Chapter 2 Data detection in MIMO-OFDM system with particle filtering method
2.9 Error mitigation method
As the equation shown above, no sorting is needed. However, Multiplications are needed for this approach. The performance will have same degradation for using approach II and III for data detection.
2.9 Error mitigation method
For approach II and III, one of the problems using particle filtering for data detection in spatial multiplexing is the error propagation problem. If the particles in previous tracings did not draw well, the estimated posterior distribution will be affected by error sampling. We can see that the top signal will be affected by all the other signals. Data detection using approach II and III for the top signal will has the worst performance as compared with other signals. We proposed a modified method for data detection in spatial multiplexing system with particle filtering. First, we consider the channel matrix and review the complex value problem of Gram-Schmidt algorithm for QR decomposition. Assume that all the entries in channel H are complex and consider the case that the number of transmitting antennas M is equal to the number of received antennas N (Assume that M=N), the channel matrix is shown as
[ .... .... ]
= 1 2 M
H h h h (2.48) Gram Schmidt process is
Step 1 : 1 1
On the other hand, we implement the Gram-Schmidt QR decomposition in reverse order as:
Step 1 : 1 M vectors in channel matrix can be expressed as following:
The new QR expression is
$ $ $
, so that channel matrix can be expressed as
another form of QR decomposition.
From the discussion above, we get two forms of QR decomposition which are
1 1 1 2 1
$ $ $ $ $ $
We observe from two equations shown above. In equation (2.57), we can use particle filtering, draw particles from the bottom signal to the top and use approach III to find the expectation value for each entry in the signal vector. On the other hand, in equation (2.58), we can use particle filtering method, draw particles from top to bottom and use approach III to find the expectation value for each entry in signal vector. Finally, we average two results, error propagation can be mitigated.
Block diagram for error propagation mitigation method
Figure 2.2 Block diagram for error propagation mitigation method
2.10 Sorted QR decomposition method
In [11], it is mentioned a method for sorted QR decomposition, which is similar to Gram- Schmidt algorithm. The idea of this method is to re-order the columns of channel matrix H for each orthogonal base searching. For Gram- Schmidt QR decomposition, we decompose the channel matrix H as shown in equation (2.55). Data detection by QR decomposition using particle filtering with approach II or III, as described before, the top signal will be affected by all the other signals. If particles in the previous stages did not draw well, the next stage signal samples will be affected by the previous stage samples. So that we need a large number of samples in order to obtain a much reliable posteriori probability. Sorted QR decomposition can improve such situation. The sorted QR decomposition combine with particle filtering use fewer particles to obtain a better performance compare with ordinary Gram Schmidt
decomposition as shown in simulations. The idea of sorted QR decomposition is to maximize the diagonal entry of channel matrix H from M to 1 by using a permutation vector p (where M is the number of transmitting antennas), such that minimizing the diagonal elements in each decomposition step in order to maximize the diagonal element in the subsequent steps.
The algorithm is shown as:
Step 1 : Let R = 0; Q = H ; p = 1 ,2,..M
qj =qj−ri j, qi (2.63) End
End
Where (M is the number of transmitting antenna , q is the lth column of orthogonal matrix l Q, r is the (i,j) entry of the upper triangular matrix R ) i j,
The procedure of MIMO-OFDM system with particle filtering and SQR decomposition
Step 1 : Using sorted QR algorithm to obtain matrix Q, R and p.
Step 2 : Multiply Q to the received signal vector. H Step 3:
For k = 1 to M (Where M is the number of transmitting antenna) For i = 1 to Np (Where Np is number of particles)
◆ Draw a particle from the importance distribution p s( k|s1:( )ik−1,y1:k)
◆ Calculate the weight by using equation (2.35)
◆ Store the new particle sk( )i to s1:( )ik−1 End For
◆ Normalized all the weights
( ) ( )
◆ Calculate the effective sample size Neff using (2.36)
◆ If Neff < Ns , then do the re-sampling scheme.
Step 4 : Detect signal using
1:
Step 5 : Reordering all the signals using permutation vector p
Chapter 3
Data detection in MIMO-OFDM with space frequency block code with particle filtering
3.1 System model:
We consider the system which has M transmitting antennas and N receiving antennas.
The transmitter architecture for MIMO-OFDM with space frequency block code system is shown in figure 3.1.
The data stream is mapped first, then these mapped signals are encoded by M/2 pairs of Alamouti code as shown in equation (3.1). For 4 transmitting antennas, 2 pairs of Alamouti code is called Double space time transmitting diversity (DSTTD) code as described in [12].
Figure 3.1 Transmitter structure for MIMO-OFDM with space frequency block code
S/P
The encoding process is shown as below :
(where FFT_len is the length of a OFDM symbol),
The modulated signal S to 1 SM fft len( _ / 2) are encoded as equation (3.2). Each column vector in matrix S represents an encoded signal vector allocated in a particular sub-carrier and each row vector in matrix S represents an encoded signal vector allocated in a particular antenna.
As the graph shown below:
* *
Then converts each row of matrix S by using Inverse Fast Fourier transform to time domain signal expressed in the next page.
* *
Adding Guard interval for each row vector, then signals in each row are transmitted from different antenna. Since the encode process is implemented in frequency domain (subcarrier).
We treat this type of code as space frequency block code.
3.2 MIMO-decoder
In receiver side, After guard interval removal and Fast Fourier transform, the received signals at n received antenna over subcarrier 1 and 2 are expressed as th
Y k : Received signal of n th received antenna at k th sub-carrier
Hmn( )k : Channel response in frequency domain for m th transmitting antenna and n th receiving antenna
S : m th mapped data m
n ( )n k : Noise at n th receiving antenna for k th subcarrier
The matrix form representation for MIMO-OFDM 4X2 with space frequency block code system (for subcarrier 1 and 2) can be expressed as
Y = HS + N (3.5)
Where H is the equivalent channel matrix, S is the original symbol vector which is one of the columns in equation (3.2) and N is the additive complex white Gaussian noise with variance σ2.Assuming that Hmn(1)≈Hmn(2) and define
Multiplying H (where ()Heq H represents Hermitian of a matrix) to the received vector we obtain
H H H
eq eq eq
Y = H Y = H HS + H N (3.8), Since we assumeHmn(1)≈Hmn(2), for 4 transmitting antennas, the equivalent channel matrix H will almost equal to H as shown eq
2 2
First, observing the matrix form shown above, we discover that H H is complex eqH eq symmetric as shown below
1
On the other hand, if the channel delay spread is large, Heq which is the average of the k and th k+1 channel is used for data detection. th
After multiplying H to received vector Y, the new expression can be shown as eqH
H H H H H From the equation shown above, there are three terms in equation (3.19). The third term is the error term, since we averaging the equivalent channel matrix, the error term is assumed to be small, so that we can ignore this term. This term will affect the performance if the error term is large.
The matrix H H is positive definite, so that we can use Cholesky decomposition to Heq eq decompose such matrix. Cholesky decomposition is a method to separate a matrix to a upper triangular matrix and its hermitian such that H H = Heq eq U UH ,where U is a upper triangular
properties from the equation(3.22) written above. First of all, the upper triangular matrix is obtained which accompanies with signal vector. Second, we consider the noise vector
H -1 H
H -1 H 2 -1 After multiplying(UH)−1 to the received signal vector, the new noise vector will become an independent white noise vector again. This method is also called whitening filter.
The matrix form is shown as
$
So that we can use particle filtering method, drawing particles from the bottom signal to the top signal to detect the signal vector S.
Block diagram of receiver structure
Figure 3.2 Receiver structure for MIMO-OFDM system with space frequency block code
3. 3 Error propagation mitigation method
In the previous section, we use Gram Schmidt decomposition to obtain two upper triangular matrixes. BER performance will be improved using particle filtering using such method in spatial multiplexing system. On the other hand, in MIMO-OFDM with space frequency block code system, this method can be implemented similar to spatial multiplexing system. In the previous section, we decompose channel matrix H into Q R and 1 1 Q R . 2 2 Now, after Cholesky decomposition, we obtain an upper triangular matrix U and the received vector is
H -1 H
Y = US + (U ) H Neq (3.30) The upper triangular matrix can be written as U=[U1 U2 .... .... UM],where U is K the kthcolumn vector in U. The upper triangular matrix U can be decomposed by
Gram-Schmidt QR decomposition and written as
Where the form of R is another form of upper triangular matrix as same as in the previous 2 section. Multiplying QH to R and obtain
=
H
Y = Q Y R S + n (3.32) 2
Since Q is an orthogonal matrix, the noise vector is still a white noise.
From the discussion above, we get two matrix forms
1 1
Now we define the upper triangular matrix
$ $ $
We observe from two equations shown above. In (3.33), we can use particle filtering, drawing particles from the bottom signal to the top one and using approach III to obtain the expectation value for each entry in signal vector. Interference will be severe inS . 1
On the other hand, in equation (3.34), draw particles from the top signal to the bottom
one and use approach III to obtain the expectation value, interference will be severe inS . M Finally, we average these two sets of soft information and make the decision of each symbol by searching the shortest distance for each entry in signal vector.
Block diagram for error mitigation method in
MIMO-OFDM with space frequency block code system
MIMO-Decoder
Whitening
Filter QR decomposition
Particle filtering using approach III
Particle filtering using approach III
Adder
Divided by 2
FFT FFT FFT
Remove CP Remove CP Remove CP
Figure 3.3 Block diagram for error propagation mitigation method in MIMO-OFDM with space frequency block code system
Chapter 4
Simulation results
4.1 Parameters for MIMO-OFDM spatial multiplexing system
Perfect channel information
Number of subcarriers 256 Length of CP 64
Channel Two paths model with (0,0)dB Particles 100 (if not mentioned in the figure) Approach I (if not mentioned in the figure)
Table 4.1 Parameters for MIMO-OFDM system
Figure 4.1 shows the BER performance for different approach for perfect CSI in 4X4 spatial multiplexing system for QPSK modulation. As can be observed from figure 4.1, Approach II and III have almost the same performance. For sorted QR decomposition using approach II, performance has 2 dB improvements as compared with unsorted QR using approach II. Approach I has the best performance as compared with approach II and approach III. However, the complexity for approach I is higher than the complexity for approach II and III.
Figure 4.2 shows the BER performance for QPSK modulation in 6X6 spatial
multiplexing system for sorted QR decomposition for approach I, QR decomposition with approach II and III with and without sorting. The result shows that the performance for sorted QR decomposition with approach I also has the best BER performance as compared with
approach II and III. Moreover, sorted QR decomposition using approach II has 2dB improvement better than without sorting QR decomposition in BER equal to 10^-2.
Figure 4.3 shows the performance of different detection scheme. Approach II is nearly to VBLAST ZF OSIC performance. Performance using sorted QR decomposition and approach II is nearly to the performance for VBLAST MMSE OSIC detection scheme. Moreover, the BER performance of iterative QR decomposition method has 3-4 dB improvement compare with VBLAST MMSE OSIC system and 1dB better than the BER performance of particle filtering using approach I.
Figure 4.4 shows the BER performance for 16QAM modulation with perfect CSI under MIMO-OFDM 4X4 system. As we can see from the figure shown, the performance for the iterative QR decomposition using approach II has 4dB improvement as compared with VBLAST MMSE OSIC. Sorted QR decomposition using approach I has better performance in this system than error propagation mitigation method using approach III.
Figure 4.5 shows the comparison between QR decomposition with and without sorting using approach I in 6X6 system, as shown in figure, the performance has 1 dB improvement under 50 particles as compared with unsorted QR decomposition method using approach I under 50 particles.
Figure 4.6 shows the performance for sorted QR decomposition using approach I in 6X6 system using 50 and 75 particles. There is a little improvement for 75 particles.
Figure 4.7 shows the comparison between the sorted QR decomposition method using approach I in 6X6 spatial multiplexing system with 16 QAM modulation, unsorted QR decomposition method using approach I in 6X6 MIMO-OFDM system and VBLAST MMSE OSIC. Sorted QR decomposition has 3-4 improvement compare with VBLAST MMSE OSIC and 1-2 dB improvement better than unsorted QR decomposition.
Figure 4.8 shows the performance for sorted QR decomposition using approach I in 6X6 system using 50 ,75 and 150 particles for 16 QAM modulation. There is a little
improvement for 150 particles and no different between 50 and 75 particles.
4.2 Parameters for MIMO-OFDM with Space frequency block code system
Perfect channel information Number of subcarriers 256 Length of CP 64
Channel Two paths model with (0,0)dB Particles 100 (if not mentioned in the figure) Approach I (if not mentioned in the figure) MIMO encoder 2 pairs of Alamouti code(DSTTD)
Table 4.2 Parameters for MIMO-OFDM with space frequency block code system
Figure 4.9 and figure 4.10 show the BER performance of 4X2 MIMO-OFDM for QPSK and 16 QAM modulation with space frequency block(2 pairs of Alamouti code) code system for five different detection schemes which are VBLAST MMSE OSIC, Cholesky
decomposition with decision feedback, particle filtering using approach I and error
propagation mitigation using approach III and ML. First of all, BER performance of particle filtering and error propagation mitigation method have 4dB better than the performance of VBLAST MMSE OSIC and 5dB better than performance of Cholesky decomposition with decision feedback. There are only 2dB worse than the performance of ML decision. For the same system using 16QAM modulation, as shown in figure 4.10, particle filtering has almost the same performance as error propagation mitigation method. Both of them have a 2dB improvement better than VBLAST performance.
In figure 4.11, for 4X4 MIMO-OFDM system with space frequency block code with QPSK modulation, the performance of particle filtering is better than the performance of VBLAST MMSE OSIC and almost the same as ML decision.
In figure 4.12, for 4X4 16QAM modulation system with space frequency block code, the performance of particle filtering is better than the performance of VBLAST MMSE OSIC and Cholesky decomposition with decision feedback.
Figure 4.1 Figure 4.1 MIMO-OFDM 4X4 QPSK modulation for different approaches
0 2 4 6 8 10 12 14 16 18
10-4 10-3 10-2 10-1 100
Es / No
BER
Sorted QR with Approach I joint probability particle =100 Approach I joint probability particle =100
Approach II marginal distribution Approach III expectation value sorted QR with Approach II
Figure 4.2 MIMO-OFDM 6X6 QPSK modulation for different approaches
0 2 4 6 8 10 12 14 16 18
10-4 10-3 10-2 10-1 100
Es / No
BER
Sorted QR decomposition with Approach II QR decomposition with Approach II Sorted QR decomposition with approach I
Figure 4.3 MIMO-OFDM 4X4 QPSK for different detection schemes
0 2 4 6 8 10 12 14 16 18
10-5 10-4 10-3 10-2 10-1 100
Es / No
BER
QR decision feedback VBLAST ZF-OSIC VBLAST MMSE-OSIC Sorted QR with particle filtering Error propagation mitigation method Sphere decoding with fixed radius ML
Figure 4.4 MIMO-OFDM 4X4 16 QAM for different detection schemes
0 5 10 15 20 25
10-4 10-3 10-2 10-1 100
Es / No
BER
QR decision feedback VBLAST MMSE-IC
Sorted QR with particle filtering Error propagation mitigation method
Figure 4.5 MIMO-OFDM 6X6 QPSK modulation with and without sorted QR decomposition
0 2 4 6 8 10 12 14 16 18
10-4 10-3 10-2 10-1 100
Es / No
BER
unsorted QR with particle filtering (particle = 50) sorted QR with particle filtering (particle = 50)
Figure 4.6 MIMO-OFDM 6X6 QPSK with particles equal to 50 and 75
0 2 4 6 8 10 12 14 16 18
10-4 10-3 10-2 10-1 100
Es / No
BER
sorted QR with particle filtering with particle = 50 sorted QR with particle filtering particle = 75
Figure 4.7 MIMO OFDM 6X6 16QAM modulation with and without sorted QR decomposition
0 5 10 15 20 25 30
10-3 10-2 10-1 100
Es / No
BER
VBLAST MMSE OSIC
sorted QR with particle filtering(particle = 50) using approach I QR with particle filtering(particle = 50) using approach I
Figure 4.8 MIMO-OFDM 6X6 16QAM modulation particles equal to 50,75 and 200
0 5 10 15 20 25 30
10-3 10-2 10-1 100
Es / No
BER
sorted QR with particle filtering( particle = 50) sorted QR with particle filtering( particle = 75) sorted QR with particle filtering( particle = 150)
Figure 4.9 MIMO-OFDM 4X2 QPSK with space frequency block code for different detection scheme
0 2 4 6 8 10 12 14 16 18 20
10-6 10-5 10-4 10-3 10-2 10-1 100
Es / No
BER
Cholesky decomposition with decision feedback VBLAST MMSE-OSIC
Particle filtering using approach I Error propagation mitigation method ML
Figure 4.10 MIMO-OFDM 4X2 16QAM with space frequency block code
0 5 10 15 20 25
10-3 10-2 10-1 100
Es / No
BER
Cholesky decomposition with decision feedback VBLAST MMSE-IC
Particle filtering using approach I Error propagation mitigation method
Figure 4.11 MIMO-OFDM 4X4 QPSK with space frequency block code
0 1 2 3 4 5 6 7 8 9 10
10-4 10-3 10-2 10-1 100
Es / No
BER
VBLAST MMSE-OSIC Particle filtering with approach I ML
Figure 4.12 MIMO-OFDM 4X4 16QAM with space frequency block code
0 2 4 6 8 10 12 14 16 18 20
10-6 10-5 10-4 10-3 10-2 10-1 100
Es / No
BER
Cholesky decomposition with decision feedback VBLAST MMSE-IC
Particle filtering using approach I
Chapter 5 Conclusion
Merits and drawbacks of particle filtering algorithm
Except for the complexity of QR decomposition and searching process mentioned in section 2.8, the complexity of particle filtering is directly proportional to three components, the scheme of modulation, the number of transmitting antennas and the number of particles.
The complexity for particle filtering is O(A*M*Np), where A is the modulation scheme, eg QPSK, 16QAM. M is number of transmitting antennas and Np is number of particles. The complexity of ML scheme exponentially increases either the number of transmitting antenna or the number of order of modulation increases. The complexity for ML decision is O(A ). M Particle filtering is a practical approach for data detection. As the simulation shown before, the performance of our proposed methods using particle filtering are close to ML decision either in spatial multiplexing system or with space frequency block code system. The complexity for QPSK modulation for 6X6 MIMO-OFDM BLAST system is only 4*4*50 = 800 trials, however, for ML decision method, number of trial is 46 =4096trials. The
complexity with particle filtering is 5 times lower than the complexity with ML decision, the BER performance of sorted QR decomposition with particle filtering using approach I is only 2 dB worse than ML decision. Moreover, for high order modulation, for example, 16 QAM for 4X4 MIMO-OFDM BLAST system, the complexity for ML will be 164 =65536 trials, however, particle filtering method only deal with 4*4*100 = 1600 trials. In conclusion, Particle filtering is a suitable approach for high modulation order and large amount of transmitting antenna system in MIMO-OFDM BLAST system.
One of the drawbacks of particle filtering is that the noise distribution is known at the receiver side. For example, if the noise distribution is white Gaussian noise, receiver need to
be estimate the variance of noise first, after that pass this variance information to particle filtering.
The second drawback is that particle filtering need the process of QR decomposition or Cholesky decomposition, the complexity will increase for when the number of transmitting antenna increases.
The third drawback is that the searching process mentioned in section 2.8 for approach I and approach II. Especially for approach I, the complexity will increase either the number of transmitting antennas or number of particles increase.
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