Transmitter and receiver optimization in multicarrier CDMA systems ...
IEEE TRANSACTIONS ON COMMUNIATIONS, VOL. 48, NO. 7, JULY 2000
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Transmitter and Receiver Optimization in Multicarrier CDMA Systems Tat M. Lok, Member, IEEE, and Tan F. Wong, Member, IEEE
Abstract—In this paper, we consider transmitter and receiver optimization in multicarrier code-division multiple-access (MCCDMA) systems under Rayleigh fading channels. Receiver optimization is performed in a decentralized manner, while transmitter optimization can be performed through either centralized or decentralized control of the powers of different carriers. Results show that when the number of users is smaller than or equal to the number of carriers, each transmitter often tends to concentrate its power on a different carrier which does not suffer deep fading. The MC-CDMA system then tends to a frequency-division multiple-access system with near-optimal frequency assignment. When the number of users gets large, each user tends to choose more than one carrier, which do not suffer deep fading, while interference suppression is performed across the chosen carriers by the corresponding receiver. Index Terms—Code-division multiple access, fading channels, multicarrier modulation, power control, transmitter–receiver optimization.
I. INTRODUCTION
R
ECENTLY, different multicarrier code-division multipleaccess (MC-CDMA) systems [1], [2] have been proposed and investigated. In [3], an MC-CDMA system with direct sequence spreading is considered to derive the benefits of both multicarrier modulation and direct sequence spreading. In [4], a blind adaptive receiver with interference suppression is proposed for the MC-CDMA system in [3]. In this paper, we assume that the knowledge at the receiver can be sent back to the transmitter for optimization of transmission. The transmitter varies the amplitude and the phase of each carrier according to the feedback information. Joint transmitter–receiver optimization has been considered by a number of researchers. In [5] and [6], joint optimization is approached under the general multiple-input–multiple-output (MIMO) framework. This general development has been specialized to tackle the problem of crosstalk in high-speed digital subscriber line applications (see [7] and [8], for example). Recently, a similar joint transmitter–receiver optimization apPaper approved by J. Wang, the Editor for Wireless Spread Spectrum of the IEEE Communications Society. Manuscript received April 16, 1999; revised August 30, 1999 and December 1, 1999. This work was supported in part by the Research Grant Council of Hong Kong. The paper was presented in part at the IEEE Wireless Communications and Networking Conference (WCNC’99), New Orleans, LA, September 1999. T. M. Lok is with the Department of Information Engineering, Chinese University of Hong Kong, Shatin, Hong Kong (e-mail: [email protected]). T. F. Wong is with the Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611 USA (e-mail: [email protected]). Publisher Item Identifier S 0090-6778(00)06161-4.
proach is proposed for wireless CDMA systems in [9]. In the context of CDMA, the approach of joint precoding and demodulation of data from all users is adopted in all the works mentioned. Although this approach can give better performance, it may not be practical in multimedia wireless communication, where users may often need to change their coding schemes to satisfy different quality-of-service (QoS) requirements. Joint precoding and joint demodulation in these situations could be very difficult. Another shortcoming of the general MIMO approach when applied to multimedia applications is that the optimization criterion is the total mean squared error (MSE) of all the users. While minimizing this criterion ensures an average performance level over all the users, it does not address the different QoS requirement characteristics of multiuser-multimedia communication. In this paper, we consider decentralized demodulation. Each receiver demodulates the data of a particular user without trying to demodulate the data of other users. Since the signals from other users are treated as (structured) noises, the criterion for receiver optimization is the signal-to-noise ratio (SNR) of that particular user (or equivalently the MSE of the user). Transmitter optimization is performed, either in a centralized or a decentralized manner, by varying the powers of the carriers of the users so that all users achieve their own target SNR’s, if it is possible. Different from [5]–[9], we consider the minimization of the total transmission power required to achieve the target SNR’s of the individual users. The coding and modulation schemes of the users are not involved in the optimization. The result is that different users can employ different data signal constellations and different error control coding schemes. The system is well suited for multirate communication. Actually, the coding scheme and the signal constellation of a user can be changed without affecting the performance of other users. In Section II, we define the system model. In Section III, we briefly discuss receiver optimization with the SNR criterion. In Section IV, we consider transmitter optimization in details. We derive the necessary and sufficient condition for simultaneous users to achieve their target performance. We also determine the optimal transmission scheme in additive white Gaussian noise (AWGN) channels. In general, closed-form solutions for optimal transmission schemes are difficult to determine in fading channels. Instead, we consider a centralized adaptive algorithm based on the method of Lagrange multiplier to solve the optimization problem. We also develop a decentralized algorithm which is obtained as an approximation of the centralized algorithm. In Section V, we use numerical examples to illustrate the performance of the optimized system. Conclusions are drawn in Section VI.
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Fig. 1.
IEEE TRANSACTIONS ON COMMUNIATIONS, VOL. 48, NO. 7, JULY 2000
Decentralized receiver for each user in the MC-CDMA system.
II. SYSTEM MODEL In this section, we describe the model of the MC-CDMA simultaneous users in the system. We assume that there are carriers. system, and each user uses the same , generates a stream of data The th user, for given by symbols (1) are random variables and are assumed to The data symbols . The data symbols need be normalized so that not be independent or even uncorrelated. Each user can apply different modulation and coding schemes to the data stream according to the type of information and the required performance. Users can also change their schemes without affecting other users. , is provided a random signature The th user, for given by sequence
therefore, equals . When , the signal on each carrier serves as a mask is not spread, and the signature sequence , the signal to shield off the data from other users. When becomes a direct-sequence multicarrier signal as defined in [3] and [4]. We now describe the channel model. We assume that the channel is a frequency-selective fading channel. By suitably and the bandwidth of [3], we can assume choosing that each carrier undergoes independent frequency-nonselective slow Rayleigh fading. We also assume the presence of AWGN . with power spectral density For simplicity, the signals in the system are assumed to be synchronized although the results can be readily generalized to an asynchronous system. The received signal in complex analytic representation is given by
(2)
(4)
is the spreading factor on each carrier, and the elwhere are modeled as independent and identically disements tributed (i.i.d.) random variables such that . The same signature sequence is used carriers of the th user. to modulate each of the The transmitted signal of the th user can be expressed as the real part of the following complex signal:
accounts for the overall effects of phase shift, path where loss, and/or fading for the th carrier of the th user, and represents AWGN. These fading coefficients are assumed to be invariant within the time interval for optimization.
(3) is the frequency of the th carrier, and is chosen where by the th transmitter to vary the amplitude and the phase of the th carrier. We assume that the chip waveform is bandlimited and the carrier frequencies are well separated so that adjacent frequency bands do not interfere with each other. We also is normalized so that . assume that The parameter is the delay between consecutive chips. Each chips on a carrier, where data symbol is modulated by is the spreading factor on each carrier. The symbol interval ,
III. RECEIVER OPTIMIZATION The optimal centralized receiver should perform joint detection (multiuser detection) for all users. Moreover, the statistics collected by the centralized receiver can be used for transmitter optimization. Despite its optimality, the optimal multiuser detector often requires too many computations in most practical cases. The situation is further complicated if the users transmit different types of information requiring different data rates and if the users are allowed to vary their coding schemes. We consider simpler decentralized receivers with interference suppression capabilities. Each receiver demodulates a different user signal according to the signal and the noise statistics collected by itself. We consider the receiver shown in Fig. 1 for the th user. It consists of branches. Each branch consists of a
LOK AND WONG: TRANSMITTER AND RECEIVER OPTIMIZATION IN MC-CDMA SYSTEMS
correlator and an appropriate weight, and is responsible for the demodulation of one carrier. The correlator on the th branch consists of a chip-matched filter and a combiner that combines the contributions from different chips according to the signature sequence of the th user. We assume that the chip waveform and the chip-matched filter are chosen to satisfy the Nyquist criterion so that there is no interchip interference. The weight vector is an -dimensional vector that combines the contributions from the branches to give the decision statistic for the th user. We consider receiver optimization without transmitter optimization by choosing the appropriate weight vectors for all users. Since we are considering decentralized receivers, optionly affects the performance of the th user. We can mizing consider optimizing the receiver for each user independently. Without loss of generality, we consider receiver optimization . The output for the first user during the symbol interval of the correlator on the th branch, due to the first user signal, where is given by (5) . We define -dimensional vectors by . These vectors can be expressed in as terms of the gain vectors
for
(6) is an diagonal matrix whose th diagonal where . element is The output of the correlator on the th branch, due to the th , is given by user signal, for (7) We also define -dimensional vectors . We denote the output of the correlator on the th branch due to AWGN by , and similarly define an -dimensional vector . The overall output of the correlators, in vector form, is given by (8) By the assumption that other noises are independent of the user is uncorrelated with for all . Moreover, it is signals, are uncorrelated for different easy to check that the vectors . Therefore, the noise and interference correlation matrix is given by E (9) denotes the conditional expectation given , for and , the superscript denotes the Hermitian operation, and denotes the identity matrix. We
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determine the optimal weight vector that maximizes the SNR defined by
E (10) In [4], it is shown that the optimal weight vector is given by (11) and can be determined by the first receiver In general, using a training sequence without the help of other receivers. , the results in [4] show that and can also When can be determined blindly by the first receiver. Actually, be obtained directly as a generalized eigenvector of a pair of matrices, which, in turn, can be estimated from the outputs of some appropriate filters. It can also be obtained directly from the outputs of the filters through a stochastic gradient descent algorithm. IV. TRANSMITTER OPTIMIZATION The results in the last section apply to any receiver by replacing the index 1 with the index of the receiver. When the th , its output SNR receiver employs its optimal weight vector is given by [4] (12) is the noise and interference matrix experienced by where the th user. We assume that each user, according to his or her type of information and modulation-coding scheme, requires certain target SNR performance. We can consider transmitter for so that the optimization by choosing total transmitted power for all users to achieve the target performance is minimized Minimize
(13)
subject to is the target SNR for the th user. We investigate this where key optimization problem in three steps. First, we derive the necessary and sufficient condition for the existence of a feasible solution. Then, we determine the optimal solution in some idealized cases. Finally, we investigate some practical adaptive methods for solving the general problem. A. Necessary and Sufficient Condition If there are a large number of users requiring high SNR targets, there may not be a feasible solution, i.e., a choice , that satisfies all the requirements. We first of consider the condition for the existence of a feasible solution. To this end, we define the total correlation matrix by
where E
(14)
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Then, for (15) Using the matrix inversion lemma, it is readily shown that (16)
The term on the left-hand side can be loosely called the signal-to-total-power ratio (STR) for the th user. Notice that is a monotonically increasing function of , and . The optimization problem can be rewritten as Minimize
where the operator takes the diagonal of a matrix to form a row vector, the inequality is interpreted as element-by-element , and . We comparisons, orthonormal would like to find that satisfies (21). We pick vectors of dimension . Each element of each vector should distinct have the same magnitude. An example would be any columns of the -dimensional discrete Fourier transform mais a trix (or the -dimensional Walsh–Hadamard matrix if multiple of 4). The vectors are scaled appropriately to form the matrix . It is readily verified that the constraints in (21) can be satisfied. The necessary and sufficient condition can be used to determine the capacity of the system in terms of the maximum number of allowable users. Suppose that the SNR requirement of each user is . Then, from (19), the number of users that can be supported is given by
(17) (22)
subject to where
, , and is 10 dB, then For example, if be smaller than 16.5, i.e., there can be at most 16 users.
must
(18) B. AWGN Channel Proposition 1, whose proof can be found in the Appendix, provides the necessary condition for the existence of a feasible solution to this problem. Proposition 1: If a feasible solution exists, then (19) The necessary condition can be used by a central controller to determine whether new users can be admitted to the system. Clearly, if the addition of a new user results in the violation of the necessary condition, the user should not be admitted to the system. We note the necessary condition in (19) is always , since for by satisfied if (18). As shown in Proposition 2, whose proof is also provided in the Appendix, the necessary condition is also sufficient to guarantee the existence of a feasible solution. Proposition 2: If the condition in (19) is satisfied, then a feasible solution exists. for To construct a feasible solution, we need to determine , so that the constraints in (17) are satisfied. The can then be determined readily corresponding gain vectors from (6). The proof of Proposition 2 illustrates how to construct a feasible solution in general. We consider a simple example , the target SNR’s , and they where satisfy (19). In this case of equal target SNR’s, the construction matrix so that the th is especially simple. Define the . Then row of the is (20) by (14). The constraints in (17) can be rewritten as (21)
Closed-form solutions for the optimization problem (17) can be very difficult to obtain except for some special cases. We and consider the special case of an AWGN channel, i.e., for . The problem can be divided into and . For , it is obvious that two cases: by assigning different carriers to different users, we can separate different users signals. Each user signal is then corrupted only by AWGN, which is clearly the best possible scenario. Actually, for are orthogonal, the same as long as situation holds. We state the result as the following proposition. , minimization is achieved in an Proposition 3: For for are orthogonal AWGN channel when vectors. is more complicated. Some users The situation for will necessarily experience interference from other user signals. In this case, the optimal scheme, for a large collection of different cases, is given by Proposition 4. A constructive proof is provided in the Appendix. , if for Proposition 4: For , minimization is achieved in an AWGN channel when the columns of are appropriate orthonormal vectors up to a constant. , if for , the For resulting power required is then given by the following formula: (23)
, , As an example, we consider the case where , dB for . By Propositions 1 and 2, the targets are achievable, i.e., with no spreading on the signal on each carrier, 16 users can be supported with 15 carriers where each user achieves an SNR of 10 dB. By Proposition 4, the columns of should be chosen as orthonormal vectors up
LOK AND WONG: TRANSMITTER AND RECEIVER OPTIMIZATION IN MC-CDMA SYSTEMS
to a constant. One possible choice is to pick all columns, except the first one, of the 16-dimensional Walsh–Hadamard matrix as after appropriate scaling. We point out the the columns of interesting fact that the rows of , which are the signal vectors for the users at the receiver, then form a simplex set. By (23), to achieve an SNR of 10 dB, the signal-to-white-noise ratio (SWNR)1 for each user has to be about 14.77 dB.
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Then, we consider an iterative approach to seek a stationary and are point of the Lagrangian function. At each step, updated according to the following relationships: (28) (29)
C. Fading Channel In most wireless communication channels, propagation loss and fading cannot be ignored. In the presence of different path losses and fading, it may be very difficult to obtain closed-form solutions for the optimization problem. Instead of trying to find exact closed-form solutions, we solve the optimization problem by the following adaptive method. We consider the Lagrange multiplier method [10] and incorporate the SNR requirements into a penalty function. We form the Lagrangian function given by
while a i.e., a gradient descent algorithm is used to update gradient ascent algorithm is used to update . We note that in order to implement this adaptive algorithm, we need to have the information from all the users. Therefore, this method is centralized in nature. In many cases, a decentralized optimization method is preferred. To this end, we can approximate the gradient descent and ascent in (28) and (29), respectively, by
(30) (31)
(24) where is the Lagrange multiplier. The Lagrangian function is considered as a function of and the components of for . The derivative of with respect to (w.r.t.) is the requirement for the target SNR’s of all the users. The derivative of w.r.t. can be obtained as follows. First, notice is not a function of while for can be that (and, hence, ) via the expressed explicitly as a function of matrix inversion lemma.
(25)
for
where (26)
Therefore, the derivative of
w.r.t.
is given by
individually. for each An interesting result has been observed from simulations of , the Lagrange multiplier method the algorithms. For often gives a solution where each user is essentially assigned a different carrier. The system then tends to a frequency-division multiple-access (FDMA) system. Therefore, we are also . In an interested in the optimal FDMA system when FDMA system, each user is assigned a different carrier. We define the optimal FDMA system as the one where the total power to achieve the target performance for all users is min, the problem of imized. Given the channel coefficients finding the optimal FDMA system can be identified as an assignment problem, which can be solved efficiently by the Hungarian method [11]. A brief outline of the method is included in the Appendix. Detailed descriptions of the method can be found in [12] with a graph theoretic approach and in [13] with a matrix approach. In spite of our observation, it should be pointed out that the following situation is conceivable. Consider a system with carriers and users. For a particular carrier, say the first carrier, for the signals from all users suffer deep fading, i.e., are very small. In this case, an MC-CDMA system with our algorithms may not tend to an FDMA system. It is because, in our algorithms, carriers are not actually assigned. In this extreme case when the same carrier fails for all users, the users could just share the remaining carriers. Of course, then, the system would not look like an FDMA system anymore. On the other hand, this extreme situation can hardly be observed (e.g., used in some of because, even for moderate the examples), the probability that all users suffer deep fading at the same carrier is very small. V. PERFORMANCE
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1See
Section V for definition.
In this section, we consider the performance of the Lagrange multiplier method in Rayleigh fading channels. We avoid the complication of different geographical distributions of different users and assume all path losses to be unity. We model
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Fig. 2. Typical performance of the centralized Lagrange multiplier method (uniform target SNR’s).
Fig. 3. Typical performance of the decentralized Lagrange multiplier method (uniform target SNR’s).
for and as i.i.d. zero-mean unit-variance complex Gaussian random variables so that the magnitude of each coefficient is Rayleigh distributed. We first consider the typical performance of the Lagrange multiplier method in achieving the target performance. Fig. 2 shows the result of a simulation run for the centralized optiand eight users mization algorithm with eight carriers . No spreading is performed on the signal on each carrier. The target performance for each user is 8 dB. As shown in the figure, the SNR of each user converges to the target. Fig. 3 shows a similar result for the decentralized optimization
algorithm except that it takes longer for the users to attain the 8-dB target. Figs. 4 and 5 show the case with nonuniform target SNR’s. Four of the users require 8-dB targets while the other four require 6-dB targets. Both the centralized and the decentralized optimization algorithms again give the desired result. Next, we consider the performance of the system as both and the number of users increase. the number of carriers , , and the target SNR for In each case, each user is 8 dB. The performance measure is the average transmitted signal-to-white-noise ratio (SWNR), defined by E , of each user needed to achieve the
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Fig. 4. Typical performance of the centralized Lagrange multiplier method (nonuniform target SNR’s).
Fig. 5. Typical performance of the decentralized Lagrange multiplier method (nonuniform target SNR’s).
8-dB SNR target. We note that the SWNR is directly proportional to the total transmitted power from all the users. As both and increase, we expect that the power for each user to achieve the target performance should decrease due to the diversity provided by the increased number of carriers. Fig. 6 shows the average result of 500 simulations. The performance determined by the (centralized and decentralized) Lagrange multiplier method is shown along with the performance determined by optimal FDMA (Hungarian method). As expected, the average power required to achieve the target performance and increase. We notice that there is a decreases as both
difference of 0.5–1 dB between the results obtained by the optimal FDMA method and the centralized Lagrangian method. It is partly due to the slow convergence of the Lagrange multiplier method as the solution approaches the optimal value,2 and is partly due to the fact that the algorithm may converge to local minima in some realizations. Moreover, there is another 1–1.5-dB difference between the centralized and decentralized Lagrangian methods. 2In Figs. 6–8, the iterative process in the Lagrange multiplier method is terminated once the targets are achieved. Therefore, the values indicated in the figures are higher than the minimum power needed.
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Fig. 6. Performance of the centralized and decentralized algorithms for K =
Fig. 7. Performance of the centralized algorithm for M = 8, N
M
.
0 1 2 and 4. ;
;
In Fig. 7, we consider the performance of the systems with eight and . carriers as the number of users increases for Only the centralized algorithm is considered in this case. As expected, the power required increases as increases until a point (depending on ) at which the 8-dB target cannot be reached. , the power only increases slowly. The centralized For Lagrange multiplier method again nearly produces the optimal FDMA solutions. The small difference of 0.5–1 dB between the results obtained by the optimal FDMA method and the centralized Lagrangian method is again due to the convergence problems
mentioned above. Notice that an average SWNR of only about 5 dB is required to achieve a target SNR of 8 dB for each user. The saving is due to the effect of diversity provided by the different carriers. Each user tends to choose a carrier with the largest gain. , interference from other users generally cannot be For avoided. The minimum power required to achieve the target performance is significantly increased. However, we can slow down this increase by increasing the spreading factor . To compare the difference between the centralized and decen, we consider the system with tralized algorithms for
LOK AND WONG: TRANSMITTER AND RECEIVER OPTIMIZATION IN MC-CDMA SYSTEMS
Fig. 8.
Performance of the centralized and decentralized algorithms for
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M = 8 and N = 2.
eight carriers and . The SWNR required by each of the two algorithms to achieve the 8-dB target for different values of is plotted in Fig. 8. The small difference of 0.5–1 dB between the results obtained by the optimal FDMA method and the centralized Lagrangian method is due to the slow convergence of the Lagrange multiplier method as the solution approaches the optimal value and the fact that the algorithm converges to local minima in some realizations. We see that the decentralized algorithm gives performance very close to the centralized algorithm gets larger, the difference befor small values of . When tween the two algorithms is about 1 dB.
APPENDIX A. Proof of Proposition 1 the
Proof: Define the is . Then
matrix
so that the th row of
(32) The constraints in (17) can be rewritten as (33)
VI. CONCLUSIONS We have considered transmitter and receiver optimization in MC-CDMA systems. Receiver optimization is performed in a decentralized manner while transmitter optimization is performed through either centralized or decentralized control of the powers of different carriers. We have derived the necessary and sufficient condition for all users to achieve their targets. We have also obtained closed-form solutions for the transmitter optimization problem under AWGN channels, and considered the Lagrange multiplier method under fading channels. Simulations show that when the number of users is smaller than or equal to the number of carriers, each transmitter often tends to concentrate its power on a different carrier which does not suffer deep fading. The MC-CDMA system then tends to an FDMA system with near-optimal frequency assignment. When the number of users gets large, each user tends to choose more than one carrier, which do not suffer deep fading, while interference suppression is performed across the chosen carriers by the corresponding receiver.
takes the diagonal of a matrix to form where the operator a row vector, the inequality is interpreted as element-by-element . If a feasible solution exists, comparisons, and the constraints have to be satisfied. In particular, the sum of the elements on both sides have to satisfy the corresponding inequality.
(34)
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where we have performed spectral factorization for into . Since is Hermitian and positive semidefinite, for and is unitary.
Init: unit coordinate vectors. 1. Set the columns of to be , 2. Calculate the current diagonal of
B. Proof of Proposition 2 , an obvious feasible solution is obProof: For tained by assigning each user a different carrier. With sufficient , we power, the target SNR can always be achieved. For matrix by the matrix construction obtain an appropriate in Section III of this Appendix. The columns of the matrix are with orthonormal and the rows have norm squares the following properties: for ; • . • The resulting matrix is scaled by a constant to give the desired matrix with the property that
3. Set the first pointer . . 4. Set the second pointer Loop: , then set . 1. If 2. Alter the th and th diagonal elements of using an appropriate rotation. that . 3. Set 4. Go to Loop.
The loop is run
times to obtain the desired
so
.
D. Proof of Proposition 4 For sufficiently large (33) can be satisfied.
, the constraints in (17) or, equivalently,
Proof: The optimization problem can be rewritten as Minimize
C. Matrix Construction . We demonstrate how to construct a We assume that matrix with orthonormal columns so that the diagis . Notice that we must have onal of and we only consider for . matrix with orthonormal Consider any given is . columns. Suppose that the diagonal of matrix with orthonormal columns We obtain another being using a and the diagonal of unitary transformation as follows. , where is the rotation matrix that Let rotates the first and second rows of . (35)
is unitary. Thus, has orthonormal columns. Notice that and , respectively. Denote the th rows of and as Then, for and (36) (37)
subject to (38) We consider a related optimization problem with a weaker constraint Minimize subject to (39) As seen before, the constraint is equivalent to (40)
are the eigenvalues of where spectral factorization
. Performing the same
(41) can vary between and by suitNote that ably choosing since cosine and sine are continuous functions. . We can alter the diagonal of Moreover, while maintaining the orthonormality of the columns of . Noand come closer together, tice that we can always make but we may not be able to make them go further apart. To construct the desired , we can use the following algoand . Without rithm. Recall that . loss of generality, we assume that
Therefore, the optimization problem reduces to Minimize
subject to
(42)
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The problem can be transformed into Minimize
subject to
(43)
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stop the iteration. An optimal assignment can be found from the locations of the zeros in . 4) Update : Find the smallest uncrossed element in . Subtract its cost from all elements of all uncrossed rows, and add its cost to all elements of all crossed columns. Remove the lines, and go to step 3. REFERENCES
by defining (44) . The solution is readily obtained by for is a convex function and a monotoninoticing that for . The criterion cally increasing function of , is minimized when . The minimum is oband thus is a scalar times the identity matrix, or tained when are orthogonal with equivalently, when the columns of the same norm. By the matrix construction in Section III of matrix whose this Appendix, we can construct a columns are orthonormal and whose rows have norm squares . This matrix is appropriately scaled to give the desired matrix that satisfies the constraint in (38) with equality. Since this choice is an optimal solution to the problem with the weaker constraint and is also a feasible solution to the original problem with the stronger constraint, it is also an optimal solution to the original problem. E. Outline of Hungarian Method The assignment problem is usually defined as follows. Assign jobs to workers with the minimum total cost, where the . All costs are assumed cost of assigning job to worker is to be nonnegative. (In the context of this paper, the problem is to carriers to users with the minimum total required assign power, where the required power when assigning carrier to user can be determined by the target SNR of user and the .) fading coefficient A brute force approach to solve the assignment problem would involve checking the costs of all possible assignments . Kuhn introduced the Hungarian with a complexity of method [11], which solves the assignment problem in polynomial time. Efficient implementation of the method requires [12], [13]. The method is quite only a complexity of elaborate. We provide only an outline here. Details can be found in [11]–[13] or other publications of operations research. cost Outline of the Hungarian Method: Let be the th element is . matrix, whose 1) Update : In each row of , find the element of the smallest cost, and subtract its cost from all elements in the row. 2) Update : In each column of , find the element of the smallest cost, and subtract its cost from all elements in the column. 3) Check for optimal solution: Cross out all the zeros in with the minimum number of straight lines through rows and/or columns. If the number of lines is equal to , then
[1] S. Hara and R. Prasad, “Overview of multicarrier CDMA,” IEEE Commun. Mag., pp. 126–133, Dec. 1997. [2] J. A. C. Bingham, “Multicarrier modulation for data transmission: An idea whose time has come,” IEEE Commun. Mag., pp. 5–14, May 1990. [3] S. Kondo and L. B. Milstein, “Performance of multicarrier DS CDMA systems,” IEEE Trans. Commun., vol. 44, pp. 238–246, Feb. 1996. [4] T. M. Lok, T. F. Wong, and J. S. Lehnert, “Blind adaptive signal reception for MC-CDMA systems in Rayleigh fading channels,” IEEE Trans. Commun., vol. 47, pp. 464–471, Mar. 1999. [5] J. Salz, “Digital transmission over cross-coupled linear channels,” AT&T Tech. J., vol. 64, no. 6, pp. 1147–1159, July/Aug. 1985. [6] J. Yang and S. Roy, “On joint transmitter and receiver optimization for multiple-input–multiple-output transmission systems,” IEEE Trans. Commun., vol. 42, pp. 3221–3231, Dec. 1994. [7] M. L. Honig, P. Crespo, and K. Steiglitz, “Suppression of near- and far-end crosstalk by linear pre- and post-filtering,” IEEE J. Select. Areas Commun., vol. 10, pp. 614–629, Apr. 1992. [8] G. D. Golden, J. E. Mazo, and J. Salz, “Transmitter design for data transmission in the presence of a data-like interferer,” IEEE Trans. Commun., vol. 43, pp. 837–850, Feb./Mar./Apr. 1995. [9] W. M. Jang, B. R. Vojcic, and R. L. Pickholtz, “Joint transmitter–receiver optimization in synchronous multiuser communications over multipath channels,” IEEE Trans. Commun., vol. 46, pp. 269–278, Feb. 1998. [10] D. P. Bertsekas, Nonlinear Programming. Belmont, MA: Athena Scientific, 1995. [11] H. W. Kuhn, “The Hungarian method for the assignment problem,” Naval Res. Logistics Quart. 2, pp. 83–97, 1955. [12] C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity. Englewood Cliffs, NJ: Prentice-Hall, 1982. [13] K. G. Murty, Operations Research: Deterministic Optimization Models. Englewood Cliffs, NJ: Prentice-Hall, 1995.
Tat M. Lok (M’95) received the B.Sc. degree in electronic engineering from the Chinese University of Hong Kong in 1991. He received the M.S.E.E. degree in electrical engineering and the Ph.D. degree in electrical and computer engineering from Purdue University, West Lafayette, IN, in 1992 and 1995, respectively. He had been a Research Assistant and a Postdoctoral Research Associate at Purdue University. Since 1996, he has been an Assistant Professor in the Department of Information Engineering, the Chinese University of Hong Kong. His research interests include code-division multiple-access systems, multiuser detection, adaptive antenna arrays, and communication theory.
Tan F. Wong (M’97) received the B.Sc. degree (first class honors) in electronic engineering from the Chinese University of Hong Kong, in 1991, and the M.S.E.E. and Ph.D. degrees in electrical engineering from Purdue University, West Lafayette, IN, in 1992 and 1997, respectively. He was a Research Engineer working on the highspeed wireless networks project in the Department of Electronics at Macquarie University, Sydney, Australia. He also served as a Postdoctoral Research Associate in the School of Electrical and Computer Engineering at Purdue University. He is currently an Assistant Professor of Electrical and Computer Engineering at the University of Florida, Gainesville. His research interests include spread-spectrum communication systems, multiuser communications, and wireless cellular networks.
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Transmitter and Receiver Optimization in Multicarrier CDMA Systems Tat M. Lok, Member, IEEE, and Tan F. Wong, Member, IEEE
Abstract—In this paper, we consider transmitter and receiver optimization in multicarrier code-division multiple-access (MCCDMA) systems under Rayleigh fading channels. Receiver optimization is performed in a decentralized manner, while transmitter optimization can be performed through either centralized or decentralized control of the powers of different carriers. Results show that when the number of users is smaller than or equal to the number of carriers, each transmitter often tends to concentrate its power on a different carrier which does not suffer deep fading. The MC-CDMA system then tends to a frequency-division multiple-access system with near-optimal frequency assignment. When the number of users gets large, each user tends to choose more than one carrier, which do not suffer deep fading, while interference suppression is performed across the chosen carriers by the corresponding receiver. Index Terms—Code-division multiple access, fading channels, multicarrier modulation, power control, transmitter–receiver optimization.
I. INTRODUCTION
R
ECENTLY, different multicarrier code-division multipleaccess (MC-CDMA) systems [1], [2] have been proposed and investigated. In [3], an MC-CDMA system with direct sequence spreading is considered to derive the benefits of both multicarrier modulation and direct sequence spreading. In [4], a blind adaptive receiver with interference suppression is proposed for the MC-CDMA system in [3]. In this paper, we assume that the knowledge at the receiver can be sent back to the transmitter for optimization of transmission. The transmitter varies the amplitude and the phase of each carrier according to the feedback information. Joint transmitter–receiver optimization has been considered by a number of researchers. In [5] and [6], joint optimization is approached under the general multiple-input–multiple-output (MIMO) framework. This general development has been specialized to tackle the problem of crosstalk in high-speed digital subscriber line applications (see [7] and [8], for example). Recently, a similar joint transmitter–receiver optimization apPaper approved by J. Wang, the Editor for Wireless Spread Spectrum of the IEEE Communications Society. Manuscript received April 16, 1999; revised August 30, 1999 and December 1, 1999. This work was supported in part by the Research Grant Council of Hong Kong. The paper was presented in part at the IEEE Wireless Communications and Networking Conference (WCNC’99), New Orleans, LA, September 1999. T. M. Lok is with the Department of Information Engineering, Chinese University of Hong Kong, Shatin, Hong Kong (e-mail: [email protected]). T. F. Wong is with the Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611 USA (e-mail: [email protected]). Publisher Item Identifier S 0090-6778(00)06161-4.
proach is proposed for wireless CDMA systems in [9]. In the context of CDMA, the approach of joint precoding and demodulation of data from all users is adopted in all the works mentioned. Although this approach can give better performance, it may not be practical in multimedia wireless communication, where users may often need to change their coding schemes to satisfy different quality-of-service (QoS) requirements. Joint precoding and joint demodulation in these situations could be very difficult. Another shortcoming of the general MIMO approach when applied to multimedia applications is that the optimization criterion is the total mean squared error (MSE) of all the users. While minimizing this criterion ensures an average performance level over all the users, it does not address the different QoS requirement characteristics of multiuser-multimedia communication. In this paper, we consider decentralized demodulation. Each receiver demodulates the data of a particular user without trying to demodulate the data of other users. Since the signals from other users are treated as (structured) noises, the criterion for receiver optimization is the signal-to-noise ratio (SNR) of that particular user (or equivalently the MSE of the user). Transmitter optimization is performed, either in a centralized or a decentralized manner, by varying the powers of the carriers of the users so that all users achieve their own target SNR’s, if it is possible. Different from [5]–[9], we consider the minimization of the total transmission power required to achieve the target SNR’s of the individual users. The coding and modulation schemes of the users are not involved in the optimization. The result is that different users can employ different data signal constellations and different error control coding schemes. The system is well suited for multirate communication. Actually, the coding scheme and the signal constellation of a user can be changed without affecting the performance of other users. In Section II, we define the system model. In Section III, we briefly discuss receiver optimization with the SNR criterion. In Section IV, we consider transmitter optimization in details. We derive the necessary and sufficient condition for simultaneous users to achieve their target performance. We also determine the optimal transmission scheme in additive white Gaussian noise (AWGN) channels. In general, closed-form solutions for optimal transmission schemes are difficult to determine in fading channels. Instead, we consider a centralized adaptive algorithm based on the method of Lagrange multiplier to solve the optimization problem. We also develop a decentralized algorithm which is obtained as an approximation of the centralized algorithm. In Section V, we use numerical examples to illustrate the performance of the optimized system. Conclusions are drawn in Section VI.
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Decentralized receiver for each user in the MC-CDMA system.
II. SYSTEM MODEL In this section, we describe the model of the MC-CDMA simultaneous users in the system. We assume that there are carriers. system, and each user uses the same , generates a stream of data The th user, for given by symbols (1) are random variables and are assumed to The data symbols . The data symbols need be normalized so that not be independent or even uncorrelated. Each user can apply different modulation and coding schemes to the data stream according to the type of information and the required performance. Users can also change their schemes without affecting other users. , is provided a random signature The th user, for given by sequence
therefore, equals . When , the signal on each carrier serves as a mask is not spread, and the signature sequence , the signal to shield off the data from other users. When becomes a direct-sequence multicarrier signal as defined in [3] and [4]. We now describe the channel model. We assume that the channel is a frequency-selective fading channel. By suitably and the bandwidth of [3], we can assume choosing that each carrier undergoes independent frequency-nonselective slow Rayleigh fading. We also assume the presence of AWGN . with power spectral density For simplicity, the signals in the system are assumed to be synchronized although the results can be readily generalized to an asynchronous system. The received signal in complex analytic representation is given by
(2)
(4)
is the spreading factor on each carrier, and the elwhere are modeled as independent and identically disements tributed (i.i.d.) random variables such that . The same signature sequence is used carriers of the th user. to modulate each of the The transmitted signal of the th user can be expressed as the real part of the following complex signal:
accounts for the overall effects of phase shift, path where loss, and/or fading for the th carrier of the th user, and represents AWGN. These fading coefficients are assumed to be invariant within the time interval for optimization.
(3) is the frequency of the th carrier, and is chosen where by the th transmitter to vary the amplitude and the phase of the th carrier. We assume that the chip waveform is bandlimited and the carrier frequencies are well separated so that adjacent frequency bands do not interfere with each other. We also is normalized so that . assume that The parameter is the delay between consecutive chips. Each chips on a carrier, where data symbol is modulated by is the spreading factor on each carrier. The symbol interval ,
III. RECEIVER OPTIMIZATION The optimal centralized receiver should perform joint detection (multiuser detection) for all users. Moreover, the statistics collected by the centralized receiver can be used for transmitter optimization. Despite its optimality, the optimal multiuser detector often requires too many computations in most practical cases. The situation is further complicated if the users transmit different types of information requiring different data rates and if the users are allowed to vary their coding schemes. We consider simpler decentralized receivers with interference suppression capabilities. Each receiver demodulates a different user signal according to the signal and the noise statistics collected by itself. We consider the receiver shown in Fig. 1 for the th user. It consists of branches. Each branch consists of a
LOK AND WONG: TRANSMITTER AND RECEIVER OPTIMIZATION IN MC-CDMA SYSTEMS
correlator and an appropriate weight, and is responsible for the demodulation of one carrier. The correlator on the th branch consists of a chip-matched filter and a combiner that combines the contributions from different chips according to the signature sequence of the th user. We assume that the chip waveform and the chip-matched filter are chosen to satisfy the Nyquist criterion so that there is no interchip interference. The weight vector is an -dimensional vector that combines the contributions from the branches to give the decision statistic for the th user. We consider receiver optimization without transmitter optimization by choosing the appropriate weight vectors for all users. Since we are considering decentralized receivers, optionly affects the performance of the th user. We can mizing consider optimizing the receiver for each user independently. Without loss of generality, we consider receiver optimization . The output for the first user during the symbol interval of the correlator on the th branch, due to the first user signal, where is given by (5) . We define -dimensional vectors by . These vectors can be expressed in as terms of the gain vectors
for
(6) is an diagonal matrix whose th diagonal where . element is The output of the correlator on the th branch, due to the th , is given by user signal, for (7) We also define -dimensional vectors . We denote the output of the correlator on the th branch due to AWGN by , and similarly define an -dimensional vector . The overall output of the correlators, in vector form, is given by (8) By the assumption that other noises are independent of the user is uncorrelated with for all . Moreover, it is signals, are uncorrelated for different easy to check that the vectors . Therefore, the noise and interference correlation matrix is given by E (9) denotes the conditional expectation given , for and , the superscript denotes the Hermitian operation, and denotes the identity matrix. We
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determine the optimal weight vector that maximizes the SNR defined by
E (10) In [4], it is shown that the optimal weight vector is given by (11) and can be determined by the first receiver In general, using a training sequence without the help of other receivers. , the results in [4] show that and can also When can be determined blindly by the first receiver. Actually, be obtained directly as a generalized eigenvector of a pair of matrices, which, in turn, can be estimated from the outputs of some appropriate filters. It can also be obtained directly from the outputs of the filters through a stochastic gradient descent algorithm. IV. TRANSMITTER OPTIMIZATION The results in the last section apply to any receiver by replacing the index 1 with the index of the receiver. When the th , its output SNR receiver employs its optimal weight vector is given by [4] (12) is the noise and interference matrix experienced by where the th user. We assume that each user, according to his or her type of information and modulation-coding scheme, requires certain target SNR performance. We can consider transmitter for so that the optimization by choosing total transmitted power for all users to achieve the target performance is minimized Minimize
(13)
subject to is the target SNR for the th user. We investigate this where key optimization problem in three steps. First, we derive the necessary and sufficient condition for the existence of a feasible solution. Then, we determine the optimal solution in some idealized cases. Finally, we investigate some practical adaptive methods for solving the general problem. A. Necessary and Sufficient Condition If there are a large number of users requiring high SNR targets, there may not be a feasible solution, i.e., a choice , that satisfies all the requirements. We first of consider the condition for the existence of a feasible solution. To this end, we define the total correlation matrix by
where E
(14)
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Then, for (15) Using the matrix inversion lemma, it is readily shown that (16)
The term on the left-hand side can be loosely called the signal-to-total-power ratio (STR) for the th user. Notice that is a monotonically increasing function of , and . The optimization problem can be rewritten as Minimize
where the operator takes the diagonal of a matrix to form a row vector, the inequality is interpreted as element-by-element , and . We comparisons, orthonormal would like to find that satisfies (21). We pick vectors of dimension . Each element of each vector should distinct have the same magnitude. An example would be any columns of the -dimensional discrete Fourier transform mais a trix (or the -dimensional Walsh–Hadamard matrix if multiple of 4). The vectors are scaled appropriately to form the matrix . It is readily verified that the constraints in (21) can be satisfied. The necessary and sufficient condition can be used to determine the capacity of the system in terms of the maximum number of allowable users. Suppose that the SNR requirement of each user is . Then, from (19), the number of users that can be supported is given by
(17) (22)
subject to where
, , and is 10 dB, then For example, if be smaller than 16.5, i.e., there can be at most 16 users.
must
(18) B. AWGN Channel Proposition 1, whose proof can be found in the Appendix, provides the necessary condition for the existence of a feasible solution to this problem. Proposition 1: If a feasible solution exists, then (19) The necessary condition can be used by a central controller to determine whether new users can be admitted to the system. Clearly, if the addition of a new user results in the violation of the necessary condition, the user should not be admitted to the system. We note the necessary condition in (19) is always , since for by satisfied if (18). As shown in Proposition 2, whose proof is also provided in the Appendix, the necessary condition is also sufficient to guarantee the existence of a feasible solution. Proposition 2: If the condition in (19) is satisfied, then a feasible solution exists. for To construct a feasible solution, we need to determine , so that the constraints in (17) are satisfied. The can then be determined readily corresponding gain vectors from (6). The proof of Proposition 2 illustrates how to construct a feasible solution in general. We consider a simple example , the target SNR’s , and they where satisfy (19). In this case of equal target SNR’s, the construction matrix so that the th is especially simple. Define the . Then row of the is (20) by (14). The constraints in (17) can be rewritten as (21)
Closed-form solutions for the optimization problem (17) can be very difficult to obtain except for some special cases. We and consider the special case of an AWGN channel, i.e., for . The problem can be divided into and . For , it is obvious that two cases: by assigning different carriers to different users, we can separate different users signals. Each user signal is then corrupted only by AWGN, which is clearly the best possible scenario. Actually, for are orthogonal, the same as long as situation holds. We state the result as the following proposition. , minimization is achieved in an Proposition 3: For for are orthogonal AWGN channel when vectors. is more complicated. Some users The situation for will necessarily experience interference from other user signals. In this case, the optimal scheme, for a large collection of different cases, is given by Proposition 4. A constructive proof is provided in the Appendix. , if for Proposition 4: For , minimization is achieved in an AWGN channel when the columns of are appropriate orthonormal vectors up to a constant. , if for , the For resulting power required is then given by the following formula: (23)
, , As an example, we consider the case where , dB for . By Propositions 1 and 2, the targets are achievable, i.e., with no spreading on the signal on each carrier, 16 users can be supported with 15 carriers where each user achieves an SNR of 10 dB. By Proposition 4, the columns of should be chosen as orthonormal vectors up
LOK AND WONG: TRANSMITTER AND RECEIVER OPTIMIZATION IN MC-CDMA SYSTEMS
to a constant. One possible choice is to pick all columns, except the first one, of the 16-dimensional Walsh–Hadamard matrix as after appropriate scaling. We point out the the columns of interesting fact that the rows of , which are the signal vectors for the users at the receiver, then form a simplex set. By (23), to achieve an SNR of 10 dB, the signal-to-white-noise ratio (SWNR)1 for each user has to be about 14.77 dB.
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Then, we consider an iterative approach to seek a stationary and are point of the Lagrangian function. At each step, updated according to the following relationships: (28) (29)
C. Fading Channel In most wireless communication channels, propagation loss and fading cannot be ignored. In the presence of different path losses and fading, it may be very difficult to obtain closed-form solutions for the optimization problem. Instead of trying to find exact closed-form solutions, we solve the optimization problem by the following adaptive method. We consider the Lagrange multiplier method [10] and incorporate the SNR requirements into a penalty function. We form the Lagrangian function given by
while a i.e., a gradient descent algorithm is used to update gradient ascent algorithm is used to update . We note that in order to implement this adaptive algorithm, we need to have the information from all the users. Therefore, this method is centralized in nature. In many cases, a decentralized optimization method is preferred. To this end, we can approximate the gradient descent and ascent in (28) and (29), respectively, by
(30) (31)
(24) where is the Lagrange multiplier. The Lagrangian function is considered as a function of and the components of for . The derivative of with respect to (w.r.t.) is the requirement for the target SNR’s of all the users. The derivative of w.r.t. can be obtained as follows. First, notice is not a function of while for can be that (and, hence, ) via the expressed explicitly as a function of matrix inversion lemma.
(25)
for
where (26)
Therefore, the derivative of
w.r.t.
is given by
individually. for each An interesting result has been observed from simulations of , the Lagrange multiplier method the algorithms. For often gives a solution where each user is essentially assigned a different carrier. The system then tends to a frequency-division multiple-access (FDMA) system. Therefore, we are also . In an interested in the optimal FDMA system when FDMA system, each user is assigned a different carrier. We define the optimal FDMA system as the one where the total power to achieve the target performance for all users is min, the problem of imized. Given the channel coefficients finding the optimal FDMA system can be identified as an assignment problem, which can be solved efficiently by the Hungarian method [11]. A brief outline of the method is included in the Appendix. Detailed descriptions of the method can be found in [12] with a graph theoretic approach and in [13] with a matrix approach. In spite of our observation, it should be pointed out that the following situation is conceivable. Consider a system with carriers and users. For a particular carrier, say the first carrier, for the signals from all users suffer deep fading, i.e., are very small. In this case, an MC-CDMA system with our algorithms may not tend to an FDMA system. It is because, in our algorithms, carriers are not actually assigned. In this extreme case when the same carrier fails for all users, the users could just share the remaining carriers. Of course, then, the system would not look like an FDMA system anymore. On the other hand, this extreme situation can hardly be observed (e.g., used in some of because, even for moderate the examples), the probability that all users suffer deep fading at the same carrier is very small. V. PERFORMANCE
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1See
Section V for definition.
In this section, we consider the performance of the Lagrange multiplier method in Rayleigh fading channels. We avoid the complication of different geographical distributions of different users and assume all path losses to be unity. We model
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Fig. 2. Typical performance of the centralized Lagrange multiplier method (uniform target SNR’s).
Fig. 3. Typical performance of the decentralized Lagrange multiplier method (uniform target SNR’s).
for and as i.i.d. zero-mean unit-variance complex Gaussian random variables so that the magnitude of each coefficient is Rayleigh distributed. We first consider the typical performance of the Lagrange multiplier method in achieving the target performance. Fig. 2 shows the result of a simulation run for the centralized optiand eight users mization algorithm with eight carriers . No spreading is performed on the signal on each carrier. The target performance for each user is 8 dB. As shown in the figure, the SNR of each user converges to the target. Fig. 3 shows a similar result for the decentralized optimization
algorithm except that it takes longer for the users to attain the 8-dB target. Figs. 4 and 5 show the case with nonuniform target SNR’s. Four of the users require 8-dB targets while the other four require 6-dB targets. Both the centralized and the decentralized optimization algorithms again give the desired result. Next, we consider the performance of the system as both and the number of users increase. the number of carriers , , and the target SNR for In each case, each user is 8 dB. The performance measure is the average transmitted signal-to-white-noise ratio (SWNR), defined by E , of each user needed to achieve the
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Fig. 4. Typical performance of the centralized Lagrange multiplier method (nonuniform target SNR’s).
Fig. 5. Typical performance of the decentralized Lagrange multiplier method (nonuniform target SNR’s).
8-dB SNR target. We note that the SWNR is directly proportional to the total transmitted power from all the users. As both and increase, we expect that the power for each user to achieve the target performance should decrease due to the diversity provided by the increased number of carriers. Fig. 6 shows the average result of 500 simulations. The performance determined by the (centralized and decentralized) Lagrange multiplier method is shown along with the performance determined by optimal FDMA (Hungarian method). As expected, the average power required to achieve the target performance and increase. We notice that there is a decreases as both
difference of 0.5–1 dB between the results obtained by the optimal FDMA method and the centralized Lagrangian method. It is partly due to the slow convergence of the Lagrange multiplier method as the solution approaches the optimal value,2 and is partly due to the fact that the algorithm may converge to local minima in some realizations. Moreover, there is another 1–1.5-dB difference between the centralized and decentralized Lagrangian methods. 2In Figs. 6–8, the iterative process in the Lagrange multiplier method is terminated once the targets are achieved. Therefore, the values indicated in the figures are higher than the minimum power needed.
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Fig. 6. Performance of the centralized and decentralized algorithms for K =
Fig. 7. Performance of the centralized algorithm for M = 8, N
M
.
0 1 2 and 4. ;
;
In Fig. 7, we consider the performance of the systems with eight and . carriers as the number of users increases for Only the centralized algorithm is considered in this case. As expected, the power required increases as increases until a point (depending on ) at which the 8-dB target cannot be reached. , the power only increases slowly. The centralized For Lagrange multiplier method again nearly produces the optimal FDMA solutions. The small difference of 0.5–1 dB between the results obtained by the optimal FDMA method and the centralized Lagrangian method is again due to the convergence problems
mentioned above. Notice that an average SWNR of only about 5 dB is required to achieve a target SNR of 8 dB for each user. The saving is due to the effect of diversity provided by the different carriers. Each user tends to choose a carrier with the largest gain. , interference from other users generally cannot be For avoided. The minimum power required to achieve the target performance is significantly increased. However, we can slow down this increase by increasing the spreading factor . To compare the difference between the centralized and decen, we consider the system with tralized algorithms for
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Fig. 8.
Performance of the centralized and decentralized algorithms for
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M = 8 and N = 2.
eight carriers and . The SWNR required by each of the two algorithms to achieve the 8-dB target for different values of is plotted in Fig. 8. The small difference of 0.5–1 dB between the results obtained by the optimal FDMA method and the centralized Lagrangian method is due to the slow convergence of the Lagrange multiplier method as the solution approaches the optimal value and the fact that the algorithm converges to local minima in some realizations. We see that the decentralized algorithm gives performance very close to the centralized algorithm gets larger, the difference befor small values of . When tween the two algorithms is about 1 dB.
APPENDIX A. Proof of Proposition 1 the
Proof: Define the is . Then
matrix
so that the th row of
(32) The constraints in (17) can be rewritten as (33)
VI. CONCLUSIONS We have considered transmitter and receiver optimization in MC-CDMA systems. Receiver optimization is performed in a decentralized manner while transmitter optimization is performed through either centralized or decentralized control of the powers of different carriers. We have derived the necessary and sufficient condition for all users to achieve their targets. We have also obtained closed-form solutions for the transmitter optimization problem under AWGN channels, and considered the Lagrange multiplier method under fading channels. Simulations show that when the number of users is smaller than or equal to the number of carriers, each transmitter often tends to concentrate its power on a different carrier which does not suffer deep fading. The MC-CDMA system then tends to an FDMA system with near-optimal frequency assignment. When the number of users gets large, each user tends to choose more than one carrier, which do not suffer deep fading, while interference suppression is performed across the chosen carriers by the corresponding receiver.
takes the diagonal of a matrix to form where the operator a row vector, the inequality is interpreted as element-by-element . If a feasible solution exists, comparisons, and the constraints have to be satisfied. In particular, the sum of the elements on both sides have to satisfy the corresponding inequality.
(34)
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where we have performed spectral factorization for into . Since is Hermitian and positive semidefinite, for and is unitary.
Init: unit coordinate vectors. 1. Set the columns of to be , 2. Calculate the current diagonal of
B. Proof of Proposition 2 , an obvious feasible solution is obProof: For tained by assigning each user a different carrier. With sufficient , we power, the target SNR can always be achieved. For matrix by the matrix construction obtain an appropriate in Section III of this Appendix. The columns of the matrix are with orthonormal and the rows have norm squares the following properties: for ; • . • The resulting matrix is scaled by a constant to give the desired matrix with the property that
3. Set the first pointer . . 4. Set the second pointer Loop: , then set . 1. If 2. Alter the th and th diagonal elements of using an appropriate rotation. that . 3. Set 4. Go to Loop.
The loop is run
times to obtain the desired
so
.
D. Proof of Proposition 4 For sufficiently large (33) can be satisfied.
, the constraints in (17) or, equivalently,
Proof: The optimization problem can be rewritten as Minimize
C. Matrix Construction . We demonstrate how to construct a We assume that matrix with orthonormal columns so that the diagis . Notice that we must have onal of and we only consider for . matrix with orthonormal Consider any given is . columns. Suppose that the diagonal of matrix with orthonormal columns We obtain another being using a and the diagonal of unitary transformation as follows. , where is the rotation matrix that Let rotates the first and second rows of . (35)
is unitary. Thus, has orthonormal columns. Notice that and , respectively. Denote the th rows of and as Then, for and (36) (37)
subject to (38) We consider a related optimization problem with a weaker constraint Minimize subject to (39) As seen before, the constraint is equivalent to (40)
are the eigenvalues of where spectral factorization
. Performing the same
(41) can vary between and by suitNote that ably choosing since cosine and sine are continuous functions. . We can alter the diagonal of Moreover, while maintaining the orthonormality of the columns of . Noand come closer together, tice that we can always make but we may not be able to make them go further apart. To construct the desired , we can use the following algoand . Without rithm. Recall that . loss of generality, we assume that
Therefore, the optimization problem reduces to Minimize
subject to
(42)
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The problem can be transformed into Minimize
subject to
(43)
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stop the iteration. An optimal assignment can be found from the locations of the zeros in . 4) Update : Find the smallest uncrossed element in . Subtract its cost from all elements of all uncrossed rows, and add its cost to all elements of all crossed columns. Remove the lines, and go to step 3. REFERENCES
by defining (44) . The solution is readily obtained by for is a convex function and a monotoninoticing that for . The criterion cally increasing function of , is minimized when . The minimum is oband thus is a scalar times the identity matrix, or tained when are orthogonal with equivalently, when the columns of the same norm. By the matrix construction in Section III of matrix whose this Appendix, we can construct a columns are orthonormal and whose rows have norm squares . This matrix is appropriately scaled to give the desired matrix that satisfies the constraint in (38) with equality. Since this choice is an optimal solution to the problem with the weaker constraint and is also a feasible solution to the original problem with the stronger constraint, it is also an optimal solution to the original problem. E. Outline of Hungarian Method The assignment problem is usually defined as follows. Assign jobs to workers with the minimum total cost, where the . All costs are assumed cost of assigning job to worker is to be nonnegative. (In the context of this paper, the problem is to carriers to users with the minimum total required assign power, where the required power when assigning carrier to user can be determined by the target SNR of user and the .) fading coefficient A brute force approach to solve the assignment problem would involve checking the costs of all possible assignments . Kuhn introduced the Hungarian with a complexity of method [11], which solves the assignment problem in polynomial time. Efficient implementation of the method requires [12], [13]. The method is quite only a complexity of elaborate. We provide only an outline here. Details can be found in [11]–[13] or other publications of operations research. cost Outline of the Hungarian Method: Let be the th element is . matrix, whose 1) Update : In each row of , find the element of the smallest cost, and subtract its cost from all elements in the row. 2) Update : In each column of , find the element of the smallest cost, and subtract its cost from all elements in the column. 3) Check for optimal solution: Cross out all the zeros in with the minimum number of straight lines through rows and/or columns. If the number of lines is equal to , then
[1] S. Hara and R. Prasad, “Overview of multicarrier CDMA,” IEEE Commun. Mag., pp. 126–133, Dec. 1997. [2] J. A. C. Bingham, “Multicarrier modulation for data transmission: An idea whose time has come,” IEEE Commun. Mag., pp. 5–14, May 1990. [3] S. Kondo and L. B. Milstein, “Performance of multicarrier DS CDMA systems,” IEEE Trans. Commun., vol. 44, pp. 238–246, Feb. 1996. [4] T. M. Lok, T. F. Wong, and J. S. Lehnert, “Blind adaptive signal reception for MC-CDMA systems in Rayleigh fading channels,” IEEE Trans. Commun., vol. 47, pp. 464–471, Mar. 1999. [5] J. Salz, “Digital transmission over cross-coupled linear channels,” AT&T Tech. J., vol. 64, no. 6, pp. 1147–1159, July/Aug. 1985. [6] J. Yang and S. Roy, “On joint transmitter and receiver optimization for multiple-input–multiple-output transmission systems,” IEEE Trans. Commun., vol. 42, pp. 3221–3231, Dec. 1994. [7] M. L. Honig, P. Crespo, and K. Steiglitz, “Suppression of near- and far-end crosstalk by linear pre- and post-filtering,” IEEE J. Select. Areas Commun., vol. 10, pp. 614–629, Apr. 1992. [8] G. D. Golden, J. E. Mazo, and J. Salz, “Transmitter design for data transmission in the presence of a data-like interferer,” IEEE Trans. Commun., vol. 43, pp. 837–850, Feb./Mar./Apr. 1995. [9] W. M. Jang, B. R. Vojcic, and R. L. Pickholtz, “Joint transmitter–receiver optimization in synchronous multiuser communications over multipath channels,” IEEE Trans. Commun., vol. 46, pp. 269–278, Feb. 1998. [10] D. P. Bertsekas, Nonlinear Programming. Belmont, MA: Athena Scientific, 1995. [11] H. W. Kuhn, “The Hungarian method for the assignment problem,” Naval Res. Logistics Quart. 2, pp. 83–97, 1955. [12] C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity. Englewood Cliffs, NJ: Prentice-Hall, 1982. [13] K. G. Murty, Operations Research: Deterministic Optimization Models. Englewood Cliffs, NJ: Prentice-Hall, 1995.
Tat M. Lok (M’95) received the B.Sc. degree in electronic engineering from the Chinese University of Hong Kong in 1991. He received the M.S.E.E. degree in electrical engineering and the Ph.D. degree in electrical and computer engineering from Purdue University, West Lafayette, IN, in 1992 and 1995, respectively. He had been a Research Assistant and a Postdoctoral Research Associate at Purdue University. Since 1996, he has been an Assistant Professor in the Department of Information Engineering, the Chinese University of Hong Kong. His research interests include code-division multiple-access systems, multiuser detection, adaptive antenna arrays, and communication theory.
Tan F. Wong (M’97) received the B.Sc. degree (first class honors) in electronic engineering from the Chinese University of Hong Kong, in 1991, and the M.S.E.E. and Ph.D. degrees in electrical engineering from Purdue University, West Lafayette, IN, in 1992 and 1997, respectively. He was a Research Engineer working on the highspeed wireless networks project in the Department of Electronics at Macquarie University, Sydney, Australia. He also served as a Postdoctoral Research Associate in the School of Electrical and Computer Engineering at Purdue University. He is currently an Assistant Professor of Electrical and Computer Engineering at the University of Florida, Gainesville. His research interests include spread-spectrum communication systems, multiuser communications, and wireless cellular networks.
