Low-Complexity Multi-User Detectors for Time. Hopping Impulse Radio Systems

Transcription

1 SUBMITTED FOR PUBLICATION TO THE IEEE TRANSACTION ON SIGNAL PROCESSING 1 Low-Complexity Multi-User Detectors for Time Hopping Impulse Radio Systems Eran Fishler and H. Vincent Poor * This work was supported in part by the National Science Foundation under grant CCR , and in part by the New Jersey Center for Wireless Telecommunications. Department of Electrical Engineering, Princeton University, Princeton 08544, USA, Tel: , Fax: , {efishler,poor}@ee.princeton.edu

2 SUBMITTED FOR PUBLICATION TO THE IEEE TRANSACTION ON SIGNAL PROCESSING 2 Abstract Ultra Wide-band UWB radio systems are currently being considered for a number of applications, such as wide personal area networks WPAN and wireless LANs. These systems promise to deliver high data rates in multiple access communication channels already occupied by other narrow-band systems. The importance of multiuser detection for achieving high data rates required from these systems has already been established in several studies. This paper studies several low complexity multi-user detectors specifically designed for UWB radio systems. It is demonstrated that many multi-user detectors developed primarily for direct sequence code division multiple access DS-CDMA systems can be used without essentially any change in UWB receivers. Further, several novel, low complexity multi-user detectors that exploit the special signal structure used for transmission by UWB systems are developed. These novel detectors are analyzed both theoretically and via simulations. It is shown that a very simple iterative multi-user detector yields performance similar to that of a single user system. I. Introdcution Ultra Wide-band UWB radio systems have drawn considerable attention among both researchers and practitioners over the past few years. This interest has recently resulted in FCC action, allowing for the wide-spread use of UWB radio systems. Impulse radio IR systems, which transmit very short pulses, typically on the order of a fraction of a nano-second, are usually used for implementing UWB radio systems. In an IR system, a train of such pulses is transmitted, and the information is conveyed either by the polarity of the transmitted signal usually referred to as Pulse Amplitude Modulation PAM [7], or by shifting the pulse starting time usually referred to as a Pulse Position Modulation PPM [19]. In addition, in order to allow many users to share the same channel, an additional random time shift, known to the receiver, is added to the pulse starting point. This way, catastrophic collisions between two users transmitting over the same channel at the same time are avoided. The transmission of digital information using short impulses was first suggested by Withington in [20]. The main ideas in this seminal paper are to use the PPM format, and to repeat the transmission of each pulse many times; that is, to use time diversity transmission. In [14], this idea is further developed to allow for multiple users to exploit the same channel. As noted above, multiple access MA is achieved by adding a known and small random time shift to each pulse. Although some collisions between two pulses transmitted from different users may occur at the receiver, this method prevents the situation in which all the pulses transmitted from two different users will always collide at the receiver. UWB radio systems

3 SUBMITTED FOR PUBLICATION TO THE IEEE TRANSACTION ON SIGNAL PROCESSING 3 based on this principle are usually referred to as time-hoping TH impulse radio systems TH-IR, and have been analyzed extensively in the past, for free space, flat fading, and frequency selective channels, and with or without narrowband interference see, among many others, [1], [9], [14], [19], [24]. Multi-user issues are also considered in [14]. By assuming strict power control, asynchronous transmissions, and a large number of users, the multi-user interference MUI is modeled as a Gaussian random variable and hence the single user matched filter is used for detecting the transmitted symbol from the user of interest. This approach, of modeling the MUI as a Gaussian random process, is further used in [19] to demonstrate that impulse radio multiple access IRMA systems are able to support large numbers of users on the order of a few hundred users with data rates of 1Mbit/s, and thousands of users at lower bit rates. However, in practical systems neither perfect power control nor large enough numbers of users to justify the use of the central limit theorem can be assumed. Moreover, the use of central limit arguments is often not justifiable when considering the tail behavior of the interference distribution [2]. In [5] a novel discrete time model for PPM IRMA systems is described. By sampling the received signal at a sub-chip rate, a discrete time, single input single output model is constructed. In [6] a similar multiple input multiple output model is described. A similar approach to the modeling of PAM IRMA systems is taken in [7], where a discrete time equivalent model is described. Also in [7], it is demonstrated that PAM IR modulation is a form of linear modulation. Digital receivers for the PAM IRMA systems are later described as well. Digital receivers for PAM IR systems, such as the ones described in [7], combined with an appropriate design of the time hopping sequence used by the various users result in deterministic and complete multi-user interference rejection. Depending on the exact system parameters, the complexity of the receivers described in [6], [7], [5] can be quite large [22]. The optimum multi-user detector MUD for IRMA systems, which requires joint detection of the transmitted symbols, can be deduced easily from [17]. This detector requires samples of the received signal at any time instant where a pulse is received. Moreover, minimizing a high dimensional non-linear function of all the samples is required as well. Computational complexity and engineering requirements prohibit the use of this optimal detector even with today s most sophisticated equipment. Simpler sub-optimal receivers, like the zero forcing receiver, are hard to implement as well due to the need to invert large matrices at the symbol rate. Although the classical algorithms for multi-user detection can by used in

4 SUBMITTED FOR PUBLICATION TO THE IEEE TRANSACTION ON SIGNAL PROCESSING 4 TH-IR systems, it is evident that simple algorithms for multi-user detection in impulse radio systems are required. In this paper we study the problem of multi-user detection in IR systems. This study, which is the first of two parts, serves two purposes. The first, is to motivate the development of low complexity multi-user detectors for IR systems; and the second is to describe a series of very low complexity multi-user detectors for IR radio in flat fading channels. Low complexity multi-user detectors for IR systems operating in frequency selective channels will be treated in the second part of this study [3]. As will be seen in the sequel, random direct sequence DS code division multiple access CDMA systems, in the sense of [8] and [16], and IR systems are very similar. Analyzing these similarities reveals that almost every multi-user detector developed for DS-CDMA systems could be used in IR system with slight, or even without any, modification. As such, our study focus on very low complexity multi-user detectors exploiting the specific signal structure of IR signals. Fortunately such detectors exists. However, while multi-user detectors designed for DS-CDMA could be used in IR systems, these new detectors designed for IR systems can not be used in what we usually refer to as DS-CDMA systems. The rest of the paper is organized as follows: in Section II we describe both the continuous-time and discrete-time signal models for the transmitted and received signal s in TH-IR systems. In Section III the relation between IR and random DS-CDMA R-CDMA systems is discussed and the two systems are compared. In Section IV the blinking receiver, which is the simplest multi-user detector to be considered here, is discussed and analyzed. The quasi-decorrelator and quasi-mmse multi-user detectors are described and discussed in Section V, while in Section VI we describe an iterative turbo like multi-user detector. In Sections VII we provide simulation results that demonstrate both the performance of the proposed algorithms, and the trade-offs between them. Some concluding remarks are included in section VIII. II. System Model A. Continuous Time Received Signal Model The transmitted signal, of say the kth user, in a TH-IR system is described by the following general model: s k tr t = j= a k j/n f w trt jt f c k j T c b k j/n f 1

5 SUBMITTED FOR PUBLICATION TO THE IEEE TRANSACTION ON SIGNAL PROCESSING 5 where T f is the nominal pulse repetition time, and w tr t is the transmitted pulse, usually referred to as the monocycle. The kth user s data is transmitted on one of the sequences {a k j/n f } and {b j/n f } in a manner defined below where N f is the number of monocycles used to transmit one information symbol and denotes the integer part. {c k j }, is a pseudo random sequence taking values in [0, 1,..., N c 1], assigned to user k. This sequence, usually referred to as the time hopping sequence, provides an additional time shift of c k j T c seconds to the jth pulse and is used to avoid catastrophic collisions between users. In order to avoid inter-pulse interference IPI, it is usually required that T c T f N c, so that an overlap between two transmitted pulses from the same user is avoided. The symbol rate, denoted by R s, is given by the following relation R s = 1 T f N f. Both PAM [7] and PPM [14] schemes have been suggested for IR systems. In PAM IR systems the information is conveyed by the series {a k j/n f } which takes values in a set {± M 1, ± M 3,..., ±1} while b k j/n f = 0 for all j and k, and M is the number of information bits per transmitted symbol. In PPM IR systems the information is conveyed by the series {b k j/n f } which take values in a set {δ 0, δ 1,..., δ M 1 }, while a k j/n f = 1 for all j and k. In general, the series {δ 0,..., δ M 1 } is chosen to satisfy some appropriate criterion, e.g., equal correlation between different symbols [10]. The transmitted signal 1 undergoes various changes due to the propagation channel and the effects of the receiving antenna on the received signal. The propagation channel can be modeled as a tapped delay line with L k taps, where L k denotes the number of propagation paths arriving at the receiver from the kth transmitter. The effect of the receiving antenna on the received signal is modeled as a differentiation [14]. These assumptions result in the following model for the received signal at the antenna output: rt = K L k 1 ak j/n f k=1 j= l=0 {β k l w rx t jt f c k j T c b k j/n f τ k l } + nt 2 where K is the total number of users in the system, nt is the additive noise process, w rx t denotes the derivative of w tx t, and β k l and τ k l represent the amplitude and the delay, respectively, of the lth path arriving at the receiver from the kth user. The additive noise is taken to be a white, zero mean, Gaussian random process, with two-sided spectral density N 0 2. The pulse shape is chosen so that the pulse duration is on the order of T c ; among the pulse shapes w tx t that have been suggested for use in IR systems, are the Gaussian pulse [14], the Rayleigh pulse, and others.

6 SUBMITTED FOR PUBLICATION TO THE IEEE TRANSACTION ON SIGNAL PROCESSING 6 B. Discrete Time Received Signal Model In what follows we describe a simplified discrete-time model for PAM IR system. For the purpose of analysis, this model is used throughout the rest of the paper. Consider a synchronous system operating under slow flat fading conditions, where the receiver tracks the channel gains, [13], [15], [21]; that is L k = 1, k, and τ1 1 = = τ 1 K = 0. Assume that the received signal, rt of 2, is passed through a linear filter matched to the received pulse w rx t, and the output of this filter is sampled every T c second. Denote by r[i] = [r incn f r incn f r i+1ncn f 1] T the vector of such samples corresponding to the ith information symbol interval. It is easily seen that r[i] is a sufficient statistic for detecting the ith information symbol of all the users. In addition, r[i] obeys the following model: r[i] = S[i]Ab[i] + n[i] 3 where n[i] is an N c N f 1 vector composed of the samples of the additive noise process at the output of the matched filter. Since the noise is assumed to be white and Gaussian, n[i] is a zero mean Gaussian random vector with correlation matrix equal σ 2 ni, where σ 2 n = N 0 2, and we assume without loss of generality that w2 rxtdt = 1. Here b[i] is a K-ary vector whose kth element is ith information symbols transmitted by kth user; A = diaga 1,..., A K is a diagonal matrix with the gains between the transmitters and the receiver on its diagonal; and S[i] is a matrix composed of zeros and ones, such that 1 if c k l = l l i 1N S[i] lk = f + Nc N c N c Nc. 4 0 otherwise Note that S[i] is simply the matrix such that the ones at the kth column are placed at indices representing the time instances where pulses from the kth user are received. Since we can consider, without loss of generality, the first transmitted information symbol we omit the dependence of the quantities r, S, b, and n on the time index, i, in the remainder of the paper. Also, for simplicity we concentrate on the case where the transmitted symbols are ±1, the extension to the case of general PAM modulation being straightforward. Note that for binary PPM modulation, a similar model could be derived, which makes the results reported here applicable to binary PPM modulation as well. Denote by r k = [r 1,..., r Nf ] the vector of samples taken at the output of the matched filter at time instances where pulses from the kth user, corresponding to the first transmitted bit, are received. In

7 SUBMITTED FOR PUBLICATION TO THE IEEE TRANSACTION ON SIGNAL PROCESSING 7 addition, denote by K k the number of users colliding with the kth user. The following model for r k can be easily deduced from 3, r k = S k A k b k + n k 5 where b k is a K k +1 1 vector, containing the information bits transmitted by the kth user and the users colliding with that user; S k is an N f K k +1 matrix such that, without loss of generality, the first column is the all-ones column, and the rest of the columns are equal to zero except the indices corresponding to time instances where the corresponding user collide with the kth user; A k is a diagonal matrix with the gains between the transmitters corresponding to the kth user and the users colliding with the kth user and the receiver; n k are the noise samples corresponding to the time instances where pulses from the kth user arrive to the receiver. It is easily seen that r k can be obtained by a linear transformation of r, and S k b k can be obtained by a linear transformation of S b followed by the deletion of all-zero columns. III. The Relation between TH-IR and R-CDMA The mathematical model for an IR system can be regarded as a random direct sequence CDMA R- CDMA system, where the kth column of S 3 can be regarded as the random spreading sequence assigned to the kth user for the transmission of the information bit. Although the mathematical model for the received signal in both TH-IR and R-CDMA system is similar, major differences between the two exists. In R-CDMA systems the elements of the spreading sequences are usually taken to be equal either 1 or 1, while in TH-IR systems these elements are equal to either 0 or 1. This property of TH-IR systems is also shared with optical CDMA systems [11]. Moreover, in R-CDMA systems the elements of the spreading sequences are usually modeled as independent and identically distributed iid binary random variables, while in TH-IR systems these elements are highly dependent. Other than this, there are no significant differences between what we usually refer to as a R-CDMA system and a TH-IR system. Nevertheless, as will be seen in the sequel, the special type of spreading sequences used by TH-IR systems allow us to develop very simple and efficient MUD algorithms. These algorithms perform quite well in scenarios were efficient MUD algorithms for R-CDMA system are not known, e.g., when the number of users is large. The close relationship between R-CDMA systems and TH-IR systems makes it easy to use almost every known MUD algorithm designed for CDMA systems. Specifically, it is easily seen from [17] that the vector

8 SUBMITTED FOR PUBLICATION TO THE IEEE TRANSACTION ON SIGNAL PROCESSING 8 y = S T r is a sufficient statistic for detecting the information bits from all users. Moreover, y can be described by the following well known model [17] y = RAb + ñ 6 where R = S T S, which can be regarded as the cross-correlation matrix between the spreading sequences of the various users. Note that we do not assume that the spreading sequences are normalized to have a norm equal to one, and as such R is not forced to have the all-one vector on its main diagonal. Alternatively, R can be viewed as the matrix with all the elements on its main diagonal equal N f, and the off-diagonal elements, say the k, lth element, equal to the number of times pulses from the kth user collide with those from the lth user. The noise vector ñ is just a linear transformation of n, and as such ñ is a zero mean Gaussian random vector, with correlation matrix σ 2 nr. It is evident from the similarity of 6 to the corresponding sufficient-statistic model for DS-CDMA that one can use essentially any multi-user detection algorithm developed for DS-CDMA systems for detecting the information symbols transmitted from the various users in IR system. Nevertheless, for the case of a large number of users, the complexity of these algorithms is very large which makes them impractical for TH-IR systems. For example, the optimal receiver has a complexity equal to O 2 K, while the decorrelator receiver has a complexity of O K 3. Note, for example, that the use of the decorrelator receiver requires the inversion of a K K matrix once per information bit, that is every N f T f seconds since R changes from one bit to the next. The rest of the paper is devoted to describing algorithms that exploit the special signal structure of IR systems to achive MUD with much lower complexity. A. The Algorithm IV. The Blinking Receiver The blinking receiver BR is a very basic multi-user detector. This receiver takes advantage of the special structure of the spreading sequences used by TH-IR systems. This makes the receiver unique in a sense that a similar approach can not be taken in wireless CDMA systems. In optical CDMA theory, this receiver is also known as the modified matched filter receiver, and as the decorrelating receiver [11]. The BR estimates the transmitted information bit of the user of interest, say the kth user, based on the samples of the matched filter at time instants when only the user of interest is received; that is, based on

9 SUBMITTED FOR PUBLICATION TO THE IEEE TRANSACTION ON SIGNAL PROCESSING 9 pulses where no collisions occurred during the reception of these pulses. The BR could be described by the following ˆbk = sgn w T r k 7 where ˆb k denotes the estimate of the kth user s binary symbol, sgn is the sign function, and w = [w 1,..., w Nf ] T is a weighting vector such that 1 if [S k ] j,2 = = [S k ] j,kk = 0 w j = 0 otherwise. 8 The intuition behind the BR receiver is very simple, detect the information bit transmitted from the user of interest based on the pulses received at time instances when no collision occurrs. It is obvious that in CDMA system where all the users are constantly transmitting energy, one cannot implement a similar receiver. Nevertheless, this receiver assumes that a sufficient amount of energy will be received from the user of interest without any interference from other users. Since the probability of collision is a monotonically increasing function of the number of users, K, and a monotonically decreasing function of the number of available slots for the hopping sequence, N c, the performance of the BR will depend primarily on these two factors. The BR is a very simple and computationally efficient algorithm. It is a linear receiver which requires sampling the received signal at times where pulses from the user of interest arrive at the receiver. Nevertheless, the receiver still needs to track the hopping sequences of the other users in the system in order to determine whether a collision has occurred or not. B. Performance Assume that the BR has recovered x pulses out of the N f transmitted pulses corresponding to the information bit, that is N f x collisions occurred during the reception of the pulses originating from the user of interest, say the kth. Following [12], the probability of error, denoted by P e k, x, is x A k P e k, x = Q σ n 9 where Q β = β α 2 e 2 2π dα. Let X denote the random variable given by the norm of w, that is X = w 2 = N f j=1 w2 j. The average probability of error, P ek, in decoding the information bit of the kth user is given by averaging P e k, x with respect to X, P e k = E {P e k, X}.

10 SUBMITTED FOR PUBLICATION TO THE IEEE TRANSACTION ON SIGNAL PROCESSING 10 The random variable X is the sum of the elements of the weight vector w. Each element, say w j, is equal to one if no pulses collide with the jth pulse of the kth user, and zero otherwise. The jth pulse of the l k user will collide with that of the kth user if and only if c l j = ck j, and the probability of that happening is easily seen to equal 1 N c. As such, the probability that the jth pulse of the l k user will not collide with that of the kth user is 1 1 N c. Since c l j is independent of ck m whenever j, l m, k, the probability that the jth pulse of the kth user will not collide with any of the pulses transmitted from other users is simply the product of the individual probabilities, and thus K 1 1 with probability 1 1 Nc w j = K 1 0 with probability Nc. 10 It is easily verifiable that the elements of the weight vector are iid random variables and as a result, K 1 X B N f, 1 1Nc 11 where Bn, p denotes a binomial random variable derived from n Bernoulli trials with probability of success in each trial equal to p. Summarizing, the average probability of error of the kth user is given by N f P e k = x=0 Nf 1 1 xk K 1 Nf x Q x N c Nc x A k σ n. 12 The average probability of error of the system provides the best description of its performance. Nevertheless, many times the average probability of error fails to fully expose some important aspects of the performance of the system. For example, the effect of the multiple access noise on the performance is better described by the receiver near-far resistance [17]. In what follows we consider the average asymptotic multiuser efficiency of the BR, which will be shown to be equal to the average near-far resistance of the BR as well. The average near-far resistance is a common performance measure used in analyzing R-CDMA systems see [8] and we adopt this measure here as well. The asymptotic multiuser efficiency of the BR, given S k and denoted by η k S k, is given by η k S k = lim σ 0 σ 2 Q 1 P e k S k N f A 2 k 13 where P e k S k denotes the probability of error in detecting the information bit of the kth user given S k, and the additional N f in the denominator serves as a normalizing factor. From the computations leading to 12 we can easily deduce that P e k S k = Q w A k σ n, where, as before, w is given by 10. Combining

11 SUBMITTED FOR PUBLICATION TO THE IEEE TRANSACTION ON SIGNAL PROCESSING 11 P e k S k with 13 results in the following simple expression, The near-far resistance of the BR given S k, denoted by η k S k is given by η k S k = w N f. 14 η k S k = inf η k S k. 15 A i >0, i k, σ 0 But since, η k S k is independent of the amplitudes of the other users, we have η k S k = η k S k = w N f. 16 The average near-far resistance is given by averaging η k S k, over S k, which gives { } { } w X η k = E { η k S k } = E = E = N f N f 1 1 N c K 1, 17 where for the last equality we have used the fact that X is distributed according to 11. Let T = N f T f denote the symbol period. Assuming, without loss of generality, that T f = N c T c, then T = N f N c T c. The system processing gain on a logarithmic scale is equal to 10 log T T c = 10 log N c N f. This processing gain can be seen as the sum of two terms. The first, 10 log N c, represents the processing gain in db due to the averaging process; and the second, 10 log N c, represents the processing gain due to transmission at only 1 N c of the total time. Assuming that T is fixed by some system constraint, an important question is how to choose N f and N c. We note that since T is fixed, N c N f should be kept fixed as well. From 17 we see that in order to maximize the average near far resistance of the system one should maximize N c, and thus he will choose N f = 1 N c = T T c. Note that, for fixed T, as N c increases the number of transmitted pulses per information bit decreases. As this happens the transmitted signal will become more and more peaky in nature, and the signal peak to average ratio will increase as well. In practical systems this might limit the ability to transmit one pulse per one information symbol, i.e., to choose N f = 1. Assume that the system processing gain is equal N, and denote by β the ratio between the number of users and the processing gain, that is, β = K N. 18

12 SUBMITTED FOR PUBLICATION TO THE IEEE TRANSACTION ON SIGNAL PROCESSING 12 In addition assume that one wishes to maximize the average near-far resistance, and as such chooses N f = 1 and N c = N. In this case the average near-far resistance is given by η k = 1 1 N Nβ 1 19 Taking N to infinity while keeping β constant, that is, taking the large network limit, we have lim N, K N =β η k = e β. 20 The average near far resistance of the BR is rather surprising. It can be seen from 20 that even when the number of users is equal to the processing gain, the average near far resistance of the BR is equal e This is in contrast to R-CDMA systems where it is well known that when the number of users is equal to the processing gain, the average near far resistance of both the MMSE and decorrelator receivers is zero [8]. V. The Quasi-Decorrelator and Quasi-MMSE Receivers A. The Quasi-Decorrelator Receivers A.1 The Algorithm In CDMA systems, the decorrelator and MMSE receivers exhibit very good performance with substantial complexity reduction compared to the optimal receiver. Nevertheless, both receivers require the inversion of a K K matrix every time the spreading sequences change. Since in TH-IR systems, the spreading sequences are changed from one information bit to the next, these receivers may be too complex when the number of users is large. Thus, it is of interest to consider a quasi-decorrelator receiver that requires the inversion of a matrix of a size smaller than the one required by the decorrelator. We recall see, e.g., [17] that the decorrelator receiver is given by, 1 ˆbk = sgn S S T S T r k 21 where S and r are, respectively, the matrix whose columns are the spreading sequences of the various users, and the vector of samples out of the matched filter. In order to reduce the complexity of the above receiver we consider using the following quasi-decorrelator receiver, ˆbk = sgn S k T 1 S k Sk T r k. 22 1

13 SUBMITTED FOR PUBLICATION TO THE IEEE TRANSACTION ON SIGNAL PROCESSING 13 where S k and r k are as in 5. Computing 22 requires the inversion of a matrix of size K k K k, where K k is the number of users colliding with the user of interest. Noting that matrix inversion has a complexity of O N 3, where N is the size of the matrix, the quasi-decorrelator receiver can lead to substantial complexity reduction. The number of users colliding with the kth user, K k, can easily be seen to be equal to K k = l k sgn s T k s l 23 where s l denotes the lth column of S. The event {sgn s T k s l = 0} occurs if and only if no collision occurs between the pulses of the kth user and the pulses of the lth user. Following the discussion leading to 10 it is easily seen that sgn s T k s l has the following distribution, sgn s T k s 1, with probability Nf Nc l =. 24 0, with probability 1 1 Nf Nc Since the TH sequences of the various users are independent from each other, we can conclude that K k is distributed as follows K k B K, 1 1 1Nc Nf, 25 from which it can be seen that the average number of users colliding with the kth user is K Nf < Nc K. As a rule of thumb we can claim that as long as 1 1 Nf N c is close to one, the number of users colliding with the user of interest is small, and hence the complexity of the quasi-decorrelator receiver is smaller than that of the decorrelator receiver. It is easily seen that N f << N c is sufficient to insure that, 1 1 N c Nf 1. Note that even if 1 << N f = N c then the number of users colliding with the user of interest is approximately 1 e 1 N u 0.66N u, which still leads to a substantial decrease in complexity. A.2 Performance Analysis In this section we analyze the performance of the quasi-decorrelator receiver. Our main result is that the quasi-decorrelator receiver is approximately the BR under mild regularity conditions. Denote by P 2 the probability that more than one pulse will collide with the kth pulse transmitted by the user of interest, say the kth user. It is easy to verify that P 2 = K 1 K K Nc Nc N c

14 SUBMITTED FOR PUBLICATION TO THE IEEE TRANSACTION ON SIGNAL PROCESSING 14 If P 2 is small compared to 1 N f then with high probability every row of S k has a weight equal to either one or two. Note that if N c > N f and N c > K then P 2 is insured to be small, and our assmptions hold. Moreover, this conditions are usually meet in IR systems. As such, with high probability S k T S k has the following structure, ] [S k T S k jl N f j = l = 1 C j l = 1, j > 1 or j = l, 1 or j = 1, l > 1 0 Otherwise 27 where C j is the number of pulses from the jth user among the K k users colliding with the kth user collide with those from the kth user. Note that due to our assumption, with high probability 27 is the exact correlation matrix. Continuing with 22, the quasi-decorrelator receiver is given by, ˆbk = sgn w T r k 28 where w equals to the first row of the matrix S k T 1 S k Sk T. In order to compute w one has to compute only the first row of S k T 1 S k which by brute force calculation can be shown to be equal to [ S k T S k 1 ] 1l = 1 N K k j=2 C j 1 N K k j=2 C j if l = 1 if l > 1 Using 29 the jth element of w, which is equal to the jth element of the first row of 29 S k T S k 1 Sk T, is equal to one if the jth row of S k has weight one and is zero otherwise. This can easily be seen because, due to our assumption, each row in the matrix S k has weight equal to one or two. If it has weight one than the jth column of S k T is equal to [1 0] T, where 0 denotes the all zero vector of appropriate length. Multiplying the first row of S k T 1 S k by [1 0] T 1 results in K N k. Otherwise, the jth column of j=2 C j S k T has the following general structure [ ]. As such, multiplying the first row of S k T 1 S k by [ ] T 1 results in zero. Thus the vector w is equal to K N k for indices where no collision occurred j=2 C j and is zero otherwise. This vector is equivalent to the weight vector used by the BR, and thus with high probability the two receivers are identical. As will be shown in the sequel, the performance of the quasi-decorrelator is practically the same as that of the BR even when the aforementioned assumptions are not valid.

15 SUBMITTED FOR PUBLICATION TO THE IEEE TRANSACTION ON SIGNAL PROCESSING 15 B. The quasi-mmse Receiver The quasi-decorrelator receiver presented in the previous subsection is based on the decorrelator receiver. In this receiver, the signal from the user of interest is decorrelated only from the portion of the signals transmitted by other users that overlap the signal transmitted by the user of interest. As shown, under appropriate conditions this receiver exhibits a large complexity reduction compared with that of the decorrelator receiver. Nevertheless, as proven in the previous subsection, the performance of the decorrelator receiver is practically equal to the performance of the BR, which perform poorly when the number of users in the system is large. This large performance degradation exhibited by the decorrelator receiver will happen even if the colliding sources are very weak, and as such the matched filter, which is nearly optimal, will outperform the decorrelator receiver considerably. In order to solve this problem, we develop the quasi-mmse receiver following the same lines leading to the quasi-decorrelator. Following 22 the quasi-mmse receiver is given by 1 ˆbk = sgn S k T S k + σ 2 A k 2 1 Sk T r k. 30 Ak The quasi-mmse detector can be analyzed using the same approach used to analyze the quasi-decorrelator receiver. The main result of this analysis is the following the complete derivation is omitted due space limitations: The quasi-mmse receiver is approximately given by ˆb k MMSE sgn w T r k where w is an N f 1 vector. The jth element of w, [w] j, is equal to 1 σ 2 n if no collision occurred during the reception of the jth pulse, and 1 σ 2 n+a 2 l otherwise, where l denotes the index of the user colliding with the kth user during the reception of the jth pulse. Several conclusions can be drawn from the above property. When the signal to noise ratio is large, the quasi-mmse receiver equals the BR. This can be easily seen by letting σn 2 approach zero, which results in a weighting vector, w, whose elements equal some finite value when collisions occurr at the corresponding time instances, and equal infinity otherwise. This should not come as a surprise because, as is well known, when the signal to noise ratio is large the MMSE and decorrelator receivers are essentially identical. Thus, since the quasi-decorrelator receiver approximately equals the BR, and since at large signal to noise ratio the quasi-mmse receiver equals the quasi-decorrelator receiver the conclusion is immediate. Another interesting conclusion that could be drawn from the above property concerns the behavior of the quasi-mmse receiver at low and moderate signal to noise ratios. At low signal to noise ratio the

16 SUBMITTED FOR PUBLICATION TO THE IEEE TRANSACTION ON SIGNAL PROCESSING 16 quasi-mmse receiver is approximately the matched filter, since all the elements of w are approximately 1 equal σ. At moderate signal to noise ratio the quasi-mmse receiver is the modified matched filter, which n 2 is optimal if the colliding users were transmitting Gaussian signals instead of binary signals. VI. Quasi-ML Approach In the previous sections we explored efficient MUD algorithms. These algorithms have a complexity which grows polynomially with K k, that is with the number of users colliding with the user of interest. In this section we present the Quasi-ML algorithm which has a complexity equal to O2 K k. In practical systems, this complexity might be much higher than the complexity required by both the BR and the quasi-decorrelator or quasi-mmse detectors. The minimum probability of error receiver is the one which estimates the transmitted symbol as the symbol which maximize the a-postriori log-likelihood [17], [23]. The ML receiver requires sampling of the received signal at any time instant where a pulse is received, and then minimizing some non-linear function of all these measurements. The computational complexity of this algorithm is enormous; thus we resort to a simpler yet similar algorithm. In this algorithm the receiver bases its decision only on samples obtained from the output of the matched filter at time instances where the pulses from the user of interest are received. The term quasi optimal stems from the fact that the receiver presented hereafter is the optimal receiver among all the receivers that base their decisions on the samples of the matched filter at these time instances. Following [17], the receiver that minimizes the probability of error in detecting the mth user from the measurements r k is given by ˆbk = arg b1 ±1 log fb r k = arg b1 ±1 b 1,1 K k r k m S k [b 1 b] T 2 31 As can be seen this receiver is the well known MAP receiver, where instead of the sufficient statistic, r, we use only a sub-vector of the sufficient statistic. This will certainly result in reduced performance compared with the optimal receiver. Nevertheless, this receiver also has reduced complexity since r k obeys a linear model in which the mixing matrix has only K k columns, and as such the complexity of the quasi-optimal receiver is O 2 K k.

17 SUBMITTED FOR PUBLICATION TO THE IEEE TRANSACTION ON SIGNAL PROCESSING 17 VII. Iterative turbo algorithm In this section we present an iterative multi-user detection algorithm for TH-IR systems that follows the turbo principle. Turbo-like iterative algorithms have been suggested as possible solutions for many problems see, [4], [18] among many other. The complexity of these iterative algorithms is very low compared with the complexity of the optimal solutions for the problems they aim to solve, while their performance tends to be very close to the performance of corresponding optimal solutions. In that sense our proposed algorithm is no different. Nevertheless, in some aspects our proposed algorithm is quite unique. Closely examining the problems in which iterative algorithms have been successfully applied reveals that these problems have a very special structure, which allows the use of iterative procedures. Take, for example, the problem of joint equalization and decoding. It is clear that the simplest algorithm will first solve the equalization problem, and then based on the equalized samples will decode the information bits. That is, the solution could be separated into two disjoint algorithms working one after the other. The same is true with the problem of joint multi-user detection and decoding of error correcting codes in CDMA systems [18]. In this problem one can use any multi-user detection algorithm or more precisely a multi-user receiver [16] that results in soft decision statistics about every channel bit. These soft decisions can be fed into any soft decoding algorithm and the result will be the estimated information bit. Turbo based algorithms provide an efficient way to iterate between the results obtained by the two constituent algorithms, where each one of these algorithms is designed to solve one part of the problem. In our problem no similar structure exists, and thus in order to use an iterative decoding algorithm, we will have to impose such structure by ignoring some of the a priori information. A. The Algorithm A.1 General structure As with other turbo like algorithms our algorithm is composed of two stages and the algorithm iterates between these stages. The first stage is denoted as the pulse detector, while the second is denoted as the bit detector. In the first stage we assume that different pulses from the same user correspond to independent information bits, while in the second stage we exploit the information that all the pulses from the same user correspond to the same information bit.

18 SUBMITTED FOR PUBLICATION TO THE IEEE TRANSACTION ON SIGNAL PROCESSING 18 Let us denote by b k j the information bit transmitted at the jth pulse from the kth user. Note that although we know a priori that b k 1 = = bk N f we choose to ignore this information at the first stage. As such, at the nth iteration the pulse detector computes the a posteriori log-likelihood ratio LLR of b k j given the received signal, the information about the transmitted bits from other users, and the a priori information about b k j posteriori LLR of b k j obtained by the bit detector. It will be shown below that at the nth iteration the a has the following form: L n 1 bk j = log Prbk j = 1 r Pr b k j = 1 r = λ n 1 bk j + λn 1 2 b k j 32 where λ2 n 1 b k j represents the a priori LLR of bk j, which is computed by the bit detector at the n 1th iteration. The extrinsic information λ n 1 bk j, which is, at least at the first iteration, independent of λn 1 2 b k j, is then fed back to the bit detector. The bit detector exploits the fact that b k 1 = = bk N f. As such, the bit detector computes the a posteriori LLR of b k j given the information from the various pulse detectors. In the sequel it will be shown that this LLR has the following general structure, L n 2 b k j = log Pr b k j = 1 λn 1 bk j, j = 1,..., N f Pr b k j = 1 λn 1 bk j, j = 1,..., N = λ n 2 b k j + λ n 1 b k j. 33 f The a posteriori LLR at the output of the pulse detector can be seen to be the sum of the prior information from the pulse detector plus the extrinsic information about b k j. This extrinsic information is obtained from the information on the pulses, other than the jth pulse, of the kth user. A.2 The pulse detector The first stage of the turbo multi-user detector is the pulse detector. The pulse detector computes the a posterior LLR of the transmitted bit modulating the jth pulse of the kth user for every j and k. Denote by L 1 b k j this ratio, L 1 b k j = log Pr b k j = 1 r. Pr b k j = 1 r 34 By using Bayes formula, L 1 b k j is equal L 1 b k j = log f r b k j = 1 f r b k j = 1 + log Prbk j = 1 f r lj,k b k j Prb k = log = 1 j = 1 f r lj,k b k j = 1 + log Prbk j = 1 Prb k j = 1 35

19 SUBMITTED FOR PUBLICATION TO THE IEEE TRANSACTION ON SIGNAL PROCESSING 19 where lj, k is the time index at which the jth pulse from the kth user was transmitted. The second equality stems from our assumption that each pulse is modulated with an independent information bit. We now turn to the computation of log fr lj,k b k j =1 fr lj,k b k j = 1. Similarly to the definition of K k, we denote by K j k the number of users colliding with the jth pulse of the kth user. It can easily be seen that K j k is the weight of the lj, kth row of S. In addition denote by fj kg ; g = 1,..., Kj k the indices of the users colliding with the jth pulse of the kth user. Following 3, r lj,m obeys the following model, r lj,m = 1 K j A j k bk j + n lj,m 36 k where 1 K j is the 1 K j k all one vector, Aj k = diag[a k A k f k j 1 A fj kkj ] is the diagonal matrix containing k the amplitudes of the pulses received from the kth user and the users colliding with the jth pulse of the kth user; and b k j = k [bk j bf j 1 j b f j kkj k j ] is the vector containing the information bits colliding with the jth pulse of the kth user. Following 36, the a priori LLR is given by, log f r lj,k b k j = 1 f r lj,k b k j = 1 = log b {±1} Kj k b {±1} Kj k j r 1 e lj,k 1A k [1 b]t 2 2σ 2 2πσ 2 j r 1 e lj,k 1A k [ 1 b]t 2 2σ 2 2πσ 2 where pb = Kj k g=1 p b f j kg j = [b] g is the a-priori probability that Following [4], p pb = log pb b {±1} Kj k b {±1} Kj k r lj,m 1A j k [1 b]t 2 e 2σ 2 pb 37 r lj,k 1A j k [ 1 b]t 2 e 2σ 2 pb [ ] b f j k1 j b f j k Kk j j = b is equal to [b] g. b f j kg j = [b] g is the prior information about b f j kg j from other sources. Simple algebraic manipulations leads to the following, p b f j kg 1 j = [b] g = 1 + [b] g tanh 2 λn 1 2 b f k j g j. 38 Combining 37 and 38 results in the following log f r lj,k b k j = 1 f r lj,k b k j = 1 = b {±1} Kj k r lj,k 1A j k [1 b]t 2 K e j 2σ 2 k g=1 1 + [b] 1 g tanh 2 λn 1 2 b f j kg j r e li,k 1Ai k [ 1 b]t 2 b {±1} Ki 2σ 2 k K j k g=1 1 + [b] g tanh 1 2 λn 1 2 b f k j g j = λ j 1 b k i 39 Combining 39 with 35 we see that the a posteriori LLR is given by 32; that is, the sum of prior information obtained from the bit detector and the extrinsic information. The complexity of computing 39 is very low. It has the complexity of O 2 Kj k which is very low. For example, if the number of users is equal to N c then the probability that K j k will be greater than one is 1 1 1/N c Nc 1 1/N c Nc

20 SUBMITTED FOR PUBLICATION TO THE IEEE TRANSACTION ON SIGNAL PROCESSING 20 Thus the probability that K j k is large very small. By recalling that in DS-CDMA systems the complexity of iterative turbo like MUD is O2 K this complexity reduction represent a major difference between DS-CDMA and IR systems. A.3 The bit detector While the pulse detector computes the a posteriori LLR of the transmitted bit corresponding to the jth pulse from the kth user, the bit detector will compute the a posteriori LLR of the transmitted information bit given the a priori LLR from the pulse detectors. The pulse detector computes the following ratio for every j = 1,..., N f and k = 1,..., K: L n 2 b k j = log Pr b k j = 1 λn 1 b k j Pr b k j = 1 λn 1 b k j j = 1,..., N f ; k = 1,..., K. 40 j = 1,..., N f ; k = 1,..., K We first note that b k j is independent of bm l for m k, and thus 40 reduces to L n 2 bk j = log Pr b k j = 1 λn 1 b k j j = 1,..., N f. Pr b k j = 41 1 λn 1 b k j j = 1,..., N f Now since b k 1 = = bk N f, Eq. 41 reduce to L n 2 bk j = log Pr b k j = 1 λn 1 b k j j = 1,..., N f Pr b k j = = log Pr b k 1 = bk N f = 1 λ n 1 b k j j = 1,..., N f 1 λn 1 b k j j = 1,..., N f Pr b k 1 = bk N f = 1 λ n 1 b k j j = 1,..., N f N f = log Pr b k l = 1 λ n 1 b k j N f l=1 Pr b k j = = log Pr b k l = 1 λ n 1 b k l N f 1 λn 1 b k Pr b k l l=1,l j l = 1 λ n +λ n 1 b k 1 b k j = λ n 1 b k l +λ n 1 b42 k j.. l l=1,l j }{{}}{{} λ n 2 b k j λ n 2 b k j As can be seen from 42, the output of the bit detector is the sum of the extrinsic information from the pulse detector about b k j, plus the information obtained from our knowledge that all the pulses form the same user bear the same information bit. λ n 2 b k j is the extrinsic information about b k j used by the pulse detector as a priori information. As can easily be seen at the first iteration, the λ n 2 b k j are independent, as expected by a turbo decoder. VIII. Simulations In this section we present simulation results demonstrating the performance of the various algorithms as a function of the SNR, the number of users in the system, and the ratio between the powers of the various users. We consider a TH-IR system with N f = 10 and N c = 20.

21 SUBMITTED FOR PUBLICATION TO THE IEEE TRANSACTION ON SIGNAL PROCESSING 21 In the first set of simulations we assume that all the sources have equal received power. Figures 1-3 depict the average bit error rate as a function of the received SNR per pulse, for 10,15, and 20 users, respectively. As can be seen from the figures, the BR and quasi-decorrelator have essentially the same performance. Moreover the theoretical and the empirical performance curves match very well. Note that these two receiver exhibit a 3 db performance degradation compared to the performance of a single user system. The quasi-mmse receiver outperforms both the BR and the quasi-decorrelator receiver, but it still suffers performance loss of about 1.5 db compared to the performance of a single user system. The iterative multiuser detector outperforms all of the other detectors; its performance is practically equal to the performance of a single user system after only two iterations, while at the first iteration its performance is equal to that of the quasi-mmse receiver P e Single User Matched Filter BR Theoretical BE q Decorrelator q MMSE Turbo 1st Turbo 2nd Turbo 3rd SNR per pulse [db] Fig. 1. The average probability of error for a system with N c = 20, N f = 10, N u = 10, as a function of the SNR, for equi-power signals.

22 SUBMITTED FOR PUBLICATION TO THE IEEE TRANSACTION ON SIGNAL PROCESSING P e Single User Matched Filter BR Theoretical BE q Decorrelator q MMSE Turbo 1st Turbo 2nd Turbo 3rd SNR per pulse [db] Fig. 2. The average probability of error for a system with N c = 20, N f = 10, N u = 15, as a function of the SNR, equi-power signals. In the next set of simulations, we present simulation results for the case of strong interfering signals. Figures 4-6 depict the average bit error rate as a function of the received SNR per pulse of the user of interest. The interfering users have an SNR per pulse 6 db higher than that of the user of interest. The total number of users is again 10,15, and 20 users, respectively. As can be seen from the figures, the BR and the quasi-decorrelator receivers are uneffected by this change. This is since their performance is invariant to the power of the interfering signals. The performance of the quasi-mmse detector approaches that of the quasi-decorrelator. This could be predicted as an immediate result of properties discussed in Section V.B, where due to the large gains compared to the noise level of the interferences the quasi-mmse become identical to the BR. Again, the iterative detector outperforms the other detectors, and after only two iterations its performance is essentially equal to that of the single channel single user case.

23 SUBMITTED FOR PUBLICATION TO THE IEEE TRANSACTION ON SIGNAL PROCESSING P e Single User Matched Filter BR Theoretical BR q Decorrelator q MMSE Turbo 1st Turbo 2nd Turbo 3rd SNR per pulse [db] Fig. 3. The average probability of error for a system with N c = 20, N f = 10, N u = 20, as a function of the SNR, equi-power signals. IX. Concluding remarks In this paper we have shown that Time Hopping Impulse Radio systems are very similar to R-CDMA systems from the viewpoint of their multi-user properties. This similarity enable the use of many multi-user detectors developed for CDMA systems in TH-IR systems without any change. Nevertheless, by exploiting the special structure of the signals transmitted by TH-IR systems, we have developed novel, simple multiuser detection algorithms for such systems. These algorithms are unique and can be used only with signal structures appearing in TH-IR systems. Of potential interest is the iterative algorithm presented in this paper, whose performance is essentially the same as the performance of a single user system, and whose complexity is very low. The analysis conducted in this paper reveals very interesting trade-offs between the two types of processing gain exploited by TH-IR systems. This leads to a fundamental question about the optimal structure