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An Approximately MAI-Free Multiaccess OFDM

System in Fast Time-Varying Channels

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

Abstract—A new multiuser orthogonal frequency division

mul-tiplexing (OFDM) transceiver with preceding, called the precoded multiuser OFDM (PMU-OFDM) system, was recently introduced by Tsai et al. (2005). The PMU-OFDM system can reduce multiac-cess interference (MAI) due to the carrier frequency offset (CFO) to a negligible amount by preceding the data of each user with a codeword selected from either even or odd Hadamard–Walsh codes. The performance of the PMU-OFDM system in a mobile environment, where the channel response varies within one OFDM symbol due to the Doppler effect, is evaluated in this paper. It is shown by analysis and simulation that the use of even or odd Hadamard–Walsh codewords can greatly reduce the MAI effect due to the Doppler effect as well. Generally, all Hadamard–Walsh codewords except for the all-one codeword result in consider-able suppression of intercarrier interference (ICI) induced by the Doppler effect. Furthermore, we discuss codeword priority schemes to minimize ICI as much as possible. Finally, simulations results are given to demonstrate the advantage of PMU-OFDM over OFDMA with a codeword priority scheme in the Doppler environment. We show that the PMU-OFDM system with odd Hadamard–Walsh codewords outperforms a half-loaded OFDMA system in a mobile environment.

Index Terms—Carrier frequency offset (CFO), Doppler effect,

fast fading, multiaccess interference (MAI), multiuser orthogonal frequency division multiplexing (OFDM), precoding, precoded multiuser OFDM (PMU-OFDM), time varying.

I. INTRODUCTION

O

RTHOGONAL frequency division multiplexing (OFDM) provides a promising technique for high data rate trans-mission applications, since an OFDM system can combat the intersymbol interference (ISI) effect due to frequency-selective fading with a simple transceiver structure [2], [4]. OFDM has been used in wireless applications such as wireless local Manuscript received May 25, 2006; revised October 12, 2006. The associate editor coordinating the review of this manuscript and approving it for publica-tion was Prof. Brian L. Evans. This work was supported in part by the Integrated Media Systems Center, a National Science Foundation Engineering Research Center, Cooperative Agreement No. EEG-9529152. This work was presented in part at the 2005 Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA. Any opinions, findings, and conclusions or recommenda-tions expressed in this material are those of the authors and do not necessarily reflect those of the National Science Foundation.

L. Tadjpour 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: tadjpour@usc.edu; cckuo@sipi.usc.edu).

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

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/TSP.2007.894260

area network standards (i.e., the IEEE 802.11 family) and the metropolitan area network standards (i.e., the IEEE 802.16 family). OFDM converts a frequency-selective fading channel into parallel flat fading channels by transmitting a block of information symbols in parallel on subcarriers. Through a careful design, a broadband channel can be efficiently shared by overlapping subchannels that have the minimum frequency separation required to maintain orthogonality while the fading effect can be mitigated by one-tap equalizers in the receiver. To mitigate effects of ISI and intercarrier interference (ICI) caused by the delay spread, each OFDM block is preceded by a cyclic prefix (CP) whose length is at least equal to the channel length.

Historically, OFDM has been designed for applications with little mobility. However, mobile OFDM technology has recently attracted a lot of attention for three reasons. First, it is desirable to provide high-quality broadband services in a mobile environ-ment when we consider the next generation wireless communi-cation system. Second, emerging wireless communicommuni-cation sys-tems are expected to lie in higher spectral bands so that they are more sensitive to the physical movement of users and their surroundings. Third, more subchannels are needed to enhance OFDM bandwidth efficiency, which implies the use of a longer OFDM block. Then, the effective channel variation rate over one OFDM block increases.

Multiuser OFDM systems have been developed to meet the need of wireless multiaccess, including time division multi-access OFDM (TDMA-OFDM), multicarrier code division multiple access (MC-CDMA) [8], and orthogonal frequency division multiple access (OFDMA) [10]. MC-CDMA inherits the ISI-resilient property from OFDM. However, its capacity is limited by multiaccess interference (MAI) in frequency-se-lective fading channels. OFDMA has been adopted by the IEEE 802.16e standard, also known as WiMax. In OFDMA, a subset of subcarriers is assigned to each user. The division of subcarriers among users is usually called frequency division multiple access (FDMA) [12]. To mitigate the frequency-selec-tive fading effect for each user, subcarriers assigned to a user can be spread across the spectrum [12]. OFDMA is MAI-free when time and frequency are perfectly synchronized. However, it is sensitive to the carrier frequency offset (CFO) and the Doppler spread, which are common in time-varying channels. The mismatch of oscillators in the transceiver or the rapid variation of the channel over one OFDM symbol destroys the orthogonality among subcarriers, and results in ICI and MAI in multiuser OFDM systems.

Since ICI degrades the performance of OFDM systems, its suppression has been intensively studied. Time domain 1053-587X/$25.00 © 2007 IEEE

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Fig. 1 Block diagram of the PMU-OFDM system.

equalizers designed based on the criterion of minimum mean squared errors (MMSE) were proposed to mitigate ICI in [15]. Lower complexity MMSE-based equalization techniques were considered in [3] and [17]. In [18] and [23], each data symbol is mapped into a group of subcarriers with predefined coefficients so that ICI among subcarriers within each group can be cancelled.

Furthermore, a time-varying channel during an OFDM block of an user degrades the performance of this user and induces MAI for others. There is little work on MAI suppression caused by the Doppler spread in MC-CDMA or OFDMA. Without effi-cient MAI suppression in a mobile environment, one may resort to a more complicated multiuser detection (MUD) technique for symbol detection, whose computational complexity increases exponentially with the number of active users [22]. Thus, a mul-tiuser OFDM system that is robust to the Doppler spread effect is very desirable.

A new multiuser OFDM transceiver, called the precoded mul-tiuser OFDM (PMU-OFDM), was introduced and analyzed in [19]. The PMU-OFDM system has three interesting properties. First, when the number of parallel transmit symbols is suf-ficiently large, it is approximately MAI-free. Second, PMU-OFDM is robust to time asynchronism. Hence, low-complexity Hadamard–Walsh codes can be used in uplink transmission. Third, PMU-OFDM is approximately MAI-free in the presence of CFO due to the mismatch of oscillators in the transceiver using even or odd Hadamard–Walsh codewords.

In this paper, we will investigate the performance of PMU-OFDM in the presence of time-varying channels. We will prove that PMU-OFDM reduces MAI due to the Doppler spread to a negligible amount with even or odd Hadamard–Walsh code-words. We will also show that PMU-OFDM employing a subset of the Hadamard–Walsh codewords with many sign changes has self-ICI cancellation in a Doppler environment. We will compare PMU-OFDM with OFDMA, and demonstrate that PMU-OFDM with odd Hadamard–Walsh codewords outper-forms half-loaded OFDMA in a Doppler environment.

The rest of this paper is organized as follows. The PMU-OFDM system and the Rayleigh time-varying channel model are reviewed in Section II. We analyze the MAI effect of PMU-OFDM due to the Doppler effect and show how to achieve the approximately MAI-free property in Section III, where the ICI effect resulting from the Doppler spread is also discussed. We also discuss the codeword priority scheme for PMU-OFDM and show that users using Hadamard–Walsh codewords with a higher sign change number experience less

ICI. The Monte Carlo simulation results are presented in Section IV and performance comparison between PMU-OFDM and OFDMA is conducted. Finally, concluding remarks are given in Section V.

II. SYSTEMMODEL

In this section, we provide a brief review to the PMU-OFDM system [19] whose block diagram is shown in Fig. 1. The input of user is an vector . Each symbol in is repeated by times to form an vector as

(1) In the next stage, is multiplied by a diagonal matrix of size whose diagonal elements are obtained by repeating the th column of an real Hadamard matrix

by times, i.e.,

(2) where is the transpose of the th column of . Since is an orthogonal matrix, the following property holds:

(3)

where is the th diagonal element of diagonal

matrix and . After passing

through , the resultant vector can be written as (4) where is the frequency index. Next, each coded vector is passed through the -point inverse DFT (IDFT) matrix. Then, following the parallel-to-serial conversion, CP are inserted at the beginning of each OFDM symbol. The th transmitted signal over a block is thus given by

(5) These symbols are fed to the multiple access channel. We con-sider the uplink scenario where each user experiences a different

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channel but under the assumption of synchronous communica-tion. The time-varying channel impulse response for any user can be expressed by [16]

(6) where and denote time and delay, respectively, is the maximum length of channel impulse response, is the channel coefficient gain at time for the th tap, and is the Kronecker delta function. Coefficient is assumed to be a complex Gaussian random variable with zero mean and unit variance. Usually, is equal to , where and are random variables with the Rayleigh and the uniform distributions, respectively. Note that is the Doppler frequency for every channel path and it is the only time-varying factor in this model. The Doppler spread effect is caused by the difference in Doppler frequencies of dif-ferent channel paths. If is the same for all channel paths, we have only the Doppler shift but not the Doppler spread. The latter is often used to model a time-invariant channel with CFO. Also, the typical wide-sense stationary uncorrelated scat-tering (WSSUS) channel model is adopted [11], [16]. Then, we have

(7) where is the variance of the th tap and is the time-autocorrelation function. One classic time-autocorrelation function is the Jakes model [11], which can be written as

(8) where is the zeroth-order Bessel function of the first kind, is the maximum Doppler frequency, and is the sampling rate. The Jakes model is adopted for the time-correlation func-tion throughout this paper. If the CP durafunc-tion is chosen such that , the received signal after removing the cyclic prefix is given by

(9) where is the discrete-time additive white Gaussian noise, is the channel complex coefficient of user , and is the number of multiple access users. Next, each block of size is converted from serial to parallel and then passed through the DFT matrix. The output of DFT can be expressed as

(10) To detect the symbol transmitted by the th user, the

output vector of DFT is multiplied by and the resultant output, denoted by , is averaged over symbols such that

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where . By substituting (5), (9), and (10) into (11), we have (12) where (13) and (14) By (1), we get (15)

where , with and

. Since all symbols of an individual user use the same

Hadamard–Walsh code, we have and

. By defining

(16) and using (15), we can rewrite (12) as

(17) Finally, to mitigate the fading effect on received symbols, is passed through a frequency equalizer (FEQ), which will be explained later.

Even though PMU-OFDM may appear to be similar to MC-CDMA in the system diagram, we should emphasize that the summation in the PMU-OFDM receiver is preformed before the gain multiplication, which is different from that of MC-CDMA systems. This difference will lead to different performance in the multipath fading channel or a CFO environ-ment [19]. For more detailed performance comparison between MC-CDMA and PMU-OFDM, we refer to [19].

III. ANALYSIS OFPMU-OFDM UNDER THEDOPPLEREFFECT The overall Doppler effect in a mobile environment can be divided into two parts. One is the interference from symbols of other users due to the time-varying channel while the other is the interference from neighboring subcarriers (i.e., Doppler ICI) of the same user to result in symbol distortions. They are called the Doppler MAI effect and the self-Doppler effect, respectively. We analyze the performance of PMU-OFDM, and derive its av-erage MAI and ICI expressions in this section.

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A. Analysis of Doppler MAI

From (17), the MAI to the th symbol of user due to user , denoted by MAI , is given by

MAI

(18)

The averaged MAI power is MAI . Let

where

(19) Assuming that and are uncorrelated, we can get from (18) that

MAI

(20) It is also assumed that cross correlation between is zero;

i.e., . Then, by (7) and (8), the

averaged power of MAI can be found as MAI

(21) Consider the case , where the resulting MAI is denoted by MAI . Then, from (21), the average power of MAI is given by

MAI

(22) It is known [9] that Hadamard-Walsh codewords can be di-vided into two groups of codewords. One is the set of even codewords satisfying

(23) and the other is the set of odd codewords given by

(24) In the following, we will prove that if only even or odd codewords of Hadamard–Walsh codewords are used,

MAI can be reduced to a negligible amount. Then, we observe that MAI is the dominating MAI by computer simulation in Section IV. Thus, by reducing MAI to a negligible amount, the total MAI is also reduced substantially.

Theorem: Suppose that only the even or the odd codewords of Hadamard–Walsh codewords are used, MAI is approximately zero if the normalized Doppler frequency is less than .

Proof: It can be easily shown that

, (also, see [21]). Then, we can rewrite (22) such that it includes this term. To do so, we can use the following:

(25) where can be any function. Note that if , we have

(26) Also, even and odd codewords have the following property [19]: (27)

Let . Using (25)–(27), for ,

can be written as

(28) By substituting (28) in (22) and making some rearrangement, we can obtain the average power of MAI as

MAI

(29) where

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Fig. 2 Approximation ofJ (z) by 1 0 ((1=4)z )=((1!) ) for z < 1.

Now, we need to show that is a constant function so that

MAI , for .

The zeroth-order Bessel function of the first kind can be ex-panded by the following power series [1]:

(31) As shown in Fig. 2, we can approximate for

by . By substituting this approximation for

with ,

(or, equivalently, ), we

obtain

(32)

where . Based on the previous

approxi-mation, we can express in (29) as

(33)

It can easily be shown that

. Also, we know from [7] that

(34)

Therefore, for and , we

get

(35) By plugging (35) in (33)

which is independent of and . By substituting this approxi-mate value of back to (29), we obtain

MAI

(36)

One can show , if the even

or odd Hadamard–Walsh codewords are used. For the detailed proof, we refer to [21]. Finally, we have

MAI if (37)

Equation (37) shows that PMU-OFDM is approximately MAI-free in a Doppler environment. This result is actually similar to that in a CFO environment [19]. We may consider a simple example for the illustration purpose. When

and , PMU-OFDM is approximately MAI-free when the normalized maximum Doppler frequency is less than . For the carrier frequency 4 GHz and the

sampling frequency 2 MHz, this maximum

Doppler frequency is equivalent to a mobile speed of 81 km/h. Since MAI is the dominating MAI in a time-invariant CFO environment [19], the suppression of MAI reduces the total MAI due to CFO greatly. We will show by simulation in Section IV that MAI is the dominating MAI in a

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Doppler environment as well so that the total MAI can be sup-pressed significantly in a Doppler environment by suppressing MAI using odd or even Hadamard–Walsh codewords.

B. Analysis of Doppler ICI and Symbol Distortion

In this section, we investigate the impairment caused by user’s self Doppler effect. Let in (17). The th detected symbol of user with no other active user is given by

(38) We can rewrite (38) as

ICI ICI (39)

where ICI , and ICI are discussed later. The distortion factor is obtained by putting and in in (38), i.e.,

(40)

for , and . If , then

. Note that

since , and is

equivalent to . Therefore, can be written as (41) That is, for every time index and any user , we take the DFT of channel coefficient over path with frequency index . The result is then averaged over one OFDM symbol to yield . When , another interpretation of is

(42)

where is the average

of the th channel tap over one OFDM block. Therefore, represents the DFT of . Often, subcarriers are much longer than the channel length and can be computed from (42). In Section IV, we assume this condition is held by adopting and in Examples 3 and 4.

We can mitigate the symbol distortion effect in the receiver using a frequency domain equalizer whose one tap gain for user is set to . To this end, channel estimation for varying channel must be preformed. When the channel is time-varying within an OFDM block, the preamble-based training method may not work well. Periodic insertion of training sym-bols during transmission of every block has been suggested for OFDM in time-varying channels. It was shown in [5] that the best set of frequency domain pilot tones are those which are equally spaced. We adopt this technique to estimate the fast

fading channel. Let be the number of equally spaced pilot

tones at subchannels , for . An

estimate of can be obtained at pilot tones via

ICI ICI

(43) Then, the estimate of is obtained through an IDFT of length as

(44) More sophisticated algorithms were suggested to improve the channel estimation performance in a Doppler environment. For example, two ICI mitigation techniques were proposed in [6] to improve the channel estimation performance in the presence of the Doppler effect. Other channel estimation methods for OFDM in time-varying channels were reported in [13] and [14]. The term ICI in (39) is the interference from subcarriers with and to the desired subcarrier ; namely

ICI

(45) Finally, the term ICI in (39) is the sum of all interferences from subcarriers , i.e.,

ICI

(46) Since the ICI average power can be obtained in a manner similar to that in deriving the MAI average power, we omit the deriva-tion and simply show the result. That is, the average power of ICI is equal to

ICI

(47) We can rewrite (25) for as

(48)

Let . Then

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By substituting (49) in (47) and using the approximate formula (32) for the Bessel function and (35), we have

ICI

(50)

The double summation term in (50) is found to be

all-one codeword

otherwise (51)

since the left term of (51) is equal to

In addition, is equal to for the all-one codeword and is 0 for all other codewords. From (50) and

(51), ICI , when

the all-one codeword is not used and .

On the other hand, if the all-one codeword is used, we

have ICI .

For instance, if ICI

for the all-one codeword and ICI for other codewords.

We can also derive the average power of ICI , denoted by

ICI

(52) As compared with the average power of ICI , the average power of ICI is negligible for all codewords for practical values of and . This can be clearly explained by the fol-lowing example.

Example: Let

, and . Then, ICI 40.5 dB

for the all-one codeword, and ICI 64 dB for other codewords. Using (52) and the same set of parameters, we compute the value of ICI to be about 10 dB for the user with the all-one codeword decibels, and between 25 and 50 dB for all other users. Clearly, ICI is much larger than the corresponding values of ICI .

Since ICI is the dominant ICI, we only need to consider this ICI effect and ignore ICI . It will also be shown later in this section and in Section IV that the total interference from subcarriers of user , i.e., ICI , with the all-one codeword is much larger than that of other codewords. It will also be shown in Section IV that the averaged ICI power depends on the assigned Hadamard–Walsh codeword.

C. Codeword Priority Schemes for ICI Cancellation

Since ICI is the dominant ICI due to the Doppler spread effect, we would like to further investigate this term in this section. It is found that codewords with a higher number of sign changes tend to lead to a smaller ICI value. This can be explained as follows. Expression in (52)

can be divided into four terms: ,

, ,

and .

Let .

Since , and by using (49), (52) can be written as ICI Define and Since , for , and

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, and by using the ap-proximate formula (32), we can express ICI for

by

ICI

(53)

where denotes the real part. We prove in the Appendix that the real part of is equal to . Based on this result, we can rewrite the ICI average power as

ICI

(54)

where is the imaginary part and can be obtained by (51) as

all-one codeword otherwise

The quantity can be interpreted as the autocorrelation of codeword . Since is a monotonically de-creasing or inde-creasing function of for given as shown in the Appendix, we can characterize the ICI values qualitatively

by based on (54). If , has a

suffi-cient number of sign changes, the

term in (54) is likely to be cancelled out after the summation over all , which leads to a smaller ICI value. Since

, for , there is a relation between the number of sign changes in the codewords and the number of sign changes in . For example, let . For the all-one codeword, is that has no sign change. For the second codeword,

with seven sign changes, is that

has six sign changes. Finally, for the seventh codeword , the associated is

and they both have two sign changes.

With the previous observation, we can adopt the following rule of thumb for codeword selection in a PMU-OFDM system: “To give a higher priority to codewords that have a higher

Fig. 3 Dominating and the residual MAI curves as a function of the normalized Doppler frequency.

number of sign-changes.” This result is similar to PMU-OFDM in the presence of a pure CFO environment [21]. Simula-tion results about the code priority scheme will be shown in Section IV.

IV. SIMULATIONRESULTS

Simulation results are presented in this section to corroborate the theoretical results derived in previous sections. In the simu-lation, we choose carrier frequency 4 GHz and sampling frequency 2 MHz. The maximum Doppler fre-quency is related to the mobile speed via , where is the speed of light and is the mobile speed. Channel co-efficients are allowed to change during one OFDM block, and they have the Rayleigh distribution with a unit variance at the same path but different time indices. The classical Jakes Doppler spectrum [11] is adopted for the time-varying fading channel, and the binary phase shift keying (BPSK) modulation scheme is considered. The simulation is conducted by the Monte Carlo method.

Example 1: Suppression of Dominating MAI: In this

ex-ample, we show that MAI is the dominating MAI as derived in Section III, and then, we investigate its suppression by using even or odd Hadamard–Walsh codewords. The

simula-tion parameters are , and . The average

MAI for user from all other users, denoted by MAI , is calculated as follows. For the th symbol of any target user, we accumulate the MAI contributed from all the th symbols of other users. This procedure is repeated for all symbols, and the MAI power is then averaged for ;

i.e., MAI MAI . This is

conducted for more than 400 000 symbols in the simulation. Similarly, the total MAI from all other symbols

of other users to user , denoted by MAI , is obtained by

averaging the value MAI .

The total MAI is plotted as a function of the maximum normal-ized Doppler frequency in Fig. 3, where MAI and MAI of all 16 users are shown by 16 solid curves and 16 dashed

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Fig. 4 Dominating and residual MAI versus the normalized Doppler frequency when onlyM=2 even or odd Hadamard–Walsh codewords are used.

curves, respectively. The solid bold curve in Fig. 3, denoted by MAI , is the average value of 16 solid curves and obtained by MAI . Similarly, the dashed bold curve, denoted by MAI , is the average value of 16 dashed curves and ob-tained by MAI . We see from this figure that the average MAI is about 10 dB more than the average MAI . Thus, we can view MAI and MAI as the dominating MAI and the residual MAI, respectively.

Now, we consider a case where only even (or only odd) Hadamard–Walsh codewords are used. The MAI performance is shown for both even and odd codewords in Fig. 4. By comparing results in this figure and Fig. 3, we see that the dominating MAI is reduced by 35–58 dB. In contrast, the residual MAI is only decreased by 5–6 dB due to the de-creased user number from 16 to 8. The results confirm that the dominating MAI can be greatly reduced using only even or odd codewords so that the total MAI, which is equal to MAI MAI , is reduced considerably.

Example 2: Doppler ICI: In this example, we consider the

ICI effect due to the Doppler spread. As mentioned previously, we focus on ICI for user , whose average power is calculated as follows. For the th symbol of the target user, we accumulate the ICI from all other symbols of the same user. The av-erage ICI power is then avav-eraged for ; namely,

ICI ICI . Fig. 5 shows the average

ICI power for each individual Hadamard–Walsh codeword used

in PMU-OFDM when , and . The

maximum Doppler frequency was chosen to be , which corresponds to the user speed of 54 km/h. We see that different users experience a different amount of ICI. Especially, the user that employs the all-one codeword suffers more ICI than all others by 15–40 dB. Intuitively, if a codeword has more sign changes, the main lobes of interfering subcarriers may cancel each other so that the ICI power is decreased. This is similar to the self-ICI cancellation technique used to mitigate ICI in OFDM [23]. Since the all-one codeword belongs to the set of

Fig. 5 ICI power as a function of user index for even and odd codewords withf T = 1 2 10 .

even Hadamard–Walsh codewords, we recommend to choose the set of odd Hadamard–Walsh codewords first, since they have the approximately MAI-free property and a lower average ICI power at the same time.

In the next several examples, we compare the performance of the PMU-OFDM and the OFDMA systems. For fair compar-ison, we keep the size of IDFT/DFT of both systems the same, i.e., NM, and consider both fully loaded and half-loaded situa-tions. For a fully loaded OFDMA system, subchannels indexed

by and are

as-signed to user . In other words, each user occupies subchan-nels which are maximally separated. To simulate the half-loaded PMU-OFDM system, the set of odd (or even) Hadamard–Walsh codewords is used. For the half-loaded OFDMA system, the th user is assigned subchannels with indices

, and . The remaining subchannels are used as the guard band.

Example 3: Performance Comparison of PMU-OFDM to OFDMA in the Doppler Environment: In this example, we

com-pare the performance of odd and even Hadamard–Walsh

code-words for PMU-OFDM with parameters ,

and . The ICI power and the MAI power for OFDMA and PMU-OFDM are plotted as functions of the normalized Doppler frequency in Fig. 6, where the average ICI power is the averaged value of eight multiple access users (i.e., ). We see that the second largest amount of interference is the ICI of PMU-OFDM with only even codewords. The reason is that the all-one codeword in the set of even codewords has a high ICI value as compared to the ICI or the MAI of all other users. On the other hand, the ICI value of PMU-OFDM with the set of odd Hadamard–Walsh codewords is only about 4 dB more than that of OFDMA. Note that subcarriers assigned to a particular user in OFDMA are spread uniformly across the available bandwidth. Hence, ICI results in less impairment in OFDMA than in a single-user OFDM. However, for OFDMA, the MAI power is significantly higher than that of PMU-OFDM. As shown in Fig. 6, we observe that the MAI value of PMU-OFDM with either even or odd Hadamard–Walsh codewords is about

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Fig. 6 Comparison of the MAI power and the ICI power as a function of the normalized Doppler frequency.

Fig. 7 BEP comparison for PMU-OFDM and OFDMA as a function of the normalized Doppler frequency.

10 dB less than that of OFDMA. Thus, we expect PMU-OFDM with only odd codewords to outperform OFDMA in the bit error probability (BEP).

Under the setting of , and

30 dB, simulation results on the BEP are shown in Fig. 7, where the BER performance is plotted as a func-tion of the Doppler frequency for six systems: fully loaded PMU-OFDM, fully loaded OFDMA, half-loaded PMU-OFDM with even codewords, half-loaded PMU-OFDM with odd codewords, PMU-OFDM with even codewords, excluding the all-one codeword, and half-loaded OFDMA. In this simulation, we assume that perfect channel knowledge is available at the receiver. An FEQ can be used to compensate the symbol distortion effect, i.e., the detected symbol is multiplied by . Thus, both systems only suffer from MAI and ICI.

Fig. 8 BEP comparison for PMU-OFDM and OFDMA as a function of the SNR value.

We see that fully loaded PMU-OFDM and fully loaded OFDMA have comparable performance for . For that corresponds to the mobile speed of 54 km/h with respect to our chosen parameters, OFDMA outperforms PMU-OFDM since OFDMA is MAI-free if time or frequency asynchronism is negligible. For half-loaded systems, we see that PMU-OFDM with odd codewords yields much lower BEP than PMU-OFDM with even codewords and OFDMA in the Doppler environment with the maximum normalized Doppler frequency ranging from to , corresponding to a mobile speed between 5.4 and 162 km/h. The poorer perfor-mance of half-loaded PMU-OFDM with even codewords is due to high ICI of the user with the all-one codeword. Thus, if we exclude the all-one codeword, PMU-OFDM with seven even codewords outperforms all other systems as shown in Fig. 7.

Now, let us consider nonperfect channel estimation in a time-varying channel. Fig. 8 compares the BEPs for the six scenarios given in Fig. 7 and two more scenarios with channel estimation. That is, we no longer assume perfect knowledge of channel at the receiver. The channel estimation was performed using (43) and (44). We used eight pilots in every OFDM block. The BEP results of all eight scenarios are plotted as a function of the SNR value , with the maximum normalized Doppler frequency fixed at . For fully loaded cases, the BEP curves of PMU-OFDM and OFDMA are close to each other. For half-loaded cases, the BEP curves of PMU-OFDM with even codewords and OFDMA are also close with each other, while PMU-OFDM with odd codewords outperforms the previous four cases considerably. PMU-OFDM with seven even codewords (excluding the all-one codeword) perform much better than half-load PMU-OFDM with all eight even codewords due to the high ICI value of the user with all-one codeword. Also, we see from Fig. 8 that the channel estima-tion penalty is about 1.5–2.0 dB for 20 dB. The performance gap between half-loaded PMU-OFDM with odd codewords and half-loaded OFDMA, however, remains the same with realistic channel estimation.

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TABLE I

ICI AVERAGEPOWER INDECIBELS FOR16 USERS

Fig. 9 BEP performance comparison as a function of the user number for OFDM and OFDMA, where the codeword priority scheme is adopted for PMU-OFDM.

Example 4: Codeword Priority: It was mentioned in

Section III that codewords with a higher number of sign changes in general have a lower amount of ICI. We conduct simulations to verify this statement here. System parameters are

chosen to be , and . The maximum

normalized Doppler frequency and the SNR value were fixed at and 30 dB, respectively.

The ICI power for 16 codewords is shown in Table I, where the corresponding number of sign changes is also provided. It is a general trend that codewords with more sign changes have less ICI.

Next, we examine how PMU-OFDM and OFDMA perform when the active number of users varies from 1 to . The codeword priority scheme for PMU-OFDM is stated later. When the system load is less than 50%, we use codewords with a higher number of sign changes among the odd codewords. When the system load is more than 50%, we choose all odd codewords plus an even codeword predetermined to have a low average ICI value based on the results given in Table I. The procedure is repeated until all 16 codewords are selected. We plot BEP as a function of the user number for PMU-OFDM and OFDMA in Fig. 9.

From Fig. 9, we see that PMU-OFDMA significantly outper-forms OFDMA in a lightly loaded (50% or less) system. When the system reaches its full loading, the BEP performance of PMU-OFDM and OFDMA becomes comparable.

V. CONCLUSION

The performance of an approximately MAI-free multiaccess OFDM transceiver, called the PMU-OFDM system, in a mobile environment was analyzed in this paper. It was shown that the MAI due to the Doppler spread can be greatly reduced using ei-ther even or odd codewords of Hadamard–Walsh code-words in the system. We also investigated the ICI performance of PMU-OFDM in the Doppler environment and showed by simulations that, excluding the all-one codeword, the average ICI power of PMU-OFDM is comparable with that of OFDMA. The codeword selection priority scheme for PMU-OFDM was also discussed. Finally, we demonstrated that PMU-OFDM has a significant performance gain over OFDMA for a wide range of system loading in a mobile environment in terms of BEP formance. As the system loading becomes higher, such a per-formance gain becomes narrower. For a fully loaded system, PMU-OFDM and OFDMA have comparable BEP performance.

APPENDIX

Our objective is to prove that the real part of is constant while the imaginary part of is a monotonic function of for a given value of . Let , where

.

First, we show that the real part is constant. The real part of can be written as

(55) With (34), we get

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if if (63) Since and for and , we have . Next, we prove that the imaginary part of is a monotonic function of given . The imaginary part of can be written as

(57) Based on the following equality given in [7]:

(58)

we can express as

(59) To prove is a monotonic function of given , we show that its derivative with respect to is either negative or positive for any given . By taking the derivative of

with respect to , we get

(60)

To understand the behavior of , we have

(61) and

(62)

for and . Thus,

and . For all

positive values of , i.e., ,

but can take either negative or positive values.

If . On the other hand, if

. Then, we conclude from (61) and (62) that (63), shown at the top of the page, holds. As a result, is an increasing function of if

and a decreasing function of if . Similarly, for , we can prove that is a decreasing function

of if and an increasing function of if

.

REFERENCES

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Layla Tadjpour (S’07) received the B.S. degree in electrical engineering from the Iran University of Science and Technology, Tehran, Iran, in 1996 and the M.S. degree in electrical engineering from the University of California, Los Angeles, in 1999. Cur-rently, she is working towards the Ph.D. degree at the Ming-Hsieh Department of Electrical Engineering, University of Southern California, Los Angeles.

From 1999 to 2005, she was with the Jet Propul-sion Laboratory, Pasadena, CA. Her current research interests are in the area of signal processing for com-munications, multicarrier systems, and mobile communications.

Shang-Ho Tsai (S’04–M’06) was born in Kaohsiung, Taiwan, in 1973. He received the B.S. degree in electrical engineering from the Tamkang University, Tamsui, Taiwan, in 1995, the M.S. degree in electrical and control engineering from the National Chiao-Tung University, Hsinchu, Taiwan, in 1999, and the Ph.D. degree in electrical engineering from the University of Southern Cali-fornia, Los Angeles, in 2005.

From 1999 to 2002, he was with the Silicon Inte-grated Systems Corporation (SiS), Taiwan, where he participated in the very large scale integration (VLSI) design for DMT-ADSL systems. From 2005 to 2007, he was with MediaTek, Inc. (MTK), Taiwan, where he participated in the VLSI design for MIMO-OFDM systems. In 2007, he joined the Department of Electrical and Control Engineering, National Chiao-Tung University. His research interests include signal processing for commu-nications, particularly in the areas of multicarrier systems and space-time pro-cessing. He is also interested in VLSI design for those topics.

Dr. Tsai was awarded a government scholarship for overseas study from the Ministry of Education, Taiwan, in 2002–2005.

C.-C. Jay Kuo (F’99) received the B.S. degree from the National Taiwan University, Taipei, Taiwan, in 1980 and the M.S. and Ph.D. degrees from the Massa-chusetts Institute of Technology, Cambridge, in 1985 and 1987, respectively, all in electrical engineering.

He was Computational and Applied Mathematics (CAM) Research Assistant Professor at the Depart-ment of Mathematics, University of California, Los Angeles, from October 1987 to December 1988. Since January 1989, he has been with the Univer-sity of Southern California, Los Angeles, where currently, he is a Professor of Electrical Engineering, Computer Science and Mathematics, and the Director of the Signal and Image Processing Institute. His research interests are in the areas of digital signal and image processing, multimedia compression, communication and networking technologies. He is the coauthor of about 120 journal papers, 680 conference papers, and seven books.

Dr. Kuo is a Fellow of The International Society for Optical Engineering (SPIE). He is the Editor-in-Chief for the Journal of Visual Communication and

Image Representation, and Editor for the Journal of Information Science and Engineering, LNCS Transactions on Data Hiding and Multimedia Security, and

the EURASIP Journal of Applied Signal Processing. He received the National Science Foundation Young Investigator Award (NYI) and Presidential Faculty Fellow (PFF) Award in 1992 and 1993, respectively.

數據

Fig. 1 Block diagram of the PMU-OFDM system.
Fig. 2 Approximation of J (z) by 1 0 ((1=4)z )=((1!) ) for z &lt; 1.
Fig. 3 Dominating and the residual MAI curves as a function of the normalized Doppler frequency.
Fig. 4 Dominating and residual MAI versus the normalized Doppler frequency when only M=2 even or odd Hadamard–Walsh codewords are used.
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