### An Approximately MAI-Free Multiaccess OFDM

### System in Carrier Frequency Offset Environment

*Shang-Ho Tsai, Student Member, IEEE, Yuan-Pei Lin, Member, IEEE, and C.-C. Jay Kuo, Fellow, IEEE*

**Abstract—The carrier frequency offset (CFO) effect of a new**

**multiaccess orthogonal frequency division multiplexing (OFDM)**
**transceiver is analyzed. We show that multiaccess interference**
**(MAI) due to CFO can be mitigated by choosing a proper set of**
**orthogonal codes. That is, if we use** **2 symmetric or **
**antisym-metric codewords of the** **Hadamard–Walsh codewords, MAI**
**can be greatly reduced to a negligible amount. The new OFDM**
**system has been shown to be approximately MAI-free when there**
**is no CFO. The new result developed here shows that the system**
**continues to be approximately MAI-free even in the presence**
**of CFO. Furthermore, we derive a close form for the reduced**
**MAI, which reveals an interesting property of the system. That**
**is, when the number of users increases, the reduced MAI and the**
**inter carrier interference (ICI) due to self-CFO will decrease.**
**Finally, it is demonstrated by simulation that the proposed OFDM**
**scheme with proper code selection outperforms the OFDMA and**
**the multicarrier code division multiplexing access (MC-CDMA)**
**systems in a CFO environment.**

**Index Terms—Carrier frequency offset (CFO), code selection,**

**frequency synchronization, MAI-free, MC-CDMA, multiuser **
**de-tection (MUD), multiuser OFDM, OFDMA.**

I. INTRODUCTION

**M**

ULTICARRIER modulation (MCM) systems have been
widely used in both wireline and wireless
communica-tions recently [6], [8], [9], e.g., the digital subscribe line (DSL),
wireless local area networks (WLANs), and digital video
broadcasting (DVB). In wireless environments, MCM systems
are usually called orthogonal frequency division multiplexing
(OFDM) systems [9]. Due to the mismatch of the oscillators
in the transceiver and the Doppler effect, OFDM systems are
sensitive to the carrier frequency offset (CFO) effect [18],
[20]. The CFO effect will destroy orthogonality of the OFDM
systems. Since the loss of orthogonality will lead to significant
performance degradation [20], research on accurate CFO
esti-mation and compensation has received a lot of attention in the
design of practical OFDM systems [4], [18], [22].
Manuscript received June 10, 2004; revised January 3, 2005. This work was supported in part by the Integrated Media Systems Center, a National Science Foundation Engineering Research Center, under Cooperative Agreement EEC-9529152. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect those of the National Science Foundation. The associate editor coor-dinating the review of this manuscript and approving it for publication was Dr. Behrouz Farhang-Boroujeny.

S.-H. Tsai and C.-C. J. Kuo are with the Department of Electrical En-gineering and Integrated Media Systems Center, University of Southern California, Los Angeles, CA 90089-2564 USA. (e-mail: shanghot@usc.edu; cckuo@sipi.usc.edu).

Y.-P. Lin is with the Department of Electrical and Control Engi-neering, National Chiao Tung University, Hsinchu, Taiwan, R.O.C (e-mail: ypl@mail.nctu.edu.tw).

Digital Object Identifier 10.1109/TSP.2005.857059

The single-user OFDM system has been thoroughly exam-ined in the past, e.g., [6], [8], and [9]. For multiuser OFDM communications, there are two popular techniques; namely, multicarrier code division multiplexing access (MC-CDMA) and orthogonal frequency division multiple access (OFDMA). When compared with CDMA systems, MC-CDMA systems inherit the advantages of multicarrier systems in combating the inter symbol interference (ISI) caused by frequency selective fading channels. Moreover, MC-CDMA systems can fully exploit the frequency diversity. MC-CDMA systems can be divided into two types [14]. In the first type, one bit is trans-mitted per time slot. The coming bit is spread into several chips, which are allocated to different subchannels. The number of subchannels equals the number of chips [7], [26]. In the second type, several bits are converted from serial-to-parallel and then each bit is spread into several chips. The chips corresponding to the same bit are allocated to the same subchannel [17]. This one is often called the MC-DS CDMA system. Two more generalized MC-CDMA system was proposed in [1], [13], and [14]. In the first scheme [13], [14], each S/P converted bit is spread into several chips and then each subcarrier is modulated with one chip, where the frequency separation corresponding to each bit is maximized to achieve frequency diversity. In the second scheme [1], similarly, each S/P converted bit is spread into several chips. Then, the chips corresponding to the same symbol are modulated in successive subcarriers. Although MC-CDMA systems spread symbols using orthogonal codes to ensure orthogonality, when used in uplink transmission, i.e., from the mobile station (MS) to the base station (BS), orthogonality may be destroyed at the receiver due to frequency selective fading, thus leading to multiaccess interference (MAI). The MAI problem cannot be solved by increasing the transmit power since increasing the transmit power for one user will also increase the interference for other users. To suppress MAI, sophisticated multiuser detection (MUD) [25] and signal processing techniques have been proposed at the receiver end [10], [12], [14]. Furthermore, due to MAI, the CFO estimation and compensation are much more complicated in MC-CDMA systems [11].

The OFDMA systems [21] are MAI-free when there is no
CFO. However, OFDMA systems are sensitive to the CFO
effect. Moreover, different CFOs of different users make CFO
estimation much more difficult than that in the single user
OFDM system due to the following reason. When any user
has a nonzero CFO, the CFO not only causes the performance
*degradation of this particular user (i.e., the self-CFO effect) but*
also leads to MAI for other users. Therefore, OFDMA systems
are no longer MAI-free in a CFO environment. CFO estimation

Fig. 1. Block diagram of the proposed system.

algorithms for OFDMA systems were proposed in [2], [5], [19]. However, all those methods demand extra computational complexity in the receiver end. Moreover, CFO compensation needs to be done in the transmitter part so that a feedback mechanism is demanded in this situation [5], [19].

An approximately MAI-free multiaccess OFDM transceiver was proposed in [23]. In this system, every user can utilize the whole bandwidth and time slots simultaneously. When the number of subchannels is sufficiently large in an environment without CFO, the proposed system has the approximately MAI-free property. Unlike the conventional MC-CDMA systems, there is no need to suppress MAI in the receiver, and the system performance improves as the transmit power increases. Moreover, this new system is robust to timing mis-match among users so that it can be conveniently used in uplink transmission [23].

In this work, we first evaluate the CFO effect of the new multiuser OFDM system and derive analytical results for MAI caused by CFO as well as self-CFO impairments. Based on the analytical results, we present a code selection scheme to mitigate MAI by choosing proper orthogonal codes. That is, if we use only the symmetric or the antisymmetric codewords of the Hadamard–Walsh codes, MAI can be greatly reduced to a negligible amount. Using this code selec-tion, we can reduce MAI caused by CFO without additional complexity such as MUD or complicated signal processing techniques. Once MAI is greatly reduced, the self-CFO im-pairment can be easily estimated and then compensated using methods developed for the single user OFDM system, e.g., [4], [18], and [22]. Since MAI is reduced to a negligible amount, we can compensate the CFO effect at the receiver end without the feedback. Furthermore, we derive a close form for the reduced MAI with code selection, which reveals an interesting property of the proposed system. That is, when the number of users increases, both the averaged reduced MAI due to others’ CFOs and the averaged inter carrier interference (ICI) due to self-CFO will decrease. Finally, it is demonstrated by computer simulation that the proposed system with proper code selec-tion has less MAI and a smaller bit error probability than the OFDMA and the MC-CDMA systems in a CFO environment. The superiority of the proposed system in MAI reduction not only facilitates the CFO estimation task but also improves the system performance.

The rest of this paper is organized as follows. The multiuser

Simulation results are given in Section IV to corroborate the an-alytical results. Furthermore, performance comparison between the proposed system, the OFDMA, and the MC-CDMA systems is given. Finally, a conclusion is drawn in Section V.

II. SYSTEMMODEL ANDITSPROPERTIES

In this section, we give a brief review of the new OFDM system and its properties [23]. The block diagram of the pro-posed system in the uplink direction is shown in Fig. 1, where is the number of users. Let the input of the th user ( ) be an 1 vector , which contains modulation symbols. As shown in Fig. 1, the transmitter consists of four stages. At the first stage, each symbol in vector is repeated times to form a new vector of size 1 as

and

(1) At the second stage, is passed through an di-agonal matrix with its diagonal elements drawn from an unitary matrix ( ), where the notation denotes conjugate-transpose of . Let the column vectors

of be . Then, is obtained by repeating

times along the diagonal, i.e., diag

where is the transpose of . Let be the th diagonal

element of , . As , these

diagonal elements satisfy the following property:

(2)

where is the conjugate of , , and

. For , , one sequence that satisfies the condition in (2) is the Hadamard–Walsh code [25]. After passing through the diagonal matrix, the th component of the 1 orthogonally coded output vector is given by

(3) At the third stage, each coded vector is passed through the -point unitary inverse discrete Fourier transform (IDFT) matrix. Finally, at the fourth stage, each transformed vector is converted from the parallel to the serial representation and the cyclic prefix (CP) of length is added, where is the maximum channel delay spread.

and then averaged. Let be the output of and be the averaged output. Then, the th element of is given by

(4) In the final stage, is passed through frequency equalization (FEQ) and ready for detection.

The new multiuser OFDM system has the approximately MAI-free property, which is described as follows [23]. Let be the th element of the -point DFT of channel path , and let be the DFT of the received noise vector of size 1. The th element of is given by [6], [8]

(5) From (3) and (5), the th element of can be expressed as

(6) Let . From (4) and (6), the th element of is given by

(7) where the second term in (7) is the MAI from other users. When

is sufficiently large, we can approximate by (8) where is the th component of the -point DFT of the th channel path. The approximation in (8) is also used in [6], [8]. Using (1) and (8), we can rewrite (7) as

by (2) (9)

Observed from (9), the MAI is approximately zero. Hence, if there is no channel noise, we can approximately

re-construct by multiplying by .

An alternative for reconstruction is multiplying by . Note that the alternative one will lead to more accurate approximation and, hence, better performance. The one-tap gain multiplication is what usually called frequency equalization. Since the proposed system is approximately MAI-free, its capacity increases as SNR in-creases. This result is very different from that of conventional MC-CDMA systems, in which increasing the transmit power of one user will also increase the MAI for other users.

Although the proposed system uses orthogonal codes such as the Hadamard–Walsh code to distinguish individual uses. It is worthwhile to point out several differences between the pro-posed system and other systems.

*A. MAI-Free*

The proposed system can achieve approximately MAI-free when is much larger than the multipath length . That is, for a fixed , as increases, the system will have less MAI. Hence, by increasing , the system can accommodate more users with negligible MAI. This MAI-free property is similar to that of the OFDMA system, which is MAI-free when time and frequency are well synchronized. Moreover, in an OFDMA system, the block duration is in general much longer than the multipath length. Such systems are usually used in wireless local area network (WLAN) applications [16]. On the other hand, CDMA or MC-CDMA systems are usually used in the cellular phone system, where co-channel interference is the major concern. Due to MAI, the number of users accommo-dated by MC-CDMA systems is much less than the spreading factor [1], [13], [14]. To increase the number of active users, should be increased accordingly. This is different from the proposed system where is increased to achieve both MAI-free and higher loading.

*B. Frequency Equalization*

Referring to Fig. 1, the proposed system performs “frequency equalization” [8] instead of “combing” before symbol detec-tion. This stands in contrast with MC-CDMA systems, where “combing” is usually used before detection [1], [13], [14]. There are several different combing methods such as MRC, EGC and ORC in MC-CDMA systems [14]. The combing tech-niques multiply every spread chips by a weighted gain and then sum up every chips before detection. Hence, there are gain multiplications for the MC-CDMA systems [1], [13], [14]. Moreover, due to MAI, multiuser detection may be involved and thus the detection of individual symbols is not independent [14]. One the other hand, from (4), the proposed system per-forms summation before the gain multiplication, i.e., frequency equalization. Thus, we only need gain multiplications in the proposed system. Furthermore, since the proposed system is approximately MAI-free, there is no need to use MUD. Thus, the detection for individual symbols is independent.

*C. Hadamard–Walsh Code in the Uplink Transmission*

In the uplink transmission, it is difficult to guarantee that every user transmits his/her signal simultaneously. This will lead to timing mismatch among users. If the Hadamard–Walsh code is used in the uplink transmission in conventional CDMA sys-tems, a small timing mismatch among users will result in great MAI even if the channel is perfect. For instance, consider the case with processing gain . If the fourth user has a delay of one chip duration while the other three users are per-fectly synchronized, the receiver cannot distinguish the fourth

user from the third one because .

There-fore, Hadamard–Walsh code is seldom used in the uplink trans-mission unless the timing mismatch problem can be well re-solved by some other mechanism. In conventional CDMA or MC-CDMA systems, quasiorthogonal codes that have less cross correlation such as the Gold code or the Kasami code are usu-ally used to mitigate the timing mismatch problem. In contrast, since the proposed system is robust to timing mismatch [23], we can adopt the low complexity Hadamard–Walsh code in the uplink transmission.

III. ANALYSIS OFCFO EFFECTS

In the proposed system, the overall CFO effect consists of two parts. One is the MAI caused by CFOs of other users. The other is the symbol distortion and the inter carrier interference (ICI) due to the user’s own CFO. They will be analyzed separately in this section.

Consider the th element of the received vector after DFT in a CFO environment, i.e.,

(10) where is the attenuated symbol of . The attenuation is caused by channel fading and the CFO effect. Suppose the th user has a normalized CFO , which is the actual CFO

normal-ized by of the overall bandwidth and ,

one can show that in (10) can be expressed by [18]

(11)

where and are given by

and (12)

The first term of (11) is the distorted symbol, and the second term is the ICI caused by the CFO. Note that when there is no CFO, equals , as defined in (5). From (4) and (10), we see that under CFO is given by

(13) where MAI is the interference due to user . In the fol-lowing, MAI and are considered separately.

*A. Analysis of Other User’s CFO Effect*

Let us consider the MAI of the th symbol of user due to user , i.e., MAI in (13). From (11) and (13), we have

MAI (14)

where

(15) and

(16) Using (1), (3), and the approximation in (8), we have

(17) Therefore, the interference term is approximately zero. Hence, only the term is of concern. The term

can be rearranged as (18), shown at the bottom of the page. Letting , (18) can be manipulated as (19), shown at the bottom of the next page. Using (1) and the approximation in (8), we have (20), shown at the bottom of the next page. We argue that the term of is the dominating MAI in (20).

When , there are pairs of to make the sine

function in (20) equal and pairs of

to make the sine function equal . On

the other hand, when or , both situations have only one pair of that makes the sine function equal and one pair of that makes the sine

function equal . Hence, the MAI term of

contributes the most to (20). If we can find ways to reduce the term of , the MAI can be greatly reduced. Intuitively speaking, the MAI in the th symbol of the target user is most seriously affected by the th symbols of other users. The farther the distance of other users’ symbols from the th symbol, the less impact they will make. Later, we will give an example (i.e., Example 3 in Section IV) to illustrate this point.

Since the MAI term of in (20) is the dominating MAI, we will rearrange this term to a form that helps us gain insights on how to reduce it. Let us extract the MAI term of in (20) and let it be denoted by , we have

(21) where (22) and (23) Referring to (22), let

When is sufficiently large, the denominator of is ap-proximately an odd function of , and the numerator is nearly

constant for all possible with , .

Hence, we can approximate as an odd function of , i.e., . Using the equality shown at the bottom of the page, and the approximation , we can rewrite (22) as

(24)

As given in (24), the quantity is determined by the property of orthogonal codewords. If , the dominating MAI term of in (20) is approximately zero. One way to achieve this is the use of only of the Hadamard–Walsh codes, which are either symmetric or antisymmetric.

*Proposition: Suppose only the* symmetric or the
antisymmetric codewords of the Hadamard–Walsh codes are

used, , and thus, .

*Proof: The* Hadamard–Walsh code can be divided into
two groups of codewords [15]. One is the set of symmetric
(even) codes satisfying

(25) and the other is the set of antisymmetric (odd) codes given by

(26) When symmetric codewords are used, from (25) and since Hadamard–Walsh code is real, we have

(27)

(19)

Let . We can rewrite (27) as

(28) Thus, in (24) is zero. As for the set of antisymmetric code-words, from (26), we again have the same equality as given in (27). This leads to (28). Therefore, the use of antisymmetric codewords also results in .

Let us give a simple example for codeword selection [24]. Suppose and is an 8 8 Hadamard matrix with column vectors . We can either choose column

vectors , which are symmetric, or column

vectors , which are antisymmetric, as the four codewords for four users.

If the proposed code selection is used, the dominating MAI can be reduced to a negligible amount. Therefore, the MAI terms of in (23) become the main MAI impairment. For convenience, with code selection, let us call it residual MAI. Next, we will investigate the residual MAI. From (20) and (22), we have (29), shown at the bottom of the page. Let

. For fixed , and , (29) can

be rewritten as (30), shown at the bottom of the page. Letting , we

have (31), shown at the bottom of the page. Using (28) and (31), we can rewrite (30), as shown in (32) at the bottom of the page. Assume that and are uncorrelated for all , and and are uncorrelated for . From the Appendix, it can be shown that the expectation value for the mean square of is

(33) where is the averaged channel gain, and is the averaged symbol power of user defined by

and

(34)

*Example 1: Assume* . Let us first consider
the CFO case of . This CFO level may be regarded

as a serious one. Let and . Fig. 2 shows

the total residual MAI, i.e., , as a

function of the user index, where the summation term accounts for in this system. As shown in Fig. 2, the worst total residual MAI is about dB, which is much smaller than the transmit power of 0 dB. Since the residual MAI

(29)

(30)

Fig. 2. Example 1: E B [k] as a function of user indexj withM = 16 for symmetric and antisymmetric codewords.

Fig. 3. Example 1:(1=T ) E B [k] as a function of CFO for differentM.

is relatively small, it will make channel estimation and CFO estimation much more accurate. Let us consider the averaged value of total residual MAI for all users, i.e., . Fig. 3 shows the averaged value of total residual MAI as a function of CFO for different with symmetric codewords. The performance with antisymmetric codewords is also similar to this figure. From Fig. 3, we see that, as increases, the averaged total residual MAI decreases. This result means that the increase of will help reduce . Note that since there are users in this system, the increase of will also increase the number of users. It implies that the increase of users can help reduce the residual MAI for a fixed CFO.

Although the codeword selection given in the Proposition de-creases the number of users from to , it reduces the dom-inating MAI greatly and the system is approximately MAI-free in the presence of CFOs. Thus, every user only has to tackle his/her own CFO problem without worrying about the CFOs of other users. This is very different from conventional multiaccess OFDM systems, where sophisticated signal processing is used to solve the multiuser CFO problem [2], [5], [11], [19]. Sig-nificant reduction of the dominating MAI using symmetric or antisymmetric codewords will only be verified by experimental results in Section IV.

*B. Analysis of Self-CFO Effect*

In this subsection, we examine the impairment caused by self-CFO. For user , the self-CFO impairment means the symbol distortion and the interference caused by his/her own CFO . From (11) and (13), we have

(35) where is the distorted symbol of due to self-CFO given by

by (3)

by (1) and (8)

by (2) (36)

and is the interference caused by ,

(37) Using the same procedure of deriving (16), (18), (19) and (20), we have (38), shown at the bottom of the page.

Using the same procedure of deriving (22) and (24), the term of in (38), which is the interference caused by the th symbol itself, can be written as

(39)

where because ,

and . Therefore, we have .

Fig. 4. Example 2:E D [k] as a function of user indexj with M = 16 for symmetric and antisymmetric codewords.

Next, consider the terms of in (38), which is the ICI caused by all other symbols except for the th symbols. Using the same procedure of deriving (29) to (32), we have (40), shown at the bottom of the page. From the derivation in the Appendix, (40) can be shown to be

(41)

*Example 2: The environment setting is the same as that*

stated in Example 1. First, let us consider the averaged ICI for each individual user, which is plotted as a function of user index in Fig. 4. We see that the worst perfor-mance is around dB for the user with codeword , i.e., the all-one code. Other codewords have performance smaller than dB. The reason for to yield significantly large ICI can be given as follows.

For convenience, let us repeat the definition of , i.e., in reference to (41). The denominator of is positive for all possible and , whereas its numerator can be approximated by 1 for all when is sufficiently large. Thus, is positive approximately. A similar argument can be applied to as well. For the all-one code,

is positive for all . Together with the fact that both

and are positive approximately, we con-clude that the all-one code results in a much larger value in than

other codewords, whose can be either

negative or positive.

Since the ICI using all other symmetric codewords are below dB (except ), this result suggests the use of symmetric codewords but excluding the all-one code to have a smaller ICI. According to (36), the distorted symbol due to self-CFO has the mean square expectation value given by

dB when and . Except ,

the amount of other users is much smaller than dB. Moreover, when compared with the residual MAI that is below dB in Example 1, the dB of the distorted symbol is still relatively large as compared with the residual MAI. Since the residual MAI and ICI are both smaller than the distorted symbol due to the self-CFO, we can estimate accu-rately and compensate it in the receiver end without demanding a feedback mechanism. We will discuss this in more detail in the next section. The averaged ICI for all users as a function

of CFO for different , i.e., , is

shown in Fig. 5. Since is excluded, there are users in the system with symmetric codewords. We see that symmetric codewords with excluded have a smaller averaged ICI than antisymmetric codewords. From the figure, we see results sim-ilar to Fig. 3. That is, the increase of helps reduce ICI for a fixed CFO. Note that for target user , the ICI due to self-CFO only affects itself. Even if is included for symmetric code-words, it does not result in significant performance degradation for other users since the MAI caused by to all other users is sufficiently small according to the discussion in Section III-A.

*C. Overall CFO Estimation and Compensation*

Here, let us consider the overall CFO effect. For convenience, let us rewrite (13) as

MAI (42)

Following the discussion in Sections III-A and B, impair-ments due to MAI and self-CFO interference are negligible if only symmetric or antisymmetric codewords of the Hadamard–Walsh codes are used. The first term of (42)

Fig. 5. Example 2:(1=T ) E D [k] as a function of CFO for differentM, where the all-one code is excluded for the set of symmetric codewords.

is much larger than all other terms (with the exception of the all-one code whose self-CFO interference term is not too small according to Example 2). Thus, we can estimate accurately and compensate it before the detection stage (i.e., every symbol multiplies before detection). The alternative is to esti-mate self-CFO of each user accurately with algorithms for single-user OFDM systems, as developed in [4], [18], and [22]. Once is estimated, we can multiply the th received symbols

of user by to remove the self-CFO effect

before passing them through the DFT matrix. The penalty is that a separate DFT is needed for each user. Note that this alternative can be used to estimate of the all-one code.

In this case, sophisticated MUD techniques and the feedback mechanism are not needed at the receiver end. On the other hand, if the MAI is not reduced, the CFO estimation for each user could be difficult and MUD techniques are needed, which imposes a heavy computational burden on the receiver. For ex-ample, CFOs of other users will cause the MAI to any target user in OFDMA systems and signal processing techniques are often used to estimate the CFO of this target user [2], [5], [19]. Fur-thermore, since OFDMA does not have a negligible MAI-free property in the CFO environment, the feedback mechanism is demanded after CFO estimation so that every user can compen-sate his/her own CFO at the transmitter end [5].

IV. SIMULATIONRESULTS

In this section, we perform computer simulation to cor-roborate the theoretical results derived in previous sections. Moreover, we will compare the performance of the proposed system, the OFDMA systems and two MC-CDMA systems under the CFO environment. We consider the performance in the uplink direction so that every user has different CFO and channel fading. We assume the channel and the CFO are quasiinvariant in the sense that it remains unchanged within one block duration. Simulations are conducted with the following parameter setting throughout this section. , and the binary phase-shift keying (BPSK) modulation is used. The

Fig. 6. Example 3: MAI effect as a function of the CFO when the fullM Hadamard–Walsh codewords are used.

channel coefficients are independently identically distributed (i.i.d.) complex Gaussian random variables with an unit vari-ance. For every individual user, the Monte Carlo method is used to run more than 500 000 symbols. We consider the worst CFO situation. That is, the CFO value of each user is randomly assigned to be either or .

*Example 3—Suppression of Dominating MAI Term: Here,*

we would like to show that defined in (22) is the dominating MAI term in (20), and it can be greatly reduced using only symmetric or antisymmetric codewords of the Hadamard–Walsh codewords. Let channel be flat and

. Then, and MAI

according to (14) and (17), where exact equalities are due to flat channel. Using similar notation rule in (21), we de-fine the MAI from the th symbol of user to user as MAI , and the MAI from all the other symbols of user

as MAI , i.e., MAI MAI MAI .

Let the number of users , i.e., a fully loaded system. The total MAI for user from the th symbol of all other users, which is denoted by MAI , is calculated as follows. For the th symbol of a target user, we accumulate the MAI contributed from the th symbol of other 15 users. The procedure is repeated, and then, the MAI power is aver-aged for from 0 to . That is, MAI is obtained by

averaging the value MAI for

more than 500 000 symbols. Similarly, the total MAI from all the other symbols ( ) of all the other users, which is denoted by MAI , is obtained by averaging the value, MAI , for more than 500 000 symbols.

The total MAI is plotted as a function of the normalized CFO value in Fig. 6, where MAI and MAI of 16 users are shown by 16 solid and 16 dashed curves, respectively. The solid bold curve in Fig. 6, which is denoted by MAI , is the averaged value of the 16 solid curves. That is, MAI is obtained via MAI . Similarly, the dashed bold curve, which

is denoted by MAI , is obtained by averaging the 16 dashed curves. That is, MAI is obtained via MAI . From this figure, we see that MAI is around 10–11 dB more than that of MAI . Hence, for each user, the MAI at the th symbol is mostly contributed from the th symbols of other users. Thus, it confirms the derived theoretical result that in (22) is the dominating MAI in (20). In the following, we call “dominating MAI” and

“residual MAI” for short.

Next, we demonstrate that the dominating MAI can be greatly reduced using only symmetric codewords of the Hadamard–Walsh codes. Let the user number decreases

from to , and only the

symmetric codewords are used. The performance is shown in Fig. 7(a). Compared with Fig. 6, the residual MAI decreases around 4 to 5 dB due to the number of users decreasing from 16 to 8. In contrast, the dominating MAI is greatly reduced by 12–47 dB. Note that, the simulation result of the residual MAI is consistent with the theoretical result in Fig. 3. Moreover, we see that using symmetric codewords, the dominating MAI is even smaller than that of the residual MAI when the CFO is less than 0.35.

The performance using antisymmetric codewords is shown in Fig. 7(b). Compared with Fig. 7(a), we see that the performance of the set of antisymmetric codewords is similar to that of the set of symmetric codewords. The result confirms that the dominating MAI can be greatly reduced using the proposed code selection scheme.

*Example 4—Comparison of CFO Effect on the Proposed*
*System, OFDMA and MC-CDMA: Here, we would like to*

compare the CFO effect on the proposed system, the OFDMA system, and two MC-CDMA systems over a flat channel. Among the two MC-CDMA schemes, one is with subcarriers uniformly allocated (called MC-CDMA/U for short) [13], [14], whereas the other is with subcarriers successively allocated (called MC-CDMA/S for short) [1]. Every user in these four systems will transmit symbols, and the DFT/IDFT size is the same, i.e., . Since all four systems transmit symbols per block and add the CP of the same length , their actual data rates are the same. We consider both fully loadded and half-loaded situations. In a fully loadded situation, the Hadamard–Walsh code is used in the proposed and the two MC-CDMA systems. For OFDMA, each user occupies subchannels which are maximally separated [21], i.e., user

will be assigned subchannels indexed by , and . In a half-loaded case, symmetric codewords of the Hadamard–Walsh code (code selection scheme) are used in the proposed system and the two MC-CDMA systems. For OFDMA, the th user will be as-signed subchannels indexed by ,

and . The remaining subchannels are

used as guard bands.

Fig. 7. Example 3: MAI effect as a function of the CFO (a) when onlyM=2 symmetric Hadamard–Walsh codewords are used and (b) when onlyM=2 antisymmetric Hadamard–Walsh codewords are used.

system, frequency equalization was discussed in Section II. For the two MC-CDMA systems, the orthogonality restoring combing (ORC) scheme is used to achieve equalization [14]. To distinguish from the MAI before equalization, we denote the MAI after equalization by MAI . For instance, in the

proposed system, MAI MAI . The

av-eraged total MAI after equalization is obtained via averaging the

value MAI for

symbols. Fig. 8 shows the averaged total MAI after equalization as a function of CFO for the four systems in fully loaded and half-loaded situations. Note that the performance of the proposed system and the MC-CDMA/S are the same for the

Fig. 8. Example 4: MAI comparison among the proposed system, the OFDMA system, and two MC-CDMA systems in a flat-fading environment.

Fig. 9. Example 4: Self-CFO impairment of the proposed system.

and consequently, the proposed system outperforms OFDMA by 10 to 11 dB. Recall that in Example 1, the dominating MAI of the proposed system is larger than the residual MAI by around 10 to 11 dB in the fully loaded situation. Using code selection, the dominating MAI is reduced to a amount that is even smaller than the residual MAI. This explains why the proposed system has similar performance with OFDMA in a fully loaded situation, whereas it outperforms OFDMA by 10 to 11 dB in a half-loaded situation with code selection.

The self-CFO impairment of the proposed system with sym-metric codewords is shown in Fig. 9. To compute the self-CFO impairment after frequency equalization, we accumulate the symbol distortion and the interference for the th symbol of a target user due to his/her own CFO. This procedure is repeated, and then, the impairment power is averaged for , . The self-CFO impairment of the eight users in the proposed system are indicated by the eight solid curves in Fig. 9. Since the self-CFO impairment of an individual user is similar, these curves are overlapping. The bold-circled curve in Fig. 9 is the averaged value of the eight solid curves. By comparing this figure with Fig. 8, we see that the self-CFO impairment is the main impairment in the proposed system.

Fig. 10. Example 4: BER comparison among the proposed system, OFDMA, and two MC-CDMA systems in a flat channel with a normalized CFOj j = 0:1.

Since the MAI is relatively small, we can estimate or ac-curately and then compensate the self-CFO effect, as discussed in Section III-C. It is worthwhile to point out that, although not shown here, the self-CFO impairment of OFDMA is very similar to that given in this figure.

Now, assume that each user has a normalized CFO of and can accurately estimate the self-CFO in these four sys-tems. We would like to find out the bit error rate (BER) when there is no feedback. The BERs for the four systems are shown in Fig. 10. For a fully loadded situation, the BER curves of the proposed system and MC-CDMA/S are overlapping. More-over, for a half-loaded situation, the BER curves of the proposed system MC-CDMA/S and MC-CDMA/U are overlapping. We see that the BER performance has a similar trend as that of the MAI performance in Fig. 8.

*Example 5—CFO Effect in a Multipath Environment: In this*

example, we examine the CFO effect when the channel has the multipath frequency-selective fading. The number of multipath is assumed to be and , whereas the other param-eters remain the same as those given in Example 4. The MAI performance of the four systems is shown in Fig. 11. In a fully loadded situation with the CFO smaller than 0.05, OFDMA has less MAI than the proposed system because OFDMA is com-pletely MAI-free when frequency and time are well synchro-nized. However, as CFO grows, the proposed system slightly outperforms OFDMA system. In a half-loaded situation, the proposed system outperforms OFDMA by around 10 dB due to the use of the code selection. The low MAI value of the proposed system with code selection is also beneficial to CFO estimation. Let us see the comparison with MC-CDMA. The MAI perfor-mance curves for the two MC-CDMA systems with ORC are not good since ORC amplifies the MAI power in the subcarrier with serious fading [14]. In a multipath environment, it is likely that the frequency selective fading contains several zeros to cause huge MAI in the two MC-CDMA systems. However, since a large MAI value may only lead to the error of certain bits, MAI may not be a good performance measure for this case. Instead,

Fig. 11. Example 5: MAI comparison among the proposed system, the OFDMA system, and two MC-CDMA systems in a multipath fading environment.

Fig. 12. Example 5: BER comparison among the proposed system, OFDMA, and two MC-CDMA systems with = 4 and j j = 0:1 in a half-loaded situation.

the BER may provide a more valuable measure for fair compar-ison. As MC-CDMA systems with MRC outperform those with ORC in a multipath environment, we will use MRC for the two MC-CDMA systems in the following discussion.

Fig. 12 shows the BER comparison among the four sys-tems with in a half-loaded situation. We see that MC-CDMA/S with the proposed code selection outper-forms OFDMA. Note that if the code selection is not used in MC-CDMA/S, instead, e.g., the first codewords of the Hadamard–Walsh code are used, the performance as shown in the star curve is much worse than that using code selection. This result shows the advantage of using code selection in certain

Fig. 13. Example 6: BER performance as a function of SNR in the presence of changing CFO within one OFDM block.

is small [13]. Finally, we observe that the proposed system with code selection outperforms OFDMA and the two MC-CDMA systems in a half-loaded situation with multipath fading.

*Example 6—Fast Variant CFO Environment: In this *

ex-ample, we consider a situation when CFO changes so rapidly that its value varies even within one OFDM block. Let the parameters remain the same as those in Example 5. We assume that the system estimates CFO per 100 blocks. We simulate three cases. The first case is that the CFO does not change within 100 blocks. The second one is that the CFO changes within 100 blocks, e.g., from 0.1 to or from to during 100 blocks. The step size is per sample in the transmitted block of size . The third one is that the CFO changes within 100 blocks. Due to the CFO variation, there is a discrepancy between the actual CFO and the estimated CFO. We want to evaluate the impairment due to this discrepancy. It is worthwhile to point out the CFO changing rate in the second and third cases are much larger than that in today’s practical operating environment. Thus, they provide extreme difficult cases to test the performance of the proposed system. For instance, let the OFDM-block duration be 256 s, carrier frequency be 4 GHz, and the mobile speed be equal to 100 km/h [16], [19]. The normalized CFO can be calculated to be around 0.09 when . If the distance between mobile station and base station is 0.02 km, the maximum normalized CFO changing rate is around per 100 ODFM blocks.

The circled curve of Fig. 13 shows the BER performance for case 1. Since the CFO does not change within 100 blocks, this curve is the same as the circled curve in Fig. 12. The cross curve shows the BER performance for case 2, where the CFO changes within 100 blocks. We see that the results of

V. CONCLUSION

The CFO effect of an approximately MAI-free multiaccess OFDM transceiver was studied in this paper. We showed ana-lytically that the MAI due to CFO can be greatly reduced using either the symmetric or the antisymmetric codewords of the Hadamard–Walsh codes in the system. As a result, it does not demand sophisticated MUD or signal processing in achieving multiuser communications, and CFO compensation can be done at the receiver side. It was demonstrated by com-puter simulation that the proposed OFDM scheme has a much better performance than the OFDMA and the MC-CDMA sys-tems in a CFO environment.

APPENDIX

PROOF(33)

Suppose that and are uncorrelated. From (32), we have (43), shown at the bottom of the page. Since

we assume the crosscorrelation between is zero, i.e., for , we can rewrite (43) as

(44)

Using the assumption in (34), we have

(45)

(43)

Now, we will show the following equality:

(46)

Using the definition of , we have

(47)

Letting , the first term on the right-hand side of (47) can be manipulated as (48), shown at the bottom of the previous page. From (47) and (48), we can reach (46). Using (46), we can rewrite (45) as in (33).

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers for their constructive suggestions, which have significantly im-proved the quality of this work.

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**Shang-Ho Tsai (S’04) was born in Kaohsiung,**

Taiwan, R.O.C., in 1973. He received the B.S. degree in electrical engineering from Tamkang University, Taipei, Taiwan, in 1995, the M.S. degree in electrical and control engineering from National Chiao-Tung University, Hsinchu, Taiwan, in 1999, and the Ph.D. degree in electrical engineering form the University of Southern California, Los Angeles, in 2005.

**Yuan-Pei Lin (S’93–M’97) was born in Taipei,**

Taiwan, R.O.C., 1970. She received the B.S. degree in control engineering from the National Chiao-Tung University, Hsinchu, Taiwan, in 1992 and the M.S. and the Ph.D. degrees, both in electrical engi-neering, from the California Institute of Technology, Pasadena, in 1993 and 1997, respectively.

She joined the Department of Electrical and Control Engineering of National Chiao-Tung Uni-versity in 1997. Her research interests include digital signal processing, multirate filterbanks, and signal processing for digital communication, particularly the area of multicarrier transmission.

Dr. Lin is a recipient of the 2004 Ta-You Wu Memorial Award. She is cur-rently an associate editor for IEEE TRANSACTIONS ONSIGNALPROCESSING, the

*EURASIP Journal on Applied Signal Processing, and Multidimensional Systems*
*and Signal Processing.*

**C.-C. Jay Kuo (F’99) received the B.S. degree from**

the National Taiwan University, Taipei, Taiwan,
R.O.C., in 1980 and the M.S. and Ph.D. degrees
from the Massachusetts Institute of Technology,
Cambridge, in 1985 and 1987, respectively, all in
electrical engineering. He is with the Department
of Electrical Engineering, the Signal and Image
Processing Institute (SIPI), and the Integrated Media
Systems Center (IMSC), the University of Southern
California (USC), Los Angeles, as Professor of
electrical engineering and mathematics. His research
interests are in the areas of digital media processing, multimedia compression,
communication and networking technologies, and embedded multimedia
system design. He has guided approximately 60 students to their Ph.D. degrees
and supervised 15 postdoctoral research fellows. Currently, his research group
at USC consists of about 30 Ph.D. students and five postdoctoral students,
which is one of the largest academic research groups in multimedia
technolo-gies. He is a co-author of more than 700 technical publications in international
*conferences and journals as well as the following seven books: Content-based*

*Audio Classification and Retrieval for Audiovisual Data Parsing with T. Zhang*

*(Boston, MA: Kluwer, 2001), Semantic Video Object Segmentation for *

*Con-tent-based Multimedia Applications with J. Guo (Boston, MA: Kluwer, 2001),*
*Intelligent Systems for Video Analysis and Access Over the Internet with*

*W. Zhou (Englewood Cliffs, NJ: Prentice-Hall, 2002), Video Content Analysis*

*Using Multimodal Information with Y. Li (Boston, MA: Kluwer, 2003), Quality*
*of Service for Internet Multimedia with J. Shin and D. Lee (Englewood Cliffs,*

*NJ: Prentice-Hall, 2003), Radio Resource Management for Multimedia QoS*

*Support in Wireless Cellular Networks with H. Chen, L. Huang, and S. Kumar*

*(Boston, MA: Kluwer, 2003), and High Fidelity Multichannel Audio Coding*
(with D. T. Yang and C. Kyriakakis (Cairo, Egypt: Hindawi, 2004).

Dr. Kuo is a Fellow of SPIE. He received the National Science Foundation
Young Investigator Award (NYI) and Presidential Faculty Fellow (PFF) Award
*in 1992 and 1993, respectively. He is Editor-in-Chief for the Journal of Visual*

*Communication and Image Representation, and Editor for the Journal of*
*Information Science and Engineering and the EURASIP Journal of Applied*
*Signal Processing. He is also on the Editorial Board of the IEEE Signal*
*Processing Magazine. He served as Associate Editor for IEEE T*RANSACTIONS
ONIMAGEPROCESSINGfrom 1995 to 1998, IEEE TRANSACTIONS ONCIRCUITS
AND SYSTEMS FOR VIDEO TECHNOLOGY from 1995 to 1997, and IEEE
TRANSACTIONS ONSPEECH ANDAUDIOPROCESSINGfrom 2001 to 2003.