Low-Complexity Multiuser Detectors for Time-Hopping ... - NYU Stern

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Low-Complexity Multiuser Detectors for Time-Hopping Impulse-Radio Systems

Eran Fishler‡ and H. Vincent Poor†

* This work was supported by the Army Research Laboratory under Contract No. DAAD 19-01-2-0011, by the National Science Foundation under Grant CCR-99-80590, and by the New Jersey Center for Wireless Telecommunications. † H. V. Poor is with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA, Tel: (201) 258-1816, Fax: (201) 258-1468, e-mail: [email protected] ‡ E. Fishler was with the Department of Electrical Engineering, Princeton University, Princeton, NJ. He is now with the Stern School of Business, New York University, NY email: March 22, 2004

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Abstract Impulse Radio (IR) and especially Time-Hopping IR (TH-IR) systems have attracted considerable attention since their introduction in the context of ultra wide-band (UWB) systems in the early 1990s. Recently, with the US Federal Communications Commission actions allowing for the wide-spread use of UWB radio systems, the interest in these systems has grown further. These systems promise to deliver high data rates in multiple access communication channels with very simple transmitter and receiver designs. The importance of multiuser detection for achieving high data rates with these systems has already been established in several studies. This paper studies several low complexity multiuser detectors specifically designed for TH-IR radio systems. It is demonstrated that many multiuser detectors developed primarily for direct sequence code division multiple access (DS-CDMA) systems can be used essentially without any change in TH-IR receivers. Further, several novel, low complexity multiuser detectors that exploit the special signal structure used for transmission by TH-IR systems are developed. These novel detectors are analyzed both theoretically and via simulations. It is shown that a very simple iterative multiuser detector yields performance similar to that of a single user system.

I. Introduction Impulse Radio (IR) systems and especially time-hopping impulse radio (TH-IR) systems have drawn considerable attention among both researchers and practitioners over the past few years. In a TH-IR system, a train of pulses is transmitted, and the information is conveyed either by the polarity of the transmitted pulse (usually referred to as Pulse Amplitude Modulation (PAM)) [9], or by shifting the pulse starting times (usually referred to as Pulse Position Modulation (PPM)) [20]. In addition, in order to allow many users to share the same channel, an additional pseudo-random time shift, known to the receiver and unique to each user, is added to each pulse starting point. This way, catastrophic collisions between two users transmitting over the same channel at the same time are avoided. The interest in TH-IR systems arises in, but it is not limited to, the context of Ultra Wide-band (UWB) radio systems, where the transmitted pulses are very short, typically on the order of a fraction of a nano-second, and the system bandwidth is very large. Nevertheless, TH-IR modulation can be used in any spread spectrum system [7], [15]. By decreasing the transmitted pulse width one can increase the transmitted bandwidth up to the point where the system become a UWB system. The transmission of digital information using this method was first suggested by Withington and Fullerton in [22]. 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 [16], this idea is further developed to allow multiple users to exploit the same channel. Time-hopping impulse radio systems have been analyzed extensively in the past, for free space, flat fading, and frequency selective channels, and with or without narrow-band

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interference (see, [2], [12], [16], [20], [27], among many others). Multiuser detection (MUD) for TH-IR systems has also been considered in many studies (see, [1], [6], [10], [9], [7], [15], [16], [21], [24], [25], among others). Scholtz [16] was the first to consider multiuser issues in TH-IR systems. By assuming strict power control, asynchronous transmission, a large number of users, and an additive white Gaussian noise (AWGN) channel, the multiple-access interference (MAI) is modeled in [16] as a Gaussian random variable and the single user matched filter detector is used for detecting the transmitted symbols of a user of interest. This approach, of modeling the MAI as a Gaussian random process, is further used in [20] to demonstrate that impulse radio 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). In [6] multiuser detection of UWB TH-IR systems transmitting over AWGN channels is considered. It is suggested to use time hopping sequences, which are based on pseudo chaotic maps, in order to reduce the MAI, when using the matched filter detector. A similar approach is suggested in [7], [9], where appropriate design of the time hopping sequences used by the various users results in deterministic and complete multiuser interference rejection even when the channel is a multi-path channel. These multiuser detection algorithms require complete knowledge of the channel impulse response between each user and the receiver, as well as the oversampling of the received signal at considerable rates, resulting in high receiver complexity [24]. In [15] the performance of IR systems transmitting through AWGN channels and using the matched filter detector is investigated using the Chernoff bound. Optimal multiuser detection for TH-IR systems using PPM modulation and transmitting over AWGN channels is investigated in [25], where it is shown that optimal multiuser detection is equivalent to optimal multiuser detection in Code-Division-MultipleAccess (CDMA) systems. Consequently, it is demonstrated that the complexity of the optimal multiuser 



detector is O 2K , where K is the number of users. The use of other multiuser detectors, like the zero forcing receiver, is suggested in [25] as well. Although the classical algorithms for multiuser detection can by used in TH-IR systems, it is evident that simple algorithms for multiuser detection in TH-IR systems are required. This paper offers novel multiuser detection algorithms specifically designed for TH-IR systems. In this paper we study the problem of multiuser detection in TH-IR systems transmitting over AWGN channels. This study serves two purposes. The first is to motivate the development of low complexity multiuser detectors for TH-IR systems; and the second is to describe a series of very low complexity multiuser detectors for TH-IR systems in flat fading channels. Low complexity multiuser detectors for IR systems operating in frequency selective channels will be treated in a subsequent study. As already noted in [25], almost every multiuser detector developed for Direct-Sequence CDMA (DS-CDMA) systems can be

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used in TH-IR system with slight, or even without any, modification. As such, our study focuses on very low complexity multiuser detectors exploiting the specific signal structure of TH-IR signals. In particular, as opposed to most multiuser detectors described in the literature, our algorithms do not make use of a bank of correlators; rather they operate directly on the samples obtained at the output of a filter matched to the received pulse. The rest of this paper is organized as follows. In Section II we describe both the continuous-time and discrete-time signal models for the transmitted and received signals in TH-IR systems. In Section III the relation between TH-IR and random DS-CDMA (RCDMA) systems is discussed and the two systems are compared. In Section IV the blinking receiver, which is the simplest multiuser detector to be considered here, is discussed and analyzed. The quasi-decorrelator and quasi-MMSE multiuser detectors are described and discussed in Section V, while in Section VI we describe and discuss the quasi-MaximumLikelihood approach. Section VII is devoted for describing a novel iterative (“turbo” like) multiuser detector. In Section VIII we provide simulation results that demonstrate both the performance of the proposed algorithms, and the trade-offs among them. Some concluding remarks are included in Section IX. II. System Model A. Continuous Time Received Signal Model The transmitted signal, of say the kth user, in a random TH-IR system is described by the following general model: sktr (t) =

∞ X

j=−∞

⌊j/Nf ⌋

bk

wtr (t − jTf − ckj Tc )

(1)

where Tf is the nominal pulse repetition time, and wtr (t) is the transmitted pulse, usually referred to as the monocycle; {bjk } is the sequence of information symbols transmitted by the kth user which are assumed throughout this paper to be binary (i.e., elements of {±1}) although the extension to more general cases is straightforward; Nf is the number of monocycles used to transmit one information symbol and ⌊·⌋ denotes

the integer part; {ckj } is a pseudo random sequence taking values in [0, 1, . . . , Nc − 1], assigned to user k, and usually referred to as the time hopping sequence, which is used to provide an additional time shift of ckj Tc seconds to the jth pulse of the kth user. In order to avoid inter-pulse interference (IPI), it is usually required that Tc ≤

Tf Nc ,

so that an overlap between two transmitted pulses from the same user is avoided.

The transmitted signal (1) undergoes various changes due to the propagation channel and the effects of the receiving antenna on the received signal. In this paper we assume that the channel is a frequency-flat March 22, 2004

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AWGN channel. Although TH-IR systems may also be subjected to frequency selective multipath, the AWGN assumption is adequate in many situations and is commonplace in studies of such systems (see, [1], [6], [15], [17], [20], [21], [23], [25], [26], among many others). The effect of the receiving antenna on the received signal is modeled as a differentiation [16]. These assumptions result in the following model for the received signal at the antenna output: r(t) =

K X

k=1

∞ X

Ak

⌊j/Nf ⌋

bk

j=−∞





wrx t − jTf − ckj Tc + n(t)

(2)

where K is the total number of users in the system, n(t) is additive noise, wrx (t) denotes the derivative of wtx (t), and Ak represents the amplitude of the signal arriving at the receiver from the kth user. In this model, for simplicity, it is assumed that the different users are synchronized, which is applicable, for example, in a down-link channel (see, [1], [6], [25], and references therein). In keeping with the AWGN model the additive noise is taken to be a white, zero mean, Gaussian random process, with two-sided spectral density

N0 2 .

The pulse shape is chosen so that the pulse duration is on the order of Tc ; among the

pulse shapes wtx (t) that have been suggested for use in TH-IR systems, are the Gaussian pulse [16], the Rayleigh pulse, and others. B. Discrete Time Received Signal Model For the purpose of analysis, it is assumed throughout the rest of the paper, that the system is a synchronous system operating under slow flat fading conditions, where the receiver tracks the channel gains, [15], [17], [6], [25]. The received signal, r(t) of (2), is passed through a linear filter matched to the received pulse wrx (t), and the output of this filter is sampled every Tc seconds.

Denote by

r[i] = [riNc Nf riNc Nf +1 . . . r(i+1)Nc Nf −1 ]T the vector of such samples corresponding to the ith information symbol interval. r[i] is a sufficient statistic for detecting the ith information symbol of all the users [25]. In addition, r[i] obeys the following model: r[i] = S[i]Ab[i] + n[i]

(3)

where n[i] is an Nc Nf × 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 σn2 I, where σn2 = R∞

2 −∞ wrx (t)dt

N0 2 ,

and we assume without loss of generality that

= 1. Here b[i] is a K-ary vector whose kth element is the ith information symbol transmitted

by the kth user; A = diag(A1 , . . . , AK ) is a diagonal matrix with the gains between the transmitters and

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the receiver on its diagonal; and S[i] is a matrix composed of zeros and ones, such that

(S[i])lk =

   1 if ck

(i−1)Nf +

  0 otherwise



l Nc



Nc

=l−

j

l Nc

k

Nc

.

(4)

Note that S[i] is simply the matrix whose non-zero elements in 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 henceforth omit the dependence of the quantities r, S, b, and n on the time index i. Note that if binary PPM is used, a similar model could be derived [8], which makes the results reported here applicable to binary PPM as well. Denote by rk = [r1 , . . . , rNf ] 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 symbol) are received. In addition, denote by Kk the number of users colliding with the kth user. The following model for rk can be easily deduced from (3), rk = Sk Ak bk + nk

(5)

where bk is a (Kk + 1) × 1 vector, containing the information symbols transmitted by the kth user and the users colliding with that user; Sk is an Nf × (Kk + 1) matrix such that the first column is the all-ones column, and the rest of the columns are equal to zero except at the indices corresponding to time instances where the corresponding user collides with the kth user; Ak is a diagonal matrix such that [Ak ]11 is the gain of the k user, and [Ak ]ii i > 1 is the gain of the (i − 1)th user colliding with the kth user; nk contains the noise samples corresponding to the time instances where pulses from the kth user arrive at the receiver. It is easily seen that rk can be obtained by linear transformation of r, and Sk and bk can be obtained by linear transformation of S and b, respectively, followed by the deletion of all-zero columns. III. The Relation between TH-IR and CDMA systems The mathematical model for a TH-IR system, described in Section II can be regarded as a generalization of an RCDMA system [11], 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 models for the received signal in both TH-IR and RCDMA systems are similar, major differences between the two exist. In RCDMA systems the elements of the spreading sequences are usually taken to be equal to either 1 or −1, while in TH-IR systems these elements are equal to either 0 or 1. Moreover, in RCDMA 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 March 22, 2004

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this, there are no significant differences between what we usually refer to as an RCDMA 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 situations where efficient MUD algorithms for RCDMA systems are not known, e.g., when the number of users is large. The close relationship between RCDMA systems and TH-IR systems makes it easy to use almost every known MUD algorithm designed for RCDMA systems. This fact was noted in [25], where a similar relationship between CDMA systems and TH-IR systems that use PPM modulation (instead of the PAM modulation assumed in this paper) was developed. Specifically, the vector y = ST r is a sufficient statistic for detecting the information bits from all users [25]. Moreover, y can be described by the following well known model [18] ˜ y = RAb + n

(6)

where R = ST S is the cross-correlation matrix between the spreading sequences of the various users; ˜ is a zero mean Gaussian random vector, with correlation matrix σn2 R. It and where the noise vector n is evident from the similarity of (6) to the corresponding sufficient-statistic model for RCDMA that one can use essentially any multiuser detection algorithm developed for RCDMA systems for detecting the information symbols transmitted from the various users in TH-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 MUD has a complexity equal to O 2K , while the decorrelating


receiver has a complexity of O K 3 , per user per bit.

IV. The Blinking Receiver A. The Algorithm The first detector we propose is the blinking receiver (BR), which is a very basic multiuser detector. This receiver takes advantage of the special structure of the spreading sequences used by TH-IR systems. This makes the receiver unique in the sense that a similar approach can not be taken in RCDMA systems. In optical CDMA, this receiver is also known as the modified matched filter detector, and as the decorrelating receiver [13]. The BR estimates the transmitted information symbol 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 pulses where no collisions occurred during the reception of these pulses. The BR can be described by

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the following: 

ˆbk = sgn wT rk BR



(7)

△ where ˆbk denotes the estimate of the kth user’s (binary) symbol, sgn (x) = x/|x| is the sign function, and

wBR = [w1 , . . . , wNf ]T is a weighting vector given by

wj =

   1 if [Sk ]j,2 = · · · = [Sk ]j,K = 0 k

.

(8)

  0 otherwise

It is obvious that in CDMA systems 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. The probability of error of the BR will depend on the transmitted energy per bit and on the probability of collision. The BR is a very simple and computationally efficient algorithm. It is a linear receiver that requires sampling the received signal at times when pulses from the user of interest arrive at the receiver. Nevertheless, the receiver still needs to track the time-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 Nf transmitted pulses corresponding to the information symbol of interest, that is Nf − x collisions occurred during the reception of the pulses originating from the user of interest, say the kth. The conditional probability of error is given by Pe (k, x) = Q △

R∞

√ Ak x σn



(9)

α2

− 2 e√ dα. Let X denote the random β 2π P N f wj2 . The average probability of ||wBR ||2 = j=1

where Q (β) = X=



variable given by the squared norm of wBR ; that is error, Pe (k), in deciding the information symbol of

the kth user is given by averaging Pe (k, x) with respect to X, Pe (k) = E {Pe (k, X)}. The random variable X is the sum of the squared elements of the weight vector wBR . Each squared element, say wj2 , 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 6= k user will collide with that of the kth user if and only if clj = ckj , and the probability of that happening is easily seen to equal

1 Nc .

As such, the probability that the jth pulse of the

l 6= k user will not collide with that of the kth user is 1 −

1 Nc .

Since clj is independent of ckm whenever

(j, l) 6= (m, k), the probability that the jth pulse of the kth user will not collide with any of the pulses

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transmitted from other users is simply the product of the individual probabilities, and thus   K−1   1 with probability 1 − 1 N wj2 = K−1 .  c   0 with probability 1 − 1 − 1

(10)

Nc

It is easily verifiable that the elements of the weight vector are iid random variables and as a result, 

1 X ∼ B Nf , 1 − Nc

K−1 !

(11)

where B(n, 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 Pe (k) =

Nf X

x=0

Nf x

!

1 1− Nc

x(K−1)



1 1− 1− Nc

(K−1) !Nf −x



√ Ak x Q σn



.

(12)

While the average probability of error of the system provides important information about the systems performance, there are times when 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 [18]. The average near-far resistance is a common performance measure used in analyzing RCDMA systems (see [11]) and we adopt this measure here as well. 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 asymptotic multiuser efficiency of the BR, given Sk and denoted by ηk |Sk , is given by σ 2 Q−1 (Pe (k|Sk )) ηk |Sk = lim n 2 →0 Nf A2k σn


2

(13)

where Pe (k|Sk ) denotes the probability of error in detecting the information symbol of the kth user given Sk , and the additional Nf in the denominator serves as a normalizing factor. From the computations leading to (12) we can deduce that Pe (k|Sk ) = Q



p

k ||wBR ||2 A σn , where, as before, w is given by (10).

Combining Pe (k|Sk ) with (13) results in the following simple expression, ηk |Sk =

||wBR ||2 . Nf

(14)

The near-far resistance of the BR given Sk , denoted by η¯k |Sk is given by η¯k |Sk =

inf

2 →0 Ai >0, i6=k, σn

ηk |Sk .

(15)

But since ηk |Sk is independent of the amplitudes of the other users, we have η¯k |Sk = ηk |Sk = March 22, 2004

||wBR ||2 . Nf

(16) DRAFT

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The average near-far resistance is given by averaging η¯k |Sk over Sk , which gives η¯k = E {¯ ηk |Sk } = E

(

||wBR ||2 Nf

)

=E

(

X Nf

)



= 1−

1 Nc

K−1

,

(17)

where for the last equality we have used the fact that X is distributed according to (11). Let T = Nf Tf denote the symbol period. Assuming, without loss of generality, that Tf = Nc Tc , then T = Nf Nc Tc . The system processing gain (on a logarithmic scale) is equal to 10 log TTc = 10 log Nc Nf . This processing gain can be seen as the sum of two terms. The first, 10 log Nf , represents the processing gain in dB due to the averaging process; and the second, 10 log Nc , represents the processing gain due to transmission at only

1 Nc

of the total time [4]. Assuming that T is fixed by some system constraint, an

important question is how to choose Nf and Nc . We note that since T is fixed, Nc Nf 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 Nc , which corresponds to the choice Nf = 1, Nc = power per bit is

Eb T

=

Eb Nc Nf Tc ,

T Tc .

while the peak transmitted power is

Note that the average transmitted

Eb Nf Tc .

As such, for fixed T , as Nf

decreases the transmitted signal will become more and more “peaky” in nature, and the peak to average power ratio will increase. Assume that the system processing gain is equal to N , and denote by β the ratio between the number of users and the processing gain, that is, β=

K . N

(18)

In addition assume that one wishes to maximize the average near-far resistance, and as such chooses Nf = 1 and Nc = N . In this case the average near-far resistance is given by 

1 η¯k = 1 − N

N β−1

.

(19)

Taking N to infinity while keeping β constant, that is, taking the large network limit, we have lim

N →∞, K =β N

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 to e−1 ≈ 0.37. This is in contrast to RCDMA 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 [11].

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V. The Quasi-Decorrelator and Quasi-MMSE Receivers A. The Quasi-Decorrelator Receiver A.1 The Algorithm In RCDMA systems, the decorrelating and MMSE receivers exhibit very good performance with substantial complexity reduction compared to the optimal MUD. 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., [18]) that the decorrelating receiver is given by, ˆbk = sgn



−1

ST S

 

ST r

(21)

k

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 suggest using the following quasi-decorrelator receiver, ˆbk = sgn



T

(Sk ) Sk

−1

T

(Sk ) rk

 

,

(22)

1

where Sk and rk are as in (5). Note that the first component in the vector bk is the data symbol of the kth user. Computing (22) requires the inversion of a matrix of size (Kk + 1) × (Kk + 1), where Kk is the number of users colliding with the user of interest. Noting that matrix inversion has a cubic complexity as a function of the dimension, the quasi-decorrelator receiver can lead to substantial complexity reduction. The number of users colliding with the kth user, Kk , is equal to Kk =

X l6=k

sgn sTk sl









(23)

where sl denotes the lth column of S. The event {sgn sTk sl = 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 seen that sgn sTk sl has the following distribution, 



sgn sTk sl =

   1,   0,



1 N Ncf

with probability 1 − 1 − with probability



1−

1 Nc

N f

.

(24)

Since the time hopping sequences of the various users are independent of each other, we can conclude that March 22, 2004

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Kk is distributed as follows 

1 Kk ∼ B K, 1 − 1 − Nc

N f !

,

(25) 



from which it can be seen that the average number of users colliding with the kth user is K 1 − 1 − 

K. As a rule of thumb we can claim that as long as 1 −

1 Nc

N f

1 Nc

N f 

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 significantly 

smaller than that of the decorrelator receiver. Thus, Nf 1 Otherwise

where Cj is the number of pulses from the jth user (among the Kk users colliding with the kth user) colliding with those from the kth user. Continuing with (22), the quasi-decorrelator receiver is given by, 

ˆb1 = sgn wT r1 deco 



where wdeco equals to the first row of the matrix (S1 )T S1 

to compute only the first row of (S1 )T S1 

T

(S1 ) S1

−1

−1 

1l

=

(27)

−1

(S1 )T . In order to compute wdeco one has

which by direct calculation can be shown to be equal to       

Nf − Nf −

P1Kk +1 j=2

P−1 Kk +1 j=1

Cj Cj

if l = 1 .

(28)

if l > 1 

Using (28) the jth element of wdeco , which is equal to the jth element of the first row of (S1 )T S1 is equal to

Nf −

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P1Kk +1 j=2

Cj

−1

(S1 )T ,

if the jth row of S1 has weight one and is zero otherwise. This can easily be seen DRAFT






1 1− Nc



1−

1 Nc

! Nf (K−2)Nf X Nf 

1

i=0

(K−1)Nf

> e−

1 1− Nc

(K−1)Nf Nc

.

i 

K −1 Nc

Nf −i

(30)

So, a sufficient condition for the probability of the event “Sk has a row with weight more than two” to be negligible, is that Nc >> (K − 1)Nf . Thus if this condition holds, then with high probability the BR will be equal to the quasi-decorrelator receiver. Note that in practical systems this condition will usually hold. In the sequel, when simulation results are presented, it is demonstrated that the performance of the quasi-decorrelator is practically equal to that of the BR. This is true even if the above condition does not hold, and Sk contains some rows having weight larger than two. B. The quasi-MMSE Receiver The quasi-decorrelator receiver presented in the previous subsection is based on the decorrelating 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, March 22, 2004

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under appropriate conditions this receiver exhibits a large complexity reduction compared with that of the decorrelating receiver. Nevertheless, as proven in the previous subsection, the performance of the quasidecorrelator receiver is practically equal to that of the BR, which performs poorly when the number of users in the system is large. This large performance degradation exhibited by the quasi-decorrelator receiver will happen even if the colliding sources are very weak, and as such the matched filter detector, which is nearly optimal, will outperform the quasi-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) we suggest the use of the quasi-MMSE receiver, which is based on the general MMSE receiver [18], and which for detecting the symbol transmitted by the kth user, is given by ˆbk = sgn



   1  T −2 −1 2 T (Sk ) Sk + σn (Ak ) (Sk ) rk . Ak 1

(31)

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 lim



T itations): The quasi-MMSE receiver is approximately given by ˆb1 |MMSE ≈ sgn wMMSE r1 where wMMSE

is an Nf × 1 vector. The jth element of wMMSE , [wMMSE ]j , is equal to the reception of the jth pulse, and

1 2 +A2 σn l

1 2 σn

if no collision occurred during

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 is approximately equal to the BR. This can easily be seen by letting σn2 approach zero, which results in a weighting vector, wMMSE , whose elements equal some finite value when collisions occur at the corresponding time instances, and equal infinity otherwise. This should not come as a surprise because, when the signal to noise ratio is large the MMSE and decorrelator receivers are essentially identical [18]. 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 can be drawn from the above property concerns the behavior of the quasi-MMSE receiver at low signal to noise ratios. At low signal to noise ratio the quasi-MMSE receiver is approximately the matched filter detector. This is due to the fact that all the elements of wMMSE are approximately equal to

1 2 σn

in this case. VI. Quasi-ML Approach

In the previous sections we explored efficient MUD algorithms. These algorithms have complexity that grows polynomially with Kk , 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 O(2Kk ). In practical March 22, 2004

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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 that estimates the transmitted symbol as the symbol that maximizes the a posteriori log-likelihood [18], [25]. The minimum probability of error receiver requires sampling of the received signal at any time instant when a pulse is received, and then minimizing a nonlinear function of all these measurements. The computational complexity of this algorithm is prohibitive; 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, as in the preceding algorithms. 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 [18], the receiver that minimizes the probability of error in detecting, say, the first user from the measurements r1 is given by ˆb1 = arg max log f (b1 |rk ) = arg max b1 ∈±1

b1 ∈±1

X

b∈{−1,1}Kk

||r1 − S1 [b1 b]T ||2 .

(32)

As can be seen this receiver is the well known maximum a posteriori probability (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 rk obeys a linear model in which the mixing matrix has only Kk columns, and 



as such the complexity of the quasi-optimal receiver is O 2Kk . VII. Iterative “turbo” algorithm In this section we present an iterative multiuser 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, [5], [14], [19] 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 multiuser detection and decoding of error correcting codes in CDMA systems [14], [19]. In this problem one can use any multiuser detection algorithm that results in March 22, 2004

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soft decision statistics about every channel symbol. These soft decisions can be fed into any soft decoding algorithm and the result will be the estimated information symbol. 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 in the first stage. Although no actual decoding occur, the term turbo detector is used, since the proposed algorithm is composed of two stages that exchange extrinsic information, similarly to many other turbo algorithms. 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 “symbol detector”. In the first stage we assume that different pulses from the same user correspond to independent information symbols, while in the second stage we exploit the information that all the pulses from the same user correspond to the same information symbol. Let us denote by bkj the information symbol transmitted at the jth pulse from the kth user. Note that although we know a priori that bk1 = · · · = bkNf 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 bkj

given the received signal, the information about the transmitted bits from other users, and the a priori information about bkj obtained from the symbol detector. It will be shown below that at the nth iteration the a posteriori LLR of bkj has the following form: Pr(bkj = 1|r)

△

Ln1 (bkj ) = log

Pr



bkj

n−1 k k  = λn (bj ) 1 (bj ) + λ2

= −1|r

(33)

where λ2n−1 (bkj ) represents the a priori LLR of bkj , which is computed by the symbol detector at the (n−1)th iteration. The extrinsic information λn1 (bkj ), which is, at least at the first iteration, independent of λ2n−1 (bkj ), is then fed back to the symbol detector. The symbol detector exploits the fact that bk1 = · · · = bkNf . As such, the symbol detector computes the a

posteriori LLR of bkj given the information from the various pulse detectors. In the sequel it will be shown that this LLR has the following general structure, △

Ln2 (bkj ) = log

March 22, 2004



Pr bkj = 1|λn1 (bkj ) , j = 1, . . . , Nf Pr



bkj

=

−1|λn1 (bkj ) , j



= 1, . . . , Nf

k n k  = λn 2 (bj ) + λ1 (bj ).

(34)

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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 bkj . 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 multiuser detector is the pulse detector. The pulse detector computes the a posterior LLR of the transmitted symbol modulating the jth pulse of the kth user for every j and k. Denote by L1 (bkj ) this ratio, L1 (bkj ) = log



L1 (bkj ) = log







f r|bkj = −1

 + log

.

(35)

Pr bkj = −1|r

By using Bayes’ formula, L1 (bkj ) can be written as f r|bkj = 1





Pr bkj = 1|r

Pr(bkj = 1) Pr(bkj = −1)

= log





f rl(j,k) |bkj = 1 

f rl(j,k) |bkj = −1

 + log

Pr(bkj = 1) Pr(bkj = −1)

(36)

where l(j, 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 symbol. f (rl(j,k) |bkj =1) . Similarly to the definition of Kk , we denote by We now turn to the computation of log f r ( l(j,k) |bkj =−1) Kkj the number of users colliding with the jth pulse of the kth user. It can easily be seen that Kkj is the weight of the l(j, k)th row of S minus one. In addition denote by fjk (g) , g = 1, . . . , Kkj the indices of the users colliding with the jth pulse of the kth user. Following (3), rl(j,m) obeys the following model, rl(j,m) = 1K j +1 Ajk bkj + nl(j,m)

(37)

k

where 1K j is the 1 × Kkj all one vector, Ajk = diag[Ak Af k (1) · · · Af k (K j ) ] is the diagonal matrix containing j

k

j

k

the amplitudes of the pulses received from the kth user and the users colliding with the jth pulse of the f k (1)

kth user; and bkj = [bkj bj j

f k (Kkj )

· · · bj j

] is the vector containing the information symbols colliding with

the jth pulse of the kth user. Following (37), the a priori LLR is given by,

log

March 22, 2004

f





rl(j,k) |bkj



=1

f rl(j,k) |bkj = −1

 = log

P

j K b∈{±1} k

P

j K k

b∈{±1}

−

e

−

e

j (rl(j,k) −1A [1 b]T )2 k 2 2σn

j (rl(j,k) −1A [−1 b]T )2 k 2 2σn

QKkj



f k (g)

j g=1 p bj

QKkj

g=1 p



f k (g) bj j

= [b]g



= [b]g



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P

= log

j K k

b∈{±1}

P

j K k

b∈{±1}

where p(b) =

QKkj



f k (g) bj j

g=1 p



fjk (g)

Following [5], p bj

−

e

j (rl(j,m) −1A [1 b]T )2 k 2 2σn

j (rl(j,k) −1A [−1 b]T )2 k − 2 2σn

e

= [b]g

= [b]g





18

p(b) (38) p(b)

is the a priori probability that



fjk (g)

f k (1) f k (K j ) bj j · · · b j j k

is the prior information about bj



= b is equal to [b]g .

from other sources. Simple algebraic

manipulations lead to the following: p



f k (g) bj j

= [b]g









1 n−1 fjk (g) 1 λ 1 + [b]g tanh bj = 2 2 2



.

(39)

combining (38) and (39) results in the following

log



f rl(j,k) |bkj = 1 

P



f rl(j,k) |bkj = −1

 =

−

e

j K k

b∈{±1}

P

Ki b∈{±1} k

−

e

j (rl(j,k) −1A [1 b]T )2 k 2 2σn

(rl(i,k) −1Ai [−1 b]T )2 k 2 2σn

QKkj

g=1 1

+ [b]g tanh

QKkj



g=1 1 + [b]g tanh

1 n−1 2 λ2





1 n−1 2 λ2

f k (g) bj j





f k (g)

bj j

△

 

j k  = λ1 bi (40)

On combining (40) with (36) we see that the a posteriori LLR is given by (33), that is, the sum of prior information obtained from the symbol detector and the extrinsic information. The complexity of 

j



computing (40) is O 2Kk and is thus very low. For example, if the number of users is equal to Nc then

the probability that Kkj will be greater than one is 1 − (1 − 1/Nc )Nc − (1 − 1/Nc )Nc −1 ≈ 0.25. Thus the probability of Kkj being large is very small.

A.3 The symbol detector While the pulse detector computes the a posteriori LLR of the transmitted symbol corresponding to the jth pulse from the kth user, the symbol detector will compute the a posteriori LLR of the transmitted information symbol given the a priori LLR from the pulse detectors. The symbol detector computes the following ratio for every j = 1, . . . , Nf and k = 1, . . . , K: Ln2 (bkj ) = log

 



Pr bkj = 1|λn1 bkj j = 1, . . . , Nf ; k = 1, ..., K 

 

Pr bkj = −1|λn1 bkj j = 1, . . . , Nf ; k = 1, ..., K

We first note that bkj is independent of bm l for m 6= k, and thus (41) reduces to Ln2 (bkj ) = log

March 22, 2004





 

Pr bkj = 1|λn1 bkj j = 1, . . . , Nf 

 



Pr bkj = −1|λn1 bkj j = 1, . . . , Nf

.

.

(41)

(42)

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19

Now since bk1 = · · · = bkNf , Eq. (42) reduces to Ln2 (bkj ) = log

=

Nf X

log

l=1

=

Nf X





 



Pr bkj = −1|λn1 bkl 

 



Pr bkj = −1|λn1 bkj j = 1, . . . , Nf

Pr bkl = 1|λn1 bkj



 


 =

Nf X

log

l=1,l6=j

|

 = log



 



Pr bk1 = · · · bkNf = 1|λn1 bkj j = 1, . . . , Nf 

 

{z

k λn 2 (bj )

}



Pr bk1 = · · · bkNf = −1|λn1 bkj j = 1, . . . , Nf  

Pr bkl = 1|λn1 bkl

  n k

+λ 1 bj k

Pr bkl = −1|λn1 bl {z

k λn 2 ( bj )



}

λn1 bkl ) +λn1 bkj .

l=1,l6=j

|

 



Pr bkj = 1|λn1 bkj j = 1, . . . , Nf

(43)

As can be seen from (43), the output of the symbol detector is the sum of the extrinsic information from the pulse detector about bkj , plus the information obtained from our knowledge that all the pulses  

from the same user bear the same information symbol. λn2 bkj is the extrinsic information about bkj used  

by the pulse detector as a priori information. As can easily be seen at the first iteration, the λn2 bkj 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 signal-to-noise-ratio (SNR), the number of users in the system, and the ratios among the powers of the various users. We consider a TH-IR system with Nf = 10 and Nc = 20. In the first set of simulations we assume that all the sources have equal received power. Figures 1 and 2 depict the average bit error rates of the BR, the quasi-decorrelator, the quasi-MMSE, and the iterative algorithm as functions of the received SNR per pulse, for 10 and 20 users, respectively. In addition, curves depicting the performance of the BR predicted by the theoretical analysis (12), the theortical performance of an uncoded single-user system employing the matched filter detector, that is Q

p



2Eb /N0 , and the 



empirical performance of the matched filter detector. Note that the matched filter is given by sgn [S]Tk: r , where [S]k: is the kth column of S. These cases are referred to as “Theoretical BR”, “Single User”, and “MF Detector” respectively. As can be seen from the figures, the BR and quasi-decorrelator have essentially the same performance. Moreover the theoretical performance of the BR, (12), and the empirical performance curves match very well. Note that these two receivers 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 a performance loss of about 1.5 dB compared to the performance of a single March 22, 2004

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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. In the next set of simulations, we present simulation results for the case of strong interfering signals. Figures 3 and 4 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 and 20, 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 receiver 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 interfering signals the quasi-MMSE receiver 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. IX. Summary and Concluding Remarks In this paper we have shown that Time Hopping Impulse Radio systems are very similar to RCDMA systems from the viewpoint of their multiuser properties. This similarity enables the use of many multiuser 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 the optimal multiuser detector, and whose complexity is very low. The results of this paper can be extended to more general IR systems. In particular, recall that the model of the transmitted signal used here assumes that all the pulses corresponding to the same information symbol are transmitted with the same sign. As noted in [4], by assigning to each user a different and known binary sequence that modulates the transmitted pulses, performance improvement occurs. Let us denote by {skj } the sequence assigned to the kth user. Upon changing the definition of S[i] from (4) to the following (S[i])lk =

   sk

(i−1)Nf +

 

0



l Nc



Nc

if ck

(i−1)Nf +

otherwise



l Nc



Nc

=l−

j

l Nc

k

Nc

.

(44)

We see that all the algorithms described in this paper can be used for MUD in such systems as well. March 22, 2004

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The study of TH-IR systems is closely related to the study of UWB communication systems. It was UWB communication systems that led to the development of the concept of TH-IR systems. When we communicate over UWB channels, the challenges might be quite different than the ones presented in this paper. The large channel delay spreads present one fundamental challenge, and the synchronization problem presents another. It is impossible to implement the methods proposed in this paper without solving these problems first. Nevertheless, in [3] some solutions to these problems are presented. As was already mentioned in Section III, the use of TH-IR modulation is not limited to UWB channels, but can be used over more conventional channels as well. It is over these more conventional channel that our proposed methods are most efficient. 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 of TH-IR systems. A major question in the future will be: for fixed system processing gain is it better to transmit a large number of pulse per bit and as a consequence to increase the number of collisions between the various pulses, or the other way around? The answer to this question is quite complicated, and, as far as we know, is still open.

Acknowledgment The authors wish to thank the associate editor Prof. H. Boelcskei and the anonymous reviewers for their constructive comments. References [1]

P. Baldi, L. De Nardis, and M.-G. Di Benedetto. Modeling and Optimization of UWB Communication Networks Through a Flexible Cost Function. IEEE Journal on Selected Areas in Communications, 20(9):1733–1744, December 2002.

[2]

I. Bergel, E. Fishler, and H. Messer. Narrow Band Interference Supression in Time Hopping Impulse Radio Systems. In Proceedings of the IEEE Conference on Ultra Wideband Systems and Technology (UWBST02), pages 303–308, Baltimore, MD, May 2002.

[3]

E. Fishler and H. V. Poor. Iterative (Turbo) Multiuser Detectors for Impulse Radio Systems Transmitting over Multipath Channels. submitted IEEE Trans. on Commun.

[4]

E. Fishler and H. V. Poor. On the Tradeoff Between Two Types of Processing Gain. In Proceedings of the 40th Annual Allerton Conference on Communication, Control, and Computing, Allerton Park, IL, October, 2002.

[5]

J. Hagenauer. The Turbo Principle: Tutorial Introduction and State of The Art. In Proceeding of the International Symposium on Turbo Codes and Related Topics, pages 1–11, Brest, France, Sept. 1997.

[6]

D. C. Laney, G. M. Maggio, F. Lehmann, and L. Larson. Multiple Access for UWB Impulse Radio with Pseudochaotic Time Hopping. IEEE Journal on Selected Areas in Communications, 20(9):1692–1700, December 2002.

[7]

C. J. Le-Martret and G. B. Giannakis. All-Digital Impulse Radio with Multiuser Detection for Wireless Cellular Systems. IEEE Trans. Commun., 50(9):1440–1450, September 2002.

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[8]

22

C. J. Le-Martret and G. B. Giannakis. All-Digital PPM Impulse Radio for Multiple-Access Through Frequency-Selective Multipath. In Proceedings of the 2000 IEEE Sensor Array and Multichannel Signal Processing Workshop (SAM2000), pages 22–26, Cambridge, MA, March 2000.

[9]

C. J. Le-Martret and G. B. Giannakis. All-Digital PAM Impulse Radio for Multiple-Access Through Frequency-Selective Multipath. In Proceedings of the 2000 IEEE Global Telecommunications Conference (GLOBECOM2000), volume 1, pages 77–81, San Francisco, CA, Nov. 2000.

[10] Q. Li and L. A. Rusch. Multiuser Receivers for DS-CDMA UWB. In Proceddings of the IEEE Conference on Ultra Wideband Systems and Technology, (UWBST02), pages 163–167, Baltimore, MD, May 2002. [11] U. Madhow and M. L. Honig. On the Average Near-Far Resistance for MMSE Detection of Direct Sequence CDMA Signals with Random Spreading. IEEE Trans. Inform. Theory, IT-45(6):2039–2045, September 1999. [12] F. Ramirez Mireles. On the Performance of Ultra-Wideband Signals in Gaussian Noise and Dense Multipath. IEEE Trans. Vehicular Technology, 50(1):244–249, January 2001. [13] L. B. Nelson and H. V. Poor. Performance of Multiuser Detection for Optical CDMA - Part I: Error Probabilities. IEEE Trans. Commun., COM-43(11):2803–2811, November 1995. [14] H. V. Poor. Turbo Multiuser Detection: A Primer. Journal of Communications and Networks, 3(3), Sept. 2001. [15] B. Sadler and A. Swami. On the Performance of UWB and DS-Spread Spectrum Communication Systems. In Proceedings of the IEEE Conference on Ultra Wideband Systems and Technologies (UWBST02), pages 289–292, Baltimore, MD, May 2002. [16] R. A. Scholtz. Multiple Access with Time-Hopping Impulse Modulation. In Proceedings of the IEEE Military Communications Conference, (MILCOM 93), volume 2, pages 447–450, Boston, MA, Oct. 1993. [17] V. S. Somayazulu. Multiple Access Performance in UWB Systems Using Time Hopping vs. Direct Sequence Spreading. In Proceedings of the IEEE Wireless Communications and Networking Conference, 2002, (WCNC2002), volume 2, pages 522–525, Orlando, FL, March 2002. [18] S. Verdu. Multiuser Detection. Cambridge University Press, Cambridge, UK, 1998. [19] X. Wang and H. V. Poor. Iterative (Turbo) Soft Interference Cancellation and Decoding for Coded CDMA. IEEE Trans. Commun., COM-47(7):1046–1061, July 1999. [20] M. Z. Win and R. A. Scholtz. Impulse Radio: How It Works. IEEE Communications Letters, 2(2):36–38, Feb. 1998. [21] M. Z. Win and R. A. Scholtz. Ultra-Wide Bandwidth Time-Hopping Spread-Spectrum Impulse Radio for wireless multiple acess Communications. IEEE Trans. Commun., COM-48(4):679–689, April 2000. [22] P. Withington and L. W. Fullerton. An Impulse Radio Communication System. In Proceedings of the International Conferrence on Ultra-Wide band, Short-Pulse Electromagnetics, pages 113–120, New York, NY, Oct. 1992. [23] L. Yang and G. G. Giannakis. Space-Time Coding for Impulse Radio. In Proceedings of the IEEE Conference on Ultra Wideband Systems and Technology, (UWBST02), pages 235–239, Baltimore, MD, May 2002. [24] L. Yang and G. G. Giannakis. Block-Spreading Codes for Impulse Radio Multiple Access Through ISI Channels. In Proceedings of the 2002 IEEE International Conference on Communication, (ICC2002), pages 807–8011, New York, NY, April 2002. [25] Y. C. Yoon and R Kohno. Optimum Multi-User Detection in Ultra-Wideband (UWB) Multiple-Access Communication Systems. In Proceedings of the IEEE International Cnferrence on Communication (ICC2002), pages 812–816, New York, NY, April 2002.

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[26] L. Zhao and A. M. Haimovich. Performance of Ultra-Wideband Communications in The Presence of Interference. IEEE Journal on Selected Areas in Communications, 20(9):1684–1691, Dec. 2002. [27] L. Zhao, A. M. Haimovich, and H. Grebel. Performance of Ultra-Wideband Communications in The Presence of Interference. In Proceedings of the 2001 IEEE International Conference on Communication (ICC2001), pages 2948–52, St. Petersburg, Russia, June 2001.

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0

10

−1

Pe

10

−2

10

−3

10

Single User MF Detector BR Theoretical BR q−Decorrelator q−MMSE Turbo 1st Turbo 2nd Turbo 3rd

−4

10

−6

−5

−4

−3 −2 SNR per pulse [dB]

−1

0

1

Fig. 1. The average probability of error for a system with Nc = 20, Nf = 10, K = 10, as a function of the SNR, for equi-power signals.

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0

10

−1

Pe

10

−2

10

−3

10

Single User MF Detector BR Theoretical BR q−Decorrelator q−MMSE Turbo 1st Turbo 2nd Turbo 3rd

−4

10

−6

−5

−4

−3 −2 SNR per pulse [dB]

−1

0

1

Fig. 2. The average probability of error for a system with Nc = 20, Nf = 10, K = 20, as a function of the SNR, equi-power signals.

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0

10

−1

Pe

10

−2

10

−3

10

Single User MF Detector BR Theoretical BR q−Decorrelator q−MMSE Turbo 1st Turbo 2nd Turbo 3rd

−4

10

−6

−5

−4

−3 −2 SNR per pulse [dB]

−1

0

1

Fig. 3. The average probability of error for a system with Nc = 20, Nf = 10, K = 10, as a function of the SNR, strong interferers.

March 22, 2004

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0

10

−1

Pe

10

−2

10

−3

10

Single User MF Detector BR Theoretical BR q−Decorrelator q−MMSE Turbo 1st Turbo 2nd Turbo 3rd

−4

10

−6

−5

−4

−3 −2 SNR per pulse [dB]

−1

0

1

Fig. 4. The average probability of error for a system with Nc = 20, Nf = 10, K = 20, as a function of the SNR, strong interferers.

March 22, 2004

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