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1318 PIERS Proceedings, Kuala Lumpur, MALAYSIA, March 2730, 2012

induced in the receiving antennas. Each antenna is designed with the intrinsic impedance of 50 and by multiplying that with the induced currents, the induced voltages across each antenna canbe found. This allows us to obtain the channel matrices, H1 for user 1 and H2 for user 2. Withthe channel matrices, the sum-rate capacity of the system can be calculated.

2. SYSTEM MODEL

We consider a narrowband MU BC MIMO system with T transmitting antennas and Rj = Rk >

1 receiving antennas. For simplicity, the receiving antennas are named as r-th antenna, r =1, 2, . . . , R, as shown in Fig. 1. We denote the signal voltages at the input ports of thet-th transmitantennas as Vin,1, Vin,2, . . . , V in,T, and the signal voltages received at the output ports of the r-threceive antennas as Vout,1, Vout,2, . . . , V out,R. The antenna terminal loads for the transmitting andreceiving antennas are denoted by ZL. The MIMO channel response can then be defined as

hij = Vout,i

Vin,j,

i= 1, 2, . . . , Rj = 1, 2, . . . , T

. (1)

From (1), the channel matrix can then be defined as

H=

h11 . . . h1T...

. . . ...

hR1 . . . hRT

. (2)

The channel matrix of each user can be found, and in the case of two users, two channel matricesH1 and H2 are obtained. With the channel matrices, the sum power iterative water filling methodcan then be used to calculate the sum-rate capacity [7]. Before moving on to discuss the steps thatthat are taken for the simulations, several assumptions have to be made as follows:

1) The transmitter and receiver are assumed to be separated by a large distance D [11].2) The systems are narrow-band and stationed in flat-fading environments [11].3) The transmit power that the transmitter used is assumed to be 10 dB.4) The two users do not communicate with each other, i.e., non-cooperative.5) There are no scatterers present in the region that is between the users.

A clearer definition of the antenna spacing and user distance is shown in Fig. 2.Assume a 2 2 2 systems which has two transmitting antennas and two receiving users with

two antennas each, the exact steps taken for the simulation are listed below

1) Using a unit voltage source Vin,j excite the j -th transmitting antenna.

Figure 1: The system model used in this paper. It has the transmitters on the left and two users of twoantennas each on the right. The users are separated by a parameter known as user distance. The scatterersexist in the far field and provide the scattering events. In between the transmitter and receivers, the signalsgo through propagation paths that are associated with a loss factor.

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Progress In Electromagnetics Research Symposium Proceedings, KL, MALAYSIA, March 2730, 2012 1319

Figure 2: This picture showing the definitions of antenna spacing and user spacing that are used in thispaper.

2) The electric far field of this antenna Ej,p is numerically determined in all the directions of thescatterers around the transmitter.

3) Each value of the far field is multiply by an independent complex Gaussian random variablewhich has a zero mean and a unit variance. The resultant far fields from all the scatterers arethen added together. This total sum is denoted as S.

4) The i-th receiving antenna is excited with a large number of plane wave sources. The number

of plane wave sources is similar to the number of scatterers around the receiver. Each of theseplane waves will be multiply with Sthat was obtained in Step 3.

5) Obtain the load current induced when excited by all the plane waves obtained in Step 4.With the load current, the voltageVout,i that is developed across the terminal loadZL can bedetermined numerically.

6) As discussed above, the channel response hij which is given by Vout,i/Vin,j can be obtained.This corresponds to a single channel realisation.

7) Repeat Step 1, expect that the unit voltage source is now applied across the (j+ 1)-th trans-mitting antenna. This will continue until the T-th transmitting antenna is being simulated.

8) With all the channel responses of the system, the channel matrices H1 and H2 can thus becalculated and obtained. H1 is obtained by considering the channel responses of Antenna 1and 2 at the transmitter and Antenna 3 and 4 at the receiver. Similarly, H2 is obtained by

considering 1 and 2 at the transmitter and Antenna 5 and 6 at the receiver.9) H1 and H2 are then used to calculate the sum capacity using the sum power iterative waterfilling algorithm

3. CAPACITY SIMULATION

3.1. Normalization

Normalization is needed to remove the actual distances of the scatterers from the transmittingantennas and from the receiving antennas and the path loss for wave propagation from the trans-mitter to the receiver. The normalization method that will be used is to [11] calculate the averagepower received by a single isolated antenna when all the transmitting antennas are transmittingin the same rich multipath environment. All the channel matrix elements are then divided by thisaverage power.

In subsequent simulations, the antennas are designed with the following parameters. Eachmonopole is designed to be of length 29.8 mm and the operating frequency is 2.4 GHz. The antennaload impedance is fixed at 50 .

3.2. Simulation Results 4 2 2 System

The first simulation is a 4 2 2 system, which has four transmitting antennas transmitting andon the receiving side, there are two users which two antennas each. In order to characterize theMU MIMO system, we vary two parameters which are the antenna spacing and the user spacing.The number of scatterers at both sides is kept at 50. Table 1 summarizes the cases that we areconsidering. Case 1 allows the investigation of antenna spacing on the sum-rate capacity. In case2, by keeping the antenna spacing constant, the effect of user spacing can be examined.

In case 1, the user spacing is kept at 1to remove correlation due to user spacing. This retainsthe impact of antenna spacing on sum-rate capacity. In addition, by keeping the user spacing toa 1, it prevents the error arising from the lack of considerations for scatterers that are present

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in between the two users. For case 2, the antenna spacing is kept at 0.4 in order to minimizethe correlation effect due to the proximity of surrounding antennas. When the antenna spacing isvaried from 0.05 to 0.4, the following sum-rate capacity is obtained. When the user spacing isvaried from 0.05to 0.7, the following sum-rate capacity is obtained.

In both cases, it can be seen that as the antenna spacing and user spacing increases, the sum-rate capacity increases. This is similar to the case of the Single User MIMO Systems. The mainreason for the increase in capacity is that when the antenna or user spacing increases, the amount

of correlation between the antennas decreases. This reduces the mutual coupling effect and bringsabout a higher capacity. When the mutual coupling effect is insignificant, the simulated sum-ratecapacity approaches the i.i.d case. In both cases, the distance between the antenna and the userspacing is found to be 0.4in order for the sum-rate capacity to converge.

3.3. Simulation Results 6 3 2 System

In this section, the number of antenna elements is increased. This will allow a deeper insight tothe changes of the sum-rate capacity when the number of antenna elements increases. The systemsimulated is 6 3 2 which has six transmitting antennas and the receiving side has two users of

Table 1: Simulation parameters for two cases.

Simulation Cases Antenna Spacing User spacing

1 Varied from 0.05 to 0.5 Kept at 1

2 Kept at 0.4 Varied from 0.05to about 0.7

Figure 3: This graph shows the effect of differentantenna spacing on the sum-rate capacity. The userspacing is kept constant at 1.

Figure 4: This graph shows the effects of varyinguser spacing on the sum-rate capacity. The antennaspacing is fixed at 0.4.

Figure 5: The graph shows the effect of differentantenna spacing of the MIMO array have on thesum-rate capacity when the number of antennas in-creases. The user spacing is kept at 1.

Figure 6: This graph shows the effect of differentuser spacing of the MIMO array have on the sum-rate capacity when there are more antenna elements.The antenna spacing is kept constant at 0.4.

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