Bryan Davis

December 3, 2000

EEL 6503

Distribution of the Forward-Link Transmitter Power using Optimal Power Control in a Multipath Environment

Introduction

In the forward-link of a CDMA cellular system, intercell interference increases the overall interference for receivers near the edge of the cell. They will therefore require more powerful transmissions than users close to the base station to achieve the same signal-to-interference ratio. When a system regulates the transmitted power from the base station in a manner where each channel is transmitted at a different power, the system is using forward-link power control. If power-control is not used on the forward-link, all signals in a cell would be transmitted at the same power. This power level would necessarily be the required power of the least-capable receiver (generally ones near the edge of the cell). This excessive power transmission causes the cell to interfere more with the neighboring cells, without increasing signal power to the receivers who need it. Since all cells assumedly use the same power-control algorithm, this would limit the forward-link capacity of the overall system as compared to a system that uses forward-link power control. The system would achieve maximum capacity, if it attempted to equalize all of the SIR (Signal-to-Inference Ratio) of the channels to at least achieve the quality of service (QoS) requirement.[3] Any set of power transmission coefficients that achieves this design heuristic is said to be optimum.

If there is a reasonably large variation in the power sent on the channels in a cell, a nonuniform modulation technique can be exploited to increase the overall throughput of the cell. [1] The amount of this increase can be measured by counting the number of pairs of receivers, where one receiver can exploit the increased power sent to another receiver. The number of pairs depends in part on the distribution of the transmission power implemented in a cell. The wider the range in power of the forward-link transmissions, the more beneficial this modulation technique becomes. While analysis has been done on the range on transmission power in a simplified system with no interference other than white noise, [4] there is no estimate of the range of transmission power for a more realistic model that includes the effect of multipath and intercell interference.

The goal of this project is to calculate the distribution in transmission power for optimum power control, when the effects of multipath and intercell interference are considered. An appropriate model for multipath and intercell interference needs to be adopted, along with reasonable simplifications about the cellular system. The multipath model used will be the wide-sense stationary uncorrelated scattering model (WSSUS) with slow, frequency selective fading [2]. Intercell interference will be modeled by considering the effects of two layers of surrounding cells (total of 18 neighboring cells), which we will assume to be symmetric in terms of the interference produced with the central cell which we are modeling. We will also assume that the number of receivers is constant and the same for each cell, and that each receiver communicates only with the cell it is closest to (no soft-handover), and that all receivers are stationary. We will consider the receivers to be uniformly distributed throughout the central cell.

Given these assumptions, analytical results will be obtained for the distribution of transmission power, wherever possible. Monte-Carlo simulations will be performed to supplement and verify the analytical results. The main result of this project will be to show a distribution of the power of transmissions, and if possible give a solution for this distribution.

Relationship Between SIR and Power Control

In a real system, we would like to configure the power control coefficients for each receiver are such that the SIR for each receiver stays below a threshold level that is predetermined based on Quality of Service (QoS) requirements. For a given system, either there is a power control configuration that satisfies this requirement, or there is not. Thus this heuristic does not give us an optimum power control configuration. One way to define optimum power control would be to maximize the average SIR for each receiver. The problem with this method is that one receiver could have a very low SIR, while the configuration has a high average SIR. Then this receiver could not meet the QoS requirements for the outcast receiver, although a configuration might exist where all receivers meet QoS requirements. The measure that we will use in this paper is to maximize the lowest SIR. This would ensure that all receivers meet the QoS requirements, if it is possible. The assumption here is that an increase in one receiver’s SIR (a decrease in that receivers power level) will result in an increase in another receiver’s SIR (a corresponding decrease in that receivers interference). [3]

In order to apply this heuristic we must calculate the SIR for each receiver in terms of power control coefficients of the entire system. Since we are assuming that the neighboring cells are symmetric with the central cell, we will also assume that the power control coefficients for the channels are identical to those in the central cell.

In this model we are assuming that the thermal noise is negligible as compared to the interference from other cells and from multipath delays in the same cell.

Intra-cell Interference

We are using a simplified version of the IS-95 forward link as our model. Specifically, we assume that each data channel is orthogonal to every other data channel on the carrier because of Walsh channel orthogonalization. [2] This orthogonality only holds when the data channels undergo a multipath spread of duration smaller than one chip (frequency non-selective fading). If the multipath spread is greater than the chip duration, then the delayed version of the signal will interfere with the LOS (line-of-sight) signal. Note that if there is no multipath, there is no interference, or the SIR is ¥. Furthermore, with the assumption that there is no frequency-selective fading, the SIR is still ¥, even under a Rayleigh fading channel. This makes sense, because we are assuming that the thermal noise is zero. If, however, we are in a frequency-selective fading channel, the interference becomes non-zero. For a given diversity path, all of the channels on the carrier are orthogonal with each other, but they will have non-zero cross-correlation values with the other diversity paths, because they are time-shifted relative to the first path by more than one-chip. We will use the Standard Gaussian Approximation (SGA), and we will assume that the interference from each diversity path is independent, zero-mean Gaussian, with a variance proportional to the square of the gain of the diversity path. For a better treatment of the SGA, see [1].

The interference I_i,j,m caused by i^th diversity path on the j^th diversity path of the channel m, using the SGA is

Ii,j,m =

| bi,m |^2 3N

K å k = 1

Pk

(1)

when i ¹ j, and 0 when i = j.

where K is the number of active channels in the cell,

N is the spreading gain,

P_i is the power sent on the i^th channel,

and b_i is the complex gain of the i^th diversity path received by the m^th receiver.

Since the interference is Gaussian, and we assume that the autocorrelation of the interference is zero when t is greater than the chip duration, we can add the interference terms from each of the multipaths non-coherently. Thus I_m, the total intra-cell interference on i^th diversity path of channel m is given by

Ii,m =

1 3N

æ è

L å j = 1 j ¹ i

| bj,m |^2

ö ø

æ è

K å k = 1

Pk

ö ø

(2)

where L is the number of diversity paths.

If we assume that the Rake receiver using maximal-ratio-combining is employed, with an arbitrary number of “fingers”, (i.e. it can resolve any number of diversity paths), then the SIR on channel m is given by

SIR_m =

3NP_m

---

K å k = 1

Pk

·

L
å
i = 1

| b_i,m |²

---

L å j = 1 j ¹ i

| bj,m |^2

(3)

where SIR_m is the signal-to-interference ratio of the m^th receiver.

So as the number of diversity paths increase (the fading becomes more frequency-selective), the amount of interference energy increases as a proportion of the total energy received, and thus the SIR decreases. If the number of diversity paths gets very large, virtually all of the energy received for any particular diversity path will be interference.

Notice as the number of diversity paths gets arbitrarily large, the outermost summation becomes 1, which is the SGA approximation for K non-orthogonal signals. [1] Thus as multipath diversity increases, the orthogonality of the channels becomes increasingly unimportant. Also, notice that the interference and the desired signal are equally affected by path loss due to the distance away from the receiver, and shadowing effects. This means that these two effects do not affect the intra-cell interference in this model.

Inter-Cell Interference

In addition to the intra-cell interference encountered because of the effects of frequency selective fading, there is also interference caused by transmitters in neighboring cells. Since we are concerned only with the forward-link, all of the interference comes from the neighboring cells’ base station. We again use the SGA for the interference from each of the neighboring cells. The interference generated by the other cell base stations is

Log-normal shadowing

The log-normal shadowing that occurs between the receiver and each of the antennas we would assume to be highly correlated. That is, if the receiver is shadowed from one base station, it is highly likely that it will also be shadowed from the other base station. Since the degree of correlation will be difficult to characterize, we will assume that the correlation is perfect in order to simplify the calculations. Since we assume that the signals from all base stations undergo the same shadowing effects, then the interference and the desired signal will be equally affected by shadowing, and thus shadowing will have no effect on the SIR.

Exponential path loss

The amount of attenuation caused by propagation is proportional to the n^th power of distance from the transmitter, where n is generally considered to be 2 to 5. [1] Here we will take n to be 4, a typical choice.

Distance to Neighboring Base Stations

Here we only consider the cells that are at most separated from the central cell by two base stations. Cells further away than this will contribute a negligible amount of interference to the central cell because of the 4^th power exponential path loss assumption.

In order determine the effects of exponential path loss on the signals transmitted from neighboring cells (which the receiver sees as interference), we must determine the distance from the receiver to the neighboring cell’s base station. We let the receiver be a distance r from the center of the central cell, at an angle q with respect to the line between the central cell and cell 1. (See the figure below). If we normalize the distance between the cells to be 2 (this makes the a circle inscribed in the cell to have a radius of 1), then using the cosine law, we get the following distances from each cell base station to the receiver, as labeled in figure 1:

Base station

Distance from receiver to this base station

0

R

1

Ö	r^2 - 4r cos q + 4

2

Ö	r^2 - 4r cos( 60^° - q ) + 4

3

Ö	r^2 + 4r cos( 60^° + q ) + 4

4

Ö	r^2 + 4r cos q + 4

5

Ö	r^2 + 4r cos( 60^° - q ) + 4

6

Ö	r^2 - 4r cos( 60^° + q ) + 4

7

Ö	r^2 - 8r cos q + 16

8

Ö	r^2 - 4Ö3 r cos( 30^° - q ) + 12

9

Ö	r^2 - 8 r cos( 60^° - q ) + 16

10

Ö	r^2 - 4Ö3 r cos( 90^° - q ) + 12

11

Ö	r^2 + 8 r cos( 60^° + q ) + 16

12

Ö	r^2 + 4Ö3 r cos( 30^° + q ) + 12

13

Ö	r^2 + 8 r cos q + 16

14

Ö	r^2 + 4Ö3 r cos( 30^° - q ) + 12

15

Ö	r^2 + 8 r cos( 60^° - q ) + 16

16

Ö	r^2 + 4Ö3 r cos( 90^° - q ) + 12

17

Ö	r^2 - 8 r cos( 60^° + q ) + 16

18

Ö	r^2 - 4Ö3 r cos( 30^° + q ) + 12

Aggregate Inter-cell Interference

With a path loss exponent of 4, the interference caused by each cell is simply the interference at a distance of 1 from the cell base station multiplied by the 4^th power of its distance from the base station. Because we are assuming that the interference generated by each cell is Gaussian, and the cells are identical, we may add the interference from the cells non-coherently. The resulting equation is too complex to warrant listing it here, but we note that it is a sum of algebraic functions of r and q , times the normalized interference term. We will simply define the summation of the 4^th power of the above listed functions as j(r, q). Thus the total inter-cell interference is I ·j( r,q ), where I is the interference contribution from a neighboring cell at a distance of 1 from the base station of that cell.

According to the SGA, the interference I is given by

I =

1 3N

K å k = 1

Pk

(4)

Accounting for both Intra-cell and Inter-cell Interference

In order to account for both intra-cell interference and inter-cell interference, we need to modify (2) to include the inter-cell interference, and to account for the path loss of intra-cell interference as the receiver gets further away from the cell. In order to do this, we simply need to multiply (2) by 1/r⁴, then add the inter-cell interference term, and we get

Ii,m =

1 3N

æ è

K å k = 1

Pk

ö ø

æ ç è

L å j = 1 j ¹ i

| bj,m |^2 r^4

+ j( r,q )

ö ÷ ø

(5)

Then our rake receiver SIR becomes

SIR_m =

3NP_m

---

K å k = 1

Pk

·

L
å
i = 1

| b_i,m |²

---

r⁴

---

L å j = 1 j ¹ i

| bj,m |^2 r^4

+ j( r,q )

        (6)

Or more simply,

SIR_m =

3NP_m

---

K å k = 1

Pk

·

L
å
i = 1

| b_i,m |²

---

L å j = 1 j ¹ i

| bj,m |^2 + r^4 j( r,q )

        (7)

Determining the Power Control Coefficients

In the above equation, we note that the SIR is dependent on the ratios of the power control coefficients, not their absolute values. Thus we can normalize the ratio by letting

K å k = 1

Pk = 1        (8)

Letting SIR’s for each channel be equal to g, and solving for P_m, we get

P_m =

g

---

3N

---

L
å
i = 1

| b_i,m |²

---

L å j = 1 j ¹ i

| bj,m |^2 + rm ^4 j( rm ,qm )

        (9)

Simulation of Random Variables

Examination of Intra-cell Interference

(9) is very complicated. Luckily it can be simplified a great deal by modeling the intra-cell interference with a simpler model. The double summation in (9) can be approximated with a simple rational expression

L
å
i = 1

| b_i,m |²

---

L å [( j = 1 ) || ( j ¹ i )]

| bj,m |^2 + rm ^4 j( rm ,qm )

»

a_0,m

---

a_1,m + r_m ⁴ j( r_m ,q_m )

        (10)

where a_0,m and a_1,m are random variables whose distribution is the sum of Rayleigh distributed random variables. These distributions are rather complicated to analyze. Also note that in (9), the g/3N term is not dependent on the user, and is therefore a common factor for all users and can be ignored, since we are concerned with relative power control coefficients at this point. If we assume that the signal attenuation due to multipath is the same for all receivers (a_0,i = a_0,j, "i,j), then

P_m µ a_1,m + r_m ⁴ j( r_m ,q_m )        (11)

Figure 2 is a plot of P_m (in decibels) versus r_m when q = 0^°, and a_1,m = ½. Note that r = 0 corresponds to the receiver being on the central base station, and r = 2, corresponds to the receiver being on the adjacent cell’s base station. r = 1 corresponds to the edge of the cell. Note on Figures 2 through 5! The absolute value of P_m doesn’t mean anything; only it’s value relative to the other power control coefficients is important. The graph here may shift vertically depending on the number of receivers in the cell. As the number of receivers increases, the following figures (2 - 5) will shift down.

Figure 2:

Plot of power control in dB wrt r on [0,1] for q = 0

Figure 3:

Plot of power control in dB wrt r on [0,2] for q = 0

Figure 3 is a plot of P_m (in decibels) versus r_m when q = 30^°, and a_1,m = ½ Note in this case at r = 2/Ö3 » 1.15, the receiver is equidistant to three base stations (see Figure 1).

Figure 4:

Plot of power control in dB wrt r on [0,1.15] for q = 30°

Figure 5:

Plot of power control in dB wrt r on [0,2] for q = 30°

Distribution of Receivers in the Cell

We are assuming that the receivers are distributed uniformly throughout the cell. To simplify calculations, we will assume that the cell is a circle with radius 1 (the circle inscribed in the hexagonal cell), and that q = 0. The distribution function for the radius will then be

fr ( r ) =

ì í î

2r,    0 < r £ 1 0,      otherwise

(12)

Distribution of Power Control

It is very difficult to determine the distribution of power control because the power control is such a complex function of r. Instead, we will approximate the distribution function using Matlab®.

Here are some plots of the power control distribution, with different values of a_1,m.

Figure 6:

Power control distribution with a_1,m = ½

Figure 7:

Power control distribution with a_1,m = 1/10

Figure 8:

Power control distribution with a_1,m = 1/100

Discussion of Results

We can see that when a_1,m = ½, the power control differs only by about 6dB at the extremes. This is not enough to utilize the method described in the introduction. As a_1,m gets smaller, however, there becomes a wide enough range of values to utilize the nonuniform modulation technique. At a_1,m = 1/100, the range is 22dB, and the power control is almost uniform across this range, except for a peak at the lowest power level. This situation would be ideally suited for the nonuniform modulation technique. More research needs to be done to determine which of the models most accurately describes a typical wireless environment.

Bibliography

[1] Michael B. Pursley, and John M. Shea, “Nonuniform Phase-Shift-Key Modulation for Multimedia Mulitcast Transmission in Mobile Wireless Networks”, IEEE Journal on Selected Areas in Communications, Vol. 17, No. 5, May 1999.

[2] Tan F. Wong, Spread Spectrum & CDMA, course notes ch.3, August 2000.

[3] Michele Zorzi, “Simplified Forward-Link Power Control Law in Cellular CDMA”, IEEE Transactions in Vehicular Technology, Vol. 43, No. 4, November 1994.

[4] Bryan Davis, “Analytical Results on Calculating the Number of Pairs in CDMA system using Nonuniform PSK Modulation”, not published.