Chapter 5 Analysis of IS-95 5.1 List of Mathematical Symbols a ij path loss and shadow fading between the zeroth BS and the ith MS in the jth microcell a 0 ( t ) path loss and shadow fading multiplicative factor b i ( t ) data sequence for the ith user b 0 value of b ( t ) over the zeroth symbol period (= 1 ) C PN ( t ) down-link pilot codes C W ( i + 1 ) ( t ) ( i + 1 ) th Walsh code c ij spreading code of of ith MS in the jth cell c i ( t ) code for the ith user cc ( n k K ) convolutional code D j distance between the zeroth BS and the adjacent jth cell site da area housing an MS d f minimal free distance ( E b = I 0 ) im SIR in the presence of power control errors E ()] expectation of () erfc ()] complementary error function of () e s sectorisation efficiency F a factor used in estimating E b = I 0 f ( r ) PDFofanMSbeinginaringofarea2πrdr f ( r j = r σ ) expectation of 10 ζ = 10 with the constraint function φ ( ζ r = r j ) 285 GSM, cdmaOne and 3G Systems. Raymond Steele, Chin-Chun Lee and Peter Gould Copyright © 2001 John Wiley & Sons Ltd Print ISBN 0-471-49185-3 Electronic ISBN 0-470-84167-2 286 CHAPTER 5. ANALYSIS OF IS-95 G c asymptotic coding gain of convolutional code G p processing gain = T b = T c g controls the rate of increase in the step size by the adptive power control algorthm g ( r j = γ σ ) variance of 10 ζ = 10 with the constraint function φ ( ζ r = r j ) h j ( t τ j ) jth component of the impulse response I total interference power at the output of the matched filter I ext intercellular interference power at the output of the matched filter I 0 ext I ext in the presence of power control errors I int intracellular interference power I 0 int I int in the presence of power control errors I j interference power from all MSs in the jth cell to the zeroth BS I MAI multiple access interference I 0 interference PSD I ( r j r ) interference power at the zeroth BS from the active MS in the jth cell J number of interfering cells K constraint length of the convolutional code k number of message bits in the convolutional code k number of active users within a cell k CDMA capacity as channels per cell per MHz L number of match filters, or number of resolvable paths by the RAKE receiver M number of chips in a spreading code (= G p ) M 0 spreading factor in the presence of convolutional coding N number of mobile users N 0 single-sided PSD of the AWGN n I m ( t ) ; n Q m ( t ) inphase and quadrature components of the multipath inter- ference n convolutional code length n ext ( t ) equivalent baseband intercellular interference n I ext ( t ) ; n Q ext ( t ) inphase and quadrature componenets of n ext ( t ) n int ( t ) equivalent baseband intracellular interference n I total ( t ) ; n Q total ( t ) inphase and quadrature components of the total interfer- ence noise N s number of sectors per cell 5.1. LIST OF MATHEMATICAL SYMBOLS 287 n ( t ) receiver noise P i transmitted power of the ith MS, or from the BS for the ith MS P ij transmitted power from the ith MS in the jth cell (up-link), or the transmitted power allocated for the ith channel at the jth BS (down-link) P m power allocated to each MS by a BS p o outage probability, i.e. probability of the BER > 10 3 P p transmitted pilot power P R received wanted signal power P T transmitted power of an MS in a power control system P tar target received signal power at a BS p b bit error probability P p transmitted power of the pilot signal Pr () probability of () Q () Gaussian Q-function R cell radius R b bit rate of the message sequence R c chip rate R dn ( t ) received signal at an MS R I ( t ) ; R Q ( t ) inphase and quadrature components of R ( t ) , the received baseband signal at the BS R 0 distance from a BS where ‘near-in’ MSs are present R up ( t ) received signal at the BS from an MS r distance of an MS from a BS S desired received power at a BS S 0 signal power at the output of the matched filter in the pres- ence of imperfect power control S I ( t ) ; S q ( t ) inphase and quadrature components of the wanted signal S p received pilot power compoment for the zeroth MS on the down-link s dn ( t ) signal transmitted from a BS s ij ( t ) transmitted signal from the ith MS to the jth BS s j dn ( t ) signal transmitted from the jth neighbouring BS s 0 ( t ) spread BPSK signal for zeroth MS T b bit duration T c chip duration 288 CHAPTER 5. ANALYSIS OF IS-95 W chip rate and bandwidth of the CDMA signal, also the width of a street in a street microcell X distance from a street microcellualr BS to the end of the microcell X b a break distance in a street microcell where the propagation path loss exponent changes x ij ( t ) transmitted baseband signal from the ith MS to the jth BS Z ext ( T b ) intercellular interference component of Z ( T b ) Z int ( T b ) intracellular interference at the output of the matched filter Z n ( T b ) receiver noise at the output of the matched filter Z ( T b ) output of the matched filter at time t = T b Z w ( T b ) wanted component of Z ( T b ) β d coefficient of the transfer function T ( D H ) of the convolu- tional code β l magnitude of the lth path of the fast fading channel impulse response L exclusive-OR operation V system parameter in the power control algorithm δ i normally distributed received error power random variable at a BS for MS i δ ij power control error for the ith MS in the jth cell δ ( t u ) delta function at time u η AWGN power at the output of the matched filter γ b E b = I 0 , or energy per bit per interference PSD γ c E c = I 0 , or energy per symbol per interference PSD γ req required ( E b = I o ) for BER < 10 3 λ ij normally distributed random variable with standard devia- tion σ and zero mean µ voice activity factor (VAF) ν i voice activity variable of the ith user ν ij voice activity variable of the ith user in the jth cell ω 2 down-link angular carrier frequency ω 1 up-link angular carrier frequency Φ fixed step size used in power control algorithm φ ij carrier phase between the interference signal from the ith MS in the jth cell and the zeroth MS in the zeroth cell φ 0 carrier phase difference ˆ θ 0 ˆ θ φ ( ζ r r j ) constraint function 5.2. INTRODUCTION 289 ρ density of MSs in a cell σ ε standard deviation of ε σ e standard deviation of δ i τ i random delay of the ith user signal at the BS on the up-link, or the random offset at the BS on the down-link τ ij relative propagation delays of the ith MS in the jth cell with respect to the zeroth MS in the zeroth cell τ p time offset of the pilot signal at the BS θ received carrier phase angle at the BS, or the transmitted phase angle of the carrier at the BS θ i random phase angle of the trnsmitted ith mobile carrier ˆ θ i change in the phase angle of the ith MS = ω 1 ( τ 0 τ i )+ θ i θ ij carrier phase of ith MS in the jth cell θ 0 overlapping angle of adjacent sectors 4 i adaptive step size used in the power control algorithm var ()] variance of () εδ j δ 0 random variable having a normal distribution ξ error in estimating P R in the power control ζλ ij λ 0 5.2 Introduction In CDMA many mobiles use the same RF bandwidth at the same time, and a CDMA re- ceiver is able to separate the wanted signal from the other mobile signals if it knows the spreading code used in the generation of the wanted CDMA signal. This demodulation process occurs in the presence of interference generated by other mobile users. This inter- ference is a major limitation on the capacity of a CDMA system. In this chapter the capacity of a CDMA system in tessellated hexagonal cells and city street microcells is investigated. The system performance in terms of outage probability for a bit error rate (BER) larger than a minimal required level is analysed. The number of users that can be supported by a cell for a given outage probability is evaluated. The corresponding capacity in terms of channels per cell per MHz is calculated according to this number of users per cell. Our discussion concentrates on the capacity evaluation rather than on other issues, such as code synchronisation. We begin by examining a single cell CDMA system before moving on to a multiple cell CDMA system. Since the arrangement of the up-link, or forward link, is different from the down-link, or reverse link, the performances of both the up-link and down-link are considered. The effect of sectorisation and channel coding on CDMA systems is also discussed. 290 CHAPTER 5. ANALYSIS OF IS-95 5.3 CDMA in a Single Macrocell Consider a single cell CDMA communication system using binary phase shift keying (BPSK) spread spectrum modulation. As shown in Figure 5.1, the BS uses the angular carrier fre- quency ω 2 on the down-link to communicate with all its mobiles, while mobiles transmit to their base stations (BSs) via the angular carrier frequency ω 1 . 5.3.1 The up-link system The CDMA single cell system consists of N mobile users transmitting to a BS receiver on the up-link. We consider a simplified mobile transmitter consisting of a BPSK modulator, formed by multiplying the data sequence for the ith user b i ( t ) , by a carrier cosω 1 t. Spread- ing occurs when the BPSK signal is multiplied by the code c i ( t ) . This is equivalent to multiplying the data signal, b i ( t ) ,byc i ( t ) and this spread data signal modulates the carrier cos ω 1 t. Figure 5.2 shows the arrangement. Let us consider a particular user, say the zeroth one. The spread BPSK signal s 0 ( t ) is applied to the radio channel shown in Figure 5.3. We have separated this channel into a part that allows for path loss and slow fading and is represented by the multiplicative factor a 0 . The fast fading is represented by a number of impulse responses h j ( t τ j ) j = 0 1 ::: L. The input of the receiver consists of: interference from the other users in the cell and is known as intracellular interference; the receiver noise n ( t ) ; and the received signal for the zeroth user. The sum of these signals, R up ( t ) , is demodulated by multiplying by a recovered carrier having the same frequency but different phase, relative to the transmitted carrier. The resulting signal is applied to a RAKE receiver that may be considered to be composed of L matched filters, one for each significant path in the impulse response of the channel. We note that in general the number of matched filters and the number of channels will not be the same, but it is desirable if there are at least as many matched filters as there are significant paths in the channel. The RAKE receiver is a maximum ratio diversity system if it can obtain accurate estimates of the complex impulse responses h j ( t τ j ) . The RAKE receiver is described in Section 2.3.2.6. A CDMA system has other attributes to combat the effects of fast fading on the signal s 0 ( t ) . These include symbol interleaving, forward error correction (FEC) coding, space diversity reception, power control, and so forth. Using this battery of techniques we can effectively compensate for the effects of fast fading. The channel model is now reduced to the multiplicative factor a 0 which accounts for path loss and slow fading. The BS receiver may now be configured for our analysis as one having despreading followed by a matched filter, i.e. single stage RAKE, which is an integrator and dump circuit for each mobile. Our simplified model of the radio channel and the BS receiver is depicted in Figure 5.4. Each user has a unique spreading code that is known to the BS. The spreading codes are 5.3. CDMA IN A SINGLE MACROCELL 291 ω 1 ω 2 ω 1 ω 1 ω 2 ω 2 BS ω 1 Figure 5.1: Single cell mobile radio communications in a hexagonal cell. c i (t) spreader modulator spreading code generator carrier generator i-th user b i (t) s i (t) cos( ω 1 t + θ i ) 2P i Figure 5.2: A mobile’s CDMA transmitter diagram. of length M chips, or an M-chip segment from the long psuedo noise (PN) sequence [1,2]. As the mobiles are in different locations within the cell, the transmission delay for each mobile is different. The signal transmitted from the ith user to its BS is s i ( t )= p 2P i b i ( t ) c i ( t ) cos ( ω 1 t + θ i ) (5.1) where P i is the transmitted power of the ith user, b i ( t ) is the data sequence of the ith user where each bit has an amplitude of 1 and a duration of T b , c i ( t ) is the spreading code sequence of ith user and each of the M chips per code has a duration T c ,andθ i is the random phase of the ith mobile carrier and is uniformly distributed in 0 2π ) . All the mobiles transmit their signals to the BS receiver over the same radio channel, and the received signal at the BS receiver is R up ( t ) = N 1 ∑ i = 0 a i s i ( t τ i )+ n ( t ) = N 1 ∑ i = 0 a i p 2P i b i ( t τ i ) c i ( t τ i ) cos ω 1 ( t τ i )+ θ i ] + n ( t ) (5.2) 292 CHAPTER 5. ANALYSIS OF IS-95 T b 1 (.)dt T b 1 (.)dt T b 1 (.)dt c 0 (t-τ 1 ) c 0 (t-τ L ) c 0 (t-τ 0 ) h 0 (t-τ 0 ) fast fading n(t) a 0 s 0 (t) R up (t) h L (t-τ L ) interference from other users 2cos(ω 1 t+θ) path loss & slow fading carrier recovery decision matched filter combiner matched filter matched filter RAKE Receiver cophaser h 1 (t-τ 1 ) + Figure 5.3: The up-link representation. where a i represents the path loss and slow fading of the ith user, τ i is the random delay of the ith user signal at the receiver and is uniformly distributed in 0 T b ) ,andn ( t ) is the additive white Gaussian noise (AWGN) of the receiver noise. The signal at the output of the zeroth matched filter is given by Z ( T b ) = 1 T b Z T b + τ 0 τ 0 R up ( t ) c 0 ( t τ 0 ) 2cos ( ω 1 t + θ ) dt = ( 1 T b Z T b + τ 0 τ 0 N 1 ∑ i = 0 a i p 2P i b i ( t τ i ) c i ( t τ i ) cos ω 1 ( t τ i )+ θ i ] c 0 ( t τ 0 ) 2cos ( ω 1 t + θ ) dt ) + 1 T b Z T b + τ 0 T 0 n ( t ) c 0 ( t τ 0 ) 2cos ( ω 1 t + θ ) dt ) (5.3) where θ is the carrier phase angle in the receiver. Note that due to the propagation delay 5.3. CDMA IN A SINGLE MACROCELL 293 T b 1 (.)dt T b 1 (.)dt T b 1 (.)dt a N-1 + n(t) channel R up (t) c i (t) c 0 (t) c 1 (t) decision decision decision b N-1 (t) i b i (t) b 1 (t) 2cos(ω 1 t+θ) Base station receiver decision b 0 (t) s 0 (t) s 1 (t) s i (t) s N-1 (t) a 0 a 1 a i c N-1 (t) matched filter matched filter matched filter matched filter + T b 1 (.)dt Figure 5.4: Simplified radio channel and BS receiver block diagram. 294 CHAPTER 5. ANALYSIS OF IS-95 of the radio path associated with the zeroth user, the integration is done from t = τ 0 to t = τ 0 + T b . Letting t = t + τ 0 in Equation (5.3), we have Z ( T b ) = ( 1 T b Z T b 0 N 1 ∑ i = 0 a i p 2P i b i ( t + τ 0 τ i ) c i ( t + τ 0 τ i ) cos ( ω 1 t + ˆ θ i ) c 0 ( t ) 2cos ( ω 1 t + ˆ θ ) dt ) + 1 T b Z T b 0 n ( t + τ 0 ) c 0 ( t ) 2cos ( ω 1 t + ˆ θ ) dt ) (5.4) where ˆ θ = ω 1 τ 0 + θ and ˆ θ i = ω 1 ( τ 0 τ i )+ θ i are the changes in the phase angle of θ and θ i , respectively. We define a relative time delay for the N 1 users with respect to the zeroth user of τ i0 = τ i τ 0 , and on substituting into Equation (5.4) we have Z ( T b ) = ( 1 T b Z T b 0 N 1 ∑ i = 0 a i p 2P i b i ( t τ io ) c i ( t τ i0 ) cos ( ω 1 t + ˆ θ i ) c 0 ( t ) 2cos ( ω 1 t + ˆ θ ) dt ) + 1 T b Z T b 0 n ( t + τ 0 ) c 0 ( t ) 2cos ( ω 1 t + ˆ θ ) dt ) : (5.5) Owing to the stationary property of the AWGN, n ( t + τ 0 ) in the above equation can be substituted by n ( t ) , and Equation (5.5) can be rewritten as Z ( T b ) = ( 1 T b Z T b 0 N 1 ∑ i = 0 a i p 2P i b i ( t τ i0 ) c i ( t τ i0 ) cos ( ω 1 t + ˆ θ i ) c 0 ( t ) 2cos ( ω 1 t + ˆ θ ) dt ) + 1 T b Z T b 0 n ( t ) c 0 ( t ) 2cos ( ω 1 t + ˆ θ ) dt : (5.6) Assuming the receiver is chip synchronised to the zeroth mobile, then for the zeroth mobile, c i ( t τ i0 ) becomes c 0 ( t ) and from Equation (5.6) c i ( t τ i0 ) for i = 0, multiplied by c 0 ( t ) yields unity, and therefore the wanted component of Z ( T b ) is Z w ( T b ) = 1 T b Z T b 0 a 0 p 2P 0 b 0 ( t ) cos φ 0 + cos ( 2ω 1 t + ˆ θ 0 + ˆ θ ) dt [...]... interfering mobile and zeroth BS, is rj = q D2 j + r2 2D j r cos(ϕ) (5.81) and ζ is the difference between λi j and λ0, the independent random variables with zero mean and standard deviation σ Hence ζ is also a random variable with zero mean and variance of σ2 = 2σ2 , while D j is the distance between the zeroth BS and the jth co-channel BS Note ζ that a2 j in Equation (5.80) is the path loss and slow fading... of 3/8 and 1/2, respectively The outage probability of the imperfect power controlled system having different standard deviations of power control error in Eb =I0 is show in Figures 5.6 and 5.7 for a VAF of 3/8 and 1/2, respectively We observe that a standard deviation of the measured Eb=I0 was found to be 1.7 dB in a particular set of measurements [7] For an outage probability of 2% and a standard... mobile is a random variable, i.e τi0 is an independent random variable that is uniformly distributed over 0 Tb) We further assume that bi (t ) represents random independent binary data, and as a consequence the intracellular interference is a stationary random process From the Central Limit Theorem, the summation of N 1 independent random process means that nint can be approximated as a Gaussian random variable... loss law and slow fading whose standard deviation was 8 dB A signal-to-AWGN ratio of 20 dB at the output of the matched filter was assumed and a BER outage threshold of 10 3 was used in the calculations Figure 5.5 shows the outage probability from Equation (5.35) for perfect power control and VAFs of 3/8 and 1/2 For an outage probability of 2%, the single cell CDMA system can support 48 users and 38 users... the ith mobile in the jth cell, and ε is a random variable p having zero mean and normal distribution with a standard deviation of σε = 2σe From Equations (5.82) and (5.100), the total intercellular interference-to-signal ratio is 0 Iext S0 = = 1 J 1 Gp ∑ j =1 ε Iext 10 10 S Z 2π Z R 0 vi j 0 r rj α ζ 10 10 φ ζ r rj ε 10 10 ρrdrdϕ (5.101) : From Equations (5.37), (5.40), and (5.101), the bit energy-to-interference... =S0 and Iext =S0 are two independent Gaussian distributed random variables, whose mean and variance may be expressed as E 0 Iint S0 + 0 Iext S0 = E var 0 Iint S0 + 0 Iext S0 = var and Iint ε 10 10 S Iext ε 10 10 S +E Iint ε 10 10 S + var Iext ε 10 10 S : (5.104) Before we derive the outage probability, we have to calculate the mean and variance of the intracellular interference-to-signal ratio, and. .. errors in Eb =I0, and the percentage decrease in users due to imperfect power control, are displayed in Table 5.1 for an outage of 2% From Table 5.1, the imperfect power control system having a standard deviation of 2 dB can only support 22 users and 28 users for VAFs of 1/2 and 3/8, respectively The reduction in capacity caused by power control error is about 6.3% and 4.3% for VAFs of 3/8 and 1/2, respectively,... differ from the target power level Ptar by δ0 dB This error power δ0 is a random variable that is normally distributed with standard deviation σe The received signal S0 at the BS for the zeroth mobile, and the intracellular interference-to-signal ratio are given by Equations (5.37) and (5.40), respectively Using Equations (5.37) and (5.80), the mobiles in area da in the jth cell produce an interfering... the up-link and the down-link communication systems 5.4.1 The up-link system The received signal at a BS includes the desired signal, intracellular interference, the AWGN at the receiver input, and intercellular interference Figure 5.11 shows the up-link communication system where the arrangement for the mobile transmitter and BS receiver are exactly the same as those shown in Figures 5.2 and 5.4, respectively... Ptar In order to track the relative path loss and slow fading variations, the transmitted power Pi j from the ith mobile in the jth cell is made inversely proportional to a2j , whence i Pi j = Ptar a2j i = Ptar rα 10 λi j 10 = Srα10 λi j 10 (5.79) where α is the path loss exponent, and λi j is a normally distributed random variable with standard deviation σ and zero mean, while r is the distance from . ) PDFofanMSbeinginaringofarea2πrdr f ( r j = r σ ) expectation of 10 ζ = 10 with the constraint function φ ( ζ r = r j ) 285 GSM, cdmaOne and 3G Systems. . a cell σ ε standard deviation of ε σ e standard deviation of δ i τ i random delay of the ith user signal at the BS on the up-link, or the random offset