Convergence Analysis

In this section, the convergence of the algorithm shown in Table II is demon-

strated. Consider an arbitrary iteration number n. First consider the in-

ner repeat loop (3) for the design of transmitter and receiver filters. Step

3c returns the receiver filters R(n)_k ,∀k, which were aimed at minimizing the

weighted sum-MSE for given transmit filters T(n_k −1),∀k, relay matrix W(n−1)

and a total power limit Pmax. Let the minimum weighted sum MSE obtained

at this stage ϵ(n,step3c).

At step 3d(i) and the design of the transmit filters U(n)_k ,∀k, the MSE

duality ensures that the same weighted sum MSE ϵ(n,step3c) can be achieved

with the same total transmit power Pmax in the downlink. In step 3d(ii),

using these transmit filters U(n)_k ,∀k, the receiver filters V(n)_k ,∀k, have been

designed so that the minimum weighted sum-MSE value should be further

reduced, i.e. ϵ(n,step3d(ii)) ≤ ϵ(n,step3c). In step 3d(v), the transmit filters Tn

k,∀k and the same receiver filters

Rn_k,∀k achieve the same weighted sum-MSE, ϵ(n,step3d(v)) = ϵ(n,step3d(ii)).

For the subsequent iteration n+1, step3c minimizes further the weighted

sum-MSE by choosing optimal R(n+1)_k ,∀k for given T(n)_k ,∀k, ϵ(n+1,step3c) ≤

ϵ(n,step3d(v)).

As a result, the weighted sum-MSE value monotonically decreases at

each iteration. However, since there is a total power limit, the weighted

sum-MSE ϵ converges to a limit as n → ∞.It was shown so far that for the inner-loop, at each iteration the sum-MSE decreases monotonically and

converges. Let us now look at the outer loop for the design of the relay.

At step 4, for a given set of transmit filters U(m_k −1),∀k and receiver filters

V(m_k −1),∀k, the optimal relay filter W(m)_{,∀k is chosen so that the sum-MSE}

value is reduced further, i.e. ϵ(m,step4) ≤ ϵ(m−1,step4). Since there is a relay power constraint, the weighted sum-MSE value ϵ must converge to a limit

Section 4.7. Simulation Results 116

as m→ ∞. This concludes the proof that the weighted sum-MSE decreases

monotonically and the iterative algorithm converges.

However, since the overall problem (4.1.8) is not convex, the global op-

timality of Algorithm I cannot be guaranteed. In general, different initial-

izations may affect the convergence speed and the minimum weighted sum-

MSE value of Algorithm I. The initial values of transmit filters Tk,∀k can

be chosen as matrices containing the right singular vectors of the channels

for better convergence instead of choosing random matrices.

4.7 Simulation Results

A number of simulations was performed to characterize the performance of

the proposed scheme under various scenarios. In all simulations, the ele-

ments of the channel matrices Fk, Hk and Gk are assumed to be circularly

symmetric complex Gaussian random variables with zero mean and unity

variance per dimension. Also, in all simulations, we have assumed two peer-

to-peer users, each employing two antennas at the transmitter and receiver,

hence allowing two simultaneous data streams for each user.

For the first simulation, the total power at the transmitters was set to

unity and the power available at the relay to 10W. A set of random channels

was generated and the achieved MMSE was computed for different num-

ber of antennas at the relay as shown in Figure 4.4. The results have been

shown for two different values of noise variances 0.01W and 0.001W. Both

relay and user terminals have identical noise variances. To generate results

for different number of relay antennas, the simulation started by choosing

random channel matrices Fk, Hk and Gk for the case of six antennas at the

relay and then the rest of the simulations has been performed by reducing

Section 4.7. Simulation Results 117 2 3 4 5 6 7 8 10−3 10−2 10−1 100 101

Number of antennas at the relay

Sum mse

ó2=0.01

ó2=0.001

Figure 4.4. The sum MMSE against number of antennas at the relay

for a network with 2 users on both sides equipped with 2 antennas each.

PT=1W, PRmax=10W

tennas required. The stopping criterion for the iteration was ξ1, ξ2 = 0.005

and approximately four to eight iterations have been observed. As seen in

Figure 4.4, the MMSE value decreases as the number of antennas at the

relay increases. As there are two users transmitting two data streams each,

the relay needs at least four antennas to perform satisfactory spatial mul-

tiplexing, as such the MMSE value drops significantly beyond the use of

four antennas. When there is adequate number of antennas at the relays,

the achieved MMSE is of the order of the variance of the noise, confirming

satisfactory performance of the proposed iterative method.

According to the proposed iterative method, the relay matrix had to

be initialized when designing the transmitter and receiver filters in the first

iteration. For the above simulation, the relay matrix W has been initialized

with randomly generated circularly symmetric complex Gaussian variables.

However, as the overall problem is non-convex, the algorithm is not expected

Section 4.7. Simulation Results 118 0 0.02 0.04 0.06 0.08 0.1 0 2 4 6 8 10 12

Minimum Mean Square Error

Frequency of Occurrence

Noise Variance = 0.001

Noise Variance = 0.01

Figure 4.5. The histogram of the final sum MMSE due to different

random initializations of the relay matrix. The results are shown for two different noise variances 0.01W and 0.001W.

order to investigate the susceptibility of the algorithm for different initial-

izations, the above simulations have been performed for 25 different random

initializations and the histogram of the final sum MMSE values obtained has

been plotted. The number of antennas at the relay is five. The histograms

are shown separately for two different noise variances 0.01W and 0.001W.

As seen in Figure 4.5, the final MMSE values differ only slightly for different

random initializations of the relay matrices. Moreover, the higher MMSE

values occur only with small probabilities.

Finally, the sum MMSE for various values of SNR has been computed.

The simulation scenario was set the same as before, i.e. two users, two an-

tennas at the transmitter and receiver and five antennas at the relay. The

SNR (SNR1) of the link from the transmitters to the relay has been set at

30dB and the SNR (SNR2) of the link between the relay and the users has

Section 4.7. Simulation Results 119 15 20 25 30 35 40 10−3 10−2 10−1 100 SNR in dB

Averaged sum MMSE

sum MMSE of user 1 and 2 sum MMSE of user 1 sum MMSE of user 2

Figure 4.6. The sum MMSE against SNR of the relay-user terminal

link, averaged over 10 set of random channels. The results are shown for sum MMSE of both users and sum MMSE of user 1 and user 2. SNR of transmitter-relay link was set to 30dB.

power and noise variance at the relay and SNR2 as the ratio between the

relay power and the noise variance at the user terminals. To obtain 30dB

for the first link (SNR1), the transmitter power was set to unity (W) and

the noise variance at the relay to 0.001 W. In order to change SNR2, the

relay power has been varied but the noise variance has been set at both user

terminals to 0.001W. Ten monte-carlo simulations have been performed for

various random channels and averaged the sum MMSE values. The average

sum MMSE of all users as well as averaged sum MMSE of each users (i.e.

summed over multiple data streams for each user) is depicted in Figure 4.6.

As seen in Figure 4.6, the sum MMSE decreases as the SNR increases. More-

over, even though sum MMSE has been considered as the design criterion,

both users attain more or less the same averaged sum MMSE values over a

Section 4.8. Conclusion 120

4.8 Conclusion

Based on the derivation of the mean square error uplink-downlink duality

for a MIMO peer-to-peer relay network, a transmitter, receiver and relay de-

sign technique has been proposed. The algorithm was based on an iterative

method and second order cone programming. The simulation results demon-

strate satisfactory performance in term of achievable mean square error and

Chapter 5

A COORDINATED