Chapter 3 Proposed Transmit Power Allocation
3.4 Computer Simulations
In this section, we simulate the proposed power allocation for MU-MIMO systems. All the simulations are measured and averaged over 1000 independent channels. Here M = and 4 N = . Fig. 3-3 shows the PDF of the ratio of 6 minimum rate to maximum rate. We can see that the ratio tend to approach to 1. That also means the proposed power allocation will tend to uniform data rates.
Fig. 3-4 simulates the minimum user data rates versus the reciprocal of channel gains in 1000 channel realizations. It compares the three schemes: waterfilling power allocation, no power allocation, and the proposed fairness power allocation. Since the channel gain
0.7 0.75 0.8 0.85 0.9 0.95
0 2 4 6 8 10 12
Min (R) / Max (R)
1000 Channels, (6,5)
Fairness scheme
Fig. 3-3 The PDF of the ratio of minimum rate to maximum rate
is inversely proportional to the square of distance between the transmitter and receiver, the horizontal axis can be regarded as the user’s distance from the base station. In the waterfilling power allocation scheme, the user may be turned off when the distance is large enough. If we choose the fairness power allocation scheme, the smallest user rate will be larger than the other two schemes. Fig. 3-5 simulates the maximum user rates versus the reciprocal of channel gains in 1000 channel realizations. In the waterfilling power allocation scheme, the user with best channel gain will be allocated with the largest power, thus the achieved rates will be higher than the other schemes.
In the fairness power allocation scheme, we sacrifice the rate of the best user and obtain more fair rates.
Min user rate, 1000 channels, (6,4)
Fairness scheme No power control Waterfilling scheme
Fig. 3-4 Minimum rates versus reciprocal of channel gains
Fig. 3-6 compares the sum rate of four users with different schemes. In the waterfilling power allocation scheme, the utility function is chosen to maximize the sum rate of all users. Thus, the sum rate of the waterfilling scheme is always higher than the other schemes. Although the sum rate of the proposed fairness scheme is lower than the scheme without power allocation, it could obtain fair rates for all users.
0 0.5 1 1.5 2 2.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1/gain
Rate
Max user rate, 1000 channels, (6,4)
Waterfilling scheme No power control Fairness scheme
Fig. 3-5 Maximum rates versus reciprocal of channel gains
-20 -10 0 10 20 30 40
Fig. 3-6 Sum rate comparison for fairness scheme
3.5 Summary
In this chapter we give a detailed description of the proposed transmit power allocation. We reformulate the nonlinear optimization problem, and apply the Interior-point method to solve it. In both SU-MIMO and MU-MIMO systems, after the proposed transmit power allocation, all users will tend to have fair transmission rates. This means that it can prevent the users with small channel gains from suffering poor data rates. We compare the sum rate of the waterfilling power allocation and the scheme without power allocation with the proposed power allocation. It’s a trade-off between the maximum average throughput and user fairness.
Chapter 4
Condition Number Discussion
In Chapter 3, we have introduced the proposed transmit power allocation with proportional fairness rates. This means that the sub-channel gains will be close to each other with the proposed power allocation applied. And the equivalent channel matrix will tend to be a well-conditioned channel matrix. In this chapter, we state that the condition number of the equivalent channel matrix is statistically smaller by observing the simulation results. If an underdetermined MIMO system is well-conditioned, the decoding complexity of the executed SDA will be reduced.
Motivated by the fairness scheme, we propose a determinant based power allocation to further reduce the condition number of the equivalent matrix. Thus the decoding complexity of the underdetermined systems can be reduced with the proposed power allocation applied. In Section 4.1, we explain why the decoding complexity can be reduced with a smaller condition number. The determinant based utility function is provided in Section 4.2. Section 4.3 shows the simulation results. Section 4.4 summarizes this chapter.
4.1 Condition Number Effect
Although SDA can reduce the decoding complexity of ML detection from exponentially increasing to polynomially increasing, its complexity still grows heavily when the condition number of the channel matrix is large. The condition number is traditionally and calculated by taking the ratio of the maximum to minimum singular values of the channel matrix. For MIMO systems, the channel condition number is calculated from the instantaneous channel matrix without the need for stochastic averaging. Small values for the condition number imply a well-conditioned channel matrix while large values indicate an ill-conditioned channel matrix.
Consider an overdetermined MIMO systems with N transmit antennas and M receive antennas. The idea of SDA is to check all the points in a hyper-sphere with radius d . It finds the nearest point from the received signal to be the estimated signal.
That is,
∑ ∑
, start from the last equation and work backward. We expand the last equation asIt means that all the constellation points satisfy (4.2) could be the candidate of x . N
The ith element of x will be bounded by
Fig. 4-1 Tree search example for 4-PAM showing sphere radius, tree levels and detection layers
1 1
N N
ij j ij j
i j i i j i
ii i ii
y d r x y d r x
r x r
= + = +
− − + −
< <
∑ ∑
. (4.3)
From (4.2) and (4.3), we know that SDA is a tree search. Fig. 4-1 is a tree search example for the 4-PAM constellation. We can search from Level 1 to Level N, and the distance for all levels should be less than the radius d. Because SDA decodes the transmit signal from the last layer xN, the boundary of xN should be as small as possible. This will reduce the number of searching points, as well as the decoding complexity.
In [22], we know that the rNN of R depends on the condition number of the channel matrix H . The lower the condition number is, the larger the rNN is. Thus, we can obtain a lower decoding complexity when the system is well-conditioned. Fig.
4-2 shows the CDF of r55 for a 5× matrix with different condition numbers. 5 We can see that when the condition number is less than 10, the value of r55 will tend
to be larger. Fig. 4-3 shows the FLOPS (Floating Point Operation Per Second) of the
SDA versus condition number. The transmitter and receiver are both equipped with four antennas, and the transmitter uses the QPSK modulation. We can see that when the condition number is larger than 5 the complexity increase rapidly. Fig. 4-4 shows the CDF (Cumulative Density Function) of the condition number with different channel correlation. Assuming that there are 1000 channel realizations in a 2× 2 MIMO system. Let R be the correlation matrix of H . A useful measure of the hh degradation in performance due to channel correlation, for a system with K diversity branches, is provided by the Kth root of the determinant of the channel correlation matrix, det(Rhh)1/K[24]. If det(Rhh)1/Kis smaller than 0.5, it can be regarded as high correlation. If det(Rhh)1/K is larger than 0.5, it can be regarded as low
condition number is larger than 10 dB. Fig. 4-4 shows that both channels with low and high correlation are probable to be ill-conditioned channels. It also shows that a MIMO system is more likely to have a lower condition number when the channel has low correlation.
0 2 4 6 8 10 12 14 16 18
6.3 6.4 6.5 6.6 6.7 6.8 6.9 7 7.1x 104
Condition Number
FLOPS
QPSK, (4,4)
SDA
Fig. 4-3 Complexity as a function of condition number
0 5 10 15 20 25 30
Fig. 4-4 CDF of condition number with different channel correlations
4.2 Proposed Utility Function for Condition Number
In Chapter 3, we know that the proposed transmit power allocation results in proportional fairness data rates. That means that the data rate for each user
log (12 SINR )
i i
R = + will be close to each other. The SINR in the log function will also be close to each other. When the data is transmitted from the ith user, the SINR at the receiver can be written as
2
vector of the channel matrix H . At the high SNR, the data rate will be interference-limited. The sub-channel gain will be close to each other with the
proposed power allocation applied. That is,
2 2 2
1 P1 ≈ 2 P2 ≈ ≈ N PN
h h h .
Thus the equivalent channel matrix will tend to be a well-conditioned channel matrix.
From the above concepts, we want to check if the condition number will be smaller with the proposed power allocation applied by computer simulations. We simulate the proposed transmit power allocation in the underdetermined SU-MIMO and MU-MIMO systems. Fig. 4-5 and Fig. 4-6 are the CDF as a function of the condition number. We simulate 1000 channel realizations, and the number of transmit and receive antennas are 6 and 4, respectively. Both SU-MIMO and MU-MIMO systems can obtain a smaller condition number with the proposed power allocation applied. By observing the simulation results, we can state that the condition number of the equivalent channel matrix is statistically smaller with the proposed power allocation applied.
Fig. 4-5 CDF of the condition number for SU-MIMO
2 3 4 5 6 7 8 9
Motivated by the fairness scheme, we want to make the condition number of the equivalent channel matrix smaller. Thus the decoding complexity of the underdetermined MIMO systems can be reduced. From the linear algebra, we know that the determinant of a matrix is the products of all eigenvalues. For example, given a full rank 5× matrix A , and 5 λ λ1, , ,2 λ are the eigenvalues of matrix A . 5 Then det( )A =λ λ1 2 λ5 and trace( )A =λ1+λ2+ +λ5. Thus we propose a power allocation utility function based on the determinant to make the eigenvalues of the equivalent channel matrix close to each other. That will lead to smaller condition number. The optimization problem can be written as
max log det
( (
HP HP( )H) )
subject to trace(
HP HP( )H)
=constant.problem. And then we can apply the interior point algorithm to solve this. Fig. 4-7 is the comparison of condition number for fairness scheme and determinant based scheme. We simulate 1000 channel realizations, and the number of transmit and receive antennas are 6 and 5, respectively. It shows that both schemes can reduce the condition number statistically. The determinant based power allocation scheme leads to smaller condition number than the fairness scheme.
Fig. 4-7 Comparisons of condition number for fairness scheme and determinant based scheme
4.3 Computer Simulations
In this section, we simulate the decoding complexity of the SSD decoder and regularization decoder for underdetermined MIMO systems with the proposed power allocation applied. We also apply the proposed power allocation with the determinant based utility function to simulate the decoding complexity. Here the complexity weights of different operations is determined according to [16]. The numerical results are measured and averaged over 1000 independent channels for various average signal-to-noise ratio (SNR).
5 10 15 20 25 30
103 104
SNR (dB)
Complexity
1000 Channels, (4,3)
SSD SSD-fair SSD-det
Fig. 4-8 Decoding complexity comparison using SSD at receiver. Transmitter has four antennas, and receiver has three antennas.
Fig. 4-8 shows the decoding complexity improvement with the proposed power allocations applied. In this simulation, we use 16-QAM modulation. The transmitter has four antennas and the receiver has three antennas. We choose the geometrical SSD to decode the underdetermined MIMO system. The SSD first finds the candidates in a slab and each candidate is followed by an SDA. Therefore, when the channel condition number is small, the decoding complexity can be reduced. Since the number of candidates in the slab depends on the noise, the larger the noise is, the more the candidate is. The SSD needs to activate more times of SDA at low SNR, thus the decoding complexity can be reduced more than at high SNR.
5 10 15 20 25 30
103 104 105
SNR (dB)
Complexity
1000 Channels, (6,5)
SSD SSD-fair SSD-det
Fig. 4-9 Decoding complexity comparison using SSD at receiver. Transmitter has six antennas, and receiver has five antennas.
Fig. 4-9 shows another simulation of decoding complexity. The transmitter has six antennas, and receiver has five antennas. The other simulation parameters are the same as Fig. 4-8. Because the transmitter and receiver have more antennas, the sizes of the channel matrix become larger. Thus the decoding complexity will increase. The applied SDA in SSD will become a more important role. We can observe that the decoding complexity of SSD can also be reduced by applying the determinant based power allocation at low SNR.
5 10 15 20 25 30
103 104
SNR (dB)
Complexity
1000 Channels, (4,3)
Regularization Regularization-fair Regularization-det
Fig. 4-10 Decoding complexity comparison using regularization method at receiver.
Transmitter has four antennas, and receiver has three antennas.
Fig. 4-10 shows the simulation of decoding complexity of the regularization method. We use 16-QAM modulation in 1000 channel realizations. The transmitter has four antennas, and receiver has three antennas. The regularization method
existing SDA could to applied on the decoding process. Fig. 4-10 shows that the decoding complexity can be reduced with the proposed power allocations applied. Fig.
4-11 shows that the decoding complexity of the regularization method with six transmit antennas and five receive antennas. The other parameters are the same as Fig.
4-10. The numerical result shows that the decoding complexity can also be reduced for a large antenna size with the proposed power allocation applied.
5 10 15 20 25 30
103 104 105
SNR (dB)
Complexity
1000 Channels, (6,5)
Regularization Regularization-fair Regularization-det
Fig. 4-11 Decoding complexity comparison using regularization method at receiver.
Transmitter has six antennas, and receiver has five antennas.
Fig. 4-12 compares the sum rate of four users with different schemes. In the waterfilling power allocation scheme, the utility function is chosen to maximize the sum rate of all users. Thus, the sum rate of the waterfilling scheme is always higher than the other schemes. Although the sum rate of the proposed fairness scheme is
lower than the scheme without power allocation, it could obtain fair rates for all users.
With the determinant based power allocation, the condition number of the equivalent channel matrix can be reduced, but the sum rate will be smaller than the sum rate without power allocation.
Fig. 4-12 Sum rate comparison for determinant based schemes
4.4 Summary
The condition number of the channel matrix is a critical factor for decoder design in underdetermined MIMO systems. Most efficient decoding algorithms of
underdetermined MIMO systems work with SDA, and thus are sensitive to the condition number. Simulation results show that the fairness power allocation can reduce the condition number of the equivalent channel matrix. Motivated by the fairness scheme, we propose a determinant based utility function to make the eigenvalues close to each other. Thus it can reduce the condition number effectively.
Simulations show that the decoding complexity of underdetermined MIMO systems can be reduced with a smaller condition number. Although the decoding complexity can be reduced in underdetermined MIMO systems, the sum rate will be smaller as a price.
Chapter 5
Conclusions and Future Works
In the beginning, this thesis reviews the development of MIMO systems, and the channel capacity of the MIMO systems is introduced. Spatial diversity and spatial multiplexing are two main techniques used in MIMO systems, and can improve the system performance. Several decoding algorithms are proposed for underdetermined MIMO systems. These decoders are developed based on the SDA. Considering uplink MU-MIMO systems, the channel gains will be small when the users are far away from the base station or blocked by obstacles in practical environments. This results in poor data rates for those users. Our goal is to achieve the fair rates for all users.
Since linear precoding techniques cannot be used in the uplink MU-MIMO, we propose a transmit power allocation scheme to achieve the goal. In Chapter 3, we give a detailed description of the proposed transmit power allocation in the MU-MIMO systems. The nonlinear optimization problem is reformulated into a modified form.
Thus we can apply the Interior-point algorithm to solve it. The proposed power allocation can also be applied in the SU-MIMO systems. Simulation results show that the fairness based power allocation could provide a higher data rate to the worst condition user than the waterfilling power allocation. This demonstrates a trade-off between the maximum sum rate and user fairness.
The discussion of condition number and the determinant based utility function are given in Chapter 4. If we use the SDA at the receiver, the decoding complexity can be reduced with a smaller condition number. The numerical results show that the fairness scheme tends to obtain a smaller condition number of the equivalent channel matrix. Thus the decoding complexity of the underdetermined MIMO systems could be reduced with the proposed power allocation applied. Motivated by the fairness scheme, we propose a determinant based utility function to reduce the condition number, further reducing the decoding complexity.
The main contributions of this thesis are as follows. We propose a transmit power allocation scheme which provides fair data rates. It improves the poor data rates for users in unfavorable situations. By reformulating the nonlinear optimization problem into a modified form, we can apply the typical Interior-point algorithm to solve it. We observe that the fairness scheme could lead to a small condition number of the equivalent channel matrix, and propose a determinant based utility function trying to equalize the eigenvalues. The decoding complexity of the SDA based decoder for underdetermined MIMO systems can thus be reduced with the proposed power allocation applied.
There are still some issues remaining to be discussed in this work. First, how to find the mathematical relation between the fairness scheme and condition numbers is a concern. Also, how to tackle with imperfect CSI at the receiver is an interesting topic. Furthermore, finding a precoding matrix to minimize the channel condition number is an important subject worthy of investigation.
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