On Sum Capacity of Continuous-Time Overloaded CDMA Systems

1

On Sum Capacity of Continuous-Time Overloaded CDMA Systems Yeo Hun Yun and Joon Ho Cho

Abstract—In this paper, the optimal design problem of overloaded code-division multiple-access (CDMA) systems is examined. Unlike previous results for chip or symbol synchronous systems, a continuous-time, band-limited, additive white Gaussian noise channel is considered for the multiple-access communications. First, non-information theoretic results are summarized, where the total transmit power is minimized, subject to lower bounds on the signal-to-interference-plus-noise ratio at the output of the linear minimum mean-squared error receivers. Second, the sum capacity is derived and shown to be the same as that of the optimal frequency-division multiple-access system, where each user’s bandwidth is upper-bounded by the cycle frequency of the corresponding CDMA system. As the non-information theoretic results, the geometric procedure called multi-user constrained water-pouring leads to the optimal system that maximizes the sum rate. It is shown that orthogonal waveforms are assigned to oversized users and continuous-time equivalents of generalized Welch bound equality sequences are assigned to non-oversized users. A method to construct an optimal codebook is also proposed to be used in a CDMA signal modulator in each transmitter. Index Terms—code-division multiple-access, cyclostationarity, frequency-division multiple-access, sum capacity, Welch bound equality sequences.

I. I NTRODUCTION

I

N code-division multiple-access (CDMA) communications, it is well known that the multiple-access interference (MAI) is the key limiting factor of the system performance. Consequently, the design of a set of signature sequences or waveforms that induce no or minimal MAI has long been an important area of research [1]–[9]. In this paper, the sequence design problem investigated in [4], [6], and [7] is revisited, where sum rate maximization is considered for overloaded CDMA systems with more users than the processing gain. In [4], the sum capacity of a symbolsynchronous CDMA system is derived with equal-power users. It is shown that the Welch bound equality sequences are optimal signature sequences that maximize the sum rate of the system. This result is extended in [6] to un-equal power cases. It is shown that the users are classified into either oversized or non-oversized users, depending on the relative signal power among users and on the processing gain. Orthogonal sequences are assigned to oversized users and generalized This work was supported in part by the Ministry of Knowledge Economy, Korea, under the grant NIPA-2009-C1090-0902-0037 for the BrOMAITRC@POSTECH supervised by the NIPA, and in part by the Ministry of Education, Science, and Technology, Korea, under the the National Research Foundation Grants 2009-0058501 and 2009-0088483. Y. H. Yun and J. H. Cho are with the Department of Electronic and Electrical Engineering, Pohang University of Science and Technology (POSTECH), Pohang, Gyeongbuk 790-784, Korea (e-mail: {yhym205, jcho}@postech.ac.kr).

Welch bound equality (GWBE) sequences are assigned to non-oversized users. In [7], the result is further generalized to chip-synchronous but not symbol-synchronous case. It is shown that the symbol asynchronous system achieves the same sum capacity as the symbol synchronous one. Similar to [6], orthogonal sequences are assigned to oversized users and generalized asynchronous WBE (GAWBE) sequences are assigned to non-oversized users. Viewed in the continuous-time model, all these results with discrete-time (DT) or vector observation model assumes a square-root Nyquist chip pulse that induces zero inter-chip interference. Signature waveform design for continuous-time band-limited overloaded CDMA system is also considered in [10]–[12]. However, strictly time-limited but root-meansquare (RMS) band-limited [10] or fractional out-of-band energy (FOBE) band-limited [11], [12], signature waveforms are designed with performance metric other than the sum capacity. Although the theory developed in [5] is completely generalized in [9] for continuous-time strictly band-limited CDMA systems, the sum capacity result in [6] is not yet extended to continuous-time strictly band-limited case. In this paper, we take a frequency-domain approach to this problem as [8] and [9]. Motivated by a recent development in the processing of shift-invariant (SI) signals and cyclostationary random processes [14], we convert the scalar multiple-access channel to a multiple-input multiple-output vector multipleaccess channel and tackle the problem. It turns out that the equivalence of an overloaded CDMA system to an optimal bandwidth-constrained FDMA system [13] is again established as [9], the multi-user constrained water-pouring optimally distributes the differential signal power, the orthogonal waveforms are assigned to oversized users, while CTE-GWBE sequences are assigned to non-oversized users. The most vivid distinction from [9] is in the codebook construction method. As shown in [9], the optimal signature waveforms for non-oversized users induce non-zero interference. Thus, a standard joint codebooks for intersymbol interference (ISI)-free channels cannot be used among users. Motivated by an equivalence between an original signal and its over-sampled and interpolated signal, we propose a method to construct joint codebooks that can achieve the capacity boundary of the multiple-access channel as if there is no ISI. II. S IGNAL M ODEL AND P ROBLEM F ORMULATION There are K active users in a single-cell uplink multipleaccess system. Each user transmits a wide-sense stationary (WSS) sequence of data symbols by using a linear modulation.

2

The kth user’s complex baseband equivalent of the transmitted signal is modeled as Xk (t) ,

∞ X

dk [m]sk (t − mT )

(1)

The objective of this work is to find the optimal signature waveforms and data sequences that maximize the sum rate of the CDMA system described above, subject to the power and the total bandwidth constraints

m=−∞

Pk ≤ pk , ∀k,

where the data sequence (dk [m])m∈Z is assumed to be a proper-complex zero-mean random process with autocorrelation function φk [l] , E{dk [m]∗ dk [m + l]},

(2)

and 1/T [symbols/sec] is the common symbol transmission rate of active users. By convention, we call the transmit waveform sk (t) of this multiple-access system in complex baseband the signature waveform of the kth user, for k = 1, 2, ..., K. The users share a strictly band-limited frequency band with W [Hz] in passband, so that all the signature waveforms (sk (t))K k=1 are assumed band-limited to the common frequency band f ∈ [−W/2, W/2). As in [5] and [6], we mainly consider an overloaded CDMA system. The channel overloading condition for K synchronous users with processing gain N is given by N < K.

K X

Xk (t) + N (t).

(9b)

for some pk > 0 and W > 0. This problem is a continuoustime version of the problem considered in [6], where the sum rate is maximized by jointly optimizing the signature sequences and the data sequences for overloaded vector or DT Gaussian multiple-access communications. III. R EVIEW OF R ELATED D EFINITIONS AND R ESULTS In this section, we first review some related definitions and results to convert the problem described in the time domain to that in the frequency domain. Then, we review non-information theoretic results on the optimal signature waveforms and associated power allocation for continuoustime overloaded CDMA systems. A. Cyclostationarity, FRESH Vectorization, and Scalarization

(4)

We start this subsection by defining a proper-complex widesense cyclostationary (WSCS) random process. Definition 1: A second-order random process Y (t) is proper-complex WSCS with cycle period T (> 0) if

for continuous-time band-limited CDMA systems. The major difference is that the bandwidth and symbol-time product W T , which corresponds to the processing gain N for synchronous case, is not necessarily an integer. It is assumed that the user signals pass through frequencyflat channels and are received in the presence of additive white Gaussian noise (AWGN) with two-sided power spectral density (PSD) N0 . Thus, the complex baseband equivalent of the received signal can be written as Y (t) =

· ¶ W W Φk (f T )|Sk (f )|2 = 0, ∀k, ∀f ∈ / − , , 2 2

(3)

As shown in [8] and [9], this assumption is equivalent to W T < K,

and

(9a)

(5)

k=1

Since the transmitted signal in (1) can be viewed as a quadrature amplitude modulation, the average received signal power of the kth user is given by [16, Ch. 4] Z Z 1 1 ∞ 2 Pk , E{|Xk (t)| }dt = Φk (f T )|Sk (f )|2 df T hT i T −∞ (6) where ∞ X φk [m]e−j2πf m (7) Φk (f ) , m=−∞

is the discrete-time Fourier transform (DTFT) of the autocorrelation function φk [m], and Z ∞ Sk (f ) , sk (t)e−j2πf t dt (8) −∞

is the continuous-time Fourier transform (CTFT) of the signature waveform sk (t).

E{Y (t)} = E{Y (t + nT )} and (10a) ∗ ∗ E{Y (t)Y (s) } = E{Y (t + nT )Y (s + nT ) }, ∀t, s ∈ R, ∀n ∈ Z. (10b) Since any integer multiple of a cycle period is also a cycle period, it is convenient to define the fundamental cycle period as the minimum of the cycle periods. Naturally, the inverse of the (fundamental) cycle period is called the (fundamental) cycle frequency. Before introducing a transformation that converts a scalar-valued signal to a vector-valued signal, we define Nyquist zones and their center frequencies. Definition 2: [14] Given a bandwidth-rate pair (W, 1/T ), the lth center frequency and Nyquist zone are defined as fl , and

l−L−1 T

(11a)

¾ 1 1 ≤ f < fl + , (11b) 2T 2T respectively, for l = 1, 2, ..., 2L + 1, where L is given by » ¼ WT − 1 L, . (11c) 2 ½

Fl ,

f : fl −

For convenience, we denote the (L + 1)th Nyquist zone as F, i.e., ¾ ½ 1 1 ≤f < . (12) F , FL+1 = f : − 2T 2T Definition 3: [14] Given a scalar-valued input signal Y (t) that is band-limited to f ∈ [−W/2, W/2), the FREquency

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SHift (FRESH)1 vectorizer with rate 1/T is defined as a single-input multiple-output (SIMO) linear time-varying system, whose output Y (t) is given by Y (t) , [Y1 (t), Y2 (t), · · · , Y2L+1 (t)]T , where the lth entry Yl (t) is defined as ¡ ¢ 1 (t), Yl (t) , Y (t)e−j2πfl t ∗ g 2T

(13a) (13b)

for l = 1, 2, ..., 2L + 1, with g1/(2T ) (t) being the impulse response of the ideal lowpass filer with bandwidth 1/(2T ), i.e., the Fourier transform G1/(2T ) (f ) , F {g1/(2T ) (t)} is given by ½ 1, ∀f ∈ F 1 (f ) = G 2T (14) 0, elsewhere. Example 1: When the kth signature waveform sk (t) is applied as the input to the FRESH vectorizer with rate 1/T , the output sk (t) , [sk,1 (t), sk,2 (t), · · · , sk,2L+1 (t)]T can be obtained by using (13b). Then, its elementwise CTFT sk (f ) is nothing but the vectorized Fourier transform (VFT), defined in [18], of sk (t). Remark 1: [14] Suppose that the rate of a FRESH vectorizer is the same as the cycle frequency 1/T of an input zeromean proper-complex WSCS random process Y (t). Then, the output Y (t) is a zero-mean proper-complex vector-valued WSS random process, i.e., E{Y (t)} = 02L+1 , ∀t, (15a) T E{Y (t + τ )Y (t) } = 0(2L+1)×(2L+1) , ∀t, ∀τ, (15b) and there exists a function rY (τ ) only of τ such that E{Y (t + τ )Y (t)H } = rY (τ ), ∀t, ∀τ,

(15c)

where 02L+1 and 0(2L+1)×(2L+1) are a (2L + 1)-by-1 and a (2L + 1)-by-(2L + 1) all-zero vector and matrix, respectively, and rY (τ ) is a (2L + 1)-by-(2L + 1) matrix-valued function.2 Proof: See [14, Proposition 1]. 2 Since the CTFT of the auto-correlation function of a WSS random process is called its PSD, it is natural to call the matrix-valued frequency function Rk (f ) the PSD of Xk (t). Actually, Rk (f ) is nothing but the matrix-valued PSD (MVPSD), defined in [18], of Xk (t). Definition 4: [14] Given a (2L + 1)-by-1 vector-valued input signal Y (t), the FRESH scalarizer with rate 1/T is defined as a multiple-input single-output (MISO) linear timevarying system, whose output Y (t) is given by Y (t) ,

2L+1 X

Yl (t)ej2πfl t ,

(16)

l=1

where fl is defined as (11a). The FRESH scalarizer defined in (16) is clearly an inverse operator of the FRESH vectorizer with rate 1/T and input bandwidth W/2. In the next section, we will use a FRESH vectorizer to generate a vector-valued sufficient statistic at the 1 The abbreviation FRESH appears first in [17], where the cyclic Wiener filtering that is a generalization of the Wiener filtering for cyclostationary random processes is proposed. 2 If Y (t) has a non-zero mean, then the mean vector, the pseudo-autocovariance, and the auto-covariance of Y (t) become, respectively, a constant vector, a zero matrix, and a matrix-valued function only of τ .

receiver, which naturally leads to at least the conceptual use of the FRESH scalarizers with rate 1/T at the transmitters. Since the output must be band-limited to the frequency band f ∈ [−W/2, W/2) of interest, we need the following necessary and sufficient condition on its input Xk (t). Remark 2: The output of a FRESH scalarizer with rate 1/T is band-limited to f ∈ [−W/2, W/2) if and only if the (2L + 1)-by-1 vector-valued input signal Xk (t) has no signal component in − for Xk,1 (t), and

1 L W ≤f < − 2T T 2

(17a)

W L 1 − ≤f < (17b) 2 T 2T for Xk,2L+1 (t), where L is defined as (11c). Proof: See [18, Sec. IV-A]. 2 Although the length of the vector-valued signal Xk (t) is 2L + 1, not every entry can take a non-zero component at a given frequency f ∈ F. Thus, the following refinements on the definitions of the VFT and the MV-PSD are useful. Definition 5: [18, Definition 7] The effective VFT of a band-limited deterministic signal sk (t) is defined as a variablelength vector-valued function of f that is obtained after removing the first entry of its VFT sk (f ) if (17a) is satisfied and the last entry of sk (f ) if (17b) is satisfied. The effective MV-PSD of a band-limited cyclostationary random process Xk (t) is defined as a variable-size matrixvalued function of f that is obtained after removing the first row and column of its MV-PSD Rk (f ) if (17a) is satisfied and the last row and column if (17b) is satisfied. For simplicity, we do not introduce new notations for the effective VFTs and MV-PSDs. Instead, we will specify whenever necessary. Since the design problems are tackled in the frequency domain, it proves to be convenient to define the degree of freedom as the length of an effective VFT. Definition 6: [8, Definition 1] The degree of freedom N (f ) as a function of f ∈ F is defined as ½ Te dW T e, for |f | < W T +1−dW 2T (18a) N (f ) , dW T e − 1, otherwise if dW T e is an odd number, and ½ e−W T dW T e − 1, for |f | < dW T2T N (f ) , (18b) dW T e, otherwise if dW T e is an even number. An interesting observation is made in [8] that the average degree of freedom always equals W T , i.e., Z T N (f )df = W T (19) F

whether dW T e is odd or even. This average degree of freedom can be interpreted as the overall signal dimension or the processing gain of the continuous-time band-limited CDMA system. Note that, unlike its DT counterpart, this value is not restricted to an integer. Using the definitions and results summarized above, we proceed to find the maximum sum rate of the CDMA system and associated optimal signature waveforms in the next two sections.

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B. Non-Information Theoretic Results on Continuous-Time Overloaded CDMA Systems The theory developed in [5] for overloaded DT or vector multiple-access channel (MAC) is first generalized to the optimal design of signature waveforms under equal power condition [8], and then under unequal power condition [9]. Unlike the information-theoretic approaches in [6] and this paper, where the sum rate is maximized under power constraints, the objective in [8] is to minimize the total mean-squared error (MSE), when the users have equal signal power. This problem is, in fact, a special case dual of the more general problem considered in [5] and [9], where the total signal power is minimized subject to pre-specified lower bounds on the signalto-interference-plus-noise ratios (SINRs) at the output of linear minimum mean squared error (LMMSE) receivers. It is assumed that the data symbols are uncorrelated with power Pk and that the signature waveforms are normalized, i.e., φk [l] = Pk T δ[l], and

sZ

(20a)

∞

|sk (t)|2 dt = 1,

ksk (t)k ,

(20b)

−∞

∀k, where δ[·] denotes the Kronecker delta function. The received signal is processed by K correlation receivers that generate the output sequences (Yk [m])m∈Z given by Z ∞ Yk [m] , gk (t − mT )∗ Y (t)dt, ∀k, (21) −∞

where and in what follows ∀k denotes ∀k ∈ {1, 2, ..., K}. We call the waveform gk (t) used in the kth correlator the receive waveform for the kth user. Then, the problem in [9] can be written as Problem 1: minimize

(Pk ,sk (t))K k=1

subject to

K X

In this subsection, every VFT and MV-PSD is an effective one. To separate the signature waveform design and the power allocation, we introduce the following definitions. Definition 7: The differential signal power Pk (f ) of the kth received signal Xk (t) is defined as 1 Pk (f ) , 2 Φk (f T )ksk (f )k2 (27) T ∀k and ∀f . This definition, which includes the effect of the data sequence, is a modified version of the differential power introduced in [9, Definition 5] defined only for the case with uncorrelated data symbols. Note that this definition is consistent to (6), since Pk (f ) integrates to Z Pk = T Pk (f )df, ∀k. (28) F

Definition 8: [9, Definition 6] The normalized VFT zk (f ) of sk (f ) is defined as ( sk (f ) for sk (f ) 6= 0, ksk (f )k , zk (f ) , arbitrary with unit norm, for sk (f ) = 0, (29) ∀k and ∀f . Thus, the optimization problem Problem 1 can be rewritten as Problem 2: Z K X minimize T Pk (f )df (Pk (f ),zk (f ))k

subject to Pk

(22a)

k=1

SINRk ≥ γk , ∀k,

(22b)

where the SINR at the output Yk [m] of the kth receiver is defined as E(|E{Yk [m]|dk [m]}|2 ) SINRk , (23) Var{Yk [m]|dk [m]} for k = 1, 2, ..., K. Since the receivers are the LMMSE receivers and the data symbols are uncorrelated as (20), the VFT of the kth receive waveform gk (t) can be shown to be p (24) gk (f ) = Pk T RY (f )−1 sk (f ), ∀f, ∀k, where and in what follows ∀f denotes ∀f ∈ F, RY (f ) denotes the matrix-valued PSD of Y (t), given by RY (f ) , N0 IN (f ) +

dk [m], the output SINR at the kth receiver can be shown to be !−1 Ã 1 R −1 . (26) SINRk = Pk T F sk (f )H RY (f )−1 sk (f )df

K X

Pk sk (f )sk (f )H ,

(25)

k=1

and IN (f ) denotes the N (f )-by-N (f ) identity matrix. With this cyclic Wiener filter [17] for the LMMSE estimation of

k=1

F

Pk (f ) ≥ 0, ∀k, ∀f, (30a) kz (f )k = 1, ∀k, ∀f, and (30b) Z k Pk (f )zk (f )H RY (f )−1 zk (f )df F

= e(γk ), ∀k,

(30c)

where the MV-PSD of the received signal RY (f ) can be rewritten as K X RY (f ) = N0 IN (f ) + Pk (f )T zk (f )zk (f )H , (31) k=1

and the effective bandwidth e(γk ) of the kth user is defined as [19] γk . (32) e(γk ) , γk + 1 One of the main results in [9] is that the optimal profile of the transmission power (Pk,opt )k is the same as that of the optimal overloaded FDMA system, which is investiaged in [13]. Remark 3: The jointly optimal transmission power of the kth user that solves Problem 1 is given, respectively, by ¢ ¡ N0 (33) e(γk ) 1 + γk + [νopt − γk ]+ , Pk,opt = T

5

for k = 1, 2, ..., K, where ν = νopt (≥ mink γk ) is the unique solution to µ ¶ K X 1 e(γk ) 1 + = WT (34) [ν − γk ]+ + γk k=1

with [x]

+

denoting the positive part of x, i.e., [x]+ , max(0, x).

(35)

Proof: See [9, Corollay 1]. 2 Interestingly, this optimal power profile is exactly the same as that of the optimal FDMA users with identical power profile and total bandwidth constraint [13], [9]. It turns out to be convenient to introduce the following definition. Definition 9: The FDMA-equivalent bandwidth Wk,opt is defined as 1 1 Wk,opt = · 1 , (36) T e(γ ) − P N0 T k k,opt which is the bandwidth allocated to the kth user that solves Problem 1 under FDMA signaling constraint. Using (33), it can be shown that 1 , (37) T where the equality holds if and only if νopt ≤ γk . Note that the upper bound 1/T is nothing but the Nyquist minimum bandwidth required for zero intersymbol interference. Similar to [5], we introduce the following definitions of oversized and non-oversized users now in terms of the FDMA equivalent bandwidth. Definition 10: Given a K-user system with power profile [P1 , P2 , ..., PK ] and symbol rate 1/T , the kth user is called oversized if Wk,opt = 1/T . Otherwise, i.e., Wk,opt < 1/T , it is called non-oversized. Let K be the index set of the oversized users. Then, the following result provides an alternate way to identify the oversized users. Remark 4: If the users are ordered to satisfy γ1 ≥ γ2 ≥ ... ≥ γK , then the number |K| of oversized users in the optimal solution is given by the unique solution to

Once the optimal power profile is obtained, the next step is to design the optimal signature waveforms. Due to (27)–(29), the VFT of the signature waveform can be rewritten as s Pk (f )T sk (f ) = zk (f ), ∀k. (40) Pk Thus, we can construct an optimal signature waveform sk,opt (f ) by finding the optimal differential signal power Pk,opt (f ) and the optimal normalized VFT zk,opt (f ) of the signature waveform, ∀k and ∀f . Another and the most non-intuitive result in [9] is that an optimal profile (Pk,opt (f ))k of the differential signal power is obtained by using a geometric procedure that optimally distributes the FDMA-equivalent bandwidth Wk,opt over f ∈ F for all k. Definition 11: [9, Definition 13] Given the bandwidth profile (Wk,opt )k of the optimal FDMA solution given by (36), a profile (Wk,opt (f ))k is called a multi-user constrained waterpouring solution if it satisfies 0 ≤ Wk,opt (f )T ≤ 1, ∀k, ∀f, Z T Wk,opt (f )df = Wk,opt , ∀k,

Wk,opt ≤

(N − |K|)e(γ|K| ) ≥

K X

e(γk ) > (N − |K|)e(γ|K|+1 ).

k=|K|+1

(38) Proof: See [13, Corollary 2]. 2 This is the same result as [5, Eq. (21)] after 3 dB adjustments of power and SINR for all k. These adjustments are required because baseband signaling with real-valued data symbols is considered in [5], while bandpass signaling with complex-valued data symbols is considered in [13], [9], and this paper. Accordingly, the optimal signal power can be rewritten as ( N 0 for k ∈ K, T · γk , (39) Pk,opt = (W T −|K|)e(γ N0 k) P · , for k ∈ / K, T (W T −|K|)− e(γ 0 ) 0 k ∈K /

k

which coincides with the result in [5, Eq. (22)] if the signal model is modified to a real baseband synchronous CDMA system.

and

(41a) (41b)

F K X

Wk,opt (f )T = N (f ), ∀f.

(41c)

k=1

Now, the optimal solution to Problem 2 is given as follows. Remark 5: The profile (Pk,opt (f ), zk,opt (f ))k is an optimal solution to Problem 2 if and only if the differential signal power is given by Pk,opt (f ) =

Wk,opt (f ) Pk,opt Wk,opt

(42)

and the normalized VFT is given by ( arbitrary orthonormal vector, for k ∈ K, PN (f )−|K| zk,opt (f ) = z˜k,l (f )el (f ), for k ∈ / K, l=1 (43) where (Wk,opt (f ))k is any constrained water-pouring solution, N (f )−|K| (el (f ))l=1 is any orthonormal basis of the (N (f ) − |K|) dimensional subspace that is orthogonal to the subspace spanned by (zk,opt (f ))k∈K , and z˜k,l (f ) is the lth entry of the generalized Welch bound equality (GWBE) sequence z˜k (f ) of length N (f ) − |K| such that ¶ µP X k∈K / Pk,opt (f ) H IN (f )−|K| . Pk,opt (f )z˜k (f )z˜k (f ) = N (f ) − |K| k∈K / (44) Proof: See [9, Eq. (85) and proof of Theorem 2]. 2 Thus, we call the signature waveforms of non-oversized users the continuous-time equivalents of GWBE (CTEGWBE) sequences. In summary, the optimal power profile is the same as that of the optimal FDMA system and, as the optimal signature waveforms, orthogonal signals are assigned to oversized users, while CTE-GWBE sequences are assigned to non-oversized users. This completes the generalization of the results in [5] to continuous-time cases.

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IV. S UM C APACITY OF BANDWIDTH -C ONSTRAINED FDMA S YSTEMS Before moving to the sum rate maximization problem for CDMA systems, we review the classical sum capacity result for FDMA systems and extend it to the case with an additional bandwidth constraint on each user. Suppose that there are K FDMA users with power profile p , [p1 , p2 , ..., pK ]T with pk > 0, ∀k. Then, the sum rate maximization problem can be formulated as Problem 3: µ ¶ K X pk maximize wk log 1 + 2 (45a) σ wk (wk )k

where N is defined as N,

(45b) (45c)

k=1

where wk is the bandwidth allocated to the kth user, wtot is the total system bandwidth, and σ 2 is the PSD of the AWGN that corrupts the complex baseband channel. It is well known [20, Ch. 15] that the solution to Problem 3 is given by pk wk,opt = PK wtot , ∀k, (46) 0 k0 =1 pk which can be viewed as the proportional-share bandwidth allocation scheme because each user is assigned bandwidth that is proportional to its signal power. A. Problem Formulation and Constraint Set Partitioning Now, we consider the same problem with an additional constraint. Problem 4: maximize C(w)

(47a)

subject to 0 ≤ wk ≤ w, ¯ ∀k, and K X wk = wtot ,

(47b)

(wk )k

(47c)

B. User Re-Ordering and Sum Rate Searching for such n requires the evaluation of the sum rate achievable by ∀w ∈ Ωn . The following proposition shows that the expression for the sum rate can be greatly simplified if we assume for notational convenience that p1 ≥ p2 ≥ ... ≥ pK ,

k=1

and w ¯ is the common upper bound on each user’s bandwidth. To proceed, we define Ω as the constraint set of every feasible solution w = [w1 , w2 , ..., wK ]T that satisfies (47b) and (47c), and wopt as an optimal solution. Lemma 1: The optimal solution wopt to Problem 4 always exists and is unique in Ω. Proof: Omitted. 2 We then define Ωn as the subset of Ω such that every w ∈ Ωn has exactly n entries satisfying wk = w. ¯ The constraint set can now be partitioned and represented as a disjoint union Ω=

N [ n=0

Ωn ,

(49)

(51)

which can be done by re-numbering the user indexes. We also assume that wtot > w, ¯ i.e., N ≥ 1, because otherwise the problem reduces to Problem 3 without the upper bound constraint. To proceed, we prove the following lemma first. Lemma 2: The optimal solution wopt satisfies pk > pk0 =⇒ wk,opt ≥ wk0 ,opt ,

(52a)

pk = pk0 =⇒ wk,opt = wk0 ,opt ,

(52b)

and where wk,opt is the kth entry of wopt . Proof: Omitted. 2 Proposition 1: If wopt ∈ Ωn for some n ∈ {0, 1, ..., N }, then the sum rate is given by n ³ X pk ´ C(wopt ) = w ¯ log 1 + 2 σ w ¯ k=1 µ ¶ K X pk + wk,opt log 1 + 2 . (53) σ wk,opt k=n+1

Proof: Omitted. 2 The following lemma provides a simple upper bound on this sum rate. Lemma 3: If wopt ∈ Ωn , then C(wopt ) ≤ C¯n ,

k=1

where C(w) denotes the sum rate of the FDMA system given by µ ¶ K X pk wk log 1 + 2 C(w) , (48) σ wk

tot

. (50) w ¯ Therefore, it becomes of interest to find in which Ωn resides the optimal solution wopt .

k=1

subject to 0 ≤ wk , ∀k, and K X wk = wtot ,

jw k

(54)

where C¯n ,

³ pk ´ w ¯ log 1 + 2 σ w ¯ k=1 Ã n X

PK

pk +(wtot − nw) ¯ log 1 + 2 σ (wtot − nw) ¯ k=n+1

Proof: Omitted.

! .

(55) 2

C. Testing Rule and Optimal Solution In this subsection, we provide a way to test whether the kth user deserves the upper bound w ¯ as the optimal bandwidth allocation or not. The testing rule we use is given by pk vk , P K (wtot − (k − 1)w) ¯ T w, ¯ (56) k0 =k pk0 where the kth user is classified as oversized if vk > w, ¯ critically-sized if vk = w, ¯ and undersized if vk < w. ¯ A user

7

passes the test if it is non-undersized while fails if undersized.3 Note that a testing result depends only on the number of users, the total bandwidth, and the relative power ratio among one another, neither the absolute power level nor the noise density. Later, it will be shown that the optimal amount of bandwidth allocated to each user exhibits the same tendency. Note that the test statistic vk is noting but the proportional share of the kth user of the remaining bandwidth (wtot − (k − 1)w), ¯ when the first k − 1 users are assigned w ¯ each. We first examine two important properties of this testing rule. Lemma 4: The test statistic vn satisfies vn ≥ w ¯ =⇒ vn0 ≥ w, ¯ ∀n0 ∈ {1, 2, ..., n}

(57a)

and 0

vn < w ¯ =⇒ vn0 < w, ¯ ∀n ∈ {n, n + 1, ..., K}.

(57b)

Proof: Omitted. 2 Proposition 2: If the nth user, for n ∈ {1, 2, ..., N − 1}, passes the test, then C(w) < C¯n , ∀w ∈

n−1 [

Ωk .

(58)

k=0

Proof: Omitted. 2 Proposition 3: If the nth user, for n ∈ {1, 2, ..., N }, fails the test, then max C(w) = C¯n−1 . (59)

that are assigned orthogonal channels.4 Moreover, the sum capacity turns out to be identical to that derived in [6] after 3 dB adjustments of power and proper scaling of system parameters, which shows that the synchronous CDMA system in [6] achieves the same sum capacity as the constrained FDMA system. In the remaining sections, we go back to the continuous-time overloaded CDMA system. The question is whether and how this identical sum capacity can be achieved by the continuous-time CDMA system. It turns out that the results in this section play important roles. V. U PPER B OUND ON S UM C APACITY OF CDMA S YSTEM In this section, we derive an upper bound on the sum rate of the continuous-time band-limited CDMA system described in Section II. It turns out that the upper bound coincides with that of the sum capacity of the contrained FDMA system considered in Section IV after proper scaling of the system parameters. A. Problem Formulation We start by FRESH vectorizing the observation Y (t) in (5).

Y (t) =

w∈Ωn−1

Proof: Omitted. 2 The following theorem shows that the optimal solution allocates wk,opt = w ¯ if and only if the kth user passes the test. Theorem 1: Let nopt ∈ {1, 2, ..., N −1} be the largest index of the users that pass the test. Then, wopt ∈ Ω0 if no such user exist, and wopt ∈ Ωnopt otherwise. In the former case, wopt as the optimal solution to Problem 4 is given by (46). In the latter case, wopt is given by  ¯ ∀k ≤ nopt  w, p k wk,opt = (wtot − nopt w), ¯ ∀k ≥ nopt + 1.  PK 0 k0 =nopt +1 pk (60) Proof: Omitted. 2 As seen in (60), for the users that fail the test, proportionalshare bandwidth allocation is performed. Thus, we call the optimal allocation scheme the constrained proportional-share bandwidth allocation scheme. Using this optimal bandwidth allocation, the maximized sum rate can also be obtained as follows. Corollary 1: If wopt ∈ Ωnopt , then the sum capacity as the maximized objective function value to Problem 4 is given by C¯nopt . Proof: Straight forward by Theorem 1. 2 There is a striking resemblance between the test (56) and that used in [6, Eq. (5)] in identifying the oversized users 3 In terms of the definitions in this paper, an oversized user in [8],[9], and [13] is actually a non-undersized user.

=

K X k=1 K X

Xk (t) + N (t) ∞ X

dk [m]sk (t − mT ) + N (t)

(61a) (61b)

k=1 m=−∞

Now, the problem becomes the joint design of the codebooks for the sequences (dk [m])k and the vector-valued signals (sk (t))k that maximize the sum rate. Still as (5), the derivation of the sum capacity with the signal model (61) is not straightforward. So, instead, we consider the Gaussian signal model that has the same second-order properties as (5), which may provide an upper bound on the sum capacity. To proceed, we introduce the following definitions. Define the diagonal matrix-valued function Φ(f T ) of f ∈ F as Φ(f T ) , diag[Φ1 (f T ), Φ2 (f T ), ..., ΦK (f T )],

(62)

where the diagonal entries are the PSDs of the data sequences (dk [m])k with proper scaling of the argument. Also define the (2L + 1)-by-K matrix-valued function S(f ) of f ∈ F as S(f ) , [s1 (f ), s2 (f ), ..., sK (f )],

(63)

where the columns are the VFTs of the signature waveforms (sk (t))k . Then, the MV-PSD of the received signal Y (t) can be written as 1 (64) RY (f ) = N0 I2L+1 + S(f )Φ(f T )S(f )H , T By applying the result in [4] on the sum rate of a synchronous CDMA system to our case at each f ∈ F, we can obtain an 4 Actually, it can be shown that non-oversized users in [6, Eq. (5)] are also assigned orthogonal channels if they are critically-sized.

8

upper bound on the sum rate of the equivalent continuous-time MISO channel as C(Φ(f T )/T, S(f )) µ Z , log det I2L+1 + F

¶ 1 S(f )Φ(f T )S(f )H df. (65) N0 T

Note that, to upper bound the sum rate by (65), the signaling constraint on each user is relaxed to include any rank-1 widesense cyclostationary process with cycle frequency equal to the symbol rate of the CDMA system and with the same second-order property as (64). Note also that the optimal input distributions must be Gaussian to achieve this upper bound. Our objective in this section is to find Φ(f T ) and S(f ) that jointly maximize this upper bound, under the power (9a) and the bandwidth (9b) constraints. Thus, the optimization problem to find an upper bound on the sum capacity of the CDMA system can be formulated as Problem 5: maximize C(Φ(f T )/T, S(f )) Z 1 subject to Φk (f T )ksk (f )k2 df ≤ pk , ∀k, T F

Φ(f T ),S(f )

(66a) (66b)

where, and in what follows, the bandwidth constraint is implicitly imposed by using only the effective ones for VFTs and MV-PSDs. Instead of finding optimal (Φk (f ))k and (sk (f ))k , we find the optimal (Pk (f ))k and (zk (f ))k , because (Pk (f ), zk (f ))k can be obtained from (Φk (f ), sk (f ))k as (27) and (29). Define the diagonal matrix P (f ) as P (f ) , diag[P1 (f ), P2 (f ), ..., PK (f )],

(67)

and the matrix-valued frequency function Z(f ) as Z(f ) , [z1 (f ), z2 (f ), ..., zK (f )].

(68)

With these definitions, we can reformulate Problem 5 as Problem 6: maximize C(P (f )T, Z(f ))

(69a)

subject to Pk (f ) ≥ 0, ∀k, Z T Pk (f )df ≤ pk , ∀k, and

(69b)

P (f ),Z(f )

(69c)

F

kzk (f )k = 1, ∀k.

(69d)

Since it is not yet guaranteed that the reverse operation is possible to obtain (Φk (f ), sk (f ))k from (Pk (f ), zk (f ))k , the equivalence of Problems 5 and 6 is not clear except that the maximized objective function of Problem 6 upper bounds that of Problem 5. This issue will be handled in the next section. At this point, it is enough to notice that the solution to Problem 6 provides an upper bound on the sum capacity of the CDMA system.

we solve Problem 6 by modifying the inequality Pk ≤ pk in the power constraint to equality Pk = Pk,opt . Then, the optimization problem Problem 6 can be rewritten in a double maximization form as Problem 7: ( max C(P (f )T, Z(f )) Z(f ) maximize (70a) P (f ) subject to kzk (f )k = 1, ∀k, ∀f subject to Pk (f ) ≥ 0, ∀k, ∀f, and (70b) Z T Pk (f )df = Pk,opt , ∀k, ∀f. (70c) F

To maximize the objective function of the inner optimization problem of Problem 7, we need to find Z(f ) that maximizes the integrand at each f ∈ F . Hence, the inner optimization problem now reduces to Sub-Problem 1: µ ¶ T maximize log det IN (f ) + Z(f )P (f )Z(f )H (71a) N0 Z(f ) subject to kzk (f )k = 1, ∀k, (71b) which needs to be solved for each f ∈ F. Note that the size of the identity matrix is changed to N (f )-by-N (f ) and the VFTs are now all effective ones. This naturally incorporates the bandwidth constraint for each f . This problem at each f , except the 3 dB adjustment and parameter scaling, is exactly the same problem considered in [6], which is also exactly the same problem considered in the previous section for the FDMA system. Thus, by replacing the parameters in Problem 4 as σ 2 = N0 , wtot = N (f ), pk = Pk (f )T, wk = Wk (f )T , and w ¯ = 1, we formulate an equivalent problem that gives the same maximum value as Sub-Problem 2: µ ¶ K X Pk (f ) maximize Wk (f )T log 1 + (72a) N0 Wk (f ) (Wk (f ))k k=1

subject to 0 ≤ Wk (f )T ≤ 1, ∀k, and K X Wk (f )T = N (f ),

Let Pk,opt be the optimal transmit power of the kth user that satisfies Pk,opt ≤ pk , ∀k. How to determine these values will be discussed later at the end of this subsection. For a while,

(72c)

k=1

where Wk (f ) is called the differential FDMA-equivalent bandwidth of the kth user at f ∈ F. Define the rate density c(Pk (f ), Wk (f )) at f ∈ F as ¶ µ K X Pk (f ) . c(Pk (f ), Wk (f )) , Wk (f )T log 1 + N0 Wk (f ) k=1 (73) Then, Problem 7 can be converted to Problem 8: Z   c(Pk (f ), Wk (f ))df max    (Pk (f ))k F maximize subject to PkZ(f ) ≥ 0, ∀k, ∀f, and (74a) (Wk (f ))k     T P (f )df = P , ∀k, k

B. Derivation of Upper Bound

(72b)

k,opt

F

subject to 0 ≤ Wk (f )T ≤ 1, ∀k, ∀f, and K X Wk (f )T = N (f ), ∀f, k=1

(74b) (74c)

9

where we converted the double maximization problem back to the joint maximization problem, and the joint maximization problem to the alternate double maximization problem. To describe the solution to the inner optimiation problem of Problem 8, we introduce the notion of FDMA-equivalent bandwidth as follows. Definition 12: Given a differential FDMA-equivalent bandwidth (Wk (f ))f of the kth user, the corresponding quantity, called the FDMA-equivalent bandwidth Wk , shortly, the equivalent bandwidth is defined as Z Wk , T Wk (f )df. (75) F

The FDMA-equivalent bandwidth, defined as the average of the differential bandwidth, is named as such because there exists an FDMA system among all optimal multiple-access systems, of which physical bandwidth assigned to each user is the same as the optimal FDMA-equivalent bandwidth. Note that the definitions of differential FDMA-equivalent bandwidth and FDMA-equivalent bandwidth of this paper look similar to those defined in [9]. However, here, these quantities have no relation to the SINR at the output of LMMSE receivers. Proposition 4: The solution to the inner optimization of Problem 8 is given by Wk (f ) Pk,opt , ∀k, ∀f, Wk and the maximum value is given by µ ¶ K X Pk,opt Wk log 1 + . N 0 Wk Pk,opt (f ) =

(76a)

(76b)

k=1

Proof: Omitted. 2 Surprisingly, given a profile of the FDMA-equivalent bandwidth (Wk )k , the maximum value (77) of the inner optimization of Problem 8 is no longer dependent on the specific choice of the frequency functions (Wk (f ))k , as far as they satisfy the constraints (74b), (74c), and (75). With the results in (76), the outer maximization in Problem 8 reduces to the following problem. Sub-Problem 3: µ ¶ K X Pk,opt maximize Wk log 1 + (77a) N0 Wk (Wk )k k=1

subject to 0 ≤ Wk ≤ K X

1 , ∀k, and T

Wk = W,

(77b) (77c)

k=1

where the first constraint (77b) is just from (74b) and (75), and the constraint (77c) is from (19) and (74c). Note that, given a fixed feasible profile (Wk )k , the objective function (77a) is a strictly monotone increasing function of Pk,opt , ∀k. Thus, we must have Pk,opt = pk , ∀k. Therefore, the optimal solution can be obtained by simply replacing Pk,opt with pk , ∀k, and solving the FDMA sum rate maximization problem, Sub-Problem 3, where the bandwidth of each user cannot exceed the symbol rate of the CDMA system. So, we do not differentiate Pk,opt and pk in what follows.

Since Sub-Problem 3 has the same solution as Problem 4 with substitutions pk = Pk,opt , w ¯ = 1/T, σ 2 = N0 , pk = Pk,opt , and wtot = W , the optimal FDMA-equivalent bandwidth Wk,opt is given by  1   , ∀k ∈ K   T Wk,opt = (78)  P 1  k,opt  ·P (W T − |K|), ∀k ∈ / K,  0 T k0 ∈K / Pk ,opt now in terms of the index set K of oversized users. Therefore, the upper bound on the sum rate is given by µ ¶ X 1 Pk,opt T log 1 + C¯ , T N0 k∈K P µ ¶ µ ¶ |K| k∈K / Pk,opt T + W− log 1 + . (79) T N0 (W T − |K|) VI. O PTIMAL S IGNATURE WAVEFORMS AND S UM C APACITY In this section, the optimal signature waveforms are derived and the tightness of the upper bound is proved by constructing a set of signature waveforms and codebooks that achieve the upper bound, which proves that the upper bound is indeed the sum capacity. A. Multi-User Constrained Water-Pouring To construct the set of optimal signature waveforms, we need to trace back the path we paved in the previous section. The first step is to find the set K of oversized users by using the testing rule described in (56). The second step is to compute the optimal profile (Wk,opt )k of the FDMA-equivalent bandwidths by using (78), after replacing Pk,opt with pk , ∀k. The third step is to construct a profile (Wk,opt (f ))k of differential FDMA-equivalent bandwidth. As discussed already after Proposition 4, any profile (Wk,opt (f ))k is optimal if it is a multi-user constrained water-pouring solution. Once an optimal profile (Wk,opt (f ))k is obtained through the water-pouring procedure, the fourth step is to construct the optimal profile (Pk,opt (f ))k of differential signal power, which can be done by (76a). Since a pair of optimal profiles (Pk,opt (f ))k and (Wk,opt (f ))k are now obtained, the rate density (73) at each f ∈ F can be computed as c(Pk,opt (f ), Wk,opt (f )). The remaining task is to construct a set of signature waveforms and codebooks that jointly achieve the same rate density as c(Pk,opt (f ), Wk,opt (f )) for every f ∈ F. B. Optimal Signature Waveforms and Tightness of Upper Bound Let us fix f0 ∈ F and interpret the main result in [6] in terms of c(Pk,opt (f0 ), Wk,opt (f0 )). Then, given Popt (f0 ) as the power profile of a DT synchronous CDMA, there exists an optimal collection Zopt (f ) of signature sequences and corresponding codebooks such that c(Pk,opt (f ), Wk,opt (f )) µ ¶ T H = log det IN (f ) + Zopt (f )Popt (f )Zopt (f ) . (80) N0

10

Let zk,0 and (dˆk,0 [m])m be the optimal signature sequence and the sequence of codeword symbols from the optimal codebook, respectively, for the kth user. Now, construct a continuous-time synchronous CDMA system for the rate-1/T FRESH vectorized received signal to be given by Y (t) =

K X ∞ X

zk,0 dˆk,0 [m]sinc (T0 (t − mT0 )) + N (t), (81)

k=1m=−∞

where T0 > T is a constant and sinc(x) , sin(πx)/(πx) is the sinc pulse. It is obvious that this continuous-time system is equivalent to a DT synchronous CDMA system in that, after matched filtering the output of the vectorizer with sinc(T0 t) and sampling at rate 1/T0 , the sampled Y [m], after proper scaling, becomes Y [m] = PK output ˆk,0 [m]zk,0 + N [m], which can be processed by the d k=1 optimal multi-user decoder to result in the maximum sum rate given by c(Pk,opt (f0 ), Wk,opt (f0 ))/T0 . The following proposition shows that exactly the same signal as (81) can be generated by transmitters with a faster symbol rate of 1/T , so that the maximum sum rate can also be achieved using the same receiver. Proposition 5: Suppose is P∞that a ˆ bandpass signal X(t) j2πfc t 1 (t − mT0 )e d[m]g generated as X(t) = , m=−∞ 2T0 ˆ ˆ and that another bandpass signal X(t) as X(t) = P ∞ 1 (t − nT ), where d[n] , X(nT ). Then, n=−∞ d[n]g 2T ˆ X(t) = X(t) if [fc − 1/(2T0 ), fc + 1/(2T0 )) ⊂ F. Proof: Omitted. 2 If the optimal differential FDMA-equivalent bandwidth Wk,opt (f ) as a function of f is smooth enough ∀k or intentionally chosen to be piecewise constant, then the frequency band F can be divided into a finite number of sub-bands. It is clear that, by applying Proposition 5 to each sub-band, we can achieve the sum capacity at each sub-band, where the optimal codebooks at each sub-band are constructed by over-sampling the sinc interpolated and frequency-shifted codeword symbol sequences from jointly designed optimal codebooks of users. Since the codeword sequences of a user for different subbands are isolated in the frequency domain, we do not need multiple number of QAM modulators that work in each subband. We need only one QAM modulator that employs the signature waveform zk (t) having the VFT zk (f ), for each user. Thus, as the optimal signature waveforms, orthogonal waveforms are assigned to oversized users and CTE-GWBE sequences are assigned to non-oversized users, as is done exactly in [9]. Then, each transmitter can generate QAM modulated signal at rate 1/T by modulating codeword symbol sequence from a single codebook, that is constructed by oversampling the sinc interpolated and frequency-shifted codeword symbol sequences. VII. C ONCLUSIONS We consider designing optimal signature waveforms and codebooks that jointly maximize the sum rate for CDMA communications over strictly band-limited, continuous-time, AWGN channels. First, we show that the sum capacity derived in [6] for a synchronous CDMA system is exactly the same

as that for a bandwidth constrained FDMA system, where an upper bound is imposed to each user’s bandwidth additional to the total bandwidth. Then, we take a frequency-domain approach based on FRESH vectorization and scalarization to derive the sum capacity and jointly optimal signature waveforms and codebooks for the continuous-time systems. Unlike [6], this continuous-time problem requires, at least conceptual, construction of the joint codebook to be used by each CDMA transmitter. For this, we propose a method to construct a single codebook per user by over-sampling the sinc interpolated joint codebooks found for vector-observation synchronous CDMA systems. R EFERENCES [1] R. Gold, “Optimal binary sequences for spread spectrum multiplexing,” IEEE Trans. Inform. Theory, vol. IT-13, no. 4, pp. 619–621, Oct. 1967. [2] L. R. Welch, “Lower bounds on the maximum cross correlation of signals,”IEEE Trans. Inform. Theory, vol. 20, pp. 397–399, May 1974. [3] P. Fan and M. Darnell, Sequence Design for Communication Applications. New York: Wiley, 1996. [4] M. Rupf and J. L. Massey, “Optimum sequence set for synchronous code-division multiple-access,” IEEE Trans. Inform. Theory, vol. 40, pp. 1261–1266, July 1994. [5] P. Viswanath, V. Anantharam, and D. N. C. Tse, “Optimal sequences, power control, and user capacity of synchronous CDMA systems with linear MMSE multiuser receivers,” IEEE Trans. Inf. Theory, vol. 45, no. 6, pp. 1968–1983, Sep. 1999. [6] P. Viswanath and V. Anantharam, “Optimal sequences and sum capacity of synchronous CDMA systems,” IEEE Trans. Info. Theory, vol. 45, no. 6, pp. 1984–1991, Sep. 1999. [7] J. Luo, S. Ulukus, and A. Ephremides, “Optimal sequences and sum capacity of symbol asynchronous CDMA systems,” IEEE Trans. Info. Theory, vol. 51, no. 8, pp. 2760–2769, Aug. 2005. [8] J. H. Cho and W. Gao, “Continuous-time equivalents of Welch bound equality sequences,” IEEE Trans. Inf. Theory, vol. 51, no. 9, pp. 31763185, Sep. 2005. [9] J. H. Cho, “Multiuser constrained water-pouring for continuous-time overloaded Gaussian multiple-access channels,” IEEE Trans. Inf. Theory, vol. 54, no. 4, pp. 1437–1459, Apr. 2008. [10] T. Guess, “User-capacity maximization in synchronous CDMA subject to RMS-bandlimited signature waveforms,” IEEE Trans. Commun., vol. 52, no. 3, pp. 457–466, Mar. 2004. [11] H. H. Nguyen and E. Shewedyk, “Bandwidth-constrained signature waveforms for maximizing the network capacity of synchronous CDMA systems,” IEEE Trans. Commun., vol. 49, no. 6, pp. 961–965, June 2001. [12] H. H. Nguyen and E. Shewedyk, “A new construction of signature waveforms for synchronous CDMA systems,” IEEE Trans. Broadcasting, vol. 51, no. 4, pp. 520–529, Dec. 2005. [13] J. H. Cho, Q. Zhang, and L. Gao, “A comparison of frequency-division systems to code-division systems in overloaded channels,”IEEE Trans. Commun., vol. 56, no. 2, pp. 289–298, Feb. 2008. [14] B. W. Han and J. H. Cho, “Cyclic water filling for cyclostationary Gaussian noise channel: Part 1–Channel capacity,” to be submitted. [15] J. H. Cho and J. S. Lehnert, “An optimal signal design for band-limited asynchronous DS-CDMA communications,” IEEE Trans. Inform. Theory, vol. 48, no. 5, pp. 1172–1185, May 2002. [16] J. G. Proakis, Digital Communications, 4th ed. New York: McGraw Hill, 2001. [17] W. A. Gardner, “Cyclic Wiener filtering: theory and method,” IEEE Trans. Commun., vol. 41, no. 1, pp. 151 -163, Jan. 1993. [18] J. H. Cho, “Joint transmitter and receiver optimization in additive cyclostationary noise,” IEEE Trans. Inf. Theory, vol. 50, no. 14, pp. 3396–3405, Dec. 2004. [19] D. N. C. Tse and S. V. Hanly, “Linear multiuser receivers: effective interfernece, effective bandwidth and user capacity,” IEEE Trans. Inform. Theory, vol. 45, no. 3, pp. 641-657, Mar. 1999. [20] T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: Wiley, 1991. [21] R. W. Yeung, A First Course in Information Theory. New York: Kluwer Academic/Plenum Publishers, 2002.