Chapter 3 Performance Analysis of a Truncated Closed-Loop Power Control Scheme for
3.6 Notation List
Notation Description
M Index of the target MS H Index of the BS that is the
home BS of target MS M m Index of an arbitrary mobile
station
C System capacity defined in (3.44)
C * Refer to (3.45)
CMD Power control command dm(t) Data stream of MS m
D MS average suspension delay defined in (3.63)
( )
E ⋅ Expectation function E MS average transmission
energy per bit defined in (3.62)
Notation Description
fm Maximal Doppler frequency fS(⋅) Probability density function
of Smb(t)
p( )
fS ⋅ Probability density function of Smb,p(t)
m( )
fz z Probability density function of zm at location z
1 2
( ,p p)( )
fγ ⋅ Pdf of a two-parameter gamma random variable γ with parameters p1 and p2
1 2
( ,p p)( )
Fγ ⋅ Cdf of a two-parameter gamma random variable γ with parameters p1 and p2
I n Multiple access and multipath interference (MAI) portion of ZM[n]
S[ ]
I n Multipath interference portion of ZM[n]
II Mean power of the intra-cell interference signal that interferes the target uplink (M,H)
IO Mean power of the other-cell interference signal that interferes the target uplink (M,H)
*
I O Normalized IO, IO* IO/PC K Number of resolvable paths
in each uplink channel k Index of power control period
Notation Description
( 0)
lcr r= X Level-crossing rate at a given strength level X 0
Lmb(t) Long-term fading of the uplink (m,b) at time t Lmb Long-term fading of the
uplink (m,b) which is approximately a constant during a medium-term period LMH Long-term fading of the
target uplink (M,H) which is approximately a constant during a medium-term period {Lmb, ( , )}∀ m b Scenario ={Lmb|∀m b, }
MS(h) Set of the MSs that are served by BS h
n Index of data symbol period NB Number of base stations in
the cellular system. I.e. cell number
NM Number of mobile stations in each cell of the cellular system
p Index of resolvable path on an uplink
Pm(t) Transmission power of MS m at time t
Pm[n] Transmission power of MS m during nth data symbol period
PM[n] Transmission power of target MS M during nth data symbol period
PO,m(t) A portion of the transmission power of MS m at time t, which is controlled by open-loop power control PO,m[n] A portion of the transmission
power of MS m at nth data symbol period, which is controlled by open-loop power control
Notation Description
PO,M[n] A portion of the transmission power of target MS M at nth data symbol period, which is controlled by open-loop power control
PC,m(t) A portion of the transmission power of MS m at time t, which is controlled by closed-loop power control PC,m[n] A portion of the transmission
power of MS m at nth data symbol period, which is controlled by closed-loop power control
PC,M[n] A portion of the transmission power of target MS M at nth data symbol period, which is controlled by closed-loop power control
PC =E P( C m, [ ])n
Q0 Desired received power level
M[ ]
Q n Received signal power at the nth data symbol period of MS M
Q = (E Q n m[ ])
For TCPC scheme, Q is obtained by (3.43)
M[ ]
Q k Estimated average signal power at the kth power
control period of target MS M in a realistic TCPC scheme r Envelope of SMH( )t
MH( ) r S t
H( )
r t The total signal received at a specific BS H
M[ ]
r n Desired signal portion of ZM[n]
R0 Data symbol rate, R0 =1/T R MS average transmission rate
Notation Description
defined in (3.57)
ℜ Average system transmission rate defined in (3.58)
sm(t) Transmitted low-pass equivalent signal of MS m S(t) Short-term fading at time t Smb,p(t) Short-term fading of the pth
resolvable path of the uplink (m,b) at time t
Smb,p[n] Short-term fading of the pth resolvable path of the uplink (m,b) during nth data symbol period
uplink (m,b) during nth data symbol period.
target uplink (M,H) during nth data symbol period.
MH[ ]
S k Estimated average short-term fading at the kth power control period on the target uplink (M,H) in a realistic TCPC scheme
T Data symbol duration Tc Chip duration
Td Power control command loop delay
Tp Power control period uplink (m,b) Uplink channel from an MS
m to a BS b ( ; )
umb τ t Time-variant impulse response of uplink (m,b)
V MS velocity
var( )⋅ Variance function
Notation Description
represents the location of BS b
X0 Preset cutoff threshold in TCPC scheme
zm A complex number that represents the location of MS m
ZM[n] Decision statistics corresponding to an MS M at nth data symbol period
Notation Description
ξ A Gaussian random variable for approximating I O* η Propagation exponent ψ Power control error
ψ= QM[k]/Q0 ΛM Outage probability of the
target uplink (M,H) defined in (3.38)
Notation Description
Λ0 Preset outage threshold Γ0 Minimum SIR per bit
required to achieve a desired bit error rate
M[ ]n
Γ SIR per bit during the nth data symbol period of target MS M at the output of RAKE receiver
∆p Step size for power adaptation
τm Transmission timing of MS m due to asynchronous
transmission among the MSs φm Random carrier phase
δ τ( ) Impulse
ϕ(s) Defined in (3.37) αmb,p(t) Power gain of the pth
resolvable path of the uplink (m,b) at time t during nth data symbol period
κ a Quantity given in (3.47) κ b Quantity given in (3.47) κ c Quantity given in (3.47)
Chapter 4
An Accurate Method for Approximating the Interference Statistics of DS/CDMA Cellular Systems with Power Control
over Frequency-Selective Fading Channels
Abstract This chapter proposes an approximation method by characteristic function (AM-CF) method to approximate the distribution of interference in DS/CDMA cellular systems. This method considers the effects of frequency-selective multipath fading; 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.
4.1 Introduction
Interference statistics of a DS/CDMA (direct-sequence/code-division multiple-access) system are essential to the understanding of the system’s dynamics. Approximating interference statistics has received a lot of attention. 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 [17], it is known that the SGA is not very accurate [18, p.1055]. In order to improve accuracy, many other methods have been proposed, such as the improved Gaussian approximation (IGA) method [18], the simplified IGA method [19], and the characteristic function method [20]. 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 chapter proposes an approximation method by characteristic function (AM-CF) to approximate the distribution of MAI (multiple access interference) signals in DS/CDMA cellular systems. The method considers the effects 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.
4.2 System Model
The system under consideration is a DS/CDMA cellular system with NB base stations (BSs) and NM mobile stations (MSs) per cell, and with frequency-selective, slowly fading multipath channels. The power gain of each resolvable path p on an uplink from an arbitrary MS m to an arbitrary BS b is statistically divided into long-term fading L and short-term fading mb Smb p, ,
wherein m∈{1, 2, ," N NB⋅ M} denotes the MS index, b∈{1, 2, ," NB} denotes the BS index, and {1, 2, , }p∈ " K denotes the resolvable path index with K being the number of resolvable paths on an uplink. For simplicity, {Smb p, , ( , , )∀ m b 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, i.e.
It is also assumed that binary phase-shift keying modulation and optimal RAKE receivers are employed. For a specific MS M communicating with its home BS H, the decision statistics ZM[n]
at the n-th data symbol (the symbol duration is T), normalized with respect to T⋅ SMH[ ]n , is obtained in four parts [78] by
[ ]Z nM =r nM[ ]+I nA[ ]+I nS[ ]+z n[ ], (4.2)
r n denotes the desired signal, I nA[ ] represents the MAI interference consisting of the intra-cell interference and other-cell interference, I n is the multipath interference, and z[n] is S[ ] the noise. Notably, [ ]I n and z[n] are usually negligible as compared to S I nA[ ] and will be ignored hereafter [17].
We consider a medium-term period 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 any one MS m, 1≤ ≤m N NB⋅ M, is
[ ] 0/( [ ])
m mh mh
P n =Q L ⋅S n , where h is the index of the BS communicating with MS m, Q is the 0
common desired received power level in the system. The term r nM[ ] is given by timing due to asynchronous transmission and has a uniform distribution over [0,1). For the case of random spreading codes with a rectangular chip waveform and an even processing gain G, it is known that E(αM p( , ')m p, [ ]) 0n = and var(αM p( , ')m p, [ ]) 2 / 3n = G [79]. On the other hand, the pdf of
(1 ) he probability functions of RVs A and B which are given by
( 1)
denotes a binomial probability with binomial coefficient C . The cdf of RV xy α is obtained by
( ) ( )
/ 2 / 2
corresponding pdf will have an impulse at these points. For | | 2 / ,x = i G i=0,1,..., / 2G , we have
The term θM p( , ')m p, [ ]n has a uniform distribution over [0,2π), and therefore the
The distribution of the MAI signal in (4.4) 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.
4.3.1 Approximating the Statistics of the Intra-cell Interference From (4.4), the intra-cell interference signal II is given by
( )
where MS(H) denotes the index set of MSs communicating with BS H. Based on the statistics of {αM p( , ')m p, }, {θM p( , ')m p, }, and {Smb p, ,∀ ∀p, ( , ) ( , )m h ≠ M H }, the mean and variance of II can be
The terms within the summation in (4.21) are obviously not iid RVs, and it is very difficult to derive its pdf. The AM-CF method modifies the profile of {Smb p, , ( , , )∀ m b p } by removing
{Smb p, ,p=2, , , ( , )" K ∀ m b } and letting ,1 ,
1 , ( , )
K
mb p mb p
S =
∑
= S ∀ m b and then treats terms within the summation in (4.21) 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 I which is defined as I* forms given in (4.17) and (4.20), respectively. For the case of an even G and a rectangular chip waveform, the pdf of χ α ϕi'= i'⋅ has a closed-form solution, denoted by i' f xχ'( ), given byThe mean and variance of χi' are zero and 1/(3G), respectively. Notably, the mean and variance of I are the same as the mean and variance of I* II, respectively. Since I is a sum of iid RVs, *I represents the Fourier transform. Accordingly, the pdf of I is obtained by I*
* *
4.3.2 Approximating the Statistics of the Other-cell Interference From (4.4), the other-cell interference signal I is given by O
( )
The parameter N is artificially introduced and L'' is given by
( ) var( )IO is measurable, the term L'' can be calculated according to its definition. By this way, the problem of measuring these quantities {L ,L }, which is practically not easy to measure, is
bypassed. Another purpose of parameter N is to tune the distribution of I . The O* {S S1,i, 2,i} are
4.3.3 Approximating the Statistics of 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
( )
1(( '( 0 ))NM 1 ( ( 0 '' )) ).Nf xI =F − Fχ Q ⋅ω − ⋅ Fψ Q L⋅ ⋅ω (4.34)
Note that the BER is obtained by BER=Pr(I < − Q0).
4.4 Results and Discussions
Figure 4.1(a) shows the cdf curves of MAI signals within the central cell using AM-CF and SGA methods, also the Monte Carlo simulation results based on (4.4), (4.5), and (4.20) with 106 samples are presented. The model of the long-term fading is the same as that in [78] with the propagation loss of 3.5 and the log-normal shadow fading standard deviation of 8dB; each MS chooses its serving BS based on measured pilot power [78]; and system parameters of (NB, G, K) are chosen to be (19, 128, 2), respectively. 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 power control. Under a perfect strength-based power control, the MS's transmission power level 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 I . In the proposed AM-CF method, as can be seen in (4.29), O I contains the factor O* 1/S2,i which reflects the power control effect and thus the AM-CF can perform better than the SGA. Also notice that the AM-CF method seems not very sensitive to the value of parameter N.
Figure 4.1(b) shows results of the situation that is the same as those in Fig. 1(a) except that the chip waveform is changed to a sinc one. The MS number NM is 45 (75) and the parameter N is chosen as 80 (130) for case (i) (case (ii)). For the situation of non-rectangular chip waveform, there is no closed-form for statistics of intra-cell and other-cell interferences. However, the Eqs.
(4.23) and (4.29) in AM-CF method remain the same and the factor αM p( , ')m p, can be numerically
calculated according to (4.5). 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%. Also, the system attains a dramatic gain in system capacity by using sinc waveform. It is because the sinc waveform is the optimal one that can minimize the interference and improve the system capacity [80].
It can be believed that the rationale of the AM-CF method is applicable to more realistic conditions, 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 that needs further study.
-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 * (Q0)0.5)
AM-CF SGA Simulation
(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 * (Q0)1/2)
AM-CF SGA Simulation
(b) Sinc chip waveform
Figure 4.1: Comparison of cdf curves of interference signals
(i) NM = 13 (ii) NM = 22
(i) NM = 45 (ii) NM = 75
4.5 Notation List
Notation Description
m Index of an arbitrary mobile station
C x Binomial coefficient dM[n] Data sequence of MS m
f x yb Binomial probability
Notation Description
F Inverse Fourier transform
'( )
Fχ ω Characteristic function of
'( ) f xχ ( )
Fψ ω Characteristic function of ( )
f xψ
G Processing gain / c G T T
IA[n] Multiple access and multipath interference (MAI) portion of ZM[n]
S[ ]
I n Multipath interference portion of ZM[n]
II Intra-cell interference signal
*
I I RV used to approximate the pdf of II
I O Other-cell interference signal
*
I O RV used to approximate the pdf of I O
K Number of resolvable paths in each uplink channel
L'' Quantity defined in (4.30) Lmb Long-term fading of the
Notation Description uplink from MS m to BS b LMH Long-term fading of the
uplink from a specific MS M to its home BS H
MS(h) Index set of the MSs that are served by BS h
n Index of data symbol period NB Number of base stations in
the cellular system. I.e. cell number
NM Number of mobile stations in each cell of the cellular system
p Index of the resolvable path on an uplink
Pm[n] Transmission power of MS m during nth data symbol period PM[n] Transmission power of
specific MS M during nth data symbol period
Q0 Desired received power level
M[ ]
r n Desired signal portion of ZM[n]
Smb,p Short-term fading of the pth resolvable path of the uplink from MS m to BS b
Smh[n] Short-term fading of the uplink from an arbitrary MS m to its home BS h during nth data symbol period
SMH[n] Short-term fading of the uplink from a specific MS M to its home BS H during nth data symbol period
SMH SMH[n] after the index n being dropped
S 1,i RV having same distribution as that of SMH
Notation Description
LMH Long-term fading of the uplink from a specific MS M to its home BS H
MS(h) Index set of the MSs that are served by BS h
n Index of data symbol period NB Number of base stations in
the cellular system. I.e. cell number
NM Number of mobile stations in each cell of the cellular system
p Index of the resolvable path on an uplink
Pm[n] Transmission power of MS m during nth data symbol period PM[n] Transmission power of
specific MS M during nth data symbol period
Q0 Desired received power level
M[ ]
r n Desired signal portion of ZM[n]
Smb,p Short-term fading of the pth resolvable path of the uplink from MS m to BS b
Smh[n] Short-term fading of the uplink from an arbitrary MS m to its home BS h during nth data symbol period
SMH[n] Short-term fading of the uplink from a specific MS M to its home BS H during nth data symbol period
SMH SMH[n] after the index n being dropped
S 1,i RV having same distribution as that of SMH
Notation Description
S 2,i RV having same distribution as that of SMH
T Symbol duration
Tc Chip duration
vi Quantity defined in (4.19) var( )⋅ Variance function
wi Quantity defined in (4.18) ZM[n] Decision statistics
corresponding to a specific MS M at nth data symbol period
z[n] Noise
ξ A Gaussian random variable for approximating I O*
τm Transmission timing of MS m due to asynchronous
transmission among the MSs τ Uniform RV over [0,1)
α RV related to the spreading code cross-correlation
between the resolvable path p in the uplink from MS M to BS H and the resolvable path p’ in the uplink from MS m to
resolvable path p in the uplink from MS M to BS H and the resolvable path p’ in the uplink from MS m to BS H
Chapter 5
Capacity Analysis of an Imperfect SIR-based
Power Control Scheme for Uplinks in DS/CDMA Cellular Systems
Abstract The chapter proposes a novel method to analyze the system capacity of an imperfect signal-to-interference ratio (SIR)-based power control scheme for uplinks in direct sequence/code division multiple access (DS/CDMA) cellular systems. Not ideally fixed at a preset level, the received SIR in the imperfect SIR-based power control scheme is a random variable affected by many practical factors such as power control loop delay and mobile velocity. Based on investigated properties of ensemble average received SIR per bit, a set of linear equations is derived to obtain the ensemble average received power on each uplink. The system capacity is then obtained according to the feasibility of the ensemble average received power vector and the ensemble average bit error rate. A closed-form solution for system capacity is approximately derived, which employs the first and second order statistics of each element in the coefficient matrix, by applying central limit theorem. Results show that numerical results are substantially matched with simulation results; this implies the novel analytical method is quite accurate.
5.1 Introduction
In a DS/CDMA system, many users transmit messages simultaneously over the same radio channel, each using a specific spread-spectrum pseudo-noise (PN) code [10]. Within a cell, the code channels in downlinks can be considered as mutually orthogonal because downlinks exhibit synchronous CDMA transmission. However, these code channels in uplinks cannot be mutually orthogonal exactly for asynchronous mobile users, and thus mutual interference occurs. In such a case, a strong signal will have good communication quality, while a weak signal may suffer from strong interference. This problem is referred to as the near-far effect and limits the CDMA system capacity [11]. Therefore, power control is an important issue affecting the system capacity in uplinks of DS/CDMA cellular systems.
Power control schemes studied in the literature can be mainly classified into two categories:
the strength-based power control [45] and the SIR-based power control [21], [27]. There were many works related to the strength-based power control scheme, such as the system capacity analysis, e.g. [3], the interference statistics, e.g. [22], the error probability, e.g. [17], etc.
Nevertheless, the SIR-based power control is in fact more important. Ariyavisitakul [21] once indicated that the SIR-based power control system has the potential for higher system capacity.
Actually, the SIR-based power control had been adopted in IMT-2000 systems as well as IS-95 system. However, the corresponding theoretical analysis for the SIR-based power control is very complicated [21] and is found significantly different from that of the strength-based power control [23]. Due to difficulty, few literature had successfully provided theoretical methods for investigating the SIR-based power control scheme, e.g. [23], [25], [26], and most works made the study via simulation, e.g. [21], [27], [28], [29], for the SIR-based power control schemes.
Kim and Sung [23] proposed a methodology for analyzing the system capacity of the SIR-based power control scheme with consideration of the voice activity, the maximum received