Chapter 2 An Overview of the Power Control Techniques for DS/CDMA Cellular Systems . 11
2.4 Concluding Remarks
This chapter provides a fundamental overview of the DS/CDMA cellular systems, the power control techniques, and the classification of these power control techniques. As a basis of a multi-user system, the multiple access techniques adopted by the cellular systems has evolved from 1G's and 2G's FDMA, TDMA to 3G's wideband CDMA. The reason is that the wideband CDMA theoretically can provide higher capacity compared with FDMA and TDMA schemes.
However, in order to achieve the high capacity, one of the crucial techniques is the power control.
This chapter further reviews the various classifications of the power controls in a DS/CDMA system. The power control can be classified as uplink/downlink power control, centralized/distributed power control, open-loop/closed-loop/outer-loop power control, strength-based/SIR-based/BER-based power control, perfect/imperfect power control, power control with fixed step size/adaptive step size. Aiming to further improve system capacity, the truncated power control and the predictive power control are also introduced. From the viewpoints of the above power control classifications, what this dissertation focuses on is the uplink, distributed, closed-loop, either truncated strength-based or SIR-based, either perfect or imperfect power control scheme with fixed step size. In the following chapters, the performance of the truncated strength-based and the SIR-based power control schemes are analyzed, respectively. Also, the problem of distribution estimation of the multiple access interference in the strength-based power control is addressed in yet another chapter.
Chapter 3
Performance Analysis of a Truncated Closed-Loop Power Control Scheme for DS/CDMA Cellular Systems
Abstract This chapter analyzes the system performance of a truncated closed-loop power control (TCPC) scheme for uplinks in DS/CDMA cellular systems over frequency-selective fading channels. In this TCPC scheme, a mobile station (MS) suspends its transmission when the short-term fading is less than a preset cutoff threshold; and otherwise, the MS transmits with power adapted to compensate for the short-term fading so that the received signal power level remains constant.
Closed-form formulae are successfully derived for performance measures, such as system capacity, average system transmission rate, MS average transmission rate, MS power consumption, and MS suspension delay. Numerical results show that the analysis provides reasonable accuracy; and the TCPC scheme can substantially improve the system capacity, the average system transmission rate, and power saving over conventional closed-loop power control schemes. Moreover, the TCPC scheme under realistic consideration of power control error due to power control step size, power control period, power control delay, and MS velocity is further investigated. A closed-form formula is obtained to accurately approximate the system capacity of the realistic TCPC scheme. A closed-form formula is obtained to accurately approximate the system capacity of the realistic TCPC scheme.
3.1 Introduction
In a DS/CDMA system, many users can 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 may exhibit synchronous CDMA transmission. However, these code channels in uplinks cannot be exactly mutually orthogonal for a set of asynchronous users, and thus mutual interference occurs among the uplinks. In such a case, a strong signal increases communication quality, and 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]. Hence, power control is an essential issue in a DS/CDMA system.
Open-loop power control, that is, the average power control, is applied to compensate for the long-term channel fading such that the average received signal power level is constant and the near-far problem is solved [12]. Closed-loop power control, however, is typically used to mitigate the short-term channel fading so that an acceptable received signal quality can be attained for the uplink communication. Several closed-loop/open-loop power control schemes have been investigated, such as: 1) the well-known perfect power control, within which MS transmission power is adjusted to the exact inverse of the short-term fading and thus the received signal power level remains constant. Such a method is also referred to as the channel inversion scheme [13], [14]; 2) combined power/rate control proposed in [15], which is the same as the perfect power control except in that MS holds its transmission power at Q0/X0 and adapts its transmission rate to S(t)⋅R0/X0 when S(t)<X0, where Q0 is the desired received power level, X0 is a preset cutoff threshold, S(t) is the short-term fading at time t, and R0 is the data symbol rate; 3) truncated average power control (TAPC) proposed in [16], which applies a truncated channel inversion scheme to conventional average power control. This truncated channel inversion scheme suspends transmission when the long-term channel fading falls below a cutoff threshold; otherwise it
adaptively controls power according to the channel inversion scheme. By suspending transmission in this way, an improvement of system capacity was reported.
In this chapter, we propose a truncated closed-loop power control (TCPC) scheme, which extends the TAPC scheme by considering the fast closed-loop power control, since mitigating the rapid variations of fading by power control mechanism had been adopted in 3G systems like cdma2000 and WCDMA. In the TCPC scheme, when the short-term fading satisfies S(t)<X0, MS suspends transmission; otherwise, MS adaptively transmits power to compensate for the short-term fading so that the received signal power level is constant. This chapter analyzes the performance of the TCPC scheme for uplinks in DS/CDMA cellular systems over frequency-selective fading channels. It first analyzes an ideal TCPC scheme in which the transmission power is continuously and instantaneously adjusted. Based on the SIR formula over a medium-term period, closed-form formulae are successfully derived for system performance including system capacity, average system transmission rate, MS average transmission rate, MS power consumption, and MS suspension delay. The upper limit of the average system transmission rate of the TCPC scheme achieved in a multi-cell system is also obtained, and it is found the same as that of the perfect power control scheme in a single-cell system. Numerical results show that the analysis is quite accurate; and the TCPC scheme is more effective than the conventional closed-loop power control schemes such as perfect power control [13], [14], combined power/rate control [15], and truncated average power control [16]. This chapter further analyzes the realistic TCPC scheme that considers power control error due to power control step size, power control command delay, and MS velocity. A closed-form formula is also derived to accurately approximate the system capacity of the realistic TCPC scheme. Numerical results indicate that the power control error has a significant impact on the system capacities of the TCPC and conventional closed-loop power control schemes.
The rest of this chapter is structured as follows. Section 2.2 introduces the system model and
derives a formula of the received SIR per bit. Section 3.3 derives the performance measures of the TCPC scheme in both ideal and realistic cases. Section 3.4 presents numerical results as well as simulation results for validation. Comparisons between TCPC and other conventional schemes are also presented. Finally, Section 3.5 remarks conclusions.
3.2 System Model
3.2.1 Channel Model
Consider NB cells in a CDMA cellular system of which the central cell is surrounded by other cells in a hexagonal-grid configuration. Each cell has a base station (BS) located at the center and has NM MSs uniformly distributed in the system. Generally, the frequency-selective multipath channel of the uplink (m,b), denoting the channel from an arbitrary MS m to an arbitrary BS b, is described by a time-variant impulse response
, ( ) 1 ,
( ; ) K ( ) jmb p t ( ),
mb p mb p c
u τ t =
∑
= α t e⋅ −θ ⋅δ τ −pT (3.1)where m∈{1, 2, ," N NB⋅ M} denotes the MS index, b∈{1, 2, ," NB} denotes the BS index, ( ; )
umb τ t is the response of the uplink (m,b) at time t, due to an impulse applied at time (t− ); Tτ c
represents the chip duration; K is the number of resolvable paths; and the random variables αmb,p(t) and θmb,p(t) are the power gain and the phase of the pth path at time t, respectively [10].
The power gain of the pth path can be divided into two parts [74],
, ( ) ( ) , ( )
mb p t Lmb t Smb p t
α = ⋅ , (3.2)
where Lmb(t) and Smb,p(t) represent long-term fading and short-term fading, respectively. The long-term fading is normally modeled as
( ) /10
( ) ( ) ( ) 10xmb t
mb m b
L t = z t −x t −η⋅ , (3.3)
where zm(t) and xb(t) are complex numbers that represent the locations of MS m and BS b,
respectively, η is the propagation exponent that depends on the environment, and xmb(t) is a zero-mean Gaussian random process with standard deviation σx. The short-term fading is widely modeled as Rayleigh fading and Smb,p(t) is exponentially distributed, i.e.
( ) 1 S p, 0 intensity profile (MIP) [10] of each uplink is assumed to be identical and every signal path has the same average received power. Without loss of generality, the overall short-term fading effect
1 ,
( ) K ( )
mb p mb p
S t
∑
= S t , (3.5)given the optimal maximal ratio combing, is normalized to have unit mean. Thus, the pdf of Smb(t), denoted by fS(⋅), is found to have a gamma distribution, which is given by
( , )
The corresponding mean and variance are respectively given by
1 2
3.2.2 Transmitter Model
If BPSK modulation is considered, the transmitted low-pass equivalent signal of MS m is given by
( ) ( ) ( ) ( ) jm, 1
m m m m m m B M
s t = P t d t⋅ −τ ⋅c t−τ ⋅e− φ ≤ ≤m N N⋅ , (3.11) where Pm(t) is MS transmission power at time t; dm(t) is the MS bipolar data stream; cm(t) is the
corresponding PN sequence whose chip waveform is rectangular; τm is a value between zero and the data symbol duration T and indicates each MS independent symbol timing due to asynchronous transmission; and φm is the random carrier phase. Under a transmission power control, Pm(t) can be conceptually divided into two parts as
, .
( ) ( ) ( )
m O m C m
P t =P t P⋅ t , (3.12)
where PO,m(t) and PC,m(t) are the transmission powers controlled by open-loop and closed-loop power control schemes, respectively. This chapter assumes perfect open-loop power control, hence for each MS m served by its home BS h, we have
, ( ) 1/ ( )
O m mh
P t = L t . (3.13)
3.2.3 Receiver Model
Here, we consider a medium-term period which is sufficiently long to make the short-term fadings be averaged out while the long-term fadings remain almost constant since MSs move only a little. Over the medium-term period, a SIR formula is derived and closed-form formulae of the performance measures are obtained accordingly. Also, the set of the long-term fading from any MS m to any BS b is referred to as a scenario and is represented as {Lmb, ( , )∀ m b } from which its time index t is dropped. Based on the models of channel impulse response given in (3.1) and the MS transmitted signal given in (3.11), the total signal received at a specific BS H can be found to be
,
where the background noise is ignored since it is much smaller than the other-cell interference.
Consider a RAKE receiver with an optimal maximal ratio combiner to take full advantage of the multipath diversity. Here, we assume slow fading such that Smb(t), Smb,p(t), Pm(t), and θmb,p(t) can be treated as constants Smb[n], Smb,p[n], Pm[n], and θmb,p[n], respectively, for all possible m, b, and p, during the nth data symbol period. That is,
, ,
The decision statistics, ZM[n], corresponding to a target MS M communicating with the specific BS H can be derived and divided into three parts
( )
The second term I nA[ ] represents the multiple access and multipath interference from the other MSs and can be obtained by
( , ')
( 1)
Over the medium-term period, the long-term fading, which is the local mean of the channel gain, is treated as constant; the short-term fading is typically modeled as a stationary random process.
Hence the channel gain, which is the product of the long-term and short-term fading, becomes a stationary random process. The transmission power under strength-based power control, which is generally a function of channel gain, will turn out also to be stationary. If both MS data streams and PN sequences are assumed to be stationary, then the interference signal I nA[ ] in (3.18) will resemble a noise-like stationary random process, and its distribution can be approximated by a Gaussian one according to the central limit theorem. Notably, I nA[ ] having a mean of zero and a variance is the mean of the interference signal power at the output of the RAKE receiver, given by
The SIR per bit during the nth data symbol period of MS M at the output of RAKE receiver, defined in [10, p.244] and denoted by ΓM[ ]n , is given by
where Q nM[ ] is the instantaneously received signal power, and II and I represent the mean O power of the intra-cell interference signals and the mean power of the other-cell interference signals, respectively. In general, the term Q nM[ ] can be obtained by,
E P n , which, for clearer notation, is denoted by
( , [ ]),
denoting the index set of the MSs served by the BS h.
3.3 Performance Analysis
In the TCPC scheme, MS adjusts its transmission power to compensate for the short-term fading when the short-term fading is above a preset cutoff threshold X0 and suspends its transmission, otherwise. Accordingly, considering an target uplink (M,H),
0 0
where Q0 denotes the preset desired received power level. If the transmission power can be adjusted continuously and immediately as shown in (3.29), the scheme is referred to as an ideal TCPC.
3.3.1 Ideal TCPC
Conditional with on {Lmb, ( , )}∀ m b , the SIR per bit in (3.22) given P nM[ ] 0> is time constant over the medium-term period. The unconditioned SIR per bit is then obtained by considering all possible {Lmb, ( , )}∀ m b . The formula of the unconditioned SIR per bit is the same as (3.22), but its denominator is now a random variable due to the randomness of {Lmb, ( , )}∀ m b . Again, applying the central limit theorem, the normalized I , represented as O
* O
O C
I I
P , (3.30)
can be approximated by a Gaussian random variable ξ and expressed as
* ( , 2)
,
The I was analyzed by Zorzi [22], and its mean and variance can be numerically computed as h
( )
denotes the pdf of a random variable X, and( )
Considering an MS M in central cell H, the outage probability of the target uplink (M,H), denoted by ΛM, is defined as
( )
where Γ is the minimum SIR per bit required to achieve a desired bit error rate at the output of 0 the RAKE receiver. The outage probability given P nM[ ] 0= is not considered, since no data is transmitted in that case. Notably, only the statistics relating to the central cell H are taken into account to avoid the corner effect. Using (3.22) and (3.31), ΛM can be approximated by
( ) ( )
variance σO2. The average transmission power PC controlled by the closed-loop power control can be obtained by0
( )
0 0
The average received power Q can be obtained by
( ) ( )
0 0 S 0 1 ( , )K K ( 0) .
Q=
∫
X∞ Q f⋅ y dy Q= ⋅ −Fγ X (3.43)Five performance measures are investigated. The first measure is the system capacity C which is defined as the maximum number of users per cell that the system can support under the constraint that the outage probability of an arbitrary MS in central cell H is less than a preset outage threshold Λ . That is, 0
where the operator ⋅ denotes the maximum integer below the argument. Substituting (3.39) into (3.44) yields C = where C* C satisfies the following equation *
(
*)
*( )
Note that C* should be always positive. Accordingly, the capacity C under the TCPC scheme is given by
Notably, when X0 = , the TCPC is reduced to the perfect power control and the above analyses 0 remain applicable. On the other hand, when X0 → ∞ , we have
( )
By (3.51) and (3.54), we have
According to (3.49), (3.52) and (3.55), when X0 → ∞ , the capacity becomes infinite, i.e.
0 0 0 0
The second measure is the MS average transmission rate R . It is,
( ) ( )
0 0 S 0 1 ( , )K K ( 0) ,
R
∫
X∞ R f⋅ y dy R= ⋅ −Fγ X (3.57)where R0 =1/T is the data symbol rate.
The third measure is the average system transmission rate ℜ which is defined as the average transmission rate multiplied by the system capacity. Thus ℜ is given by
2 2 (3.58) corresponds to the average system transmission rate of the perfect power control scheme and is the lower limit of ℜ. By (3.57), it is obvious that
0
By (3.58) and (3.60), the upper limit of the average system transmission rate in a multi-cell system under TCPC is
( )
which is just the upper limit of the average system transmission rate in a single cell system under perfect power control.
The fourth measure is the MS average transmission energy per bit E which is defined as the average ratio of MS transmission power to its transmission rate when data is being transmitted.
E is given by
The last measure is the MS average suspension delay D , which is defined as the expected duration of a suspension period. It is in fact the average fade duration defined as [75, p.189]
( ) ( )
Envelope r follows a Nakagami distribution since the overall short-term fading SMH( )t follows a
Considering that MS transmission power is adjusted discretely according to finite-bit power control commands and factors of power control error (PCE) due to step size for power adaptation
∆p, power control command loop delay Td, MS velocity V, and power control period Tp, the TCPC scheme is referred to as a realistic TCPC scheme.
Figure 3.1 presents the functional blocks of the realistic TCPC scheme. The transmitted signal s (t) with power P (t) from MS M will pass through the channel u (τ;t) and suffer
k : power control period
M[ ]
Figure 3.1: Functional blocks of the realistic TCPC scheme
interference from other MS transmissions. The received signal rH(t) at the BS H is then fed into the optimal RAKE receiver to extract the averaged signal power Q k over the kth power M[ ] control period
and estimate the average short-term channel fading SMH[ ]k
( 1)
The BS H has a power control command unit (PCCU) to generate the power control command, CMD, periodically;
The CMD is then transmitted through downlink channel to the destination MS. The MS has a transmission control unit (TCU) which adjusts PM(t) according to the received CMD. It suspends the transmission temporarily if the received CMD is 'suspend'; and adapts its transmission power by an amount of CMD⋅∆p dB, otherwise.
In the realistic TCPC scheme, the received power is no longer a time constant. Conditional with QM[n]>0, the power control error (PCE) is defined as
0 M[ ] Q n
ψ Q . (3.69)
Thus, the outage probability ΛM turns out to be the average of (3.38) over all possible PCE values.
Using (3.31), ΛM can be approximated by
( ) ( )
As shown in (3.70), ΛM equals an arithmetic average of Fψ(x) with weighting function of
2
Based on the system capacity definition given in (3.44), the system capacity can then be approximated by
From (3.74), it is found that the factor G/Γ0 is usually a number larger than other factors, thereby the PCE ψ plays an important role in affecting the system capacity C. The higher the dispersion degree of PCE is, the lower the factor Fψ−1(Λ0) and the system capacity will be. Note that, merely the factor Fψ−1(Λ0) is enough to estimate the system capacity, and the knowledge of the whole distribution of PCE ψ is not necessary. On the other hand, the factor µO has nothing to do with closed-loop power control and is the same as that given in (3.32). The PC and Q in the
realistic TCPC scheme depend on several factors including cutoff threshold X0, step size ∆p, power control loop delay Td, number K of resolvable paths, and MS velocity V. Hence, it is difficult to derive general formulae for PC and Q . The simulation shows that the PC and Q in the realistic TCPC scheme can be adequately approximated by the analytical results given in (3.40) and (3.43), respectively. Based on the above approximations and by assuming that the factor Fψ−1(Λ0) is obtainable, the system capacity can be calculated.
Additionally, we assume that the average short-term channel fading SMH[ ]k is still Rayleigh distributed. Therefore, the performance measures R , E , and D have the same formulae given in (3.57), (3.62), and (3.65), respectively. Finally, ℜ is computed according to its definition. Notably, as X0=0, the realistic TCPC becomes a realistic perfect power control and the above analyses remain applicable.
The realistic TCPC has an issue with how the transmitter can understand when SMH[ ]k turns out to be good after a bad period. Using the real TCPC applied to a WCDMA system [77] as an example. The WCDMA system has two types of uplink dedicated physical channels (DPCHs):
the uplink dedicated physical data channel (DPDCH) and the uplink dedicated physical control channel (DPCCH). Each connection is allocated a DPCH including one DPCCH and several DPDCHs. The DPDCH is used to send user data, and its transmission is suspended when the received CMD is 'suspend'. The DPCCH is used to carry control information, including pilot
the uplink dedicated physical data channel (DPDCH) and the uplink dedicated physical control channel (DPCCH). Each connection is allocated a DPCH including one DPCCH and several DPDCHs. The DPDCH is used to send user data, and its transmission is suspended when the received CMD is 'suspend'. The DPCCH is used to carry control information, including pilot