28 IEEE COMMUNICATIONS LETTERS, VOL. 9, NO. 1, JANUARY 2005
An Accurate Method for Approximating the
Interference Statistics of DS/CDMA
Cellular Systems with Power Control over
Frequency-Selective Fading Channels
Chieh-Ho Lee, Student Member, IEEE, and Chung-Ju Chang, Senior Member, IEEE
Abstract— This letter proposes an approximation method bycharacteristic function (AM-CF) method to approximate the distribution of interference in DS/CDMA cellular systems. This method considers the effects of frequency-selective multipath fad-ing; it also assumes perfect power control and a rectangular/sinc chip waveform. The AM-CF method can yield results that fit the Monte Carlo simulation results more accurately than the conventional standard Gaussian approximation method.
Index Terms— CDMA, frequency-selective fading, interference
statistics, power control.
I. INTRODUCTION
I
NTERFERENCE statistics of a DS/CDMA(direct-sequence/code-division multiple-access) system are essen-tial to the understanding of the system’s dynamics. Approx-imating interference statistics has received a lot of atten-tion. In the literature, the most widely used method is the SGA (standard Gaussian approximation) method. Although the SGA is easy to use and applicable to a complicated circumstance, e.g. the cellular system over the frequency-selective fading channel in [1], it is known that the SGA is not very accurate [2]. In order to improve accuracy, many other methods have been proposed, such as the improved Gaussian approximation (IGA) method [2], the simplified IGA method [3], and the characteristic function method [4]. These methods have better accuracy, however they are only applicable to the limited circumstance of a single cell system over the AWGN channel. Therefore, an approximation method that has better accuracy and can be applied to a complicated circumstance is still desirable. This letter proposes an approximation method by characteristic function (AM-CF) to approximate the dis-tribution of the MAI (multiple access interference) signals in DS/CDMA cellular systems. The method considers the effect of a frequency-selective multipath fading channel; it also assumes perfect power control and a rectangular/sinc chip waveform. Using this method, the distribution of the MAI signals is more accurately approximated.
II. SYSTEMMODEL
The system under consideration is a DS/CDMA cellular system with NB base stations (BSs) and NM mobile stations
Manuscript received March 23, 2004. The associate editor coordinating the review of this letter and approving it for publication was Dr. P. Spasojevic.
C.-H. Lee and C.-J. Chang are with the National Chiao Tung University, Hsinchu, Taiwan (email:gyher.cm86g@nctu.edu.tw; cjchang@ cc.nctu.edu.tw).
Digital Object Identifier 10.1109/LCOMM.2005.01001.
(MSs) per cell, and with frequency-selective, slowly fading multipath channels. The power gain of each resolvable path p on a link from any MS m to any BS h is statistically divided into long-term fading Lmh and short-term fading Smh,p. For simplicity, {Smh,p} are assumed to be Rayleigh
fadings that have an independently and identically distributed (iid) exponential pdf with mean 1/K and variance 1/K2, where K is the number of resovable paths on a link. It is also assumed that binary phase-shift keying modulation and optimal RAKE receivers are employed. For an MS M communicating with BS H, the decision statistics ZM[k] at
the k-th data symbol (the symbol duration is T ), normalized with respect to T·SMH[k], is obtained in four parts [5] by
ZM[k] = rM[k] + I[k] + IS[k] + n[k] (1)
where SMH[k] =∆ Kp=1SMH,p[k] is a unit-mean gamma
RV with variance 1/K, and rM[k], I[k], IS[k], and n[k]
denote the desired signal, the MAI interference, the multipath interference, and the noise, respectively. Notably, IS[k] and
n[k] are usually negligible as compared to I[k] and will be ignored hereafter [1].
We consider a time interval over which the short-term fading varies while the long-term fading is constant. And the perfect strength-based power control is further assumed to be such that the transmission power of MS M is PM[k] =
Q0/(LMH·SMH[k]), where Q0is the desired received power
level. The term rM[k] = dM[k]·√Q0, where dM[k] represents
the MS M ’s data sequence that has values {±1} with equal probability. The term I[k] is obtained by
I[k] = K p=1 SMH,p[k] SMH[k] m=M K p=1 Pm[k]LmHSmH,p[k] ·α(m,pM,p)[k] · cos θM,p(m,p)[k] (2) where α(m,pM,p)[k] and θ(m,pM,p)[k] are RVs related to the spread-ing code cross-correlation and the phase difference of the signal paths, respectively. The term α(m,pM,p)[k] is given by
α(m,pM,p)[k] = 1 T (k+1)T +(p+τM)Tc kT +(p+τM)Tc dm(t − (p+ τm)Tc) · cm(t − (p+ τm)Tc) · cM(t − (p + τM)Tc) dt (3)
where Tc is the chip duration, cm(t) is MS m’s
spread-ing code signal, and τm reflects the different timing due
LEE, CHANG: APPROXIMATING THE INTERFERENCE STATISTICS OF DS/CDMA CELLULAR SYSTEMS 29
to asynchronous transmission and has a uniform distribu-tion over [0, 1). For the case of random spreading codes with a rectangular chip waveform and an even process-ing gain G, it is known that E(α(m,pM,p)[k]) = 0 and var(α(m,pM,p)[k]) = 2G/3 [6]. On the other hand, the pdf of α(m,pM,p)[k] equals to that of (τGi=1dm[i] · cm[i] · cM[i]/G +
(1 − τ)Gi=1dm[i] · cm[i] · cM[i + 1]/G), where τ is a
uni-form RV over [0, 1), and cm[i] is the random spreading code. It
has the same pdf as that of (τ A+(1−τ)B) where A and B are iid discrete RVs having identical pdf of Pr(A = x) = Pr(B = x) = fb((x + 1)G/2; G), x = −1, −1 + 2/G, ..., 1 − 2/G, 1,
where fb(k; n)= C∆ kn/2n denotes a binomial probability with
binomial coefficient Ckn. After derivation, the pdf fα(x) of
α(m,pM,p)[k] has a closed-form solution given by
fα(x) = gn· δ(0), |x| = 2nG, n∈ {0, 1, · · · ,G2}, hn, 2(n−1)G <|x| < 2nG, n∈ {1, · · · ,G2}, (4) where gn = fb2(n + G/2; G) and hn= G n−1 i=−G/2 G/2 j=n fb(i + G/2; G) · fb(j + G/2; G) j− i . (5)
The term θM,p(m,p)[k] has a uniform distribution over [0, 2π), and therefore the cos(θ(m,pM,p)[k]) is a zero-mean RV with variance 1/2, and its pdf, denoted by fϕ(x), is given by
fϕ(x) = 1
π√1 − x2,|x| < 1. (6) III. THEAM-CF METHOD
The distribution of the MAI signal in (2) is too complicated to be computed directly. We propose an AM-CF method to estimate the distribution of the MAI signal, which is composed of the intra-cell and other-cell interference signal. To help make things clearer, the time index k is neglected hereafter. A. Approximating the Statistics of the Intra-cell Interference From (2), the intra-cell interference signal II is given by
II = K p=1 S MH,p SMH m∈MS(H) m=M K p=1 Q0·SmH,p SmH · α(m,pM,p)· cos θM,p(m,p) (7)
where M S(H) denotes the set of MSs communicating with BS H. Based on the statistics of {α(m,pM,p)}, {θ(m,pM,p)}, and {Smh,p,∀(m, h) = (M, H)}, the mean and variance of II are
obtained to be zero and Q0· (NM − 1)/(3G), respectively.
The terms within the summation in (7) are obviously not iid RVs. Therefore, it is very difficult to derive its pdf. The AM-CF method modifies the profile of{Smh,p} by removing
{Smh,p, p= 2, · · · , K} and letting Smh,1=Kp=1Smh,p, and
further treating terms within the summation in (7) as mutually independent RVs. By doing so, the overall characteristic function is obtained by simply multiplying the characteristic function of each term. As will be shown later, based on the
AM-CF method, the pdf of II can be approximated without
significant distortion by the pdf of II∗ which is defined as
II∗=∆ NM−1 i=1 Q0· αi· ϕ i. (8) The terms {αi, ϕ
i} are mutually independent RVs, and the
pdfs of αi and ϕ
i have the forms given in (4) and (6),
respectively. For the case of an even G and a rectangular chip waveform, the pdf of χi= α
i· ϕ
i has a closed-form solution,
denoted by fχ(x), given by fχ(x) = fα(s) · fϕ xs |s|1ds = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ g0· δ (x) , x = 0, 2hn π logn+ √ n2−G2·x2/4 G·x/2 + G/2 m=n+1 2hm π log m+ √ m2−G2·x2/4 (m−1)+√(m−1)2−G2·x2/4+ G/2 m=n gmG π√m2−(G·x/2)2, 2(n−1) G <|x| ≤2nG, n∈ {1, 2, · · · ,G2}. (9)
The mean and variance of χi are zero and 1/(3G),
respec-tively. Notably, the mean and variance of II∗ are the same as
the mean and variance of II, respectively. Since II∗ is a sum
of iid RVs, we have FI∗
I(ω) = (Fχ(
√
Q0· ω))NM−1, where
Fχ(ω) = F(fχ(x)) is the characteristic function of the pdf
fχ(x) , and F(·) represents the Fourier transform.
Accord-ingly, the pdf of II∗ is obtained by fI∗
I(x) = F−1(FII∗(ω)),
where F−1(·) denotes the inverse Fourier transform.
B. Approximating the Statistics of the Other-cell Interference From (2), the other-cell interference signal IO is given by
IO = K p=1 SMH,p SMH m∈MS(h) h=H K p=1 Q0·LmH·SmH,p Lmh·Smh ·α(m,pM,p)· cos θM,p(m,p) . (10)
The mean and variance of IO are zero and
m∈MS(h),h=H(LmH/Lmh) · K/(K − 1) · Q0/(3G),
respectively. By dividing the average other-cell interference power var(IO), from the NB − 1 cells, into N equal
quantities, the AM-CF method estimates the pdf of IO by the pdf of IO∗ which is defined as IO∗ =∆ N i=1 Q0· L· S1,i S2,i · α i · ϕ i. (11)
The parameter N is artificially introduced and L is given
by L = (∆ m∈MS(h),h=HLmH/Lmh)/N so that the mean
and variance of IO∗ equal to those of IO. Once the N is chosen and assume var(IO) is measurable, the term Lcan be
calculated according to its definition. By this way, the problem of measuring each Lmh, which is practically not easy to do,
is bypassed. Another purpose of parameter N is to tune the distribution of IO∗. The{S1,i, S2,i} are iid RVs with the same
30 IEEE COMMUNICATIONS LETTERS, VOL. 9, NO. 1, JANUARY 2005
distributions as that of SMH, and the pdf of βi=∆
S1,i/S2,i, denoted by fβ(x), is obtained by fβ(x) = fS s· x2 · fS(s) · 2s · x · ds = 2 Γ(2K)·x2K−1 Γ2(K)·(1+x2)2K (12) where fS(x) denotes the pdf of SMH. The pdfs of αi and
ϕi have the form given by (4) and (6), respectively, and
the pdf of χi = α∆
i · ϕ
i is the same as the fχ(x) given
by (9). Consequently, the pdf of ψi = βi · χ
i has an
identical pdf, denoted by fψ(x), which is obtained by fψ(x) =
+1
−1 fχ(s) · fβ(x · s−1) · |s|−1ds. Since IO∗ is a sum of iid
RVs,Q0· L·ψi, we have FI∗
O(ω) = (Fψ(
Q0· L·ω))N, where Fψ(x) is the characteristic function of the pdf fψ(x).
C. Approximating the Statistics of the MAI Interference Since the intra-cell and other-cell interference signals are independent under a strength-based power control, the pdf of the MAI signal I can be approximated by fI(x) =
F−1((F
χ(√Q0·ω))NM−1·(Fψ(Q0· L·ω))N). Note that
the BER is obtained by BER = Pr(I <−√Q0). IV. RESULTS ANDDISCUSSIONS
Fig. 1(a) shows the cdf curves of the MAI signals within the central cell using AM-CF and SGA methods, as well as the Monte Carlo simulation results based on (2), (3), and (6) with 106 samples and (NB, G, K) = (19, 128, 2). The model
of the long-term fading is the same as that in [5] with the propagation loss of 3.5 and the log-normal shadow fading standard deviation of 8dB, and each MS chooses its serving BS based on measured pilot power [5]. The MS number NM
is 13 (22) and the parameter N is chosen as 20 (34) for case (i) (case (ii)) where the corresponding BER is around 10−3 (5 · 10−2). It is found that with proper N, AM-CF curves fit
the simulation results better than SGA curves. For example, in case (i), BER based on the AM-CF method deviates from the simulation result by 4%, while BER based on the SGA method deviates by as high as 73%. The reason why the SGA does not work well is highly related to the power control. Under a perfect strength-based power control, the MS’s transmission power might become pretty high due to a deep fade, which will induce a high other-cell interference power to others and therefore a shallow falloff of the pdf of IO. In the proposed
AM-CF method, as shown in (11), IO∗ contains the factor 1/S2,i, which reflects the power control effect, so that the AM-CF could perform better than the SGA. Also notice that the AM-CF seems not very sensitive to the parameter N .
Fig. 1(b) shows results of the situation which is the same as those in Fig. 1(a) except that the chip waveform is changed to a sinc one. The NM is 45 (75) and the N is chosen as 80
(130) for case (i) (case (ii)). The Eqs. (8) and (11) in AM-CF method remain the same and the factor α(m,pM,p) can be numerically calculated according to (3). Results show that the AM-CF curves still fit the simulation results better than SGA curves. For example, in case (i), BER based on the AM-CF method deviates from the simulation result by 8%, while BER based on the SGA method deviates by as high as 64%. Note that, the system attains a dramatic gain in capacity by using
-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 10-4 10-3 10-2 10-1 100 x Pr( I<=x * (Q 0 ) 1 /2) (i) NM=13 AM-CF SGA Simulation (ii) N M=22
(a) Rectangular chip waveform
-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 10-4 10-3 10-2 10-1 100 x Pr ( I<=x * (Q 0 ) 1/2 ) (i) NM=45 AM-CF SGA Simulation (ii) N M=75
(b) Sinc chip waveform Fig. 1. Comparison of cdf curves of interference signals.
sinc waveform. It is because the sinc waveform is the optimal one that minimizes interference and improves the capacity [7]. It can be believed that the rationale of the AM-CF method is applicable to more realistic conditions considering, such as short-term fadings with non-equal average power, power control error, power control period, power control step, power control command delay, MS velocity, etc. However further systematic work is needed to study these extensions of the AM-CF method. Moreover, the determination of the parameter N is still an issue which needs further study.
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