On Capacity of CDMA System Shishir Pandey Tata Institute of Fundamental Research, Mumbai April 18, 2009 Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 1 / 30
Introduction I will be talking about Capacity of CDMA Systems By Capacity we refer to the number of users supported in a cell FDMA and TDMA are band limited while CDMA is only interference limited Capacity for forward link and downward link is treated We compare results with FDMA systems. Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 2 / 30
Single Cell CDMA Capacity For N users, each cell-site demodulator processes a composite received waveform: Containing desired signal having power S and N 1interfering signals each also of power S Hence, signal-to-noise(interference) power is (No other noise considered) SNR = S (N 1)S = 1 N 1 (1) Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 3 / 30
Bit Energy to Noise Density Ratio Bit Energy to Noise Density ration - is obtained as follows E b /N 0 = S/R (N 1)S/W = W /R N 1 (2) The above eqaution background noise, η, (over entire bw) due to spurious interference and thermal noise contained in total BW, W. Including this we have E b /N 0 = W /R (N 1) + (η/s) (3) Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 4 / 30
Number of Users From above eqn., the number of users supported are N = 1 + W /R E b /N 0 η S (4) W /R is referred to as processing gain. Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 5 / 30
Sectorization We want to increase the number of users. Capacity can be increased by reducing other user interference Use of sectorization and voice-activity monitoring For. Eg. If there are 3 Antenas per cell site, each with beamwidth of 120 o each antenna has in interference by (N 1)/3 users and hence, the total number of users per cell can be 3N. N s denotes the number of users per sector. Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 6 / 30
Voice Activity Monitoring Voice activity can be monitored and transmission can be suppressed. From studies either speaker active only 35% to 40% of time. (Assume voice-activity factor, α = 3/8) Ē b /N 0 = W /R (N s 1)α + (η/s) (5) The improvement might look to be 8/3 times but it is only 2 times. Since, the number of calls per sector are limited and hence, there is nonnegligible probability that and above avg. number of users are talking at once. Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 7 / 30
Power Control Each subscriber monitors total received signal power from the cell site. For the forward link, no power control is required in a single cell system. Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 8 / 30
Reverse Link Capacity for Multiple Cell CDMA The path loss between subscriber and the cell site is proportional to 10 (ξ/10) r 4. r is the distance between the subscriber and cell site. ξ is the Gaussian random variable with standard deviation σ = 8 and zero mean. Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 9 / 30
Interference For transmitters within the cell, power is controlled by the same cell site. Interference from transmitters within the users cell is never greater than (N 1)S. On average this value is reduced by the voice-activity factor. Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 10 / 30
Reverse Link Geometry Figure: Fig 1: Reverse Link Geometry Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 11 / 30
Interference from other cells If the interfering subscriber is at a distance r m from its cell site and r 0 from the cell site of the desired user. The other user when active, produces an interference ( ) I (r 0, r m ) 10 (ξ 0/10) ( ) r 4 = m S 10 (ξm/10) r 4 0 (6) = ( rm r 0 ) 4 10 (ξ 0 ξ m)/10 1 Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 12 / 30
1 st term is attenuation caused by distance and blockage. 2 nd term is the effect of power control of the out-of-cell interferer. ξ 0 and ξ m are independent so the difference has mean zero and variance 2σ 2 The above expression is less than unity. Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 13 / 30
Assuming uniform density of subscribers and normalizing hexagonal cell radius to unity. The density of users in ρ = 2N 3 3 = 2N s 3 per unit area. (7) Hence, the total other-cell to interference ratio is I /S = ψ ( rm r 0 ) 4 {10 (ξ 0 ξ m)/10 }.Φ(ξ 0 ξ m, r 0 /r m )ρda (8) Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 14 / 30
Terms I /S = ψ ( rm r 0 ) 4 {10 (ξ 0 ξ m)/10 }.Φ(ξ 0 ξ m, r 0 /r m )ρda (9) m is the cell site index for which r 4 m10 ξm = min k 0 r 4 k 10 ξ k and Φ(ξ 0 ξ m, r 0 /r m ) = { 1, if (r m /r 0 ) 4 10 (ξ 0 ξ m)/10 1 0, otherwise (10) and ψ is the voice activity variable, which equals 1 with probability α and 0 with probability (1 α). Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 15 / 30
On r m Calculations are simplified and the results are only sligthly increased if Instead of the least attenuation r m = min k 0 r k (11) Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 16 / 30
Mean of the interference-to signal ratio where f and ( rm r 0 ) r 4 E(I /S) = α m f r 0 ( rm r 0 ) ρda (12) { [ ( ) 40 = exp[(σln10/10) 2 ] 1 Q.log r0 10 2σ 2σ 2 r 2 ln10 ]} m 10 (13) Q(x) = 1 e y 2 /2 dy (14) 2π x Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 17 / 30
For each point in space the value of r 0 and r m needs to be found before computing the above function. The above function is evaluated numerically. For σ = 8dB is E(I /S) 0.247N s Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 18 / 30
Variance where g ( rm r 0 ( ) 8 [ rm V (I /S) αg r 0 ) [ (σln10 ) ] 2 = exp 5 { [ 40 1 Q.log 10 2σ 2 ( r0 r m ( rm r 0 ) α 2 f ( rm r 0 ) ( ln10 2σ 2 5.log 10 ( r0 r m The result for σ = 8dB is var(i /S) 0.078N S )] ρda (15) )]} ) 2σ 2 ( ln10 5 ) (16) Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 19 / 30
W /R E b /N 0 = Ns 1 i=1 χ i + (I /S) + (η/s) where N s is the users/sector I is the total interference from users outside the desired user s cell The above equation follows from (17) Ē b /N 0 = W /R (N s 1) + (η/s) The N s 1 same sector noramalize power users, are now random variables χ i with distributions { 1, with probability α χ i = (18) 0, with probability 1 α Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 20 / 30
Performance Analysis Adequate performance (BER < 10 3 ) is achivable on the reverse link with E b /N 0 5(7dB) Hence, the required probability is achieved with probability P = Pr(BER < 10 3 ) = Pr(E b /N 0 5) 1 P = Pr(BER > 10 3 ) = Pr ( Ns ) χ i + I /S > δ i=1 (19) Where, δ = W /R E b /N 0 η S, E b/n 0 = 5 Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 21 / 30
χ i has binomial distribution I /S is Guassian with E(I /S) 0.247N s and var(i /S) 0.078N S All variables are mutually independent, then 1 P = = N s 1 k=0 N s 1 k=0.q Pr(I /S > δ k χ i = k)pr( χ i = k) ( Ns 1 C k ) α k (1 α) Ns 1 k ( ) δ k 0.247Ns 0.078Ns (20) Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 22 / 30
Figure: Fig 2: Reverse Link capacity per sector (W=1.25MHz, R=8kbps, voice activity =3/8, δ = 30) Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 23 / 30
Multiple Cell Forward Link Capacity In single no power control is required. With multiple cells near boundaries of cell considerable interference can be received from other cell-site transmitters. Power control would mean power allocation according to the needs of the individual subscribers. Cell measures its relative SNR, ratio of power from own cell site to the total power received. Sends S T1 and k i=1 S T i, where S T1 > S T2 >... > S Tk > 0 (21) are the powers received by the given mobile from the cell site sector facing it. Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 24 / 30
i th subscriber will receive a fraction of S T1. The remainder of S T1 and other cell site powers as noise. Thus ( Eb N 0 ) i βφ i S T1 /R [( k j=1 S T j )i + η ] /W (22) Where β - is fraction of total cell-site power for subscribers (assume β = 0.8) Φ i - is that fraction of this devoted to subscriber i. If interference is more then more Φ i needs to be allocated. Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 25 / 30
where From above eqn. we have [ Φ i (E b/n 0 ) i βw /R 1 + N s i=1 ( K j=2 S T j S T1 ) i ] + η (S T1 ) i (23) Φ i 1 (24) Relative received cell-site power K f i 1 + S Tj /S T1, i = 1,..., N s (25) j=2 Then N s i=1 f i Ns βw /R (E b /N 0 ) η δ (26) S T1i i=1 Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 26 / 30
Capcity if a random variable whose distribution is obtained from the distribution of f i. N s 1 P = Pr(BER > 10 3 ) = Pr( f i > δ ) (27) Monte Carlo simulations done. i=1 Figure: Forward link capacity/sector. (W=1.25MHz, R=8kbps, voice activity=3/8, pilot power =20%) Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 27 / 30
Conclusions W = 1.25MHz is 10% of total spectal allocation for a provider. S/η = 1dB reflects a reasonable subscriber transmitter level. 36 users/sector or 108users/cell, with 10 3 bit error rates better than 99% of time. If entire cellular allocation is devoted to CDMA, numbers are increased 10 folds. Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 28 / 30
Comparision with FM/FDMA Cellular system 30KHz Channel allocation assuming 3 sectors/cell Frequency reuse factor of 1/7 Number of channels in a 1.25MHz band is slightly less than 42 This gives less than 6 users/cell. CDMA offers 18 fold increase in capacity. Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 29 / 30
Reference 1 On the Capacity of a Cellular CDMA System, Gilhousen et. al., IEEE Trans. on Vehicular Technology, May 1991 Shishir Pandey (TIFR) On Capacity of CDMA System April 18, 2009 30 / 30