Simplified Multiaccess Interference Reduction for MC-CDMA With Carrier Frequency Offsets

Simplified Multiaccess Interference Reduction for

MC-CDMA With Carrier Frequency Offsets

Layla Tadjpour, Shang-Ho Tsai, and C.-C. Jay Kuo, Fellow, IEEE

Abstract—Multicarrier code-division multiple-access

(MC-CDMA) system performance can severely be degraded by multiac-cess interference (MAI) due to the carrier frequency offset (CFO). We argue that MAI can more easily be reduced by employing complex carrier interferometry (CI) codes. We consider the sce-nario with spread gain N , multipath length L, and N users, i.e., a fully loaded system. It is proved that, when CI codes are used, each user only needs to combat 2(L− 1) (rather than N − 1) interferers, even in the presence of CFO. It is shown that this property of MC-CDMA with CI codes in a CFO channel can be exploited to simplify three multiuser detectors, namely, parallel interference cancellation (PIC), maximum-likelihood, and decor-relating multiuser detectors. The bit-error probability (BEP) for MC-CDMA with binary phase-shift keying (BPSK) modulation and single-stage PIC and an upper bound for the minimum error probability are derived. Finally, simulation results are given to corroborate theoretical results.

Index Terms—Carrier frequency offset, complexity reduction,

decorrelating detection, interferometry codes, maximum likelihood detection, multiaccess interference, multicarrier code-division multiple-access (MC-CDMA), multiuser detection, orthogonal codes, parallel interference cancellation.

I. INTRODUCTION

M

ULTICARRIER code-division multiple access (MC-CDMA) has emerged as a promising multiaccess tech-nique for high-data-rate communications [2], [10]. MC-CDMA is inherently more robust to intersymbol interference than a conventional CDMA system due to the use of the orthogonal frequency-division multiplexing (OFDM) structure. However, the multipath and/or the carrier frequency offset (CFO) effects tend to destroy orthogonality among users and lead to multiac-cess interference (MAI). Thus, the performance of MC-CDMA can greatly degrade.

There has been research on MAI suppression using single-user detection (SUD) techniques. For example, the structural

Manuscript received December 10, 2008; revised August 2, 2009; accepted January 8, 2010. Date of publication February 8, 2010; date of current version June 16, 2010. This work was supported in part by the Integrated Media Systems Center, a National Science Foundation Engineering Research Center, under Cooperative Agreement EEG-9529152, and in part by the National Science Council, Taiwan, under Cooperative Agreement 97-2221-E-009-071-MY2. The review of this paper was coordinated by Prof. J. Lie.

L. Tadjpour is with Information System Laboratories, Vienna, VA 22182 USA (e-mail: ltadjpour@isl-inc.com).

S.-H. Tsai is with the Department of Electrical Engineering, National Chiao Tung University, Hsinchu 300, Taiwan (e-mail: shanghot@mail.nctu.edu.tw).

C.-C. J. Kuo is with the Department of Electrical Engineering and Integrated Media Systems Center, University of Southern California, Los Angeles, CA 90089-2564 USA (e-mail: cckuo@sipi.usc.edu).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2010.2042473

differences of interfering users caused by CFO were exploited at the receiver to suppress MAI in [12]. However, this MAI-suppression technique imposes a computational burden on the receiver since a discrete Fourier transform (DFT) of size larger than N is required due to the oversampling of the received signal in the frequency domain. Another way to reduce MAI is achieved by code design while keeping the structure of MC-CDMA unchanged [19]. In [19], a code-design method based on real Hadamard–Walsh (HW) codes was proposed and shown to achieve zero MAI in a multipath environment in MC-CDMA. However, not all users can enjoy MAI-free communications with this design. In addition, suppression of the MAI due to the CFO effect was not considered in this paper.

Multiuser-detection (MUD) techniques have been developed to mitigate MAI. However, the complexity of MUD techniques is generally high. Much effort has been made to reduce the complexity of the multiuser detectors. Cai et al. [4] proposed to assign a set of subcarriers to a group of users while preserving the frequency diversity of MC-CDMA as much as possible. A new maximum-likelihood (ML) MUD scheme called sphere decoding was proposed for MC-CDMA whose complexity is a polynomial function of the user number [3]. However, when the user number is large, the sphere-decoding ML algorithm is cumbersome to perform. Moreover, neither of these techniques are shown to be effective in the presence of CFO.

In this paper, we show how to suppress MAI with simplified techniques for MC-CDMA in CFO environments using carrier-interferometry (CI) codes. CI codewords were introduced to MC-CDMA in [17], which showed that two sets of orthogonal CI codewords can increase user capacity from N to 2N in MC-CDMA with negligible performance degradation in a multipath fading channel. CI codes were also used as training sequences for channel estimation to decouple the interantenna interference in a CFO-free MIMO-OFDM system [14].

Here, we first show how to completely eliminate MAI by employing CI codes in MC-CDMA with CFO. That is, for CI codes, when the number of active users is less than or equal to

N/G, where G is a power of 2 with G≥ L, N is the spread

gain, and L is the multipath length, we demonstrate that a proper choice of CI codewords allows MC-CDMA systems to enjoy MAI-free communication.

Second, for a fully loaded MC-CDMA system with CI codes (CI-MC-CDMA), we prove that for L≤ N/2, each user receives interference from 2(L− 1) users only (instead of other N − 1 active users) in CFO and multipath-fading environments.

Third, we demonstrate that by exploiting the sparsity of the cross-correlation matrix of CI-MC-CDMA, we can lower

Fig. 1. Block diagram of the uplink transmission of the ith user in an MC-CDMA system.

the complexity of three MUD techniques, namely, parallel interference cancellation (PIC), optimum ML detection, and decorrelating detection, as explained in the succeeding discus-sion. The complexity for fully loaded PIC (i.e., N active users) is linearly proportional to the channel multipath length L rather than the user number N . Since N is, in general, much larger than L, the complexity is substantially reduced. We also show that it is possible to lower the complexity of ML detectors such that its complexity exponentially grows with L instead of N . Moreover, we explain that, for MC-CDMA with N− L + 1 users, the cross-correlation matrix can be transformed to a band matrix, which, in turn, can be used to reduce the com-plexity of the decorrelating detector to O(2(N− L + 1)(L − 1)2), which is far less than the O((N− L + 1)3/3) of the

decorrelating detector for a general dense matrix of similar dimension.

Finally, we analyze the bit-error-probability (BEP) perfor-mance of fully loaded MC-CDMA with single-stage PIC using CI codes. Although CI-MC-CDMA with PIC has previously been considered [21], a detailed BEP analysis of PIC in the presence of CFO has yet to be studied. Moreover, we derive an upper bound on the minimum error probability for an MC-CDMA system in a CFO environment.

The rest of this paper is organized as follows: The MC-CDMA system model is reviewed, and the CFO-induced MAI is derived in Section II. CI codeword schemes are presented, and their MAI-free properties are proved in Section III. The single-stage PIC is described, and its BEP is analyzed in Section IV. We describe the ML detector for MC-CDMA with and without CFO in Section V. In addition, the sparsity of the cross-correlation matrix of CI codes, along with its associated tail-biting trellis (TBT), is studied in this section. In addition, an upper bound on the minimum error probability is derived for MC-CDMA in a CFO environment. In Section VI, we discuss the reduced-complexity decorrelating detector by using a proper subset of CI codes, resulting in a band cross-correlation matrix. Finally, simulation results are shown in Section VII.

II. SYSTEMMODEL

Consider K users in an MC-CDMA system. The block diagram of the uplink transmission of the ith user is shown in Fig. 1. As shown in the figure, symbol xiis spread by an N× 1

codeword in the frequency domain to yield vector

yi[k] = wi[k]xi, 0≤ k ≤ N − 1 (1)

where wi[k] is the kth component of the ith orthogonal code.

The kth component of the DFT output ˆy can be expressed by

ˆ

y[k] =

K−1_ j=0

rj[k] + n[k] (2)

where n[k] is the DFT of additive noise, and rj[k] is the

received signal contributed by the jth user due to the channel fading and CFO effects. We suppose that user j has a normal-ized CFO j, i.e., the actual CFO normalized by 1/N of the

overall bandwidth, and −0.5 ≤ j ≤ 0.5. rj[k] can be written

as [15], [19] rj[k] = αjλj[k]yj[k] + βj N_−1 m=0, m=k {λj[m]yj[m]gj[m− k]} (3) where λj[m] is the mth component of the N -point DFT of the

channel impulse response of user j, i.e.,

αj = sin πj N sinπj N ejπjNN−1 βj = sin (πj)ejπj N−1 N gj[m− k] = e−jπm−kN N sinπ(m−k+j) N .

It is apparent that when there is no CFO, i.e., j= 0, rj[k] =

λj[k]yj[k]. However, if CFO exists, there are two terms as there

are in (3). The first term is λj[k]yj[k] distorted by αj, and

the second term is the intercarrier interference (ICI) caused by CFO. Since βjgj(0) = αj, and by (1) and (3), we have

ˆ

y = Cx + n (4)

where the element in the ith row and the jth column of C is

C(i, j) = βj N−1_ m=0 gj(m− i)λj[m]wj[m] (5) x = (x0, x1, . . . , xK−1)T ˆ

y = (ˆy[0], ˆy[1], . . . , ˆy[N− 1])T (6)

n = (n[0], n[1], . . . , n[N− 1])T (7) which is a circularly symmetric complex Gaussian random vector with zero mean and covariance matrix σ2_{I. Maximum}

ratio combining (MRC) is usually used to detect the symbol transmitted by the ith user as

ˆ zi= N_−1 k=0 ˆ y[k]λ∗_i[k]w∗_i[k] = si+ K−1_ j=0, j=i MAIi←j + N_−1 k=0 n[k]λ∗_i[k]w∗_i[k] (8) where siconsists of the distorted chip, and ICI caused by CFO

for the desired user and can be written as

si= N_−1 k=0 ri[k]λ∗i[k]w∗i[k] = αi N_−1 k=0 |λi[k]|2+ βi N_−1 m=0, m=k · {λi[m]wi[m]λ∗i[k]w∗i[k]gi[m− k]} (9) MAIi←j = N−1_ k=0 rj[k]λ∗i[k]wi∗[k]. (10)

MAI_i←j is the MAI of user i due to the jth user’s CFO and multipath fading channels. With (3) and (10), we can show that the MAI term is given by

MAIi←j= βjxj N_−1 k=0 N_−1 m=0 gj(m−k) · {λj[m]wj[m]λ∗i[k]wi∗[k]} (11) where we used the fact that βjgj(0) = αj. Equation (10) can be

expressed in matrix form as [19] MAIi←j = βjxj N_−1 p=0 gj(−p) · h(p)_i _† F†₀  W(p)_i _† WjF0hj  (12) where W(p)_i = diag (wi[p]· · · wi[N− 1]wi[0]· · · wi[p− 1])T F0= F  IL 0  N×L h(p)_i =  hi(0)e−j 2π0p N · · · h i(L− 1)e−j 2π(L−1)p N T . (13)

IL is an L× L identity matrix, and F is the N × N DFT

matrix whose element at the kth row and the nth column is [F]k,n= (1/

√

N )e−j(2π/N)kn. In addition, † in (12) denotes the matrix Hermitian operation.

III. ORTHOGONALCODES FORMULTIACCESS

INTERFERENCE-FREEMULTICARRIERCODE-DIVISION

MULTIPLEACCESSWITHCARRIERFREQUENCYOFFSET A. Requirements on MAI-Free Codes

Theoretical requirements on codes to produce an MAI-free MC-CDMA system in the presence of CFO are implied by (12).

That is, to have zero MAI in a frequency-selective channel with CFO, we demand MAIi←j = 0. Define

R(p)_i,j =  W_i(p) _† Wj, D(p)ij = F†R (p) i,jF.

It is well known that D(p)_ij is a circulant matrix [9]. Therefore, its first column, i.e., (di,j[0]· · · di,j[N− 1])T, is the N -point

inverse DFT (IDFT) of r(p)_i,j, where r(p)_i,j[k] = w_i(p)[k]w_j∗[k], and w_i(p)[k] = wi[((N− p + k))N], for k = 0, 1, . . . , N− 1,

where ((n))N denotes n modulo N . It was shown in [19] that

condition MAIi←j = 0 is equivalent to



di,j[n] = 0, 0≤ n ≤ L − 1

di,j[N− n] = 0, 1 ≤ n ≤ L − 1. (14)

B. CI Orthogonal Codes

In this section, we study the CI codes of size N , which is of the following form:

wi[k] = ej

2π

Nki_{, k, i = 0, 1, . . . , N}− 1. ₍₁₅₎ Then, the MAI-free property of this code can be stated as follows.

Theorem 1: Let the channel length be L, and let G = 2q_≥

L. There exist N/G CI codewords such that the corresponding

MC-CDMA is MAI free in a CFO environment.

Proof: Consider two codewords with indices i and i. We would like to show that the condition in (14) is met. By taking the IDFT of r(p)_i,j, we have

r_i,i(p)(n) = 1 √ N N−1 m=0 r(p)_i,i[m]ej 2π Nmn. (16) Note that we use (.) in the time domain and [.] in the fre-quency domain. Let m = k + gN/G, 0≤ k ≤ N/G − 1 and 0≤ g ≤ G − 1. We can rewrite (16) as r_i,i(p)(n) = 1 √ N N/G_−1 k=0 G−1_ g=0 · r_i,i(p)[k + gN/G]ej 2π N(k+gN/G)n . (17) Since w(p)_i [k] = ej(2π/N )(N−p+k)i _{and r}(p)

i,i[k + gN/G] = w_i(p)[((k + gN/G))N]wi∗[((k + gN/G))N], we have r_i,i(p)[k + gN/G] = ej 2π N(N−p+k+gN/G)ie−j 2π N(k+gN/G)i = ej2πN(N−p+k)i_e−j2πNki_ej2πGg(i−i)_. If i− i= mG, where m can be any nonzero integer,

ej(2π/G)g(i−i)_{= e}j2πmg_{= 1. Then, we have}

r_i,i(p)[k + gN/G] = ej 2π

N(N−p+k)i_e−j2πNki _{= r}(p)

i,i[k]. (18)

Using (18), we can rewrite (17) as

r(p)_i,i(n) = 1 √ N N/G_−1 k=0 r_i,i(p)[k]ej 2π Nkn G−1_ g=0 ej2πGgn. (19)

Since G−1_ g=0 ej2πGgn=  G, n = 0,±G, . . . 0, otherwise (20) r_i,i(p)(n) is equal to  G √ N N/G−1 k=0 r (p) i,i[k]ej 2π Nkn, n = 0,±G, . . . 0, otherwise.

Furthermore, for i= i, we have

r(p)_i,i(0) = 1 √ N N_−1 k=0 r(p)_i,i[k] = N_−1 k=0 ej2π(iN−i)(N−p+k)= 0. (21) Thus, (14) holds, and MAIi←j = 0. Since there are N/G codewords such that i− i= mG, m = 1, 2, . . ., the total num-ber of MAI-free codewords from N CI codes is N/G.  The MAI-free property of CI codes in a CFO environment can be expected since a multiuser system with the CI spreading codes in the frequency domain is equivalent to a time-division multiple-access system in the time domain.

Theorem 2: For two CI codewords with indices i and i, i,

i= 0, 1, . . . , N− 1, we have MAI_i←i = 0 in a CFO environ-ment if ((|i − i|))N−(L−1)≥ L, where L is the channel length.

Proof: For p = 0, 1, . . . , N− 1, we have r_i,i(p)(n) = 1 √ N N−1 k=0 ej2πN(N−p+k)i· e−j2πNikej 2π Nkn =√1 N N−1 k=0 ej2πN(i−i+n)ke−j 2π Nip. Therefore, r(p)_i,i(n) can be written as

 √

N e−j2πNip, ((i− i+ n))

N = 0

0, otherwise. (22)

To meet the condition ((|i − i|))N−(L−1)≥ L, |i − i|

should take values from {L, L + 1, . . . , N − L}. Consider the case where i− i≥ 0. Under the conditions where i − i is equal to a value in this set, r(p)_i,i(n)= 0, for n = N −

L, N− L − 1, . . . , N by the preceding equation. On the other

hand, if i− i≤ 0, r_i,i(p)(n)= 0, for n = N + L, N + L + 1, . . . , 2N− L. Since r_i,i(p)(n + N ) =√1 N N_−1 k=0 ej2πN(N−p+k)i· e−j2πNik_ej2πNk(n+N ) =√1 N N−1_ k=0 ej2πN(N−p+k)i· e−j2πNik_ej2πNk(n) = r_i,i(p)(n)

r_i,i(p)(n) is periodic with period N . We conclude that ri,i(p)(n)= 0 for n = L, L + 1, . . . , N− L, whereas r(p)_i,i(0) = r(p)_i,i(1) =

· · · = r(p)

i,i(L− 1) = 0, and r(p)i,i(N− L + 1) = r(p)i,i(N− L +

2) =· · · = r(p)_i,i(N− 1) = 0, for ((|i − i|))N−(L−1)≥ L. In

other words, (14) is satisfied, and MAIi←i = 0. 

Corollary: For an MC-CDMA system in a CFO

environ-ment consisting of N active users with CI codes, if L≤ N/2, each user has 2(L− 1) interfering users only.

Proof: The condition L≤ N/2 guarantees that there are i

and isatisfying ((|i − i|))N−(L−1)≥ L. For every codeword

with index i, there are L− 1 codewords such as codeword

i for which i− i≥ 0 and ((i − i))N−(L−1)≥ L, and L − 1

codewords such as codeword i for which i− i< 0 and

((i− i))_N−(L−1)≥ L. Therefore, the number of interferers is

2(L− 1). 

IV. REDUCED-COMPLEXITYPARALLELINTERFERENCE

CANCELLATION FORCARRIERINTERFEROMETRY

MULTICARRIERCODE-DIVISIONMULTIPLEACCESS

The receiver for the ith user in an MC-CDMA system with PIC is depicted in Fig. 2. It is assumed that the exact knowledge of channel gains and CFO values of all users is available in the receiver. First, initial bit estimates for all users are derived from the SUD receivers, which is basically the same as that depicted in Fig. 1. We call this stage as stage 0 of the PIC detector and denote detected symbols by ˆx0i, i = 0, . . . , K− 1. Let F0hj=

λj and F0h(p)i = λ

(p)

i . From (3)–(9) and (12), we can express

the detected symbol for user i at the zeroth stage of PIC as ˆ x0i= N−1_ k=0 |λi[k]|2xi+ K−1_ j=0, j=i xjγi,j+ ˆni (23) where γi,j = βj N−1_ p=0 gj[−p]  λ(p)_i _† W(p)_i _† Wjλj and ˆni = N−1

k=0 n[k]λ∗i[k]wi∗[k]. Tentative hard decisions are

made on ˆx0j, j = 1, 2, . . . , K, j= i, to produce initial bit estimates, namely, sgn[ {ˆx0j}]. Then, the MAI estimate for the desired user i is generated and subsequently subtracted from its received signal ˆy. The new detected symbol at stage 1 of the

PIC detector is given by ˆ x1i= ˆx0i− K−1_ j=0, j=i sgn [Re{ˆx0j}] βj · _N−1  p=0 gj[−p]  λ(p)_i † W_i(p) † Wjλj+ ˆni . (24) We know from Theorem 2 and its corollary that, if MC-CDMA employs CI codes in fading and CFO environments with multipath length L, every user only has 2(L− 1) interfer-ers. Hence, the PIC complexity with CI codes linearly increases with 2L− 1 instead of N. Since L N, in practice, this implies huge savings in the computational cost by employing the CI codes in association with the PIC detector.

If a correct decision is made on a particular interferer’s bit, the interference from that user to the ith user can completely be cancelled. On the other hand, if an incorrect decision is made,

Fig. 2. MC-CDMA receiver with single-stage PIC for the ith user.

the interference from that user will be enhanced rather than cancelled. By substituting ˆx0ifrom (23) in (24), we obtain

ˆ x1i= xi N_−1 k=0 |λi[k]|2+ K_−1 j=0, j=i γi,j(xj−sgn [ {ˆx0j}])+ˆni. (25)

If xj is a binary phase-shift keying (BPSK) symbol, (xj−

sgn[ {ˆx0j}]) is a three-valued random variable (0, 2, −2) whose magnitude represents whether a tentative decision is correctly made on the jth user’s bit at the previous stage. It can easily be shown that, given λi, ˆni is a circularly

symmet-ric zero-mean Gaussian random variable with variance equal to σ2N−1

k=0 |λi[k]|2. The random vector λi consists of N

correlated Rayleigh random variables. In other words, λi is

a multivariate Gaussian random vector with zero mean and covariance matrix Riwhose elements are given by

Ri(k, k) =E {λi[k]λ∗i[k]} = L−1  l=0 L−1  l=0 E{hi(l)h∗i(l)}· e−j2πNklej 2π Nkl . (26)

We assume a uniform model for the multipath intensity (power-delay) profile. In addition, the typical wide-sense sta-tionary uncorrelated scattering channel model [13] is adopted. Then, we haveE{hi(l)h∗i(l)} = σ2hiδ(l− l

_{), and} Ri(k, k) = L−1  l=0 σ_h2_ie−j2πN(k−k)l =  σ2 hiL, k = k  σ2 hi 1−e−j2π(k−k)L/N 1−e−j2π(k−k)/N , k= k _.

Since γi,j is a linear transform of λj given λi, γi,j|λi is a

circularly symmetric complex Gaussian random variable with zero mean and variance

var[γi,j|λi] =|βj|2 N_−1 p,p=0  W_i(p)λ(p)_i _† · WjRjW†j  W(p_i )λ(p_i )  gj(−p)g∗j(−p) . (27) The probability of error for user i at state 0 of the PIC detector using (23) can be written as

P [sgn [ {ˆx0i}]) = xi] = P ⎡ ⎣ ⎧ ⎨ ⎩ˆni+ K−1  j=0, j=i γi,jxj ⎫ ⎬ ⎭ − N_−1 k=0 |λi[k]|2> 0  . (28)

Due to the presence of xj terms,

K−1

j=0, j=ixjγi,j in (23) is

not Gaussian given λi. However, if it is conditioned on all

pos-sible xj, j= i, we will have a collection of Gaussian random

variables that can be approximated by the Gaussian distribution [22]. Hence, we can see that I0i= Re{ˆni+

_K−1

j=0, j=iγi,jxj}

in (28) is zero-mean Gaussian conditioned on λi. By using

e0i to denote (xi− sgn[Re{ˆx0i}]), the probability of error conditioned on λiis simply given by

P [e0i= 0|λi] = Q ⎛ ⎝ Nk=0−1|λi[k]|2 K−1 j=0, j=iσ2γi,j+ σ 2 n ⎞ ⎠ (29)

where Q(.) is the well-known Q-function, and σ_γ2_i,j =E ( {γi,j})2λi =1 2var[γi,j|λi] σ_n2 =E ( {ˆni})2 = 1 2σ 2 N_−1 k=0 |λi[k]|2.

Then, the BEP can be obtained by averaging P [e0i= 0|λi]

over λias P [e0i= 0] = ∞  0 Q ⎛ ⎝ Nk=0−1|λi[k]|2 K−1 j=0, j=iσγ2i,j+ σ 2 n ⎞ ⎠ P [λi] dλi. (30) This integral can be calculated by the Monte Carlo method [18]. Now, we are ready to derive the BEP of the detected symbol after stage 1 of PIC. From (25), we have

P [ {ˆx1i} > 0|xi=−1] = P ⎡ ⎣ ⎧ ⎨ ⎩nˆi+ K_−1 j=0, j=i γi,je0j ⎫ ⎬ ⎭> N_−1 k=0 |λi[k]|2 ⎤ ⎦ . To get the closed form for BEP is difficult since e0j’s are dependent with given λi. However, following the arguments in

[7] for the BEP derivation of direct-sequence CDMA with PIC, we can assume that e0j’s are actually independent.

A. Derivation of the BEP With a Gaussian Residual Interference Model

Using the aforementioned simplifying assumptions, we can assume that the distribution of the total residual interfer-ence (after cancellation) converges to the Gaussian distribu-tion for a sufficiently large number K of users. Under this assumption, we can derive the conditional BEP by noting (E{e0j {γi,j}|λi})2= 0, since

E{e0j {γi,j}|λi}

= 

k∈{0,2,−2}

{P [e0j= k]E{e0j {γi,j}|λi, e0j= k}} = 2P [e0j= 2]E { {γi,j}|λi}

− 2P [e0j =−2]E { {γi,j}|λi} = 0.

In addition, we can use the fact that

E e20j( {γi,j})2λi = 4P [e0j= 0]E ( {γi,j})2λi E e0j {γi,j}e0j {γi,j}λi ! = 0.

Therefore, the conditional BEP is obtained by

P [e|λi] = ⎛ ⎝ Nk=0−1|λi[k]|2 K−1 j=0, j=i4P [e0j = 0]σ2γi,j+ σ 2 n ⎞ ⎠ (31) and the BEP is given by

P [e] =

∞

 0

P [e|λi]P [λi] dλi. (32)

B. Derivation of the BEP Using a Non-Gaussian Model for Residual Interference

The assumption of Gaussian residual interference is not true when the number of interfering users is not sufficiently large. For example, as proved before, every user encounters only 2(L− 1) nonzero interference terms if the CI codewords are used. Thus, for small values of L, the Gaussian assumption for residual interference is not reasonable. The distribution of residual interference must be derived, and therefore, a new BEP formula can be obtained.

To simplify the derivation, we first derive the BEP formula for CI codewords and a channel of length L = 2, where the number of interferers is equal to 2, and then extend the result to a generalized case. Suppose users i and i are the two interfering users for user i. The detected received symbol for user i is given by ˆ x1i = xi N−1 k=0 |λi[k]|2+ e0iγi,i+ e0iγi,i+ ˆni. (33)

Let e0iγi,i = u1, e0iγi,i = u2, and ˆni= η. Then, the

probability density function (pdf) of u1 conditioned on λi,

which is denoted by fU1(u1|λi), can be obtained by

fU1(u1|λ1) = fU1(u1|e0i = 2λi)P [e0i = 2|λi] + fU1(u1|e0i =−2λi)P [e0i =−2|λi] + fU1(u1|e0i = 0λi)P [e0i = 0|λi] = fU1(2γi,i|λi)P [e0i = 2|λi] + fU1(−2γi,i|λ1)P [e0i =−2|λi] + δ(u1)P [e0i = 0|λi]. (34)

We use N (a, b) to denote the Gaussian distribution, with mean a and variance b. Since fU1(2γi,i|λi) =

fU1(−2γi,i|λi) = N (0, 4σ 2 γi,i), we obtain fU1(u1|λi) = P [e0i = 0|λi]N  0, 4σ2_γ i,i  + (1− P [e0i = 0|λi]) δ(u1) (35) where P [e0i = 0|λi] = Q ⎛ ⎝ Nk=0−1|λi[k]|2 _K−1 j=0, j=iσ2_γ i ,j+ σ 2 n ⎞ ⎠ . Similarly fU2(u2|λi) = P [e0i= 0|λi]N  0, 4σ_γ2_i,i  + (1− P [e0i= 0|λi]) δ(u2). (36) Again, e0i and e0i are assumed to be independent. Thus,

u1and u2are independent, and given the independence among

η and u1 and u2, the pdf of their sum can be obtained by convolving their individual pdf’s. Thus, the conditional BEP

after one stage of PIC detector, with CI codewords and L = 2, is given by

P [e|λi] = P [e0i = 0|λi]P [e0i= 0|λi]

· ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Q ⎛ ⎜ ⎜ ⎝ _N−1 k=0|λi[k]|2 $ 4  σ2 γi,i+ σ 2 γi,i  + σ2 n ⎞ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ + P [e0i = 0|λi] (1− P [e0i= 0|λi]) · ⎧ ⎨ ⎩Q ⎛ ⎝_Nk=0−1|λi[k]|2 4σ2 γi,i+ σ 2 n ⎞ ⎠ ⎫ ⎬ ⎭ + (1− P [e0i = 0|λi]) P [e0i= 0|λ1] · ⎧ ⎨ ⎩Q ⎛ ⎝_Nk=0−1|λi[k]|2 4σ2 γi,i+ σ 2 n ⎞ ⎠ ⎫ ⎬ ⎭ + (1− P [e0i = 0|λi]) (1− P [e0i = 0|λi]) ·  Q &N−1 k=0 |λi[k]|2 σn ' . (37)

Now, we extend the aforementioned results to any num-ber I of interferers and any set of codewords. Letting the set of all interfering users’ indices for user i be ζi=

{ζi[1], ζi[2], . . . , ζi[I]}, then we have

ˆ

x1i = xi N−1

k=0

|λi[k]|2+ e0ζi[1]γi,ζi[1]

+ e0ζi[2]γi,ζi[2]+· · · + e0ζi[I]γi,ζi[I]+ ˆni (38) and the conditional BEP can be obtained as the equation shown at the bottom of the page, and

P [e] =

∞

 0

P [e|λi]P [λi] dλi. (39)

The analysis of the performance of multiple-stage PIC is even more complicated than the performance analysis given in this section. In Section VII, we shall evaluate the performance of multiple-stage PIC by computer simulation.

V. COMPLEXITYREDUCTION INMAXIMUM

LIKELIHOODMULTIUSERDETECTION

We consider the ML-detection technique based on the re-ceived signal given in (4) in this section. Again, it is assumed

that the receiver has the perfect knowledge of channel co-efficients and CFO values. We will separately examine the multipath and CFO effects. We will show in this section how the MAI-free property of CI codes can significantly reduce the complexity of the ML-MUD receiver.

A. ML-MUD in a Multipath Fading Channel

For given transmitted signal x, we would like to maximize the likelihood of the received signal. From (4), and since noise

n is Gaussian, the ML estimate can be written as

ˆ

x = arg min

x ˆy − Cx

2_{. (40)}

By expanding the right-hand side of (40) and noting that

ˆy 2_{is independent of x, we can reformulate the optimization} problem as ˆx = arg minxΩ(x), where

Ω(x) = Cx 2− 2 {Cx, ˆy}

., . is the inner product of two vectors, and C is defined

in (5) in a CFO environment. If there is no CFO, (5) can be written as ( C(k, j) = wj[k]λj[k]. (41) Thus, we have  (Cx, ˆy = x†C(†y =ˆ K−1_ j=0 xj N−1 k=0 ˆ yj[k]w∗j[k]λ∗j[k].

Note thatN_k=0−1yˆj[k]w∗j[k]λ∗j[k] is actually the estimate of

the input signal for user j obtained by MRC (i.e., ˆzj). On the

other hand, (Cx 2_{= x}†_{Hx, where (}( _{H = (C}†_{C. Then, the ML}( optimization problem is equivalent to minimizing

Ω(x) = x†Hx( − 2 (x†, ˆz)!

with respect to x, where ˆz is the output vector of MRC with its

ith element given in (8). We denote the MAI from user j to the

desired user i without CFO by )MAI_i←j. Then, we have from (11) that ) MAIi←j = xj N−1_ k=0 λj[k]wj[k]λ∗i[k]w∗i[k].

It can easily be shown that (H(i, j) = )MAIi←j/xj. In fact, (H

can be viewed as the cross-correlation channel matrix.

P [e|λi] = 1  r1=0 1  r2=0 · · · 1  r_ζi[I]=0 P*e0ζi[1]= 0|λi +r1_{· · · P}* e0ζi[I]= 0|λi +r_ζi[I], 1− P*e0ζi[1]= 0|λi +-1_−r1_{· · ·} × ,1− P*e0ζi[I]= 0|λi +-1_−rζi[I] Q ⎛ ⎜ ⎜ ⎝ N−1 k=0 |λi[k]|2 $ 4  r1σγ2_i,ζi[1]+· · · + rζi[I]σ 2 γ_i,ζi[I]  + σ2 n ⎞ ⎟ ⎟ ⎠

Fig. 3. Cross-correlation matrix of CI codewords with N = K = 16 and L = 2.

B. ML-MUD in a Multipath Fading Channel With CFO

We rewrite (8) in vector format as ˆ

z = Hx + ˆn (42)

where the (i, j)th entry of H is equal to

H(i, j) =

 _N−1

k=0 |λi[k]|2, i = j

MAIi←j

xj , i= j.

The ML detector has the following form: ˆ x = arg min x Ω(x) (43) where Ω(x) = ˆz − Hx 2= K_−1 i=0 |ˆzi− hix|2 (44)

and where hiis the ith row of H.

C. VA for the TBT

We know from Theorem 2 that, for K = N active users with CI codewords and L≤ N/2, each user has only 2(L − 1) (instead of N− 1) interfering users. Both H and (H are sparse

matrices so that ML-MUD can be performed with a much lower complexity. As shown in Fig. 3, for N = 16, K = 16, and

L = 2, the nonzero elements (indicated by black squares) of

H (or (H for the case without CFO) are concentrated along the

three diagonal lines. Elements in the off-diagonal region with

|i − j| ≥ L are all equal to zero, except for two corners.

The well-known Viterbi algorithm (VA) can be used to solve the ML optimization problem. Generally speaking, its complexity is proportional to the number of states. It turns out that, regardless of how we define the states, the complexity of ML-MUD is O(2K) for a general nonsparse cross-correlation channel matrix and BPSK modulation. On the other hand, by exploiting the sparsity of H, we can show that the complexity

of the VA exponentially grows with 2L− 1, as explained in the succeeding discussion.

The structure of H (or (H for the case without CFO) implies a

trellis that is defined on a circulant time axis (or called the TBT [5]). The TBT was defined and discussed for error-correcting codes in [5]. The TBT also arises in the context of ML detection in overloaded array processing [11]. A method for trellis con-struction for a similar matrix structure was proposed in [11], as explained in the following discussion.

We assume that K = N and denote the state of the trellis at stage i and the state space at the ith stage by s[i] and

Si, respectively. We use V [i] to denote the column indices of

nonzero elements on the ith row of H. The ith state is defined as [11]

s[i] ={xu|u ∈ V [((i − 1))N]∩ V [i]} . (45)

Using the preceding definition, we obtain

s[i]∪ s [((i + 1))_N] ={V [((i − 1))_N]∩ V [i]}

∪ {V [i] ∩ V [((i + 1))N]}

= V [i]. (46)

In other words, the state sequence {s[i]} for the TBT is defined such that, during the ith stage of the VA recursion, V [i] corresponds to symbol indices in both s[i] and s[i + 1]. From the sequence of states defined by (45), we can construct the trellis by listing state values at stage i and connect the valid state transition from stage i to stage i + 1. Fig. 4 shows the trellis for the example in Fig. 3. Some approximate ML algorithms with less complexity and satisfactory results for decoding the TBT were discussed in [5] and [11]. In particular, a less-complicated approximate ML algorithm, called the iterative tail-biting VA (ITB-VA), was proposed in [11] that applies the VA iteratively around a TBT multiple times without excluding paths that are not closed. This approach is taken in our work.

We define Nstgs=NroundN, where Nround> 1 is a real number. Nround and Nstgs are in fact the desired numbers of iterations and stages around the TBT, respectively. After going around the TBT Nroundtimes, the optimum path is chosen, and the estimated sequence is translated into a sequence of symbol estimates ˆx0, . . . , ˆxN−1. To do this, we recall that the VA has the property that all surviving paths merge after a certain stage Δ [6]. Thus, the optimum surviving path at the ith stage is used to estimate the (i− Δ)th information bits, where Δ is called the

traceback depth or the truncation length [6]. A more detailed

description of the ITB-VA can be found in [11].

The number of state transitions per stage determines the com-plexity of the VA. By assuming that all components of sparse matrix H (or (H) can be precomputed and their computational

complexity is negligible, as compared with the complexity of the ITB-VA, the complexity of the ITB-VA for CI-MC-CDMA and BPSK modulation is O(22L−1) at each stage. Since there are NroundN stages, where Nround is usually less than 2 [5], the total complexity for all stages is O(NroundN 22L−1_{), which}

is far less than the complexity of a conventional ML-MUD technique [i.e., O(N 2N_)].

Fig. 4. TBT for the case with N = K = 16 and L = 2. D. Upper Bound on the Minimum Error Probability

It is difficult to obtain a closed-form solution of the minimum error probability for MC-CDMA. However, we can derive its upper bound for BPSK transmitted symbols. We take a similar approach to the procedure for synchronous CDMA in the AWGN channel [22] and the fading channel [23] and extend it to synchronous MC-CDMA in a CFO environment. We define

Ei to be the set of error vectors that affect the ith user in the

form of Ei={e ∈ {−1, 0, 1}K, ei= 0}, where ei= xi− ˆxi.

The set of admissible error vectors that are compatible with the transmitted vector x∈ {−1, 1}K _{is denoted by A(x) =}

{e ∈ E, ei = xior 0} = {e ∈ E, 2e − x ∈ {−1, 1}K}, where

E =∪K_i=1Eiis the set of nonzero error vectors. We can upper

bound the error probability for user i, which is denoted by

Pi(e), in the AWGN channel by

Pi(e)≤



e∈Ei

P{Ω(x − 2e) ≤ Ω(x), e ∈ Ai(x)} (47)

where Ai(x) = A(x)∩ Ei, and we have used the fact that if

x− 2e is the most likely vector, it is more likely than x. It can

easily be shown that when no CFO is present, we have Ω(x−2e)−Ω(x) = 4 {eTxˆ} + 4eTHe( −2xTHe( −2eTHx(

= 4eTHe+4( {eTnˆ}

where the second equality is true since ˆz = (Hx + ˆn. We can

see that this event is dependent on noise ˆn only while e∈ A(x)

depends on x only. Thus, we conclude that these two events are independent. Extending (47) to the fading channel, we can express the error probability as

Pi(e| (H)≤



e∈Ei

P{Ω(x−2e)−Ω(x)≤0| (H}·{P {e ∈ A(x)}, }

which follows from the fact that the admissibility of e is independent of (H. For equally likely transmitted bits, we have

P{e ∈ A(x)} =

K_.−1 i=0

P{(xi− ei)ei = 0} = 2−w(e)

where w(e) =K_i=1|ei|. To compute P {Ω(x−2e) − Ω(x) ≤

0| (H}, we note that, since ˆn is a proper (circularly symmetric)

complex Gaussian random vector with zero mean and co-variance matrix σ2_H,( _{E{( {e}T_nˆ})2_{} = (1/2)E{e}T_nˆˆn†_{e} =}

(1/2)σ2_eT_{He. Thus, the error probability for user i is}(

bounded by Pi(e| (H)≤  e∈Ei 2−w(e)Q &/ 2eT_He( σ ' . (48)

When CFO is present, Ω(x− 2e) − Ω(x) = 4eT_H†_{He +}

4 {eT_H†_{nˆ}. By taking a similar approach, we can show that}

the upper bound for the error probability with CFO is

Pi(e|H) ≤  e∈Ei 2−w(e)Q & √ 2eT_H†_He σ / eT_H†_HHe( ' . (49)

The unconditional BEP for user i can be obtained by

Pi(e) = ∞

 0

Pi(e|H)P {H} dH. (50)

This integral can be calculated using the Monte Carlo method [18].

VI. COMPLEXITYREDUCTION IN

DECORRELATINGMULTIUSERDETECTION

In this section, we consider the complexity reduction for the decorrelating multiuser detector using the CI codes. From Fig. 3, we can see that the channel matrix can be converted to a band matrix if the corner values are reduced to zero. A band matrix is a matrix whose nonzero elements are confined to a diagonal band comprising the main diagonal and several subdiagonals. For band matrix A with A(i, j) = 0 if i− j >

mland j− i > mu, integers mland muare called the lower

and upper bandwidths, respectively, and m = ml+ mu+ 1 is

the total bandwidth.

Channel matrix H (or (H for the case without CFO) can be

converted to a band matrix by reducing the number of users to N− (L − 1) and employing CI codes wi[k] = ej2πki/N,

k = 0, 1, . . . , N− 1, with the set of indices i, which is either {0, 1, . . . , N − L} or {L, L + 1, . . . , N}. The bandwidth of

the resulting band matrix is L− 1 + L − 1 + 1 = 2L − 1. To give an example, for a channel of length L = 3, its cross-correlation matrix with N = 16 can be transformed into a band matrix of size N− (L − 1) = 14 and bandwidth 2L − 1 = 5, as shown in Fig. 5, by omitting the first two CI codes, i.e.,

Fig. 5. Conversion of a channel matrix into a band matrix with N = 16 and L = 3 by reducing the user number to 14.

Band matrices are usually stored by recording only the diag-onal entries in the band. Therefore, a band matrix system can be solved by LU decomposition faster and with less storage space than a general dense matrix of the same dimension. Consider an MC-CDMA system with a spreading gain of N and Nb=

N− (L − 1) users employing codewords with indices i = L, L + 1, . . . , N (or 0, 1, . . . , N− L) in a CFO environment.

By using MRC in the receiver and disregarding the additive noise vector ˆn in (42), the received signal ˆz becomes ˆz = Hbx,

where Hb is the corresponding cross-correlation matrix. The

preceding equation is in fact a linear system equation involving complex band matrix Hb, which can be solved by the Gaussian

elimination algorithm with partial pivoting.

The Gaussian elimination algorithm first factors Hbinto the

product of an upper triangular matrix U and a lower triangular matrix L, namely, Hb= LU. Next, the solution of the system

Hbx = ˆz can be rewritten by L(Ux) = ˆz, which demands

forward and backward substitutions. The total number of op-erations required to solve Hbx = ˆz depends on the number of

pivoting required. Generally speaking, if Nb ml+ mu, the

number of operations required by the factorization in Hb=

LU is O(Nbml(ml+ mu)), whereas the number of operations

required by solving x in L(Ux) = ˆz by forward/backward

substitutions is about O(Nb(2ml+ mu)) [8].

Note that, for the cross-correlation matrix Hbof our interest,

the lower and upper bandwidths are both equal to L− 1. Hence, the complexity of the factorization process and the solution process is equal to O(2(N− L + 1)(L − 1)2) and O(3(N−

L + 1)(L− 1)), respectively. In contrast, to solve a general

linear system of equations with Gaussian elimination with a dense matrix of size (N− L + 1) × (N − L + 1), it demands

O((N− L + 1)3_{/3) for the LU factorization and O((N− L +} 1)2_{) for forward and backward substitutions. The complexity} of matrix inversion with fast algorithms is O((N− L + 1)2_). Thus, the complexity of the decorrelating MUD technique for MC-CDMA has considerably been reduced with CI codes in practical channel scenarios, where N 2(L − 1)2_.

In the absence of CFO, we denote the cross-correlation matrix by (Hb, where (Hb= (C†C, and (( C(k, j) is given by (41).

(

Hb is a Hermitian positive definite banded matrix with m =

ml= mu= L− 1, for which there is an even faster

Gaussian-elimination algorithm for solving the linear system. The total number of operations is approximately equal to O((N−

L + 1)(L− 1)2_{/2− (L − 1)}3_{/3) for the LU factorization} and O(2(N− L + 1)(L − 1) − (L − 1)2_{) for the forward and} backward substitutions [8]. In contrast, the complexity of solv-ing a general dense matrix of the same size is O((N− L + 1)3/6) for the LU factorization and O((N− L + 1)2) for the forward and backward substitutions, and the complexity of matrix inversion with fast algorithms is O((N− L + 1)2).

A. Error Probability for Decorrelating MUD

The detected symbol for the aforementioned MUD technique is given by

ˆ

x = x + H−1_b nˆ (51)

where ˆn is an (N− L + 1) × 1 Gaussian random vector

with zero mean and covariance matrix σ2_{H. Hence,}( E{H−1

b nHˆ −1b nˆ†} = E{H−1b nˆˆn†(H−1b )†} = σ2H−1b H(H( −1b )†.

Let Hn = H−1b H(H( −1b )†. Since ˆn is a proper complex random

vector, E{ {ˆnˆn†}} = (1/2)E{ˆnˆn†} = (1/2)σ2Hn. Then,

under BPSK modulation, the BEP for the ith user is equal to

Pi(e) =E  Q &0 2 σ2_[H n]i,i ' .

VII. SIMULATIONRESULTS

The Monte Carlo simulation was conducted to corroborate the theoretical results derived in the previous sections. In the simulation, channel taps were generated as independent identically distributed random variables with zero mean and

Fig. 6. Analytical and simulated BEP results versus Eb/N0with N = 16,

L = 2, and CFO =±0.1.

unit variance. Every user had his/her own CFO value. To compute the analytical BEP, the Monte Carlo integration method was used [18]. That is, random variables λi[k], k =

0, 1, . . . , N− 1 are generated by taking the DFT of complex Gaussian-distributed channel taps M times. Then, computer-generated samples of λi[k], k = 0, 1, . . . , N− 1 are substituted

in the Q-function, and the sum of trials is divided by M .

Example 1: Theoretical Versus Simulated BEP of Fully Loaded MC-CDMA With Single-Stage PIC: In this example,

we evaluate the BEP performance of MC-CDMA with PIC in the presence of CFO and examine both analytical and simulated BEP results. We employ two orthogonal codes, namely, orthog-onal HW codes and CI codes. Fig. 6 depicts the analytical and simulated BEP results for a fully loaded MC-CDMA system with CI and HW codewords as a function of the SNR value

Eb/N0 under the setting of N = 16, L = 2, K = 16, and CFO =±0.1. To shorten the simulation time, only the BEP results for the first users were compared. Since L = 2, we can use (37) as the analytical BEP expressions of CI-MC-CDMA. For this case, we observe close agreement between the analytical BEP expression and simulation results. For MC-CDMA with HW codes, the approximate BEP expression, as shown in (31), was used. Due to a higher number of interfering users employing HW codes, the analytical Gaussian model for the total residual user interference can be used. However, the analytical and simulation results for HW codes (the last two curves in the figure) do not have strong agreement, particularly in the high-SNR regime, as shown in Fig. 6.

Example 2: ML-MUD BEP Versus Minimum Probability of Error: Fig. 7 shows the upper bound to the BEP as a function

of the SNR value Eb/N0for CI-MC-CDMA under the setting of

N = 8, L = 2, and K = 8 for both zero CFO and CFO =±0.3

cases. The upper bound curves in each case are plotted against their corresponding simulated BEP. To obtain the simulated BEP, Nround = 1.5 and Δ = 2 for zero CFO, and Nround = 1.5 and Δ = 3 for nonzero CFO values. To shorten the simulation and computation time, only the BEP for the first user was com-puted. We can see close agreement between the performance

Fig. 7. Upper bound and simulation of the BEP for N = 8, L = 2, and CFO = 0,±0.3.

Fig. 8. ML-MUD BEP performance versus Eb/N0 with N = 16, L = 2,

and CFO =±0.3.

of the ML-MUD BEP and the upper bound for the minimum BEP. It is also clear from the figure that ML-MUD performs better in the absence of CFO than in the presence of CFO. This can be explained by noting the fact that the denominator in (49) for the upper bound on the minimum BEP with CFO is

/

eT_H†_{HHeσ, as opposed to just σ in the denominator of}(

(48) for the upper bound on the minimum BEP with no CFO.

Example 3—ML-MUD Performance: Fig. 8 shows a

sig-nificant performance improvement of the ML detector, where the performance of CI-MC-CDMA with ML-MUD is com-pared with CI-MC-CDMA with single-user MRC detection. The parameters for the simulated CI-MC-CDMA system were

N = 16, L = 2, and K = 16. As compared with Fig. 7 with N = 8, we can see that ML-MUD performs better since there

were more pairwise MAI-free users. Separate simulations were performed to acquire the BEP performance for CFO = 0 and CFO =±0.3. We can see that the BEP achieved by ML for both systems is very low when the SNR is close to 10 dB. Again,

Fig. 9. Analytical and simulated BER performance of the decorrelating detector with the reduced-complexity Gaussian elimination algorithm for MC-CDMA with CFO =±0.5.

we can see that the performance of ML-MUD is better with no CFO that in the presence of CFO.

Example 4—Decorrelating Detector Performance: Fig. 9

shows the theoretical and simulated BEP results as a function of the SNR value Eb/N0in the presence of CFO =±0.5 with a decorrelating detector with N = 21, L = 4, and K = 18. To shorten the simulation time, only the BEP for the first user was computed. We can see that simulated and analytical BEP results are in good agreement. The average BEP performance of CI-MC-CDMA and with N = 16, L = 3, and K = 14 is also shown in Fig. 9. We can see that, as the number of users increases from 14 to 18, the BEP performance degrades up to 2.5 dB for Eb/N0< 8 dB and around 5 dB for Eb/N0> 8 dB.

Example 5: Performance Comparison of PIC, ML, and Decorrelating Detectors: Fig. 10 compares the BEP

perfor-mance of CI-MC-CDMA employing second-stage PIC, ML, and decorrelating detectors with the MRC detector as the benchmark for N = K = 16, L = 2, and CFO = 0. As ex-pected, the optimum ML-MUD detector greatly outperforms all other detectors. We also observe that the decorrelating detector with N− L + 1 = 15 users outperforms the second-stage PIC detector when Eb/N0> 15 dB.

It is worthwhile to mention that we concentrated on the performance of CI-MC-CDMA with the simplified multiuser detectors in this work. Although we observed in Fig. 6 that the PIC detectors with CI and HW codes have comparable performance, the simulation results were presented in this section to demonstrate that CI-MC-CDMA with the simplified MUD techniques can successfully suppress MAI, as compared with single-user detection. The performance of MC-CDMA with other codes was compared with CI and HW codes in the previous literature. Particularly, in [17], CI-MC-CDMA was compared with HW, Gold, and quadriphase codes and found to have a better BEP performance for all users. In addition, the BEP performance of MC-CDMA with a special form of HW codes was compared with Shi and Latvaaho codes in [19]. It was shown that the proposed scheme can significantly outperform Shi and Latvaaho codes.

Fig. 10. Comparison of various MUD techniques for N = 16, L = 2, and CFO = 0.

VIII. CONCLUSION

We have shown in this paper that, for an MC-CDMA system with spread gain N , multipath length L, and N users, when CI codes are used, a proper subset of CI codes leads to a completely MAI-free MC-CDMA system in a CFO environ-ment. We proved that each user only has to combat 2(L− 1) (rather than N− 1) interferers, even in the presence of CFO. We analyzed the BEP of MC-CDMA in a CFO environment with PIC, ML, and decorrelating multiuser detectors. We also demonstrated that the sparse cross-correlation matrix of the CI codes can be used to considerably reduce the complexity of the aforementioned multiuser detectors. Finally, simulation results were given to corroborate derived theoretical results.

ACKNOWLEDGMENT

The authors would like to thank J. Hicks of Aerospace Corporation for his invaluable help with conducting simulations using the ITB-VA.

REFERENCES

[1] M. K. Varanasi and B. Aazhang, “Multistage detection in asynchronous code-division multiple-access communications,” IEEE Trans. Commun., vol. 38, no. 4, pp. 509–519, Apr. 1990.

[2] J. A. C. Bingham, “Multicarrier modulation for data transmission: An idea whose time has come,” IEEE Commun. Mag., vol. 28, no. 5, pp. 5–14, May 1990.

[3] L. Brunel, “Multiuser detection techniques using maximum likelihood sphere decoding in multicarrier CDMA systems,” IEEE Trans. Wireless Commun., vol. 3, no. 3, pp. 949–957, May 2004.

[4] X. Cai, S. Zhou, and G. B. Giannakis, “Group-orthogonal multicarrier CDMA,” IEEE Trans. Commun., vol. 52, no. 1, pp. 90–99, Jan. 2004. [5] A. R. Calderbank, G. D. Forney, Jr., and A. Vardy, “Minimal tail-biting

trellises: The Golay code and more,” IEEE Trans. Inf. Theory, vol. 45, no. 5, pp. 1435–1455, Jul. 1999.

[6] L. Lin and D. Costello, Error Control Coding., 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 2003.

[7] D. Divsalar and M. K. Simon, “CDMA with interference cancellation for multiprobe missions,” Jet Propulsion Lab., Pasadena, CA, TDA Progress Rep. 42-120, Feb. 1995.

[8] J. J. Dongarra, J. R. Bunch, C. B. Moler, and G. W. Stewart, LINPACK Users’ Guide., 2nd ed. Philadelphia, PA: SIAM, 1980.

[9] R. M. Gray, “On the asymptotic eigenvalue distribution of Toeplitz matri-ces,” IEEE Trans. Inf. Theory, vol. IT-18, no. 6, pp. 725–730, Nov. 1972.

[10] S. Hara and R. Prasad, “Overview of multicarrier CDMA,” IEEE Commun. Mag., vol. 35, no. 12, pp. 126–133, Dec. 1997.

[11] J. Hicks, S. Bayram, W. H. Tranter, R. J. Boyle, and J. H. Reed, “Over-loaded array processing with spatially reduced search joint detection,” IEEE J. Sel. Areas Commun., vol. 19, no. 8, pp. 1584–1593, Aug. 2001. [12] B. Hombs and J. S. Lehnert, “Multiple-access interference suppression for

MC-CDMA by frequency oversampling,” IEEE Trans. Commun., vol. 53, no. 4, pp. 677–686, Apr. 2005.

[13] W. C. Jakes, Microwave Mobile Communications. New York: Wiley, 1974.

[14] Y. Li, “Simplified channel estimation for OFDM with multiple transmit antennas,” IEEE Trans. Wireless Commun., vol. 1, no. 1, pp. 67–75, Jan. 2002.

[15] P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE Trans. Commun., vol. 42, no. 10, pp. 2908–2914, Oct. 1994.

[16] S. Moshavi, “Multi-user detection for DS-CDMA communications,” IEEE Commun. Mag., vol. 34, no. 10, pp. 124–136, Oct. 1996.

[17] B. Natarajan, Z. Wu, C. R. Nassar, and S. Shattil, “Large set of CI spreading codes for high-capacity MC-CDMA,” IEEE Trans. Commun., vol. 52, no. 11, pp. 1862–1866, Nov. 2004.

[18] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes, the Art of Scientific Computing. New York: Cambridge Univ. Press, 1986.

[19] S. H. Tsai, Y. P. Lin, and C.-C. J. Kuo, “MAI-free MC-CDMA systems based on Hadamard Walsh codes,” IEEE Trans. Signal Process., vol. 54, no. 8, pp. 3166–3179, Aug. 2006.

[20] L. Tadjpour, S. H. Tsai, and C.-C. J. Kuo, “Orthogonal codes for MAI-free MC-CDMA with carrier frequency offsets (CFO),” in Proc. IEEE GLOBECOM, San Fransisco, CA, Nov. 2006, pp. 1–5.

[21] V. Thippavajjula and B. Natarajan, “Parallel interference cancellation techniques for synchronous carrier interferometry/MC-CDMA uplink,” in Proc. IEEE 60th Veh. Technol. Conf., Sep. 2004, pp. 399–403.

[22] S. Verdu, Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, 1998.

[23] Z. Zvonar and D. Brady, “Multiuser detection in single-path fading channels,” IEEE Trans. Commun., vol. 42, no. 2–4, pp. 1729–1739, Feb.–Apr. 1994.

Layla Tadjpour received the B.S. degree in

electri-cal engineering from the Iran University of Science and Technology, Tehran, Iran, in 1996, the M.S. degree in electrical engineering from the University of California, Los Angeles, in 1999, and the Ph.D. degree in electrical engineering from the University of Southern California, Los Angeles, in 2008.

From 1999 to 2005, she was a System Engineer for deep-space communication systems with the Jet Propulsion Laboratory, Pasadena, CA. In 2008, she was with Wilinx Co., Carlsbad, CA, where she par-ticipated in the system design for multiband orthogonal frequency division multiplexing systems. She is currently with Information System Laboratories, Vienna, VA, working on algorithm development and high-fidelity modeling and simulation for radar and communications systems. Her research interests include signal processing for communications and software-defined radios.

Shang-Ho Tsai received the Ph.D. degree in

elec-trical engineering from the University of Southern California, Los Angeles, in 2005.

From June 1999 to July 2002, he was with Silicon Integrated Systems Corporation, where he partic-ipated in the very-large-scale integration (VLSI) design for Discrete Multi-tone Asymmetric Digi-tal Subscriber systems. From September 2005 to January 2007, he was with MediaTek, Inc., where he participated in the VLSI design for multiple-input– multiple-output-orthogonal frequency division mul-tiplexing (MIMO-OFDM) systems. Since February 2007, he has been with the Department of Electrical Engineering (formerly the Department of Electrical and Control Engineering), National Chiao Tung University, Hsinchu, Taiwan, where he is currently an Assistant Professor. His research interests include signal processing for communications, particularly the areas of OFDM and MIMO systems. He is also interested in ultrawideband and VLSI design related to the aforementioned topics.

Dr. Tsai received a government scholarship for overseas study from the Ministry of Education, Taiwan, during 2002–2005.

C.-C. Jay Kuo (S’83–M’86–SM’92–F’99) received

the B.S. degree from the National Taiwan Uni-versity, Taipei, Taiwan, in 1980 and the M.S. and Ph.D. degrees from the Massachusetts Institute of Technology, Cambridge, in 1985 and 1987, respec-tively, all in electrical engineering.

From October 1987 to December 1988, he was a Computational and Applied Mathematics Research Assistant Professor with the Department of Mathe-matics, University of California, Los Angeles. Since January 1989, he has been with the University of Southern California, Los Angeles, where he is currently a Professor of electrical engineering, computer science, and mathematics and the Director of the Signal and Image Processing Institute. He is a coauthor of about 170 journal papers, 800 conference papers, and ten books. He has guided about 100 students to their Ph.D. degrees. His research interests are in the areas of digital signal and image processing, multimedia compression, communication, and networking technologies.

Dr. Kuo is a Fellow of the International Society for Optical Engineers. He is the Editor-in-Chief of the Journal of Visual Communication and Image Representation and an Editor of the Journal of Information Science and Engineering, LNCS Transactions on Data Hiding and Multimedia Security, and the EURASIP Journal of Applied Signal Processing. He was the recipient of the National Science Foundation Young Investigator Award and the Presidential Faculty Fellow Award in 1992 and 1993, respectively.