Total transmit power: The graph in Figure 3.8 is also proved that the proposed

SPO2 technique has produced the optimal value approximately with the other two

techniques.

- Average calculation time: Figure 3.9 shows that the proposed technique "SPO2"

has the calculation time faster than the "SPO1" technique from 2 to 7 times.

Specifically, Table 3.5 shows that the rate of calculation time of SPO1 compared

with SDR is from 0.52 to 1.05 times, while the ratio of calculation time between

SPO2 and SDR techniques ranges from 0.19 to 0.29 times. The ratio of calculation

time of SPO2 technique to SPO1 is from 0.15 to 0.56. Thereby, the time for

calculating comparisons between SPO2 and SPO1 techniques has decreased from

44% to 85%.

- Number of average iterations: Figure 3.10 shows the number of average

iterations of SPO1 and SPO2 techniquefor in each SINR threshold. It can be seen

that the proposed SPO2 technique only needs the number of iterations less than 5

and maintains no more than 5 when SINR changing from 2 dB to 10 dB. As can be

seen from Table 3.6, when the SINR threshold increases, the average number of

iterations between SPO2 and SPO1 techniques decreases from 0.54 to 0.21 times.

That means the average number of iterations of the SPO2 optimization technique

has decreased from 46% to 79% as the SINR increased.

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C
min f
X
X (1.46)
subject to
( ) 1rank X (1.47)
where X is a symmetric matrix PSD and f(X) is convex function. The constrain
condition (1.47) of the problem (1.40) is a non-convex function, the problem
(1.46) is the optimal-NP-hard problem. Condition (1.47) can be transformed into
xe( ) ( ) 0matrac X X (1.48)
where
x ( )ma X is the largest eigenvalue of matrix X. If xe( ) λ ( )matrac X X then
xe( ) ( )matrac X X , this means that only one unique eigenvalue of X satisfies:
x x x( )
H
ma ma maX λ X x x (1.49)
Where
xmax is the unit eigenvalue vector x( 1)ma x of X corresponding to the
maximum eigenvalue
x ( )ma X . Because x ( )ma X is a convex function on the set of
the Hermitian matrix, therefore, constrain condition
xe( ) ( )matrac X X is concave
function. In order to constraint condition to satisfy (1.48),
xe( ) ( )matrac X X
should be small enough. Based on the theory of the penalty function, the problem
(1.46) is put into the form of problem using the penalty coefficient:
N N ma0 C
( ) [ ( ) ( )]xmin f μ trace λ
X
X X X (1.50)
in which the penalty value 0 should be determined as appropriate to achieve a
condition
xe( ) ( )matrac X X small enough. To solve the problem (1.50), it is
necessary to optimize both components ( )f X and
xe( ) ( )matrac X X . Regarding
the speed of convergence during the process of formulating algorithm, the choice
of coefficients and the steplength of penalty factors after each iteration will affect
the convergence rate.
1.7. Computational complexity of the optimization problem
In fact, the minimum power problems for wireless transmission models presented
in section 1.4 are combined optimal problems belonging to non-deterministic
polynomial-time hard (NP-hard) problem. Problems with a small number of
operations can be solved by directly search methods. Problems with great
complexity using previously proposed optimization techniques often cannot find
out the optimal solution or the determined value has a large tolerence with the real
optimal value. The complexity of the problem may be the number of calculations
(including the number of memory accesses, or write to memory), but it also can be
the total program execution time or the capacity of memory that needs to be
allocated to run the program (spatial complexity). The procesing time for
computer in algorithm performing is not only depending on the algorithm but also
computer’s configuration. In order to evaluate the effectiveness of an algorithm,
considering the number of calculations that must be processed by algorithm
running. The effectiveness of an optimal technique is considered the optimal
algorithm of construction techniques in many aspects:
1. Effective application on problems: convex / non-convex problem, constraint/
non-constraint.
2. The speed of convergence of the algorithm: calculation time/ number of
iterations.
3. Algorithm's solution quality: The results are obtained through stopping
criterium or local/global solutions of the cost function.
The problems of optimal transmit power in multi-antenna wireless transmission
network belong to the NP-hard problem. Therefore, in order to analyze the
computational complexity of algorithms, this section we use the complexity
analysis formula of SDP problem. SDP optimization problem with constrain
condition is stated follow
( )
NR
min trace
x
CX (1.47)
subject to
( ) 0i itrace , , i = 1, 2. . . , M. A X b X (1.48)
where C and Ai are symmetric matrices, then the complexity of the problem is
determined: ( log log( ))n n n/ε , with ε is the parameter of the algorithm. In order
to estimate the complexity of optimization problem SDP, it is necessary to
determine the array parameter n of matrices in the object function and constraint
condition. Number of calculations depend on the size of the problem,. In the
survey models of the thesis, the complexity of problem depend on: the number of
transmit antenna at base station, number of antennas at the relay node, number of
transmitters and receivers, SINR threshold.
1.8. Conclusion of chapter 1
Research, development and apply effective optimization techniques for solving
minimum power transmission problems in multi-antenna wireless transmission
network with great scientific and practical significance. In particular, the penalty
function technique is selected to determine the minimum transmit power while
improving the convergence speed compared with random, SDR optimization
techniques,. Based on the outstanding issues, the thesis proposes sollutions for the
following chapters:
1. Improvement the calculation speed for the minimum power transmission
problem for the base station wireless transmission model by minimum the penalty
coefficient.
2. Improvement the calculation speed for the minimum power transmission
problem for the transitional wireless transmission model with AF processing by
adding additional variables in the penalty function of the object function.
The results in chapter 1 have been published in scientific papers [3].
CHAPTER 2. IMPROVEMENT THE CALCULATION SPEED FOR
MINIMUM TOTAL TRANSMIT POWER AT BASE STATION
2.1 Minimum problem of transmit power at base station
2.2. Formulating transmit power minimum problem at base station
The multi-antenna base station transmission broadcasting multipoint model is
shown in Figure 2.2. In particular, the base station is equipped with N transmit
antennas that broadcast the same information (without interference) to the
destination M users.
Figure 2.2: Model of multi-antenna base station transmission
broadcasting multipoint
where 1 2[ ]
T
i i i iNh ,h ,...,hh is the channel vector between the base station and the i
th
user. The problem need to design an optimal weight vector x at the base station
with the signal xS that transmitted to all users on the receiver. Then, the total
transmit power of the antennas at the base station is determined by
( )HTP trace xx (2.1)
The objective of the problem is to find out optimal vectors x that to minimized
total radiated power of the antennas at the base station with SINR constraint
condition greater than the threshold i . The signal to noise ratio at the i
th user is
determined by
2
( )
;
H
i i
i
H
i itraceSINR i 1, 2, .., M
xxh h
(2.2)
with 2i is the additive white Gaussian noise of the i
th user on the receiving side.
The problem (2.6) is converted to the SDR problem through transformation
HX x x , Hi i iH h h with X is a PSD matrix. The problem of minimum transmit
power at the base station is determined by
( )
NC
min trace
X
X (2.3)
subject to:
2
( )i
i i
i
Htrace
SINR , i 1, 2..., M.
HX
(2.4)
HX xx (2.5)
To solve the problem (2.3), it is necessary to add the constraint condition for X
satisfying the condition of rank (X) = 1. So the problem (2.3) is stated as follow
( )
NC
min trace
X
X (2.6)
subject to:
2
( )i
i i
i
Htrace
SINR , i 1, 2..., M
X H
(2.7)
HX xx (2.8)
( ) 1rank X (2.9)
If problem (2.7) determines
optX that satisfied the condition ( ) 1rank X then
optX = xoptxopt
H is considered the optimal value of the problem. In case matrix
optX
is analysed in SVD form to determine the optimal vector xopt.
2.3. Nonsmooth optimization technique used penalty function
Constrain condition of the problem (2.7) can be expressed:
max( ) 0trace X X (2.12)
If trace(X) λmax (X) is always true for all values 0X , then (2.12) becomes
max( )trace X X (2.13)
This means that only a specific value of X other than 0 satisfies the condition
max max max
HX X x x (2.14)
where
xmax is the eigenvector of X corresponding to the maximum eigenvalue
x ( )ma X . Based on (2.12), the problem (2.7) is presented by
0
( )
N NC
min trace
X
X (2.15)
subject to
2
( )i
H
i i
i
trace
SINR , i = 1, 2,..., M.
HX
(2.16)
max( ) 0trace X X (2.17)
When trace(X) - λmax (X) is small enough, then max max max
HX X x x , 1/2x x( )ma ma X x
satisfies the constraint condition SINR of the problem (2.7). Therefore, the target
of the problem should be optimally implemented so that trace(X) − λmax (X)
reaches the minimum value. Under these conditions, using the penalty function
technique combined with the objective function, then the problem (2.15) becomes
max
0
( ) ( ) ( )
N NC
min f trace trace
X
X X X X (2.18)
subject to
2
( )i
H
i i
i
trace
SINR , i = 1, 2,..., M.
HX
(2.19)
On the other hand, the component gradient is x x
H
ma max x , by using mathematical
properties:
max max max max( ) ( ) (( ) ), 0
H Htrace Y X Y X x x Y (2.20)
Based on the repeatability property, if X (k) is the optimal value of the kth iteration
of the problem (2.18) with the largest eigenvalue λmax(X(k)) corresponding to the
eigenvector X(k) . Then, set up the problem (2.21) at the k + 1 iteration,:
ma
( ) ( )
x
)
0
( ) ( H( ) ( ) ( ) (( ) )
N N
k k kH
C
kmin trace trace trace
X
X X X xxX X (2.21)
subject to
2
( )i
H
i i
i
trace
SINR , i = 1, 2,..., M.
HX
(2.22)
become an SDP problem as (2.23)
( ) ( )
0
H( ) ( ) ( )
N N
kH k
C
min trace trace trace
X
x xX X X (2.23)
subject to
2
( )i
H
i i
i
trace
SINR , i = 1, 2,..., M.
HX
(2.24)
When applying the Nonsmooth non-convex optimization technique, the
penalty function needs to be performed in two stages to solve the problem (2.21).
Specifically, at the initialization stage, choose an appropriate set of values ( ,
(0)
X ). Proceeding to determine the optimal value (k 1)X that satisfied the
constraint condition and given value through solving the problem (2.21). The
optimal stage selects the optimal value (k 1)X and fixes the value of µ from the
initialization stage, continues to solve the problem (2.21) for determining the Xopt
that satisfied the constrain condition. The key steps of algorithm for Nonsmooth
technique combined penalty function approach are outlined in Algorithm 1.
Algorithm 1
------------------------------------------------------------------------------------------------
% Initialization stage:
- Initial step: Initialize proper and (0)X to satisfy (2.21), set k = 0
- Step k: Solve (2.21) to obtain its optimal solution (k 1)X
+ if ( 1)
xe( ) ( )
k
matrac
X X (rank-one solution found) then
Reset (0) ( 1): kX X . Terminate, and output and (0)X .
+ else if ( 1)
xe( ) ( )
k
matrac
X X (no improved solution found, no rank-one
result)
then
Reset : 2 and return to the initial step.
else
Reset : 1k k and ( ) ( 1)k kX X for the next iteration.
end if
% Optimization stage:
Set k = 0. Solve (2.21) to obtain its optimal solution ( 1)kX .
if ( 1) ( )e( ) e( )k ktrac trac X X then
Terminate, and output
( 1)k
opt
X X .
else
Reset : 1k k and ( ) ( 1)k kX X Continue to the next iteration.
end if
Output the final solution for problem (2.21).
-----------------------------------------------------------------------------------------------
Many previous publications when applying algorithm 1, the authors often choose
µ empirically but they did not calculate specifically. There are many cases where
the optimal value is determined, or some cases where the optimal value cannot be
determined because selections does not proceed to the convergence area. There
always exists a value so that the problem (2.19) will be finished after only a few
iterations. The value is determined by:
0
1
Tmax
ν
λ ν (2.25)
However is very difficult to be found as the dual Lagrangian formulation of
(2.25) is also NP-hard. Hence the idea of finding the optimal value after only one
iteration is almost impossible. Fortunately, if is obtained by the Lagrangian
relaxation of (2.25) it is still useful as this value will help reduce significantly the
number of iterations in NSM method. The Lagrangian relaxation of (2.25) is
established as,
2
0
1
M
i i i
i
= max
λ (2.26)
subject to
( ) 0,HN i i i = 1, 2,..., M. I h h (2.27)
The key steps of our newly-devised approach are outlined as follow:
Step 1: Obtain (0)X from (2.21) and 0 from (2.26).
Step 2: Solve (2.21) and update with very small amount. The solution
is also updated ( 1) ( )k k X X .
Step 3: When a rank-one solution is found, stop updating µ. X(k) is still
updated and converges after very few iterations.
2.4. Simulation algorithm performance
2.4.1. SDR optimization algorithm performance
2.4.2. Random optimization algorithm performance
2.4.3. NSM1 optimization algorithm performance
2.4.3. NSM2 optimization algorithm performance
2.5. Numerical Results
• Simulation 1
Simulation 1 compares SDR, random optimization techniques with Nonsmooth
techniques (NSM1 and NSM2) with two criterias: the minimum total transmit
power and the number of average iterations. The simulation parameters are shown
in Table 2.1.
Table 2.1: Simulation parameters 1
No Parameter Value
1 The number of users M on the receiver 16, 24
2 The number of antenna N at base station 8
3 The number of iterations for determining penalty
factor µ
30
4 The number of iterations at the optimal stage 20
5 Number of iterations ITE at each SNR threshold 500
6 SNR threshold (dB) 2, 4, 6, 8, 10
7 Initialized power value Pw for random engineering 2000
8 Stopping criterium 1 for NSM1 optimization
technique
10-6
9 Stopping criterium 2 for NSM2 optimization
technique
10-6
10 Penalty factor µ in the initialization stage for NSM1
optimization technique
0.5
11 The steplength of penalty coefficient after each loop
changes according to the exponential rule for NSM1
technique
µ(k+1) = 1,5µ(k)
- Total transmit power: Observing the simulation results in Figure 2.9 and Figure
2.10 shows that the Nonsmooth technique combined with the penalty function
results closer to the lower boundary of SDR and better than the random technique.
In particular, the results of the total minimum transmit power of NSM1 and NSM2
techniques are approximately value(the graphs of NSM1 and NSM2 overlap). In
the case of Figure 2.9, when the SNR threshold is less than 4 dB, the optimal
techniques give approximately results, but when the SNR is greater than 4 dB, the
effectiveness of the Nonsmooth technique with the penalty function has been
demonstrate advantages. This is also evident in the results of the graph of Figure
2.10 when increasing the number of users (M = 24). Detail data is shown in Tables
2.2 and 2.3.
- Number of average iterations: Figures 2.11 and 2.12 describes the evaluation
results, comparing the number of average iterations of the NSM2 technique with
NSM1. Specifically, in the case of M = 16 and N = 8 when the threshold SNR ≤ 2
dB, the graphs of NSM1 and NSM2 in Figure 2.11 show that the number of
average iterations of the two cases is the same. Observing the results in the case of
Figure 2.12, the number of average iterations of the NSM2 technique remains
stable around the value of 5 times. However, the NSM1 graph at the threshold
SNR 6 dB, the number of average iterations tended to decrease because the
number of average iterations in the simulation to get the average result was not
enough (about a few hundred times). Simulation results do not reflect the theory.
Remark: Through the results in simulation 1, it is confirmed that the advantages of
Nonsmooth combined with penalty function technique compared to the random
Figure 2.9: Comparison of total transmitted
power in the case of M = 16, N = 8
Figure 2.9: Comparison of total transmitted
power in the case M = 24, N = 8
Figure 2.11: Comparison of average
number of iterations between NSM1 and
NSM2 techniques (M = 16, N = 8)
Figure 2.11: Comparison of average number
of iterations between NSM1 and NSM2
techniques (M = 24, N = 8)
optimization technique and move towards optimal value of SDR technology. The
proposed optimization technique NSM2 has a fast convergence compared to the
NSM1 optimization technique. However, the number of iterations ITE run for
each SNR threshold is not large enough. That is reason why transmit power and
the number of average iterations does not reflect the advantages of the proposed
optimization technique.
• Simulation 2
In order to demonstrate the effectiveness of the proposed NSM2 optimization
technique, we conducted a simulation where the number of antennas at the base
station N = 8 and the number of users in destination varied among the three cases:
M = 16, M = 24, M = 32. In particular, increase the number of iterations for each
SNR threshold is 1000 times. The simulation parameters are shown in Table 2.4.
Table 2.4: Simulation parameters 2
No Parameter Value
1 The number of users M on the receiver 16, 24, 32
2 The number of antenna N at base station 8
3 The number of iterations itex for determining
penalty factor µ
30
4 The number of iterations at the optimal stage 20
5 Number of iterations ITE at each SINR threshold 100
6 SNR threshold (dB) 2, 4, 6, 8, 10
7 Initialized power value for random engineering 2000
8 Stopping criterium 1 for NSM1 optimization
technique
10-6
9 Stopping criterium 2 for NSM2 optimization
technique
10-6
10 Penalty factor µ0 in the initialization stage for
NSM1 optimization technique
0.5
11 The steplength of the penalty coefficient after
each loop changes according to the exponential
rule for NSM1 technique
µ(k+1) = 1,5 µ(k)
- Total transmit power: Observing the results on the graph in Figure 2.13 and
detailed data in Tables 2.5, 2.6, 2.7 shows that the minimum power value of the
proposed NSM2 technique is better in all cases at each SNR threshold. Based on
the calculation of the power ratio in Table 2.8, it was shown that when increasing
the number of users (M = 32), the transmit power ratio of PNMS2/PNMS1 fluctuated
stably from 0.93 to 0.95. When M = 16 and M = 24 the PNMS2/PNMS1 ranges from
0.96 to 1.0. Thus, the total transmit power has had good results when using the
optimal technique proposed NSM2.
Table 2.8: Transmit power ratio of PNMS2 / PNMS1
SNR (M,N) = (16,8) (M,N) = (24,8) (M,N) = (32,8)
2 dB 1.00 0.97 0.93
4 dB 0.98 0.97 0.94
6 dB 0.98 0.97 0.94
8 dB 0.98 0.96 0.94
10 dB 0.97 0.96 0.95
Table 2.12: Number of average iteration ratio ITENSM2/ITENSM1
SNR (M,N) = (16,8) (M,N) = (24,8) (M,N) = (32,8)
2 dB 0.26 0.30 0.27
4 dB 0.27 0.27 0.29
6 dB 0.23 0.22 0.24
8 dB 0.22 0.21 0.24
10 dB 0.21 0.20 0.21
- Number of average iterations: The graph in Figure 2.14 and detailed data in
tables 2.9, 2.10, 2.11 also demonstrate that the proposed NSM2 optimization
technique has decreased significantly when compared to the NSM1 optimization
technique. Specifically, from the graph in Figure 2.14, the number of average
iterations of the proposed technique is reduced by 3.5 to 6 times and reaches a
steady value when changes SNR threshold. Table 2.12 shows the number of
average iterations of ITENSM2/ITENSM1 between NSM2 and NSM1 techniques.
These data have shown that the ratio of ITENSM2 / ITENSM1 varies from 0.21 to
0.30 for the cases. In particular, in case of M = 32, the number of average
iterations of the NSM2 optimization technique has decreased from 71% to 79%
compared to the NSM1 technique.
Remark: Through simulation results have shown that proposed techniques is not
only determining the optimal value but also requiring additional calculations. In
the case of small scale problem (M, N is small number), increasing the
computational complexity in the case of optimal parameter µ does not improve the
Figure 2.13: Comparison of total transmitted power
between NSM1 and NSM2 techniques
Figure 2.14: Comparison of average number of
iterations between NSM1 and NSM2 techniques
convergence speed too much. However, in case of large scale problem (M, N is
large number), the optimization of penalty parameter µ become to greatly
improves the convergence speed in determining the minimum transmit power.
2.6. Conclusion of chapter 2
In chapter 2, the thesis achieved some results as follow:
- Formulated simulation algorithms using software Matlab in combination with
SDPT3 and Yalmip tools to evaluate the results of the proposed optimization
techniques compared to SDR and random optimization techniques.
- The simulation results have proved that SDR technique gives better results than
random techniques. In addition, the application of Nonsmooth technique combined
with penalty function has results is not only asymptotic to SDR technology but
also better the convergence speed. However, the random selection of penalty
coefficients and the steplength of the coefficients will affect the speed of
convergence.
- The proposed optimization technique NSM2 has performed the optimal
calculation of the penalty coefficient is not only determining the optimal value but
also improving the speed of convergence. The effectiveness of the proposed
solution is even more evident when the scale of the problem increases in the
number of users on the destination as well as the number of antennas at the base
station at each SNR threshold. In particular, the number of average iterations of
NSM2 technique has decreased from 70% to 80% compared to NSM1 techniques.
The results in chapter 2 have been published in two scientific papers [5] and [6].
CHƯƠNG 3. IMPROVEMENT CALCULATION SPEED FOR
TOTAL TRANSMIT POWER MINIMUM PROBLEM IN MULTI-
ANTENA WIRELESS RELAY TRANSMISSION NETWORK
3.1. The transmit power minimum problem in multi-antenna wireless
transmission network.
3.2. Multi-antenna wireless relay transmission model with AF processing
protocol
3.2.1. Amplifier and forward method AF
3.2.2. Mathematical basis for formulating the total transmit power minimum
problem
3.2.3. Formulating minimum problem for the multi-antenna relay model
Consider a scenario of MIMO relaying communication as shown in Figure 3.2,
where M sources communicate in pairs to other M receiver destinations with
the assist of an N -antenna MIMO relay. Since the use of the MIMO relay is to
increase the coverage of communications links, it is reasonable to assume that
the direct links between sources and destinations cannot be established. A dual-
hop MIMO relay operating in the amplify-and-forward protocol mode is
considered. Let 1 2[s ,s ,...,s ]
T M
Ms be the vector of signals sent by M
sources, which are assumed to be zero mean and component-wise independent
with variance
22 [ ]s iE s . The MIMO Rayleigh flat-fading channels are
considered in the chapter. Let 1 2[h ,h ,...,h ] , 1,2,...,
T N
i i i iNh i M be the
uplink channel vector between source i and the relay while
1 2[l ,l ,...,l ] , 1,2,...,
T N
j j j jNl i M denotes the downlink channel vector
between the relay and destination j. The signal vector
1 2[s ,s ,...,s ]
T M
M C s is
sent by M sources which assumed to be zero mean independent and variance
22 [ ]s iE s . For 1 2[h ,h ,...,h ]
T N
i i i iN C h as the uplink channel vector between the
user ith at transmitter and relay node, 1 2[l ,l ,...,l ]
T N
j j j jN C l is the downlink
channel vector between jth the relay and user jth at destination.
Figure 3.2: MU-MIMO wireless relay model with AF processing method
The signal is processed from the sources to the destination as follows:
+ On the first hop, all sources simultaneously transmit the signals to the relay.
The received signal at the relay is given by
up r y Hs n (3.13)
where H is the uplink channel matrix that contains all the channel vectors as its
columns and 1 2[n ,n ,...,n ] , 1,2,..., N
T N
r r r rNn i represents the additive
noise at the relay receivers, which is assumed to be white Gaussian noise with
zero mean and variance
22 [ ]r rnE n ,
+ The second stage, relay transmits signal after processing to the users in
destination. Then, X is the optimal weight matrix is multiplied by the receiver
signal at the relay. Therefore, the relay sends the following signals to the
destinations:
relay up r y Xy XHs Xn (3.14)
Accordingly, the received signal vector at the destinations is
d r dy LXHs LXn n (3.15)
where
dn is additive white Gaussian noise at the destination with element variance
2
d and L is the

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