CDMA Capacity Slides

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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

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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 interferencelimited

Capacity for forward link and downward link is treated

We compare results with FDMA systems.




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Single Cell CDMA Capacity

For N users, each cell-site demodulator processes a compositereceived 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 noiseconsidered)

SNR = S

(N − 1)

S =

1

N − 1

(1)


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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 tospurious interference and thermal noise contained in total BW, W .Including this we have

E b /N 0 =

W /R

(N − 1) + (η/S ) (3)


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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”.


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Sectorization

We want to increase the number of users.

Capacity can be increased by reducing other user interference

Use of sectorization and voice-activity monitoringFor. Eg. If there are 3 Antenas per cell site, each with beamwidth of 120o 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.


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Voice Activity Monitoring

Voice activity can be monitored and transmission can be suppressed.

From studies either speaker active only 35% to 40% of time. (Assumevoice-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 aretalking at once.


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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 cellsystem.




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Reverse Link Capacity for Multiple Cell CDMA

The path loss between subscriber and the cell site is proportional to10(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 andzero mean.




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Interference

For transmitters within the cell, power is controlled by the same cellsite.

Interference from transmitters within the users cell is never greaterthan (N − 1)S .On average this value is reduced by the voice-activity factor.


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Reverse Link Geometry

Figure:

Fig 1: Reverse Link Geometry




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Interference from other cells

If the interfering subscriber is at a distance r m from its cell site and r 0from the cell site of the desired user.

The other user when active, produces an interference

I (r 0, r m)

S =

10(ξ0/10)

r 40

r 4m

10(ξm/10)

(6)

= r m

r 04

10

(ξ0−ξm)/10

≤ 1


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1st term is attenuation caused by distance and blockage.

2nd 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 2σ2

The above expression is less than unity.


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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 r m

r 0 4

{10(ξ0−ξm)/10}.Φ(ξ 0 − ξ m, r 0/r m)ρdA (8)


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Terms

I /S =

ψ

r m

r 0

4{10(ξ0−ξm)/10}.Φ(ξ 0 − ξ m, r 0/r m)ρdA (9)

m is the cell site index for which r 4m10−ξm = mink =0 r

4

k

10−ξk

and

Φ(ξ 0 − ξ m, r 0/r m) =

1, if (r m/r 0)410(ξ0−ξm)/10 ≤ 1

0, otherwise (10)

and ψ is the voice activity variable, which equals 1 with probability α and0 with probability (1 − α).


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On r m

Calculations are simplified and the results are only sligthly increased if

r m = mink =0 r k (11)

Instead of the least attenuation


M f h i f i l i

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Mean of the interference-to signal ratio

E (I /S ) = α

r 4m

r 0f

r m

r 0

ρdA (12)

where

f

r m

r 0

= exp [(σln10/10)2]

1 − Q

40√

2σ2.log 10

r 0

r m

−√ 2σ2 ln10

10

(13)

and

Q (x ) =

1

√ 2π ∞x

e −y 2/2

dy (14)


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For each point in space the value of r 0 and r m needs to be foundbefore computing the above function.

The above function is evaluated numerically.

For σ = 8dB isE (I /S ) ≤ 0.247N s


V i

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Variance

V (I /S ) ≤

r m

r 0

8

αg

r mr 0

− α2f

r mr 0

ρdA (15)

where

g

r mr 0

= exp

σln10

5

2

1 − Q

40√ 2σ2

.log 10

r 0

r m

−√

2σ2

ln10

5

.log 10

r 0r m

−√ 2σ2 ln105

(16)

The result for σ = 8dB is var (I /S ) ≤ 0.078N S


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E b /N 0 = W /R

N s −1i =1 χi + (I /S ) + (η/S )

(17)

whereN s is the users/sectorI is the total interference from users outside the desired user’s cell

The above equation follows from

Ē b /N 0 = W /R

(N s − 1) + (η/S )The N s − 1 same sector noramalize power users, are now randomvariables χi with distributions

χi =

1, with probability α

0, with probability 1 − α (18)


P f A l i

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Performance Analysis

Adequate performance (BER < 10−3) is achivable on the reverse linkwith E b /N 0 ≥ 5(7dB)Hence, the required probability is achieved with probabilityP = Pr(BER < 10−3) = Pr(E b /N 0 ≥ 5)

1 − P = Pr(BER > 10−3) = Pr

N s i =1

χi + I /S > δ

(19)

Where,

δ = W /R E b /N 0

− ηS

, E b /N 0 = 5


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χi 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

Pr(I /S > δ − k |χi = k )Pr(χi = k )=

N s −1k =0

N s −1C k

αk (1 − α)N s −1−k

.Q δ − k − 0.247N s √ 0.078N s (20)


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Figure:

Fig 2: Reverse Link capacity per sector (W=1.25MHz, R=8kbps, voice

activity =3/8, δ = 30)


Multiple Cell Forward Link Capacity

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Multiple Cell Forward Link Capacity

In single no power control is required.With multiple cells near boundaries of cell considerable interferencecan 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 tothe total power received.

Sends S T 1 and

k i =1 S T i , where

S T 1 > S T 2 > ... > S T k > 0 (21)

are the powers received by the given mobile from the cell site sector facingit.


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i th subscriber will receive a fraction of S T 1 .

The remainder of S T 1 and other cell site powers as noise.

Thus E b

N 0

i

≥ β Φi S T 1 /R

k j =1 S T j i + η /W (22)

Whereβ - is fraction of total cell-site power for subscribers (assume β = 0.8)Φi - is that fraction of this devoted to subscriber i .

If interference is more then more Φi needs to be allocated.


F b h

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From above eqn. we have

Φi ≤ (E b /N 0)i β W /R

1 +

K j =2 S T j

S T 1

i

+ η

(S T 1 )i

(23)

whereN s i =1

Φi ≤ 1 (24)

Relative received cell-site power

f i

1 +

K

j =2S T j /S T 1

, i = 1, ..., N s (25)

ThenN s i =1

f i ≤ β W /R (E b /N 0)

−N s i =1

η

S T 1i δ (26)


Capcity if a random variable whose distribution is obtained from the

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p ydistribution of f i .

1

−P = Pr(BER > 10−3) = Pr(

N s

i =1

f i > δ ) (27)

Monte Carlo simulations done.

Figure:

Forward link capacity/sector. (W=1.25MHz, R=8kbps, voiceactivity=3/8, pilot power =20%)


Conclusions

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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 areincreased 10 folds.


Comparision with FM/FDMA Cellular system

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Comparision with FM/FDMA Cellular system

30KHz Channel allocation

assuming 3 sectors/cell

Frequency reuse factor of 1/7Number 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.


Reference

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Reference

1 “On the Capacity of a Cellular CDMA System”, Gilhousen et. al.,

IEEE Trans. on Vehicular Technology, May 1991

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