On the Performance of Multicarrier DS-CDMA
With Imperfect Power Control and
Variable Spreading Factors
Li-Chun Wang, Senior Member, IEEE, and Chih-Wen Chang, Student Member, IEEE
Abstract—Multicarrier direct-sequence code-division multiple
access (MC-DS-CDMA) becomes an attractive technique for the future fourth-generation (4G) wireless system because it can flexibly adapt transmission rates by changing both time and frequency spreading factors and possesses many physical-layer advantages in dispersive fading channels. However, power control errors (PCE) and the complete multiple access interference (MAI) from all the intersubcarriers may significantly degrade the per-formance of the MC-DS-CDMA system. In this paper, we propose an analytical method to evaluate the joint effects of the PCE and the complete MAI on the multirate MC-DS-CDMA system. From analysis and simulation, we obtain some important insights into the performance issues of the MC-DS-CDMA system. First, the effect of PCE can exacerbate the impact of the complete MAI on the MC-DS-CDMA system, or vice versa. ForBER = 10 3in a considered case, the joint effect of the complete MAI and PCE fur-ther degrades the performance by 2.1 dB compared with the sum of the degradation from the complete MAI and the PCE individu-ally. Second, increasing frequency- or time-domain spreading gain can improve the performance of the MC-DS-CDMA system, but the system also becomes more sensitive to power control errors. Third, a larger PCE can possibly make the frequency-domain diversity diminish faster than the gain obtained from the time-do-main spreading although an MC-DS-CDMA system with a larger frequency-domain spreading gain ( ) is usually better than that with a larger time-domain spreading gain ( ). In our example, for the standard deviation of PCE ( ) equal to 0 dB, the BERs with( ) = (4, 16) and (16, 4) are 9.3 10 4and 3.7 10 5, respectively, while for = 4 dB, the BER performances of the two cases are all in the order of 10 3.
Index Terms—Multicarrier direct-sequence code-division
mul-tiple access (MC-DS-CDMA), power control, power control errors.
I. INTRODUCTION
M
ULTICARRIER direct-sequence code-division multiple access (MC-DS-CDMA), a combination of multicarrier modulation and spread-spectrum, has become an important technique for the fourth-generation (4G) wireless systems [1]–[4]. The advantages of the MC-DS-CDMA system include the immunity to the dispersive fading channel, less peak-to-av-erage power ratio compared with the orthogonal frequency-Manuscript received April 1, 2005; revised October 1, 2005. This work was supported in part by the National Science Council and in part the Programdivision multiplexing (OFDM) system, and the flexibility of assigning spreading factors in both the frequency and time domains. Pure frequency spreading, pure time spreading, and joint time and frequency spreading are three spreading methods for the MC-DS-CDMA systems. It has been reported that MC-DS-CDMA using both time and frequency spreading codes outperforms the pure time or frequency spreading methods [5].
Although MC-DS-CDMA has been extensively studied recently [6]–[10], to our knowledge, the effects of power control errors (PCE) on MC-DS-CDMA seem to be neglected. Improper power control may significantly degrade the perfor-mance of the MC-DS-CDMA system. The key question is how to quantitatively analyze the impact of power control errors on the MC-DS-CDMA system. Unlike most current papers in the subject of MC-DS-CDMA assuming perfect power control, the first goal of this paper is to investigate the impact of open-loop power control errors on the MC-DS-CDMA systems. In gen-eral, power control schemes can be divided into two categories: the closed-loop power control and the open-loop power control. The former technique is to combat the small scale fading such as the Rayleigh-fading caused by multipath propagation, while the latter one aims to resolve the near–far effect, owing to the large-scale fading caused by path loss and shadowing, in a multiuser CDMA system. Note that with open-loop power control to overcome the near–far effect, each subcarrier may still experience severe multipath fast fading.
The second objective of this paper is to analyze the effect of the complete multiple access interference (MAI) on top of power control errors for the multirate MC-DS-CDMA system. To support various types of services in the 4G system, MC-DS-CDMA becomes an attractive technique because it can adapt transmission rates by dynamically changing time and fre-quency spreading factors. However, most analytical models for the MC-DS-CDMA system are mainly focused on the single rate case. In a multirate MC-DS-CDMA system, the MAI issue becomes more involved because the users with different transmission rates may cause different levels of the MAI. In addition, asynchronous transmissions among users is another
we are motivated to develop an analytical model to evaluate the effects of the complete MAI of asynchronous multirate users for the MC-DS-CDMA system.
In the literature, the studies on the performance of the MC-DS-CDMA system subject to power control errors and MAI can be summarized twofold. On the one hand, from the standpoint of MAI, the authors in [11] analyzed the MAI’s ef-fect from the main subcarrier for a single-rate MC-DS-CDMA system. In [12], the authors analyzed the effect of the other intersubcarriers’ MAI in the MC-DS-CDMA system, but only for the single-rate case. The authors in [13] analyzed the per-formance of the multirate multicarrier DS-CDMA systems, but considered only the effect of the MAI from the main subcarrier. On the other hand, from the view point of power control errors, most studies for the MC-DS-CDMA systems were performed with the assumption of perfect power control [11]–[13]. Al-though power control has been extensively studied for the single-carrier DS-CDMA system in the literature [14]–[18], extending these results from the single-carrier case to the multicarrier case is a nontrivial task because the effect of power control errors in an MC-DS-CDMA system has to be jointly evaluated with the complete MAI from all intersubcarriers.
In summary, in this paper, we develop an analytical model to characterize the joint effects of both the PCE and the com-plete MAI for the multirate MC-DS-CDMA system. Applying the developed analytical model, we obtain some important insights into the performance issues of the MC-DS-CDMA system. First, when both PCE and the complete MAI are jointly considered, the effect of PCE can exacerbate the impact of complete MAI on MC-DS-CDMA, or vice versa. That is, the joint impact of complete MAI and PCE is actually severer than the summation of the performance degradation from the complete MAI and PCE individually. Second, increasing the maximum total spreading gain by either increasing frequency-or time-domain spreading gain can enhance the errfrequency-or rate performance for the multirate MC-DS-CDMA system, but it is necessary to pay attention to the side effect of higher sensitivity to power control errors. Third, for the same total time and frequency spreading gain, an MC-DS-CDMA system with a larger frequency-domain spreading factor outperforms the system with a larger time-domain spreading factor. However, the performance differences between the two are shrunk as power control errors increase.
The rest of this paper is organized as follows. In Section II, we introduce the signal and system models of the MC-DS-CDMA system using both time- and frequency-domain spreading codes. In Section III, we derive the error rate probability for the multi-rate MC-DS-CDMA system taking into account of the complete MAI. In Section IV, we discuss the impact of PCE on the
multi-Fig. 1. The transmitter structure of the MC-DS-CDMA system using time-and frequency-domain spreading codes.
rate MC-DS-CDMA system. Numerical results are provided in Section V. Section VI gives concluding remarks.
II. SYSTEMMODEL
A. Transmitted Signal
The transmitter structure in the MC-DS-CDMA system using both time- and frequency-domain spreading codes is shown in Fig. 1. First, the serial-to-parallel process converts a data stream with the duration of to slower parallel with bit
dura-tion , where is the user index and the user
group will be defined later. After the serial-to-parrel processing, unlike that the original data stream is trans-mitted in a frequency-selective fading channel, each substream is now transmitted in a frequency flat (or nondispersive) fading channel. In the next step, the data in each substream are spread by a time-domain spreading code . Being copied to subcarriers, each substream is multiplied by a
frequency-do-main spreading code .
The transmitted signal of user in group can be ex-pressed in (1) shown at the bottom of the page, where and , respectively, represent the transmitted power and
the PCE, while and represent the subcarrier
frequency and the initial phase for the th carrier of the th substream, respectively. The waveform in the th substream consists of a sequence of independent rectangular pulses with duration , where with equal probability. The time-domain
spreading code
resents the chip sequence of the rectangular pulses of duration
, where with equal probability. Note that
the time-domain spreading factor of user in the group is .
The user group is defined as follows. Let and be the time-domain spreading code and frequency-domain spreading code of the reference user, respectively. As in [11], we categorize the interfering users into three groups.
1) Group A: the user utilizing a time-domain spreading code and a frequency-domain spreading code .
2) Group B: the user utilizing a time-domain spreading code and a frequency-domain spreading code .
3) Group C: the user utilizing a time-domain spreading code and a frequency-domain spreading code .
Noteworthily, (or ) can also be
the case with the same particular pseudonoise sequence offset by two different values.
B. Received Signal
Since each substream is sent in the flat Rayleigh-fading channel, the received signal of the reference user (denoted by ), can be expressed in (2) shown at the bottom of the page, where and , respectively, represent the transmitted power and the PCE of the reference user; and are the channel’s amplitude of the reference user and that of the th user in group , respectively; is the propagation delay for the reference user which is assumed to be without loss of generality; is the misalignment with respect to the ref-erence user, which is uniformly distributed in the time interval
; is the number of users in the group ; and is the white Gaussian noise with double-sided power spectrum
density of . In (2), and
are uniformly distributed
in , where and are the initial phase and the
channel’s phase of the reference user, and and are those for the th user in group , respectively.
The receiver structure of the MC-DS-CDMA system using time-domain and frequency-domain spreading codes is shown in Fig. 2. Without loss of generality, let the bit of interest be the first bit in the th substream of the reference user (denoted by ). After time-domain despreading, the output variable in
Fig. 2. The receiver structure of the MC-DS-CDMA system using time- and frequency-domain spreading codes.
the th subcarrier of the th substream for the reference user can be expressed as
(3)
where and are the transmit power and the bit duration of the reference user; is the weights for a certain com-bining scheme; and denote the MAI from the main subcarrier and that from other intersubcarriers, respectively; and is the white Gaussian noise with zero mean and variance
of , where is the bit
energy of the reference user. Combining subcarriers, the de-cision variable of for the reference user becomes
(4) where is given in (3).
C. Assumption
To make have the maximal diversity combining gain, we assume that the associated subcarriers in (4) experience independent fading. For this purpose, any two neighboring subcarriers in the same substream are at least separated by the maximal coherent bandwidth, denoted by . Denote the minimal frequency spacing between any two adjacent substreams with the same subcarrier index. Then, (the frequency of the th subcarrier of the th substream) can be determined by the following rule:
(5) where is the main carrier frequency. In order to achieve the maximum diversity combining gain in (4), the following condition:
(6) should be satisfied, where is the size of the serial-to-par-allel process. Note that the system bandwidth is proportional to . For a fixed total number of subcarriers ( ), should be large enough to satisfy the condition of (6). Oth-erwise, the assumption of independent subcarriers may become less realistic.
III. EFFECT OF COMPLETE MAI ON BIT-ERRORRATE(BER) PERFORMANCE
A. Motivation
The complete MAI in a multirate MC-DS-CDMA system is defined as the sum of the other users’s interference form their main subcarrier and all the other intersubcarriers. For any two synchronous transmissions at subcarriers and with the initial phase and , respectively, the orthogonality requirement
(7) can be fulfilled if the following condition is sustained:
(8) where is nonzero positive integer and is the chip duration.
Fig. 3. An illustrative example of intersubcarrier interference for asynchronous users, where the misalignment between the reference user o and the interfering user k in group X by .
However, for asynchronous users, the orthogonality condi-tion between subcarriers can not be held anymore. Assume that the reference user and the interfering user in group is mis-aligned by , where . As shown in Fig. 3, we con-sider the th chip of the reference user, during which the interfering user sends part of the th chip and part of the th chip. Recall that denotes the th chip of the spreading code of the reference user and denotes the th chip of the spreading code of the interfering user. Sup-pose that the reference and the interfering users transmit data at subcarriers and , respectively, and define the unit step
function as
(9) We can express the cross correlation between the two subcar-riers and during the th chip as shown in (10) at the bottom of the page, where is a nonzero value. Clearly, the orthogonality between subcarriers of the two users can no longer be ensured for asynchronously transmitted data. Thus, in addi-tion to the MAI from the main subcarrier in an MC-DS-CDMA system, it is also important to evaluate this kind of other inter-subcarriers’ MAI.
B. BER Performance
We denote the MAI from the main subcarrier by and the MAI from other intersubcarriers, by , respectively. As in [7] and [12], both types of MAIs are modeled by a zero-mean Gaussian random variable. Because can be obtained from by letting , we first analyze . To analyze the MAI in the case of multirate transmissions, we further consider two scenarios according to the relationship between (the bit
when
duration of the reference user) and (bit duration of the interfering user).
1) Other Intersubcarriers’ MAI From Low Data Rate Users: In this case . Since is uniformly distributed in the interval , may fall into two possible regions:
a) and b) . Clearly, the
probability of falling in region a) is
(11) and that of falling in region b) is
(12) Denote and the MAI from other intersubcarriers in cases a) and b), respectively. Let be the bit of interest. Referring to (3) and (10), the MAI from the th subcarrier of the th substream ( ) to the desired th subcarrier of the th substream ( ) can be expressed in (13) shown at the bottom of
the page, where is uniformly distributed
in
(14)
(15) Similarly, we can express as
(16) where
(17) Note that (16) only exists for the multirate transmissions, which is not seen in the single rate case. Because it is assumed that
the MAI can be approximated by a Gaussian distributed random variable, we express the variance of and as follows:
(18) and
(19)
The detail derivations of , , and
can be found in Appendices A and B. Note that for the flat Rayleigh-fading channel. The dis-tributions of power control errors among all the users are as-sumed to be the same. Denote the average power control error. After some derivations, we can have
(20) From (11), (12), (19), and (20), it is followed that:
(21) (22) 2) Other Intersubcarriers’ MAI From High Data Rate Users:
In this case, . Assume , where is
a positive integer. Similarly, be referring to (3) and (10), we express the MAI from the th subcarrier of the th substream ( ) to the desired th subcarrier of the th substream ( ) as follows in (23) shown at the bottom of the next page, where (24)–(25) are shown at the bottom of the next page. Note that (23) exists only for the multirate transmissions. Following the procedures of deriving (20), we obtain
(26)
3) Main Subcarriers’ MAI: Let in
(26). Referring to [12], we can obtain the variance of the main subcarriers’ MAI from the low or high data rate users as
Note that in (22), (26), and (27), the effects of multirate
transmissions and PCE are included in and ,
respectively.
C. The Statistics of the Decision Variable
Consider the maximum ratio combining scheme and choose the weighting factor ( ) as the complex conjugate of the channel response of the reference user ( ). The decision variable is also a Gaussian distributed random variable since the MAI is modeled by a Gaussian distributed random variable. Combining (3), (4), (26), and (27) and averaging over and , we can express the mean and the variance of with a frequency-domain spreading factor as follows:
(28) and
(29) where
(30) Define the received signal to noise ratio (denoted by ) as
(31)
Substituting (28) and (29) into (31), we can have
(32) where
(33)
is a parameter to characterize the joint effects of the complete MAI and power control errors and
(34)
is the central chi-square distributed random variable with degrees of freedom. From [19], the probability density function (pdf) of is given by
(35)
Like [13] and [20], we let in (29) and
(33), where and are the transmission rates of the refer-ence user and user in group , respectively. This implies that a user needs more power to support higher transmission rate.
For binary phase-shift keying modulation with coherent de-tection, the conditional error probability for a given and is equal to
(36)
(23)
(24)
where . Referring to [7], [15], and averaging (36) over the pdf of from (35), we can further simplify the conditional error probability for a given PCE as
(37)
IV. EFFECT OFPCEONBER PERFORMANCE
Now, we consider the effect of power control errors. Ac-cording to [15] and [21], the open-loop PCE can be modeled as a log-normally distributed random variable with standard
devi-ation in the decibels domain and mean ,
where . Let . Then, becomes a
normal distributed random variable with zero-mean and stan-dard deviation . Averaging of (37) over the pdf of , we can obtain the total error probability as in (38) shown at the bottom of the page, where
(39) Based on the Hermite polynomial approach of [22], the integra-tion for a funcintegra-tion can be computed by
(40)
where and are the abscissas and the weight factor of the Hermite polynomials with order , respectively. Letting
in (38), we can further simplify the total error proba-bility to (41) shown at the bottom of the page. For the single-carrier DS-CDMA, the influence of the PCE was investi-gated by means of the first-order Taylor expansion method in [15]. However, this method is only suitable for a small PCE ( dB) and small diversity order ( ). Our approach can evaluate the effect of a larger PCE (i.e., dB) and a higher diversity order for the multicarrier DS-CDMA system.
Unlike that the diversity order in the single-carrier DS-CDMA system means path diversity, the diversity order in the MC-DS-CDMA system means the frequency-domain diversity.
V. NUMERICALRESULTS
In this section, we apply the developed analytical models to evaluate the joint impact of the complete MAI and the PCE on the error probability and capacity performances of a mul-tirate MC-DS-CDMA system. In order to have the complete frequency diversity gain, the frequency separations for the subcarriers carrying the same data bit follow the assumption of (6). We adopt the following parameters in most examples unless they are defined again.
1) The total number of subcarriers is 512 ( ), where is the frequency-domain spreading factor and
is the size of the serial-to-parallel process.
2) Variable data rates are achieved by changing the time-domain spreading factors in a set of for a fixed frequency-domain spreading factor .
3) There exist total 12 asynchronous users in the system, where three equal-numbered groups are formed. These three groups transmit data rates with
frequency-do-main and time-dofrequency-do-main spreading factor ,
, , respectively.
4) The maximum total spreading gain is 256, i.e., .
5) The Hermite polynomials order used in (41) is 20. A. Discussions
Here, we first qualitatively discuss the the error probability by observing the received signal-to-noise and interference ratio (SINR) defined in (31). For the sake of convenience, we as-sume that the bit energy ( ) and the is large enough to ignore the influence of noise on the error probability. Note that
(i.e., is fixed).
Thus, we can rewrite (31) as
(42)
In this paper, variable data rates are achieved by changing the time-domain spreading factors in a set of
Fig. 4. Cumulative density functions of = G j j for (M; G ) = (4; 2) and (M; G ) = (8; 1).
for a fixed fre-quency-domain spreading factor . Therefore, the ratio of different transmission power is fixed for various values of
, i.e., is fixed. Note that
[13], [20]. According to (42), we summarize the impacts of the power control error ( ), the transmission powers ( and ), the frequency-domain spreading gain ( ), and the time-domain spreading gain ( ) on the error probability in the following.
1) With fixed and , a high data rate user can have better error rate performance because the high data rate user experience small amount of interference produced by low data rate users transmitting at lower power, i.e., is smaller.
2) With the same conditions as above, a larger time-domain spreading factor ( ) can lead to a lower error probability because of the larger spreading gain.
3) With fixed and , increasing the frequency-do-main spreading gain ( ) can result in a better error rate performance because of the larger frequency-diversity gain.
4) With fixed , a larger can make error probability more sensitive to power control errors because the larger can magnify a minor change of . Thus, a better error rate performance thanks to a larger comes at the cost of being more sensitive to power control errors. 5) Similar to the above phenomenon, a large can also raise
the sensitivity of the error probability to power control errors, although larger frequency-domain spreading gain can result in a lower error probability.
6) With a fixed product of , frequency-domain spreading ( ) in the MC-DS-CDMA system is more sensitive to power control errors than time-domain
spreading ( ). To compare and (8, 1),
we let . Fig. 4 shows the
cumula-tive density functions of for and (8,
1). From the figure, one can find the 90th percentile of
for is 4.6 which is larger than 3.3 for
Fig. 5. The impact of the joint effect of complete MAI and PCE on the error rate performance of the MC-DS-CDMA system with(M; G ) = (8; 8), (8,16), and (8,32), whereM 2 max(G ) = 256 and the standard deviation of PCE = 1:5 dB.
. Thus, the case of
results in larger diversity gain than .
That is, the case of can amplify a small
change of more than the case of .
Hence, is also more sensitive to power
control errors.
B. Impact of Complete MAI and PCE on Bit Error Probability Fig. 5 shows the joint effect of complete MAI and PCE on the error rate performance of the MC-DS-CDMA system. Assume that 12 asynchronous users be equally partitioned into three groups with the frequency-domain and time-domain spreading factors , (8, 16), and (8, 32), respectively. First, without power control, we have following important observations.
• Compared with the effect of the main-subcarrier-only MAI, the complete MAI from all the intersubcar-riers significantly degrades the BER performance. For
10 and , the required
is 10 dB in the main-subcarrier-only MAI case, while the required is increased to 12 dB in the complete MAI case.
• High data rate users have better BER performance com-pared with low data rate users. This observation confirms the first point of qualitative analysis in Section V. • Note that both the main-subcarrier-only MAI and the
complete MAI result in a floor of error probability in the region of high . In the former case, the error floor is about 1.95 10 , while in the latter case the error floor is increased to 1.47 10 .
Now, we consider the impact of PCE. In the figure, the stan-dard deviation of PCE dB. For comparison, we also show the case with the single user and perfect power control (the curve with the legend “Neither complete MAI nor PCE”) and the case with the single user and imperfect power control (the curve with the legend “PCE only”). We observe that PCE exacerbates
Fig. 6. The impact of PCE on the error rate performance of the MC-DS-CDMA system with a fixed frequency-domain spreading factors (M = 8) and various time-domain spreading factors G = 8, 16, and 32 for M 2 max(G ) = 256, 512, and 1024, respectively, when E =N = 25 dB. the effect of the complete MAI. That is, the increase of the re-quired due to the joint PCE and complete MAI is indeed higher than the summation of those due to PCE only and com-plete MAI only individually. For 10 , the required are 8.2, 9.1, 13.8, and 16.8 dB for the cases of “Neither complete MAI nor PCE, ” “PCE only,” “Complete MAI only,” and “Joint complete MAI and PCE, ” respectively. Compared with the based-line case of “Neither complete MAI nor PCE, ” the effect of power control errors increases the required by 0.9 dB (i.e., the difference of 8.2 and 9.1 dB), while the effect of the complete MAI increases the required by 5.6 dB (i.e., the difference of 8.2–13.8 dB). However, the joint effect of the complete MAI and PCE increases the required by 8.6 dB (i.e., the difference of 16.8 and 8.2 dB). As a result, the joint effect of complete MAI and PCE further degrades the per-formance by 2.1 dB compared with the sum of the degradation from complete MAI only and PCE only individually.
C. Impact of Various Time-Domain and Frequency-Domain Spreading Factors
Fig. 6 shows the impact of various time-domain spreading factors and PCE on the multirate MC-DS-CDMA system
when frequency-domain spreading factor and
dB. The error rate performances of three
groups of spreading factors for , 512, and
1024 are compared. To ease illustration, we only show the per-formances with the highest data rate in each group, i.e., (8,8),
Fig. 7. The impact of PCE on the error rate performance of the MC-DS-CDMA system with a fixed time-domain spreading factors (G = 16) and various frequency-domain spreading factors M = 4, 8, and 16 for M 2 max(G ) = 256, 512, and 1024, respectively, when E =N = 25 dB. of transmission power from surrounding multirate users. In
Fig. 6, the group with means that four
users are allocated with the rates , (8, 16), (8, 32), respectively. Thus, the user with in this group belongs to the low rate user and will experience the interference from three low-rate users with ,
four medium-rate users with , and four
high rate user with . Similarly, a user with
in the group with
is classified to a high rate user, who will be interfered by three other high-rate users with , four medium-rate
users with , and four low rate users with
. Due to less interference, a user with in the group of
indeed outperforms the user with in the
group of . Our developed analytical
model (i.e., (34)) can accurately calculate the MAI in the mul-tirate MS-DS-CDMA system. Furthermore, due to the reason of the second point of the qualitatively analysis in Section V,
the high rate user with in the group of
can outperform the high rate user with
in the group of , as
shown in Fig. 5.
However, an MC-DS-CDMA system with a larger time-do-main spreading factor also becomes more sensitive to PCE.
When and increasing from 0 to 4 dB, the
error probability is increased by about an order of 10 (i.e.,
Fig. 8. The impact of PCE on the error rate performance of the MC-DS-CDMA system with various combinations of(M; G ) = (4; 16), (8,8), and (16,4), whereM 2 max(G ) = 256 and E =N = 25 dB.
( ) and changing the frequency-domain spreading fac-tors when dB. Consider three sets of spreading
factors for , 512, and 1024. Here, we
only show the performance with for
, (8, 16) for , and (16, 16)
for . In the figure, as stated in the third
and fifth points of the qualitative analysis in Section V, a larger maximum total spreading gain by increasing the frequency-do-main spreading factor results in better performance due to a larger frequency-domain diversity gain but the sensitivity of the system to PCE also becomes higher. For example, when and increases from 0 to 4 dB, the error probability changes from the order of 10 to 10 , whereas for the error probability changes from the order of 10 to 10 . When the frequency-domain spreading factor ( ) increases, the ag-gregated PCE’s among subcarriers become quite significant, and therefore reduce the advantage of frequency diversity over the system with a smaller frequency-domain spreading factor.
Interestingly, by comparing Figs. 6 and 7, one can find that frequency-domain spreading ( ) in the MC-DS-CDMA system is more sensitive to power control errors than time-do-main spreading ( ). This phenomena can be explained by the sixth point of the qualitative analysis. To further confirm the above interesting observation, we perform simulations and evaluate the BER of MC-DS-CDMA systems according to (38) to illustrate the impact of PCE ( ) with various combinations of frequency-domain and time-domain spreading factors, as shown in Fig. 8. We let the maximum total spreading gain in this example. First, it is shown that the Hermite approach can accurately match both the analytical integration results and the simulation results. Second and more
importantly, comparing the result of with
, we find that the performance improvement thanks to a larger frequency spreading factor is reduced as increases. For dB, changing the spreading factor
from to can improve
the bit error rate 10 to 10 .
Fig. 9. The impact of complete MAI and PCE on the capacity of the multirate MC-DS-CDMA system withf(M; G )g = f(8; 8); (8; 16); (8; 32)g.
However, in the case of dB, 10
for and 10 for
. Although the system with a larger frequency-domain spreading factor still performs better than that with a smaller frequency spreading factor, the difference between the two becomes quite insignificant in the presence of a large power control error.
D. Impact of Complete MAI and PCE on Capacity
Fig. 9 shows the impact of the complete MAI and PCE on the capacity of the multirate MC-DS-CDMA system with time-do-main and frequency-dotime-do-main spreading factors in the group of
, and dB. In the
figure, the total number of users is three times the abscissa and the only the curve of is drawn for each case.
For a given 10 requirement and dB, the
numbers of acceptable users in each group are eight and five for the main-subcarrier-only MAI and the complete MAI, re-spectively. The capacity decreases by 37.5%. However, for PCE dB and the complete MAI, is the number of acceptable users becomes two. That is, the joint effect of the PCE and the complete MAI can further decrease the capacity by 60% (from 5 to 2). Clearly, the effect of PCE worsens the impact of the complete MAI on the capacity of the multirate MC-DS-CDMA system. This phenomenon can be explained by observing the parameter (denoting the average power control error ) in of (33), where the joint effect of PCE and complete MAI is analyti-cally modeled by the term
.
VI. CONCLUDINGREMARKS
In this paper, we have developed an analytical model to quan-titatively evaluate the performance of multirate MC-DS-CDMA systems using time- and frequency-domain spreading codes subject to the PCE and the complete MAI from all subcarriers. Applying the developed analytical model, we have obtained the following observations.
• When both PCE and the complete MAI are jointly consid-ered, the effect of PCE can exacerbate the impact of com-plete MAI on the MC-DS-CDMA system, or vice versa. • A larger maximum total spreading gain either by
in-creasing frequency- or time-domain spreading factors can enhance the performance of the multirate MC-DS-CDMA system. However, the sensitivity of these performance gains to power control errors cannot be ignored.
• A larger PCE ( ) can possibly make the frequency-do-main diversity gain diminish faster than the gain obtained from the time-domain spreading. Thus, although for the same maximum total time and frequency spreading gain, the MC-DS-CDMA system with a larger frequency-do-main spreading factor results in better performance than that with a larger time-domain spreading factor, the per-formance difference between the two become less signif-icant as power control errors increase.
The developed analytical model and the above observations can provide some important insights into the performance is-sues of multirate MC-DS-CDMA systems. Possible interesting research topics that can be extended from this work include: 1) to analyze the multirate MC-DS-CDMA system under other types of fading channels [12], [23] and other distortion [24] sub-ject to power control errors and complete MAI and 2) to de-velop resource allocation mechanisms including code assign-ment, subcarrier allocation [25], scheduling [26], and power control schemes [27] for the multirate MC-DS-CDMA system.
APPENDIX I
In this appendix, we derive . To ease the
notation, let and . Then, we can calculate
by
(43)
Recall that in the case of ( ), is uniformly
distributed in . Hence, we assume that
, where and is the
time-do-main spreading factor of the reference user. Following the same procedure as [7], we have (44) shown at the bottom of the page. Then, the expectation of (44) with respective to over can be obtained by calculating (45)–(46) shown at the bottom of the page. Similarly, we can obtain
(47) (48)
APPENDIX II
Here, we derive . Let and
without loss of generality. Then, can be cal-culated as
(49) Recall that and are uniformly distributed in and
, respectively. Assume , where
; and are the time-domain
spreading factors of the reference user and the interfering user, respectively. Then, referring to (10), we can express as
(50) where
(51)
(56)
(57)
After some deductions, we obtain
(52)
where
(53) (54) Taking the expectation of respective to over ,
, and , where and are assumed to be
with equal probability, we can have
(55)
Finally, substituting (55) into (49), we can get by calculating (56)–(57) shown at the top of the page.
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Li-Chun Wang (M’96–SM’05) received the B.S.
degree from the National Chiao Tung University, Hsinchu, Taiwan, R.O.C., in 1986, the M.S. degree from National Taiwan University, Taipei, Taiwan, R.O.C., in 1988, and the M.Sci. and Ph.D. degrees from the Georgia Institute of Technology, Atlanta, in 1995, and 1996, respectively, all in electrical engineering.
From 1990 to 1992, he was with the Telecommu-nications Laboratories, Ministry of Transportations and Communications, Taiwan (currently the Telecom Laboratories of Chunghwa Telecom Company). In 1995, he was affiliated with Bell Northern Research of Northern Telecom, Inc., Richardson, TX. From 1996 to 2000, he was with AT&T Laboratories, where he was a Senior Technical Staff Member in the Wireless Communications Research Department. Since August 2000, he has been an Associate Professor in the Department of Communica-tion Engineering, NaCommunica-tional Chiao Tung University. He holds a U.S. patent and three more pending. His current research interests are in the areas of cellular ar-chitectures, radio network resource management, cross-layer optimization, and cooperation wireless communications networks.
Dr. Wang was a corecipient of the 1997 IEEE Jack Neubauer Best Paper Award for his paper “Architecture Design, Frequency Planning, and Perfor-mance Analysis for a Microcell/Macrocell Overlaying System,” with G. L. Stuer and C.-T. Lea, in the IEEE TRANSACTIONS ONVEHICULARTECHNOLOGY(best systems paper published in 1997 by the IEEE Vehicular Technology Society). Currently, he is serving as an Associate Editor for the IEEE JOURNAL ON SELECTEDAREAS INCOMMUNICATIONS: Wireless Communications Series.
Chih-Wen Chang (S’02) received the B.S. and M.S.
degrees in electrical engineering from the National Sun Yat-Sen University, Kaohsiung, Taiwan, R.O.C., in 1998 and 2000, respectively, the Minor M.S. degree in applied mathematics and the Ph.D. degree in communication engineering from the National Chiao-Tung University, Hsinchu, Taiwan, in 2005 and 2006, respectively.
His current research interests include wireless communications, wireless networks, and cross-layer design.
