Prolate spheroidal wave functions induce ... - Semantic Scholar
Prolate spheroidal wave functions induce Gaussian chip waveforms Yutaka Jitsumatsu and Tohru Kohda Dept. of Computer Science and Communication Engineering, Kyushu University, Motooka 744, Nishi-ku, Fukuoka, 819-0395, Japan. Email: {jitumatu, kohda}@csce.kyushu-u.ac.jp Abstract— Slepian’s prolate spheroidal wave function (PSWF) is optimum in energy concentration within given frequency bandwidth and time-width. However it is not employed in communications mainly because its inter-symbol interference is unfavorable. We show that introducing Markovian spreading codes makes PSWF be a strong candidate for the chip waveform in CDMA systems. For design tractability, Gaussian waveform is recommended.
I. I NTRODUCTION It is not possible to generate completely band-limited signals in finite time; they are physically unrealizable. Defining energy concentration ratios in a time duration T and a frequency bandwidth W , respectively, as α2 and β 2 , Slepian and Pollak [1] and Landau and Pollak [2] showed that the maximum attainable α2 and β 2 are given by the largest eigenvalue of a Fredholm integral equation with sinc kernel whose eigenfuntions are known as prolate spheroidal wave functions (PSWFs). The completely bandlimited signals are physically unrealizable, and therefore we should assume the channels are approximately bandlimited. Code division multiple access (CDMA) has become a standard method to realize the multiplex of users. Sum capacity of synchronous and time-discrete CDMA system is maximized by Welch bound equality (WBE) codes [3]. The aim of this paper is to give a rigorous formulation of the approximately band-limited and time-continuous CDMA channels from the Slepian’s standpoint [4], [5]. The most important assumption of this paper is that we cannot make the time delays between different users’ signals infinitesimal and there exists unavoidable time delays. Thus, we consider time delays are continuous-valued random variables. Spreading codes are currently generated by linear feedback shift registers (LFSRs). In this paper, they are considered to be generated from a stochastic process and are denoted by {Xn }. We assume they have correlations such that EX [Xn Xm ] = λ|n−m| with −1 < λ < 1, where EX [·] denotes the expectation with respect to codes. As a model of noise process and interferences, such an exponentially decaying autocorrelation has been employed [6], [7]. PSWFs do not satisfy the Nyquist condition; they have intersymbol-interference (ISI) as well as inter-chip interference (ICI). We therefore analyze the signal-to-interference-plusnoise ratio (SINR) of the receiver output. It is observed SINR performance of the eigenfunction corresponding to the largest
sampling at lT
message m
coder
~ φ(ω)
channel
decoder
m
bandlimiting filter
Fig. 1.
A model of time-continuous and bandlimited single-user channel.
message m1
coder
message m2
coder
message mK
coder
bandlimiting filter ~ φ(ω) delay
~ φ(ω)
delay
~ φ(ω)
delay
m1
sampling
decoder channel
m2
mK
Fig. 2. A model of time-continuous and bandlimited multiple-access channel.
eigenvalue is very poor and decreasing β 2 improves the SINR. This implies the spill-over is not a worthless thing but it turns to be advantageous for decreasing interferences. Hence, we should look for a chip waveform whose (α2 , β 2 ) is in the interior of the Slepian’s attainable region to enhance SINR. II. T IME -C ONTINUOUS BAND - LIMITED CDMA C HANNEL A time-continuous and bandlimited single-user channel model [8] is depicted in Fig. 1. The channel noise is assumed to be additive and Gaussian. Shannon’s capacity formula W log(1 + P0 /N0 W ) is based on the ideal low-pass filter 1 |ω| ≤ 2πW, ˜ φ(ω) = 2W (1) 0 |ω| > 2πW, √ ∞ ˜ where φ(ω) = −∞ φ(t)e− −1ωt dt and P0 is the signal power, N0 is the spectral density of Gaussian noise, and W is 2πW t the bandwidth. However, the chip waveform φ(t) = sin2πW t is physically unrealizable. Wyner [8], [9] gave a rigorous proof of the capacity formula, based on physically possible filters and Gaussian noise, where Slepian’s prolate spheroidal wave functions play an important role. Consider a baseband equivalent model of asynchronous direct sequence/code division multiple access (DS/CDMA) systems with K active users (Fig.2). The i-th user’s signature
waveform is defined as (i) xφ (t)
=
N −1
1
x(i) n φ(t − nTc ),
(2)
0.95
n=0 (i)
r(t) =
K ∞
0.9
2
0.85
β
−1 N where {xn }N n=0 ∈ {+1, −1} is a spreading sequence of the i-th user with spreading factor N and φ(t) is a chip waveform. We assume the chip duration Tc = 1 and that chip waveform has tails outside this interval. Signature waveform has excess energy outside the data duration Td = N Tc . (i) Let {dp }∞ p=−∞ be a data sequence of the i-th user. Then the received signal is
0.8
0.75
0.7
(i)
Ai d(i) p xφ (t − pTd − ti ) + n(t),
(3)
boundary rectangular pulse sinc pulse root raised cosine Gaussian pulse
0.65
i=1 p=−∞
where Td = N Tc is the data duration, Ai and ti are an amplitude and a time delay of the i-th user’s signal, and n(t) is an additive white Gaussian noise with two-sided noise spectrum N20 . For simplicity, Ai = 1 is assumed for i = 1, . . . , K. The matched filter output is given by (N −1)Tc +Tc /2 (i) r(t + pTd + ti )xφ (t)dt (4) Zp(i) =
0.6 0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
α2
Fig. 3. Possible combinations of (α2 , β 2 ) for φ(t) with 2W T = 1, where α2 is the energy contained in t ∈ [− T2 , T2 ] and β 2 is the energy contained in ω ∈ [−2πW, 2πW ]. We set T = Tc = 1 for rectangular, sinc, root raised cosine pulses.
−Tc /2
(i)
= Sp(i) + η (i) + ξp(i) + IK,p ,
(5)
(i)
where Sp is the i-th user’s signal component, η (i) is noise K (i) (i) (i,j) denotes component, ξp is ISI, and IK,p = j=1,j=i Ip MAI from other K − 1 users. Assume the noise, ISI and MAI are uncorrelated. The signal-to-noise plus interference ratio (SINR) of the singleuser receiver is SINR(i) =
(i)
{Sp }2 (i,j) 2 } j=i {Ip
(i)
+ E[{η (i) }2 ] + {ξp }2
,
(6)
∞ (i) where E[{η (i) }2 ] = N20 xφ (t)2 and φ2 = −∞ |φ(t)|2 dt. For multiuser detection (MUD) receivers, one can select sum capacity as the performance criterion [3], [10]. III. P ROLATE S PHEROIDAL WAVE F UNCTIONS It is well-known that any waveform has uncertainty tradeoff in frequency and time domains and that Gaussian pulse is the optimum one when the variance of the energy distribution function is concerned. On the other hand, Slepian and Pollak [1] and Landau and Pollak [2] discussed this uncertainty problem in a different manner, which is suitable for communication engineering. They introduced linear operators BΩ and DT for limiting the bandwidth and time-width, respectively, i.e. Ω √ 1 −1ωt ˜ dω (7) BΩ φ(t) = φ(ω)e 2π −Ω φ(t) − T2 ≤ t ≤ T2 , DT φ(t) = (8) 0 |t| > T2 .
They investigated the achievable energy concentration ratios α2 and β 2 , in a given angular frequency bandwidth, Ω = 2πW and a given time interval T . α2 =
DT φ2 , φ2
β2 =
B2πW φ2 . φ2
(9)
There are two fundamental questions: for bandlimited waveforms what is the maximum energy concentration ratio in time slot T and for time-limited waveforms what is the maximum energy concentration ratio in frequency band W . The answers to these questions are, respectively, given by the maximum eigenvalues of the following integral equations DT B2πW ψα = γα ψα , B2πW DT ψβ = γβ ψβ . (i)
(i)
Each set of eigenfunctions {ψα } and {ψβ } thogonal, i.e. ∞ ψ˜α(i) (f )ψ˜α(j) (f )df = δi,j , −∞ ∞ (i) (j) ψβ (t)ψβ (t)dt = δi,j , −∞ W ψ˜α(i) (f )ψ˜α(j) (f )df = γα(i) δi,j , −W T /2 (i) (j) (i) ψβ (t)ψβ (t)dt = γβ δi,j . −T /2
(10)
is doubly or⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(11)
Because B2πW and DT are dual with respect to Fourier (i) (i) transform, they have same eigenvalues γα = γβ (= γ (i) ) (i) (i = 0, 1, . . .). The eigenfunctions have a relation ψβ = 1 (i) (γ (i) )− 2 DT ψα . Therefore it is sufficient to consider one of the two integral equations.
4
Any chip waveform should have its energy concentration ratio in this region. Fig. 3 shows the possible combinations for 2W T = 1, as well as rectangular, sinc, and root raised cosine pulses with roll-off factor 0.22. The dashed curve shows t2 (α2 , β 2 ) for Gaussian waveforms φ(t) = a exp(− 2τ 2 ) where τ is a variable parameter and a is a normalization factor. The left equation in (10) is expressed as T /2 T sin 2πW (t − s) ds (|t| ≤ ). (12) γφ(t) = φ(s) π(t − s) 2 −T /2 This equation has isolated positive eigenvalues 1 > γ0 > γ1 , . . .. The eigenfunctions, denoted by ψ (i) , are referred to as prolate spheroidal wave functions, which are scaled versions d (1 − t2 ) dφ of the solutions of the differential equation dt dt + 2 2 (χ − c t )φ = 0 (t ∈ [−1, 1]), where c = πW T and χ = χ(c) is the eigenvalue of this equation [1]. A rigorous proof of Shannon’s capacity formula was given by Wyner [8], [9], where a code word (x0 , x1 , . . . , xn−1 ) is n−1 (i) modulated into i=0 xi ψ (t) with n = 2T W . In spite of its rigorousness, such a modulation, however, is not used, probably because n 1 filters are needed in parallel. Instead, n−1 we consider simple modulation in the form of i=0 xi φ(t−i), where φ(t) is called chip waveform. Let us consider the band-limitedness and time-limitedness of the signature waveforms in CDMA systems in a way similar to Slepian [1], [2]. The signature waveform is Xφ (t) = N −1 N −1 N −1 n=0 Xn φ(t − nTc + 2 Tc ), where the time shift 2 Tc is added so that the time slot [0, N Tc ] is replaced by [− N2Tc , N2Tc ]. Assume the chip duration is Tc = 1 and the symbol duration is Td = N . The bandwidth of the spreading codes is W = 1/(2Tc) = 1/2. Note that we consider the bandwidth is W = 1/2 for both φ and Xφ but time slots are Tc = 1 and Td = N , respectively. Then α2 =
DN Xφ 2 , Xφ 2
β2 =
Bπ Xφ 2 . Xφ 2
(13)
Obviously, the optimum chip waveform depends on spreading codes X. We can see π 2 ˜ ˜ |X(ω)| |φ(ω)|2 dω 2 β = −π (14) ∞ 2 ˜ 2 ˜ −∞ |X(ω)| |φ(ω)| dω ˜ where X(ω) is the Fourier transform of {Xn }. In order to see the effect of codes, let us consider that spreading codes are random variables with exponentially vanishing autocorrelation: EX [Xn Xm ] = λ|n−m| ,
(n, m = 0, 1, . . . , N − 1)
(15)
where EX [·] denotes the expectation with respect to codes and −1 < λ < 1 is a parameter. The average power spectrum of
3.5
λ=0.6
λ=−0.6 3
~ EX[|X( f )| 2 ]
Landau and Pollak [2] showed that the possible combinations of (α2 , β 2 ) is the intersection of the unit square 0 ≤ α2 ≤ 1 and 0 ≤ β 2 ≤ 1 and the elliptical region cos−1 α + cos−1 β ≥ cos−1 γ (0) .
2.5
2
λ=0.2
λ=−0.2
1.5
λ=0.0
1
0.5
0 -0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
frequency [f]
Fig. 4.
The power spectrum of Markov sequences with a parameter λ.
the spreading codes is given by N −1 N −1 √ 1 1 2 ˜ EX [|X(f )| ] = EX [Xn Xm ]e−2π −1(n−m)f N N n=0 m=0
≈
1+
λ2
1 − λ2 , − 2λ cos(2πf )
(16)
when N 1, which is a periodic function with period 2W = 1 and is illustrated in Fig. 4. This figure shows the spreading codes with positive λ works as a low pass filter, whereas the codes with negative one works as a high pass filter. This function is applied in (14) therefore β 2 increases (decreases, respectively) if positive (negative) λ is used. The case λ = 0 implies i.i.d. codes which has flat power spectrum. Note that sequences generated by a Markov chain have the above exponential vanishing autocorrelation functions. R.E.Kalman found that such a Markov chain can be embedded into a chaotic dynamic of piecewise-linear Markov map [11]. Mazzini et. al reported that Markovian spreading codes improves the bit error rate of chip-asynchronous CDMA systems [12]. Landolsi and Stark [13] assumed that chip waveforms are time limited within one chip duration to design inter-chip interference (ICI)-free solitary waves. For symbol-synchronous CDMA systems, two modulation methods using PSWFs are analyzed in [14]. One method is i xi ψ (i) (t), where ψ (i) (t) are completely timelimited PSWFs in [0, Td] and the other is i xi φ(t − i), where φ(t) is completely timelimited PSWF in [0, Tc ]. The former method is optimum in terms of fractional out-of-band energy bandwidth and the latter one is suboptimal and found better than rectangular pulse and half sine waves. In [15], a chip waveform design which allows ICI is presented to improve BER performance. We give new integral equations corresponding to (13): Lemma 1: For an absolutely bandlimited chip waveform, i.e., φ(t) = BΩ φ(t), the maximum energy concentration of Xφ (t) in t ∈ [−N/2, N/2] is given by the largest eigenvalue of the integral equation: 1/2 2 ˜ γX,α EX [|X(f )| ]ψX,α (f ) = Kα (f, f )ψ˜X,α (f )df −1/2
(17)
TABLE I
4
PERFORMANCES FOR I . I . D . CODES AND SEVERAL WAVEFORMS WHEN 3.5
2W Tc = 1.
β2 K−1 SIR N
(1)
(2)
ψX,α 0.991136 1 0.8134
ψX,α 0.971059 1 0.7153
ψX,α 0.956526 1 0.7023
(0)
ψX,β 1 0.2050 2.7210
(1)
ψX,β 1 0.0113 3.6310
ψX,β 1 0.7833 1.4563
α2 β2 K−1 SIR N
(2)
sinc 0.7737 1 1
rrc 0.8387 0.9647 1.0582
rect 1 0.7737 1.5
2.5
~ (0) ψ X,α ( f )
(0)
α2
(0)
(1)
γX,α 0.9329 0.9674 0.9710 0.9644 0.9215
(2)
γX,α 0.9103 0.9508 0.9565 0.9510 0.9093
λ=0.4 λ=−0.4
λ=0.0
1
0.5
T HE EIGENVALUES OF CODE AVERAGED KERNEL FOR COMPLETELY BAND - LIMITED AND TIME - LIMITED SIGNALS . γX,α 0.9498 0.9838 0.9911 0.9942 0.9959
λ=−0.8 2
1.5
TABLE II
λ -0.8 -0.4 0 0.4 0.8
λ=0.8
3
(0)
γX,β 0.5824 0.6888 0.7833 0.8710 0.9536
(1)
γX,β 0.4070 0.2974 0.2050 0.1213 0.0434
(2)
γX,β 0.0103 0.0134 0.0113 0.0074 0.0028
0 -0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
frequency [ f ]
0.3
0.4
0.5
0.6
0.7
(0) Fig. 5. Code modulated prolate spheroidal chip waveforms ψ˜X,α (f ) bandlimited to f ∈ [−1/2, 1/2]. 1.4
λ=0.8 λ=0.4
1.2
) is a Fourier transform of {Xn } and where X(f )X(f ˜ )] sin N π(f − f ) . Kα (f, f ) = EX [X(f π(f − f )
(18)
On the other hand, for an absolutely time-limited chip waveform in one chip duration, i.e. φ(t) = DTc φ(t), the maximum energy concentration of Xφ (t) in frequency f ∈ [−1/2, 1/2] is given by the largest eigenvalue of the integral equation: 1/2 1 ψX,β (s)Kβ (t, s)ds, (|t| < ) (19) γX,β ψX,β (t) = 2 −1/2 N −1 N −1 1 π(t−s+n−m) Kβ (t, s) = EX [Xn Xm ] sinπ(t−s+n−m) . N n=0 m=0
(20)
Note that if EX [Xn Xm ] = δn,m , the kernel Kβ reduces to the Slepian’s sinc kernel. (0) (1) (0) (1) Let γX,α > γX,α > . . . and γX,β > γX,β > . . . (0) be the eigenvalues of Eqs. (17) and (19) and let ψX,α (t), (1) (0) (1) ψX,α (t), . . . and ψX,β (t), ψX,β (t), . . . be the corresponding eigenfunctions. These eigenfunctions depend on the spreading codes. The largest eigenvalue changes as shown in Table II. (0) (0) The dominant eigenfunctions ψ α (f ) and ψβ (t) are depicted in Fig. 5 and 6, respectively, for λ = 0, ±0.4, ±0.8. The chip (0) waveform ψβ (t) time-limited to [−0.5, 0.5] approaches to a rectangular pulse as λ becomes large and to the shape like half sine pulse as −λ becomes large. IV. I NTERFERENCES Aa far as energy concentrations (α2 , β 2 ) are concerned, spreading code with positive λ is better and optimum choice (0) (0) of chip waveforms is φ(t) = c0 ψX,α (t) + d0 ψX,β (t) for some constants c0 and d0 . However, we do not recommend
ψX,(0)β (t)
1
λ=0.0
0.8
λ=−0.4
0.6
λ=−0.8 0.4
0.2
0 -0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
time [t] (0)
Fig. 6. Code modulated prolate spheroidal chip waveforms ψX,β (t) timelimited to t ∈ [−0.5, 0.5].
such spreading codes and chip waveforms for CDMA system, because this pair of codes and chip waveforms has very strong multiple-access interference (MAI), which limits the (0) capacity of CDMA system. Table I shows that ψX,α and (0) ψX,β for i.i.d. codes case are worse than root raised cosine (rrc) with roll-off factor 0.22 and rectangular (rect) pulses, respectively. This table implies we should use the second and third eigenfunctions, too. Hence the chip waveform is expressed by φ(t) =
∞ i=0
(i)
ci ψX,α (t) +
∞
(i)
di ψX,β (t).
(21)
i=0
The coefficients ci and di should be chosen so as to achieve compromise between the energy concentrations (α2 , β 2 ) and the MAI. PSWFs do not satisfy the Nyquist condition, therefore we should analyze the SIR. Table I suggests that there is a tradeoff between SIR and β 2 . Hence we should use negative λ, which sacrifices β 2 but improves SIR performance. Our
14
desired chip waveform is chosen so as to be in the interior of the Slepian’s region, but not on the boundary of it.
12
10
1 SIR
6
,
λ=−0.8 λ=−0.9 λ=−0.4
4
λ=−0.0 λ=0.4 λ=0.6
0
λ=0.8
-2
-4 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 τ
1.1
1.2
R EFERENCES [1] D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty-I,” Bell Syst. Tech. J., vol. 40, no. 1, pp. 43–63, Jan. 1961. [2] H.J.Landau and H.O.Pollak, “Prolate spheroidal wave function, Fourier analysis and uncertaintry - II,” Bell Syst. Tech. J., pp. 65–84, Jan. 1961. [3] M. Rupf and J. L. Massey, “Optimum sequence multisets for synchronous code-division multiple-access channels,” IEEE Trans. Inform. Theory, vol. 40, no. 4, pp. 1261–1266, 1994. [4] D. Slepian, “On bandwidth,” Proc. IEEE, vol. 64, pp. 292–300, 1976.
1.4
1.5
1.6
1.7
1.8
1.9
2
10
rrc τ=1.0
sinc
(23)
N −1 where Eb = k=1−N (N − k)λ|k| Φ(k). Eq. (22) implies that we need to calculate the fourth order quantity of φ(t) to evaluate SIR. Double orthogonality (11) (i) (i) holds for ψX,α and ψX,β , but orthogonality does not hold (i) (j) between ψX,α and ψX,β . These facts make the choice of ci and di bothersome. In order to avoid choosing ci and di and to simplify the problem, a Gaussian chip waveform isexamined. A Gaussian t2 , where a is a pulse is defined by φG (t) = a exp − 2τ 2 normalization factor and τ expresses the standard deviation of the energy distribution in time domain. Note that β 2 increases (thus, bandwidth of the signal decreases) as τ increases. In Fig. 7, SIR performance is plotted against a parameter τ . Interestingly, SIRs have peaks around τ = 0.85, 0.95, 1.05 for λ = −0.7, −0.8, −0.9, respectively, while SIRs are monotone decreasing for positive λ. The curve of SINR is obtained by shifting up the curve of SIR according to (23). Fig. 8 illustrates the SIR performance versus −1 < λ < 1. For τ = 0.9, the optimum SIR is achieved by λ = −0.76 with energy concentrations (α2 , β 2 ) = (0.9853, 0.9980). SIR performance as well as energy concentrations are better than that of root raised cosine pulses. Optimization of the chip waveform can be performed in terms of ci and di in (21), which is left as a future work.
1.3
Fig. 7. SIR performance of Gaussian pulse with parameters of τ and Markov codes with several λ’s (N = 127, K = 31). SINRs of root raised cosine (rrc) and sinc pulses are indicated.
8
N0 2Eb
λ=−0.7
rrc sinc
2
τ=0.3 6
τ=0.7
τ=0.9 SIR [dB]
SINR(λ, N, K, N20 , Φ) =
1 +
λ=−0.6
8
SIR [dB]
Lemma 2: For chip-asynchronous DS/CDMA systems with K active users, spreading factor N , and chip waveform φ(t), in an additive white Gaussian noise with two-sided spectrum density N0 /2, the signal to interference ratio of single user receiver output with Markov codes having eigenvalue λ is
2 1 |k| (N − k)λ Φ(k) SIR(λ, N, K, Φ) = N k 1 + λ2 · (K − 1) |k| + λ|k| Φ ∗ Φ(k) 1 − λ2 k −1 1 + E[{ξp(i) }2 ] , (22) N where Φ(t) = φ(t )φ(t− t )dt and Φ∗ Φ(t) = Φ(t )Φ(t− t )dt . Then SINR is given by
τ=0.5
4
2
τ=1.5
τ=1.3
τ=1.2
τ=1.1
0
-2
-4
-1
-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1
0
0.1
0.2 0.3
0.4
0.5 0.6
0.7 0.8
0.9
1
λ
Fig. 8. SIR performance of Gaussian pulse with parameters of λ with τ from 0.3 to 1.5 (N = 127, K = 31). SIRs of rrc and sinc pulses are indicated. [5] ——, “Some comments on Fourier analysis, uncertainty and modeling,” SIAM Review, vol. 25, no. 3, pp. 379–393, 1983. [6] ——, “Estimation of signal parameters in the presence of noise,” Transaction of the I.R.E, vol. 3, pp. 68–89, Mar. 1954. [7] N. S. Jayant and P. Noll, Digital Coding of Waveforms. Pentice-Hall, INC., 1984. [8] A. D. Wyner, “The capacity of the band-limited Gaussian channel,” Bell Syst. Tech. J., vol. 45, pp. 359–395, Mar. 1966. [9] ——, “A note on the capacity of the band-limited Gaussian channel,” Bell Syst. Tech. J., vol. 55, no. 3, pp. 343–346, Mar. 1976. [10] J. Luo, S. Ulukus, and A. Ephremides, “Optimal sequences and sum capacity of symbol asynchronous CDMA systems,” IEEE Trans. on Information Theory, 2005. [11] R. E. Kalman, “Nonlinear aspects of sampled-data control systems,” in Proc. Symp. Nonlinear Circuit Analysis VI, 1956, pp. 273–313. [12] G. Mazzini, R.Rovatti, and G. Setti, “Interference minimisation by autocorrelation shaping in asynchronous DS/CDMA systems: Chaos based spreading is nearly optimal,” Electronics Letters, IEE, vol. 35, pp. 1054–55, 1999. [13] M. A. Landolsi and W. E. Stark, “DS-CDMA chip waveform design for minimal interference under bandwidth, phase, and envelope constraints,” IEEE Trans. Commun., vol. 47, no. 11, pp. 1737–1746, Nov. 1999. [14] H. H. Nguyen and E. Shwedyk, “Bandwidth-constrained signature waveforms for maximizing the network capacity of synchronous CDMA systems,” IEEE Trans. Comm., vol. 49, no. 6, pp. 961–965, June 2001. [15] W. Gao, J. H. Cho, and J. S. Lehnert, “Chip waveform design for DS/SSMA systems with aperiodic random spreading sequences,” IEEE Trans. Wireless Commun., vol. 1, no. 1, pp. 37–45, Jan. 2002.
I. I NTRODUCTION It is not possible to generate completely band-limited signals in finite time; they are physically unrealizable. Defining energy concentration ratios in a time duration T and a frequency bandwidth W , respectively, as α2 and β 2 , Slepian and Pollak [1] and Landau and Pollak [2] showed that the maximum attainable α2 and β 2 are given by the largest eigenvalue of a Fredholm integral equation with sinc kernel whose eigenfuntions are known as prolate spheroidal wave functions (PSWFs). The completely bandlimited signals are physically unrealizable, and therefore we should assume the channels are approximately bandlimited. Code division multiple access (CDMA) has become a standard method to realize the multiplex of users. Sum capacity of synchronous and time-discrete CDMA system is maximized by Welch bound equality (WBE) codes [3]. The aim of this paper is to give a rigorous formulation of the approximately band-limited and time-continuous CDMA channels from the Slepian’s standpoint [4], [5]. The most important assumption of this paper is that we cannot make the time delays between different users’ signals infinitesimal and there exists unavoidable time delays. Thus, we consider time delays are continuous-valued random variables. Spreading codes are currently generated by linear feedback shift registers (LFSRs). In this paper, they are considered to be generated from a stochastic process and are denoted by {Xn }. We assume they have correlations such that EX [Xn Xm ] = λ|n−m| with −1 < λ < 1, where EX [·] denotes the expectation with respect to codes. As a model of noise process and interferences, such an exponentially decaying autocorrelation has been employed [6], [7]. PSWFs do not satisfy the Nyquist condition; they have intersymbol-interference (ISI) as well as inter-chip interference (ICI). We therefore analyze the signal-to-interference-plusnoise ratio (SINR) of the receiver output. It is observed SINR performance of the eigenfunction corresponding to the largest
sampling at lT
message m
coder
~ φ(ω)
channel
decoder
m
bandlimiting filter
Fig. 1.
A model of time-continuous and bandlimited single-user channel.
message m1
coder
message m2
coder
message mK
coder
bandlimiting filter ~ φ(ω) delay
~ φ(ω)
delay
~ φ(ω)
delay
m1
sampling
decoder channel
m2
mK
Fig. 2. A model of time-continuous and bandlimited multiple-access channel.
eigenvalue is very poor and decreasing β 2 improves the SINR. This implies the spill-over is not a worthless thing but it turns to be advantageous for decreasing interferences. Hence, we should look for a chip waveform whose (α2 , β 2 ) is in the interior of the Slepian’s attainable region to enhance SINR. II. T IME -C ONTINUOUS BAND - LIMITED CDMA C HANNEL A time-continuous and bandlimited single-user channel model [8] is depicted in Fig. 1. The channel noise is assumed to be additive and Gaussian. Shannon’s capacity formula W log(1 + P0 /N0 W ) is based on the ideal low-pass filter 1 |ω| ≤ 2πW, ˜ φ(ω) = 2W (1) 0 |ω| > 2πW, √ ∞ ˜ where φ(ω) = −∞ φ(t)e− −1ωt dt and P0 is the signal power, N0 is the spectral density of Gaussian noise, and W is 2πW t the bandwidth. However, the chip waveform φ(t) = sin2πW t is physically unrealizable. Wyner [8], [9] gave a rigorous proof of the capacity formula, based on physically possible filters and Gaussian noise, where Slepian’s prolate spheroidal wave functions play an important role. Consider a baseband equivalent model of asynchronous direct sequence/code division multiple access (DS/CDMA) systems with K active users (Fig.2). The i-th user’s signature
waveform is defined as (i) xφ (t)
=
N −1
1
x(i) n φ(t − nTc ),
(2)
0.95
n=0 (i)
r(t) =
K ∞
0.9
2
0.85
β
−1 N where {xn }N n=0 ∈ {+1, −1} is a spreading sequence of the i-th user with spreading factor N and φ(t) is a chip waveform. We assume the chip duration Tc = 1 and that chip waveform has tails outside this interval. Signature waveform has excess energy outside the data duration Td = N Tc . (i) Let {dp }∞ p=−∞ be a data sequence of the i-th user. Then the received signal is
0.8
0.75
0.7
(i)
Ai d(i) p xφ (t − pTd − ti ) + n(t),
(3)
boundary rectangular pulse sinc pulse root raised cosine Gaussian pulse
0.65
i=1 p=−∞
where Td = N Tc is the data duration, Ai and ti are an amplitude and a time delay of the i-th user’s signal, and n(t) is an additive white Gaussian noise with two-sided noise spectrum N20 . For simplicity, Ai = 1 is assumed for i = 1, . . . , K. The matched filter output is given by (N −1)Tc +Tc /2 (i) r(t + pTd + ti )xφ (t)dt (4) Zp(i) =
0.6 0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
α2
Fig. 3. Possible combinations of (α2 , β 2 ) for φ(t) with 2W T = 1, where α2 is the energy contained in t ∈ [− T2 , T2 ] and β 2 is the energy contained in ω ∈ [−2πW, 2πW ]. We set T = Tc = 1 for rectangular, sinc, root raised cosine pulses.
−Tc /2
(i)
= Sp(i) + η (i) + ξp(i) + IK,p ,
(5)
(i)
where Sp is the i-th user’s signal component, η (i) is noise K (i) (i) (i,j) denotes component, ξp is ISI, and IK,p = j=1,j=i Ip MAI from other K − 1 users. Assume the noise, ISI and MAI are uncorrelated. The signal-to-noise plus interference ratio (SINR) of the singleuser receiver is SINR(i) =
(i)
{Sp }2 (i,j) 2 } j=i {Ip
(i)
+ E[{η (i) }2 ] + {ξp }2
,
(6)
∞ (i) where E[{η (i) }2 ] = N20 xφ (t)2 and φ2 = −∞ |φ(t)|2 dt. For multiuser detection (MUD) receivers, one can select sum capacity as the performance criterion [3], [10]. III. P ROLATE S PHEROIDAL WAVE F UNCTIONS It is well-known that any waveform has uncertainty tradeoff in frequency and time domains and that Gaussian pulse is the optimum one when the variance of the energy distribution function is concerned. On the other hand, Slepian and Pollak [1] and Landau and Pollak [2] discussed this uncertainty problem in a different manner, which is suitable for communication engineering. They introduced linear operators BΩ and DT for limiting the bandwidth and time-width, respectively, i.e. Ω √ 1 −1ωt ˜ dω (7) BΩ φ(t) = φ(ω)e 2π −Ω φ(t) − T2 ≤ t ≤ T2 , DT φ(t) = (8) 0 |t| > T2 .
They investigated the achievable energy concentration ratios α2 and β 2 , in a given angular frequency bandwidth, Ω = 2πW and a given time interval T . α2 =
DT φ2 , φ2
β2 =
B2πW φ2 . φ2
(9)
There are two fundamental questions: for bandlimited waveforms what is the maximum energy concentration ratio in time slot T and for time-limited waveforms what is the maximum energy concentration ratio in frequency band W . The answers to these questions are, respectively, given by the maximum eigenvalues of the following integral equations DT B2πW ψα = γα ψα , B2πW DT ψβ = γβ ψβ . (i)
(i)
Each set of eigenfunctions {ψα } and {ψβ } thogonal, i.e. ∞ ψ˜α(i) (f )ψ˜α(j) (f )df = δi,j , −∞ ∞ (i) (j) ψβ (t)ψβ (t)dt = δi,j , −∞ W ψ˜α(i) (f )ψ˜α(j) (f )df = γα(i) δi,j , −W T /2 (i) (j) (i) ψβ (t)ψβ (t)dt = γβ δi,j . −T /2
(10)
is doubly or⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(11)
Because B2πW and DT are dual with respect to Fourier (i) (i) transform, they have same eigenvalues γα = γβ (= γ (i) ) (i) (i = 0, 1, . . .). The eigenfunctions have a relation ψβ = 1 (i) (γ (i) )− 2 DT ψα . Therefore it is sufficient to consider one of the two integral equations.
4
Any chip waveform should have its energy concentration ratio in this region. Fig. 3 shows the possible combinations for 2W T = 1, as well as rectangular, sinc, and root raised cosine pulses with roll-off factor 0.22. The dashed curve shows t2 (α2 , β 2 ) for Gaussian waveforms φ(t) = a exp(− 2τ 2 ) where τ is a variable parameter and a is a normalization factor. The left equation in (10) is expressed as T /2 T sin 2πW (t − s) ds (|t| ≤ ). (12) γφ(t) = φ(s) π(t − s) 2 −T /2 This equation has isolated positive eigenvalues 1 > γ0 > γ1 , . . .. The eigenfunctions, denoted by ψ (i) , are referred to as prolate spheroidal wave functions, which are scaled versions d (1 − t2 ) dφ of the solutions of the differential equation dt dt + 2 2 (χ − c t )φ = 0 (t ∈ [−1, 1]), where c = πW T and χ = χ(c) is the eigenvalue of this equation [1]. A rigorous proof of Shannon’s capacity formula was given by Wyner [8], [9], where a code word (x0 , x1 , . . . , xn−1 ) is n−1 (i) modulated into i=0 xi ψ (t) with n = 2T W . In spite of its rigorousness, such a modulation, however, is not used, probably because n 1 filters are needed in parallel. Instead, n−1 we consider simple modulation in the form of i=0 xi φ(t−i), where φ(t) is called chip waveform. Let us consider the band-limitedness and time-limitedness of the signature waveforms in CDMA systems in a way similar to Slepian [1], [2]. The signature waveform is Xφ (t) = N −1 N −1 N −1 n=0 Xn φ(t − nTc + 2 Tc ), where the time shift 2 Tc is added so that the time slot [0, N Tc ] is replaced by [− N2Tc , N2Tc ]. Assume the chip duration is Tc = 1 and the symbol duration is Td = N . The bandwidth of the spreading codes is W = 1/(2Tc) = 1/2. Note that we consider the bandwidth is W = 1/2 for both φ and Xφ but time slots are Tc = 1 and Td = N , respectively. Then α2 =
DN Xφ 2 , Xφ 2
β2 =
Bπ Xφ 2 . Xφ 2
(13)
Obviously, the optimum chip waveform depends on spreading codes X. We can see π 2 ˜ ˜ |X(ω)| |φ(ω)|2 dω 2 β = −π (14) ∞ 2 ˜ 2 ˜ −∞ |X(ω)| |φ(ω)| dω ˜ where X(ω) is the Fourier transform of {Xn }. In order to see the effect of codes, let us consider that spreading codes are random variables with exponentially vanishing autocorrelation: EX [Xn Xm ] = λ|n−m| ,
(n, m = 0, 1, . . . , N − 1)
(15)
where EX [·] denotes the expectation with respect to codes and −1 < λ < 1 is a parameter. The average power spectrum of
3.5
λ=0.6
λ=−0.6 3
~ EX[|X( f )| 2 ]
Landau and Pollak [2] showed that the possible combinations of (α2 , β 2 ) is the intersection of the unit square 0 ≤ α2 ≤ 1 and 0 ≤ β 2 ≤ 1 and the elliptical region cos−1 α + cos−1 β ≥ cos−1 γ (0) .
2.5
2
λ=0.2
λ=−0.2
1.5
λ=0.0
1
0.5
0 -0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
frequency [f]
Fig. 4.
The power spectrum of Markov sequences with a parameter λ.
the spreading codes is given by N −1 N −1 √ 1 1 2 ˜ EX [|X(f )| ] = EX [Xn Xm ]e−2π −1(n−m)f N N n=0 m=0
≈
1+
λ2
1 − λ2 , − 2λ cos(2πf )
(16)
when N 1, which is a periodic function with period 2W = 1 and is illustrated in Fig. 4. This figure shows the spreading codes with positive λ works as a low pass filter, whereas the codes with negative one works as a high pass filter. This function is applied in (14) therefore β 2 increases (decreases, respectively) if positive (negative) λ is used. The case λ = 0 implies i.i.d. codes which has flat power spectrum. Note that sequences generated by a Markov chain have the above exponential vanishing autocorrelation functions. R.E.Kalman found that such a Markov chain can be embedded into a chaotic dynamic of piecewise-linear Markov map [11]. Mazzini et. al reported that Markovian spreading codes improves the bit error rate of chip-asynchronous CDMA systems [12]. Landolsi and Stark [13] assumed that chip waveforms are time limited within one chip duration to design inter-chip interference (ICI)-free solitary waves. For symbol-synchronous CDMA systems, two modulation methods using PSWFs are analyzed in [14]. One method is i xi ψ (i) (t), where ψ (i) (t) are completely timelimited PSWFs in [0, Td] and the other is i xi φ(t − i), where φ(t) is completely timelimited PSWF in [0, Tc ]. The former method is optimum in terms of fractional out-of-band energy bandwidth and the latter one is suboptimal and found better than rectangular pulse and half sine waves. In [15], a chip waveform design which allows ICI is presented to improve BER performance. We give new integral equations corresponding to (13): Lemma 1: For an absolutely bandlimited chip waveform, i.e., φ(t) = BΩ φ(t), the maximum energy concentration of Xφ (t) in t ∈ [−N/2, N/2] is given by the largest eigenvalue of the integral equation: 1/2 2 ˜ γX,α EX [|X(f )| ]ψX,α (f ) = Kα (f, f )ψ˜X,α (f )df −1/2
(17)
TABLE I
4
PERFORMANCES FOR I . I . D . CODES AND SEVERAL WAVEFORMS WHEN 3.5
2W Tc = 1.
β2 K−1 SIR N
(1)
(2)
ψX,α 0.991136 1 0.8134
ψX,α 0.971059 1 0.7153
ψX,α 0.956526 1 0.7023
(0)
ψX,β 1 0.2050 2.7210
(1)
ψX,β 1 0.0113 3.6310
ψX,β 1 0.7833 1.4563
α2 β2 K−1 SIR N
(2)
sinc 0.7737 1 1
rrc 0.8387 0.9647 1.0582
rect 1 0.7737 1.5
2.5
~ (0) ψ X,α ( f )
(0)
α2
(0)
(1)
γX,α 0.9329 0.9674 0.9710 0.9644 0.9215
(2)
γX,α 0.9103 0.9508 0.9565 0.9510 0.9093
λ=0.4 λ=−0.4
λ=0.0
1
0.5
T HE EIGENVALUES OF CODE AVERAGED KERNEL FOR COMPLETELY BAND - LIMITED AND TIME - LIMITED SIGNALS . γX,α 0.9498 0.9838 0.9911 0.9942 0.9959
λ=−0.8 2
1.5
TABLE II
λ -0.8 -0.4 0 0.4 0.8
λ=0.8
3
(0)
γX,β 0.5824 0.6888 0.7833 0.8710 0.9536
(1)
γX,β 0.4070 0.2974 0.2050 0.1213 0.0434
(2)
γX,β 0.0103 0.0134 0.0113 0.0074 0.0028
0 -0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
frequency [ f ]
0.3
0.4
0.5
0.6
0.7
(0) Fig. 5. Code modulated prolate spheroidal chip waveforms ψ˜X,α (f ) bandlimited to f ∈ [−1/2, 1/2]. 1.4
λ=0.8 λ=0.4
1.2
) is a Fourier transform of {Xn } and where X(f )X(f ˜ )] sin N π(f − f ) . Kα (f, f ) = EX [X(f π(f − f )
(18)
On the other hand, for an absolutely time-limited chip waveform in one chip duration, i.e. φ(t) = DTc φ(t), the maximum energy concentration of Xφ (t) in frequency f ∈ [−1/2, 1/2] is given by the largest eigenvalue of the integral equation: 1/2 1 ψX,β (s)Kβ (t, s)ds, (|t| < ) (19) γX,β ψX,β (t) = 2 −1/2 N −1 N −1 1 π(t−s+n−m) Kβ (t, s) = EX [Xn Xm ] sinπ(t−s+n−m) . N n=0 m=0
(20)
Note that if EX [Xn Xm ] = δn,m , the kernel Kβ reduces to the Slepian’s sinc kernel. (0) (1) (0) (1) Let γX,α > γX,α > . . . and γX,β > γX,β > . . . (0) be the eigenvalues of Eqs. (17) and (19) and let ψX,α (t), (1) (0) (1) ψX,α (t), . . . and ψX,β (t), ψX,β (t), . . . be the corresponding eigenfunctions. These eigenfunctions depend on the spreading codes. The largest eigenvalue changes as shown in Table II. (0) (0) The dominant eigenfunctions ψ α (f ) and ψβ (t) are depicted in Fig. 5 and 6, respectively, for λ = 0, ±0.4, ±0.8. The chip (0) waveform ψβ (t) time-limited to [−0.5, 0.5] approaches to a rectangular pulse as λ becomes large and to the shape like half sine pulse as −λ becomes large. IV. I NTERFERENCES Aa far as energy concentrations (α2 , β 2 ) are concerned, spreading code with positive λ is better and optimum choice (0) (0) of chip waveforms is φ(t) = c0 ψX,α (t) + d0 ψX,β (t) for some constants c0 and d0 . However, we do not recommend
ψX,(0)β (t)
1
λ=0.0
0.8
λ=−0.4
0.6
λ=−0.8 0.4
0.2
0 -0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
time [t] (0)
Fig. 6. Code modulated prolate spheroidal chip waveforms ψX,β (t) timelimited to t ∈ [−0.5, 0.5].
such spreading codes and chip waveforms for CDMA system, because this pair of codes and chip waveforms has very strong multiple-access interference (MAI), which limits the (0) capacity of CDMA system. Table I shows that ψX,α and (0) ψX,β for i.i.d. codes case are worse than root raised cosine (rrc) with roll-off factor 0.22 and rectangular (rect) pulses, respectively. This table implies we should use the second and third eigenfunctions, too. Hence the chip waveform is expressed by φ(t) =
∞ i=0
(i)
ci ψX,α (t) +
∞
(i)
di ψX,β (t).
(21)
i=0
The coefficients ci and di should be chosen so as to achieve compromise between the energy concentrations (α2 , β 2 ) and the MAI. PSWFs do not satisfy the Nyquist condition, therefore we should analyze the SIR. Table I suggests that there is a tradeoff between SIR and β 2 . Hence we should use negative λ, which sacrifices β 2 but improves SIR performance. Our
14
desired chip waveform is chosen so as to be in the interior of the Slepian’s region, but not on the boundary of it.
12
10
1 SIR
6
,
λ=−0.8 λ=−0.9 λ=−0.4
4
λ=−0.0 λ=0.4 λ=0.6
0
λ=0.8
-2
-4 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 τ
1.1
1.2
R EFERENCES [1] D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty-I,” Bell Syst. Tech. J., vol. 40, no. 1, pp. 43–63, Jan. 1961. [2] H.J.Landau and H.O.Pollak, “Prolate spheroidal wave function, Fourier analysis and uncertaintry - II,” Bell Syst. Tech. J., pp. 65–84, Jan. 1961. [3] M. Rupf and J. L. Massey, “Optimum sequence multisets for synchronous code-division multiple-access channels,” IEEE Trans. Inform. Theory, vol. 40, no. 4, pp. 1261–1266, 1994. [4] D. Slepian, “On bandwidth,” Proc. IEEE, vol. 64, pp. 292–300, 1976.
1.4
1.5
1.6
1.7
1.8
1.9
2
10
rrc τ=1.0
sinc
(23)
N −1 where Eb = k=1−N (N − k)λ|k| Φ(k). Eq. (22) implies that we need to calculate the fourth order quantity of φ(t) to evaluate SIR. Double orthogonality (11) (i) (i) holds for ψX,α and ψX,β , but orthogonality does not hold (i) (j) between ψX,α and ψX,β . These facts make the choice of ci and di bothersome. In order to avoid choosing ci and di and to simplify the problem, a Gaussian chip waveform isexamined. A Gaussian t2 , where a is a pulse is defined by φG (t) = a exp − 2τ 2 normalization factor and τ expresses the standard deviation of the energy distribution in time domain. Note that β 2 increases (thus, bandwidth of the signal decreases) as τ increases. In Fig. 7, SIR performance is plotted against a parameter τ . Interestingly, SIRs have peaks around τ = 0.85, 0.95, 1.05 for λ = −0.7, −0.8, −0.9, respectively, while SIRs are monotone decreasing for positive λ. The curve of SINR is obtained by shifting up the curve of SIR according to (23). Fig. 8 illustrates the SIR performance versus −1 < λ < 1. For τ = 0.9, the optimum SIR is achieved by λ = −0.76 with energy concentrations (α2 , β 2 ) = (0.9853, 0.9980). SIR performance as well as energy concentrations are better than that of root raised cosine pulses. Optimization of the chip waveform can be performed in terms of ci and di in (21), which is left as a future work.
1.3
Fig. 7. SIR performance of Gaussian pulse with parameters of τ and Markov codes with several λ’s (N = 127, K = 31). SINRs of root raised cosine (rrc) and sinc pulses are indicated.
8
N0 2Eb
λ=−0.7
rrc sinc
2
τ=0.3 6
τ=0.7
τ=0.9 SIR [dB]
SINR(λ, N, K, N20 , Φ) =
1 +
λ=−0.6
8
SIR [dB]
Lemma 2: For chip-asynchronous DS/CDMA systems with K active users, spreading factor N , and chip waveform φ(t), in an additive white Gaussian noise with two-sided spectrum density N0 /2, the signal to interference ratio of single user receiver output with Markov codes having eigenvalue λ is
2 1 |k| (N − k)λ Φ(k) SIR(λ, N, K, Φ) = N k 1 + λ2 · (K − 1) |k| + λ|k| Φ ∗ Φ(k) 1 − λ2 k −1 1 + E[{ξp(i) }2 ] , (22) N where Φ(t) = φ(t )φ(t− t )dt and Φ∗ Φ(t) = Φ(t )Φ(t− t )dt . Then SINR is given by
τ=0.5
4
2
τ=1.5
τ=1.3
τ=1.2
τ=1.1
0
-2
-4
-1
-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1
0
0.1
0.2 0.3
0.4
0.5 0.6
0.7 0.8
0.9
1
λ
Fig. 8. SIR performance of Gaussian pulse with parameters of λ with τ from 0.3 to 1.5 (N = 127, K = 31). SIRs of rrc and sinc pulses are indicated. [5] ——, “Some comments on Fourier analysis, uncertainty and modeling,” SIAM Review, vol. 25, no. 3, pp. 379–393, 1983. [6] ——, “Estimation of signal parameters in the presence of noise,” Transaction of the I.R.E, vol. 3, pp. 68–89, Mar. 1954. [7] N. S. Jayant and P. Noll, Digital Coding of Waveforms. Pentice-Hall, INC., 1984. [8] A. D. Wyner, “The capacity of the band-limited Gaussian channel,” Bell Syst. Tech. J., vol. 45, pp. 359–395, Mar. 1966. [9] ——, “A note on the capacity of the band-limited Gaussian channel,” Bell Syst. Tech. J., vol. 55, no. 3, pp. 343–346, Mar. 1976. [10] J. Luo, S. Ulukus, and A. Ephremides, “Optimal sequences and sum capacity of symbol asynchronous CDMA systems,” IEEE Trans. on Information Theory, 2005. [11] R. E. Kalman, “Nonlinear aspects of sampled-data control systems,” in Proc. Symp. Nonlinear Circuit Analysis VI, 1956, pp. 273–313. [12] G. Mazzini, R.Rovatti, and G. Setti, “Interference minimisation by autocorrelation shaping in asynchronous DS/CDMA systems: Chaos based spreading is nearly optimal,” Electronics Letters, IEE, vol. 35, pp. 1054–55, 1999. [13] M. A. Landolsi and W. E. Stark, “DS-CDMA chip waveform design for minimal interference under bandwidth, phase, and envelope constraints,” IEEE Trans. Commun., vol. 47, no. 11, pp. 1737–1746, Nov. 1999. [14] H. H. Nguyen and E. Shwedyk, “Bandwidth-constrained signature waveforms for maximizing the network capacity of synchronous CDMA systems,” IEEE Trans. Comm., vol. 49, no. 6, pp. 961–965, June 2001. [15] W. Gao, J. H. Cho, and J. S. Lehnert, “Chip waveform design for DS/SSMA systems with aperiodic random spreading sequences,” IEEE Trans. Wireless Commun., vol. 1, no. 1, pp. 37–45, Jan. 2002.
