UWB signals, chip-rate and frame rate sampling are not very practical for such systems. In order to have a lower sampling rate, the received signal can be correlated with template signals which enable symbol rate sampling of the output [11]. The template signal for the lth path of the incoming signal can be expressed as for the ith information symbol, where we consider user 1 without loss of generality. In other words, by using a correlator for each multipath component that we want to combine, we can just use symbol rate sampling at each branch, as shown in Figure 2.
Note that the use of such template signals results in equal gain combining (EGC) of different frame components. This may not be optimal under some conditions (see [12] for (sub)optimal schemes). However, it is very practical since it facilitates symbol-rate sampling. Since we consider a system that employs template signals of the form (3), i.e. EGC of frame components, it is sufficient to consider the problem of selection of the optimal paths just for one frame. Hence, we assume N f = 1 without loss of generality.
Let L = {l 1 , . . . , l M } denote the set of multipath components that the receiver collects ( Figure 2). At each branch, the signal is effectively passed through a matched filter (MF) matched to the related template signal in (3) and sampled once for each symbol. Then, the discrete signal for the lth path can be expressed, for the ith information symbol, as 2 which can be expressed as a sum of the desired signal part (SP) and multiple-access interference (MAI) terms: where the kth elements can be expressed as with I (k) l,m being the indicator function that is equal to 1 if the mth path of user k collides with the lth path of user 1, and 0 otherwise.

III. PROBLEM FORMULATION AND OPTIMAL SOLUTION
The problem is to choose the optimal set of multipath components, L = {l 1 , . . . , l M }, that minimizes the bit error probability (BEP) of the system. In other words, we need to choose the best samples from the L received samples r l , l = 1, . . . , L, as shown in (4). Using the selection matrix X, we can express the vector of received samples from M multipath components as where n is the vector of thermal noise components n = [n 1 · · · n L ] T , and S is the signature matrix given by S = [s 1 · · · s L ] T , with s l as in (5).
Using (5)- (7), (8) can be expressed as where S (MAI) is the MAI part of the signature matrix S.
Then, the linear MMSE receiver can be expressed aŝ where the MMSE weight vector is given by [13] with R being the correlation matrix of the noise term: The SINR of the system can be expressed as Xα (1) . (13) Hence, the optimal path/finger selection problem can be formulated as where X has the previously defined structure.
Note that the objective function to be maximized is not concave and the optimization variable X takes binary values, with the previously defined structure. In other words, two major difficulties arise in solving (14) globally: nonconvex optimization and integer constraints. Either makes the problem NP-hard. Therefore, it is an intractable optimization problem in this general form.

IV. CONVENTIONAL ALGORITHM
Instead of the solving the problem in (14), the "conventional" finger selection algorithm chooses the M paths with largest individual SINRs, where the SINR for the lth path can be expressed as for l = 1, . . . , L. This algorithm is not optimal because it ignores the correlation of the noise components of different paths. Therefore, it does not always maximize the overall SINR of the system given in (13). For example, the contribution of two highly correlated strong paths to the overall SINR might be worse than the contribution of one strong and one relatively weaker, but uncorrelated, paths. The correlation between the multipath components is the result of the MAI from the other users in the system.

V. RELAXATIONS OF OPTIMAL FINGER SELECTION
Since the optimal solution in (14) is quite difficult, we first consider an approximation of the objective function in (13). When the eigenvalues of 1 σ 2 n XS (MAI) A 2 (S (MAI) ) T X T are considerably smaller than 1, which occurs when the MAI is not very strong compared to the thermal noise, we can approximate the SINR expression in (13) as follows 3 : which can be expressed as Note that the approximate SIN R expression depends on X only through X T X. Defining x = [x 1 · · · x L ] T as the diagonal elements of X T X, x = diag{X T X}, we have x i = 1 if the ith path is selected, and x i = 0 otherwise; and L i=1 x i = M . Then, we obtain, after some manipulation, Then, we can formulate the finger selection problem as follows: Note that the objective function is convex since P is positive definite, and that the first constraint is linear. However, the integer constraint increases the complexity of the problem. The common way to approximate the solution of an integer constraint problem is to use constraint relaxation. Then, the optimizer will be a continuous value instead of being binary and the problem (19) will be convex. Over the past decade, both powerful theory and efficient numerical algorithms have been developed for nonlinear convex optimization. It is now recognized that the watershed between "easy" and "difficult" optimization problems is not linearity but convexity. For example, the interior-point algorithms for nonlinear convex optimization are highly efficient, both in worst case complexity (provably polynomial time) and in practice (very fast even for a large number variables and constraints) [14]. Interior-point methods solve convex optimization problems with inequality constraints by applying Newton's method to a sequence of equality constrained problems, where the Newton's method is a kind of descent algorithm with the descent direction given by the Newton step [14].
We consider two different relaxation techniques in the following subsections.

A. Case-1: Relaxation to Sphere
Consider the relaxation of the integer constraint in (19) to a sphere that passes through all possible integer values. Then, the relaxed problem becomes Note that the problem becomes a convex quadratically constrained quadratic programming (QCQP) [14]. Hence it can be solved for global optimality using interior-point algorithms in polynomial time.

B. Case-2: Relaxation to Hypercube
As an alternative approach, we can relax the integer constraint in (19) to a hypercube constraint and get where the hypercube constraint can be expressed as x 0 and x 1, with y z meaning that y 1 ≥ z 1 , . . . , y L ≥ z L . Note that the problem is now a linearly constrained quadratic programming (LCQP), and can be solved by interior-point algorithms [14] for the optimizer x * .

C. Dual Methods
We can also consider the dual problems. For the relaxation to the sphere considered in Section V-A, the Lagrangian for (20) can be obtained as where λ ∈ R and ν ∈ R + . After some manipulation, the Lagrange dual function can be expressed as Then, the dual problem becomes which can be solved for optimal λ and ν by interior point methods. Or, more simply, the unconstraint problem (24) can be solved using gradient descent algorithm, and then the optimizerν is mapped to ν * = max{0,ν}.
After solving for optimal λ and µ, the optimizer x * is obtained as Note that the dual problem (24) has two variables, λ and ν, to optimize, compared to L variables, the components of x, in the primal problem (20). However, an L × L matrix needs to be inverted for each iteration of the optimization of (24). Therefore, the primal problem can be preferred over the dual problem in this case.
Similarly, the dual problem for the relaxation in Section V-B can be obtained from (21) as minimize σ 2 It is observed from (27) that there are 2L + 1 variables and also L × L matrix inversion operations for the solution of the dual problem. Therefore, the simpler primal problem (21) is considered in the simulations.

D. Selection of Finger Locations
After solving the approximate problem (19) by means of integer relaxation techniques mentioned above, the finger location estimations are obtained by the indices of the M largest elements of the optimizer x * .
Both the approximation of the SINR expression by (16) and the integer relaxation steps result in the suboptimality of the solution. Therefore, it may not be very close to the optimal solution in some cases. However, it is expected to perform better than the conventional algorithm most of the time, since it considers the correlation between the multipath components. However, it is not guaranteed that the algorithms based on the convex relaxations of optimal finger selection always beat the conventional one. Since the conventional algorithm is very easy to implement, we can consider a hybrid algorithm where the final estimate of the convex relaxation algorithm is compared with that of the conventional one and the one that minimizes the exact SINR expression in (13) is chosen as the final estimate. In this way, the resulting hybrid suboptimal algorithm can get closer to the optimal solution.

VI. SIMULATION RESULTS
The simulation results are performed to evaluate the performance of different finger selection algorithms for an IR-UWB system with N c = 20 and N f = 1. There are 5 equal energy users in the system (K = 5) and the users' TH and polarity codes are randomly generated. We model the channel coefficients as α l = sign(α l )|α l | for l = 1, . . . , L, where sign(α l ) is ±1 with equal probability and |α l | is distributed lognormally as LN (µ l , σ 2 ). Also the energy of the taps is exponentially decaying as E{|α l | 2 } = Ω 0 e −λ(l−1) , where λ is the decay factor and L l=1 E{|α l | 2 } = 1 (so ). For the channel parameters, we have λ = 0.1, σ 2 = 0.5 and µ l can be calculated from µ l = 0.5 ln( 1−e −λ 1−e −λL ) − λ(l − 1) − 2σ 2 , for l = 1, . . . , L. We average the overall SINR of the system over different realizations of channel coefficients, TH and polarity codes of the users.
In Figure 3, we plot the average SINR of the system for different noise variances when M = 5 fingers are to be chosen out of L = 15 multipath components. As is observed from the figure, the convex relaxations of optimal finger selection result in SINR values reasonable close to those of the optimal exhaustive search scheme. Note that the gain by using the proposed algorithms over the conventional one increases as the thermal noise decreases. This is because when the thermal noise get less significant, the MAI becomes dominant, and the conventional technique gets worse since it ignores the correlation between the MAI noise terms when choosing the fingers.
Next, we plot SINR of the proposed suboptimal and conventional techniques for different finger numbers in Figure  4, where there are 50 multipath components and E b /N 0 = 20. The number of chips per frame, N c , is set to 75, and all other parameters are kept the same. In this case, the optimal algorithm takes a very long time to simulate since it needs to perform exhaustive search over many different finger combinations (therefore not implemented). The improvement using convex relaxations of optimal finger selection over the conventional technique decreases as M gets large since the channel is exponentially decaying and the most of the significant multipath components are already combined by all the algorithms.
Finally, we consider a MAI-limited scenario, where there are 10 users with E 1 = 1 and E k = 10 ∀k, and all the parameters are as in the previous case. Then, as shown in Figure 5, the improvement by using the suboptimal finger selection algorithms increase significantly. The main reason for this is that the suboptimal algorithms consider, although approximately, the correlation caused by MAI whereas the conventional scheme simply ignores that.

VII. CONCLUDING REMARKS
Optimal and suboptimal finger selection algorithms for MMSE-SRake receivers in an IR-UWB system are consid-   ered. Since UWB systems have large number of multipath components, only a subset of those components can be used due to complexity constraints. Therefore, the selection of the optimal subset of multipath components is important for the performance of the receiver. We have shown that the optimal solution to this finger selection problem requires exhaustive search which becomes prohibitive for UWB systems. Therefore, we have proposed approximate solutions of the problem based on the Taylor series approximation and integer constraint relaxations. Using two different integer relaxation approaches, we have introduced two convex relaxations of the optimal finger selection algorithm. Implementing these suboptimal algorithms on top of the conventional scheme, we can get close to the optimal solution, with much lower complexity.
The two contributions of the paper are the formulation of the optimal problem and the convex relaxations. In the first, the formulation is globally optimal but the solution methods for non-convex nonlinear integer constrained optimization must use heuristics to get to a locally optimal solution because otherwise computational load for global optimality is too much. In the second, the formulation is relaxed, but the interior-point methods efficiently computes the globally optimal solution for these relaxations.