Chapter 1 Introduction
1.3 About the thesis
This thesis is organized as following. Chapter 2 describes the use of particle filtering
method in data detection in spatial multiplexing system. Then we describe two modified methods to mitigate the error propagation problem in particle filtering. Chapter 3 presents the use of particle filtering for data detection in MIMO-OFDM with space frequency block code system. Then also presents a method to mitigate the error propagation. Chapter 4 shows all the simulations for each detection scheme in both spatial multiplexing in MIMO-OFDM and MIMO-OFDM with space frequency block code system. Finally, conclusions are introduced in the last chapter.
Chapter 2
Data detection in MIMO-OFDM system with particle filtering method
2.1 Spatial multiplexing system description:
Figure 2.1 Spatial multiplexing system
In MIMO-OFDM spatial multiplexing system, we consider the system shown in figure 1.
We assume that there are M transmitting antennas and N receiving antennas. At the
transmitter side, bit stream is divided into M data layers and mapped each data layer to be M modulated signal streams. M modulated signal streams in M layer pass through IFFT, add cyclic prefix and then transmit parallel through M transmitting antennas. At the receiver side, there are N receiving antennas, after cyclic prefix removal and pass through FFT, the received signal vector X can be expressed as
X = HS + N (2.1) Tx
Where X is an N by 1 received signal vector, H is a N by M channel matrix, S is a M by 1 Tx transmitted signal vector and N is a N by 1 noise vector. The channel matrix H is assumed to be full rank. The received signal vector is passed through the data detection scheme as shown in figure 1. There are several schemes for data detection in MIMO-OFDM BLAST system.
One of them is VBLAST- OSIC.
V-BLAST Zero-forcing OSIC [3] scheme is widely used in spatial multiplexing system.
The procedure of the V-BLAST can be mainly divided into following steps: first, ordering the received signal according to signal to noise (SNR) ratio in descending order, then detects the first signal that belongs to the highest order of SNR. After detecting the first signal, then treats this signal as interference and cancelled out from the received signal vector, then starts to detect the second highest SNR signal. This process keeps moving until all the data are
detected. In spatial multiplexing system, the optimum solution is to use Maximum likelihood (ML) detection. However, ML detection is an exhaustive search, the complexity increases either the number of transmitting antennas or order of modulation increases.
On the other hand, Maximum a posteriori (MAP) detection also give an optimum solution, therefore, if we can obtain the posteriori pdf (probability density function) or pmf(probability mass function), then MAP detection can be used for data detection. MAP approach is as same as ML approach. As describes above, the received signals can be expressed as
X = HS + N , (2.2) Tx
All the elements in vector X, H, S and N are complex number. In this thesis, we only Tx consider the case that the number of transmitting antennas is equal to or less than the number of receiving antennas.
Assuming that the channel matrix H is full rank such that it can be decomposed using QR decomposition as shown below
H = QR, (2.3) where R is an upper triangular matrix and Q is an orthogonal matrix.
Multiply QH (where ()Hdenoted as Hermitian of a matrix) to X and the system model can be expressed as
H
X = Q X = RS + N (2.4)Tx
We consider the case that the number of transmitted antennas are equal to the received antennas(M=N), so that
We re-define some parameters, first of all, we define three vectors Y, S and n as the reverse order of X ,S and N where Tx
The new expression can be shown as
11 12 1
Assume that there are M transmitting signal layers from M transmitting antennas, if we
obtain the posteriori distribution and assume that each entry in the noise vector is independent Gaussian distribution, zero mean and varianceσ . 2
The posteriori distribution will be expressed as:
1: 1: / 2 2 / 2 2 From equation (2.10), MAP decision needs to test all the possible combinations and choose the minimum distances. The complexity is related to two factors: first, the modulation scheme, for example, QPSK, 16QAM, and second, the number of transmitting antennas. The
complexity increases exponentially as one of the factors increases. So that the complexity is O(A ), where M is the number of transmitting antennas and A is the modulation scheme. For M example, QPSK with 4 transmitting antennas, number of trials will become 44 =256.
Furthermore, if modulation change to 16QAM, number of trials will become 164 =65536. MAP decision is not practical in this case.
2.3 Monte Carlo method
Before we mention the detail of particle filtering or called sequential Monte Carlo method algorithm, first we take a look on how a posteriori distribution can be approximated by a set of random samples.
p s( 1:M =s1:M |y1:M)=
∫
p s( 1:M |y1:M) (δ s1:M −s1:M)ds1:M (2.11) If Np is large and then the desired posteriori distribution can be approximated as :( ) As the equation mentioned above,
1: 1
{ }
k
i Np
s i= denoted a set of samples drawn from a desired posteriori distribution function, the posteriori distribution function can be approximated by
( ) Monte Carlo approach is one of the methods to construct the approximation of high
dimensional posteriori distribution. If we can draw samples directly from the desired
posteriori distribution p s( 1:M |y1:M), so that the posteriori distribution can be constructed by all the samples 1:
( )
{ M} 1 i Np
s i= (where Np represents the number of samples) drawn from the desired posteriori distribution and this approximation will converge to the true posteriori distribution as there are infinite number of samples.
2.4 Importance sampling
Importance sampling is a method to approximate the desired posteriori distribution by drawing samples {s( )i }iNp=1 from a trial function called importance distribution q s( 1:M |y1:M) if the desired posteriori distribution cannot be drawn directly. This importance distribution is tractable for sampling. The different between Monte Carlo method and importance sampling is that importance sampling needs to compute the weights of the corresponding i th sample
using
The approximation of the posteriori distribution can be derived as :
1: 1:
If Np is large and then the posteriori distribution can be approximated as : (1/Np)
=
∑
as normalized weight corresponding to i th sample, then the posterioridistribution can be approximated as
1: 1: ( ) 1: 1:( ) The importance distribution can be chosen freely, however, the variance will be
increased if the importance function is not highly related to the true posteriori distribution.
2.5 Particle filtering Methods [4]
If we need to draw samples directly from the posteriori distribution, we need to know the joint posteriori distribution first. The complexity is same as or higher than MAP decision.
Now, if we do not have any information about the desired posteriori distribution, however, we have the conditional probability distribution p(y | s ) , particle filtering or called sequential k 1:k
monte carlo method described in [5] and [6] provides a new method to obtain the posteriori distribution with low complexity by using the idea of importance sampling. The main idea is to estimate the desired posteriori distribution by drawing a set of random samples from importance distribution and to update the corresponding weights recursively.
Let’s take a look on how all the samples can be drawn recursively. After finishingk−1th
importance distribution q s( 1:k|y1:k) by sampling Np sampled sequences with length k-1(from 1 to k-1 )
{ }
s1:( )ik−1 Npi=1 from q(s1:k−1|y1:k−1) and by sampling a new set of samples{ }
s( )ki i=Np1 The posteriori distribution can be approximated using{ }
1:( ) 1i Np k i=
s recursively as equation
(2.21) and updated weights{wk( )i−1}Npi=1 recursively using equation (2.26), then normalize all the weights, the posteriori distribution can be approximated as
( ) ( ) Since it is an approximation method, increasing the number of samples will increase the accuracy of the approximation. In the jargon of particle filtering, these samples in each tracking are called particles.
The problem is how to choose the importance distributionq(s sk | 1:k−1,y1:k). In [7], it is mentioned, in order to minimize the variance of the approximation, the importance function is chosen as:
( ) ( )
1: 1 1: 1: 1 1:
( k| ik , k) ( k| ik , k)
q s s − y = p s s − y (2.28) If we choose the importance function as equation (2.28),
1: 1 th tracking after normalization. From the deviation of equation (2.35), we can observe that the
weight in i th particle at k th tracking depends on two factors: The previous weights of i th
{wki−}iNp= .Then the new particles can be drawn from the importance distribution
( ) ( )
1: 1 1: 1: 1 1:
( k | ik , k) ( k| ik , k)
q s s − y = p s s − y and then update the corresponding weight using equation (2.35). After that normalize all the weights at M th tracking by
( ) ( )
particle filtering, this procedure is called Sequential importance sampling (SIS) scheme.
The procedure of k th tracking is summarized as following:
－For i = 1 to Np
◆ Draw a particle from the importance distribution (2.28)
◆ Calculate the weight by using equation(2.35)
◆ Store the new particle sk( )i to s1:( )ik−1
－End For
◆ Normalized all the weights
2.6 Degeneracy
After several tracking, the variance of the estimator will increases as shown in [7], since some of the particles have negligible weights and do not have any contribution to the process.
This problem is called degeneracy. In [10], resampling algorithm is used to overcome this problem. The main idea is to replace some small weighted samples by some large weighted samples. In [8] and [9]. Both papers mention that one of the methods to measure degeneracy is to calculate the effective sample size Neff . Neff can be obtained by
( ) 2 So that we can set a threshold sample size called Ns. Ns is set as 60% of Np in our simulation.
If Neff <Ns, resampling algorithm is needed.
Algorithm for resampling
For i = 1 to Np
Generate a random variable U with uniform distribution from [0 1]
For j = 1 to Np
After resampling, new set of particles are obtained, the connection with previous samples is broken and their weights at k th tracking are all equal. In the jargon of particle filtering, this procedure is called Sequential importance sampling (SIS) with resampling scheme.
The procedure of k th tracking is summarized as following:
－For i = 1 to Np
◆ Draw a particle from the importance distribution from equation (2.28)
◆ Calculate the weight by using equation (2.35)
◆ Store the new particle sk( )i to s1:( )ik−1
－End For
◆ Normalized all the weights by using
( ) ( )
◆ Calculate the effective sample size Neff using (2.36)
◆ If Neff < Ns , then do the resampling scheme.
2.7 Data detection scheme in MIMO-OFDM BLAST system with particle filtering:
In MIMO-OFDM spatial multiplexing system, we assume that the channel matrix H is full rank such that it can be decomposed using QR decomposition as shown H = QR and
H
X = Q X = RS + NTx (2.40) Since R is an upper triangular matrix, one of the methods for data detection is to use decision feedback method that detects signals from the bottom to the top.
First, compute the probability of (p stxM |yM). For example, the distribution of noise in each entry is complex Gaussian distribution then detection stxM using minimum distance. The next step is to compute p s( txM−1|stxM,yM−1) and detectstxM−1. The process keeps moving until all the signals are detected. However, this method has error propagation problem and the SNR of each signal mainly depends on the diagonal. On the other hand, since QH is an
orthogonal matrix, so that after multiplying QH to the initial noise vector, the new noise vector is also independent white noise vector. As mentioned in section 2.1, we define three new vectors Y, S and n, and the relationship between y is also dependent on k s and 1:k n k which is yk =R sk k k, +Rk k, +1sk−1+...Rk M, s1+ . We assume that the noise before multiplying nk
QH to the received signal vector is white noise. Hence, after multiplying QHto the received signal vector, the noise vector is still a white noise vector. We treat each noise entry n as an k independent white noise. A particle sk=a is drawn from the importance function k
( ) in signal constellation) and variance is equal toσ2 and the second term in numerator is assumed to be equally likely. Finally, we can draw samples from p s( k |s1:( )ik−1,y1:k)which is equal to equation (2.36). For example, in QPSK modulation, the set of $a is {M 1
(1 )
2 − } also there is 16 combinations for 16-QAM modulation. j For example, for the QPSK modulation, the particle filtering (SIS) is shown below:
In k-th tracking:
( ) 2 Generate a uniform distribution U between [0 ,1]
If α1>U > 0, then ( ) 1
Example : For MIMO-OFDM 4X4 system with BPSK modulation.
After QR decomposition, the signal model become
4 11 12 13 14 4 4
In order to draw particles for s , first of all, calculate the probability for 1
1 1
two probabilities. For example, assume that 1 1 first tracking. Assuming that the five particles are {1 1 1 -1 -1}. In order to draw particles for
the second tracking s2( )i , first calculate the
random variable U, if U<0.3, then s2(2)=1 ,otherwise s2(2)=-1 ,and the corresponding weight for 2nd particle for 2nd tracking is
From the equation shown above, we observe that the i th particle at 2 th tracking is related to the previous i th particle and weight. After getting five new particles and update five corresponding weights for i th particle at 2nd tracking. Assume that they are {-1 -1 -1 -1 1}, attach these five particles to the first five particles, we can get
1 1 1 1 1 row represents the second tracking particles(s2( )i ). Assume that the corresponding weights at
2nd tracking is w2(1:5) = {0.3 0.3 0.3 0.05 0.05}. Keep moving until 4th tracking is done. We can
First, consider the first three columns, we discover that the first three columns are identical to each other which is s1:4( )i = {1 -1 1 1}, the 4th and 5th column are different to the first 3 columns,
Approach I : Sequence detection This process needs to find all the same sequences and adds all the weights which belong to the same sequence This process will increase the complexity if the sequence is too long, which means that if the number of tracking increases, the complexity will increases.
Approach II : Detect directly from the marginal posteriori probability
( ) ( )
the detection scheme needs to find one dimension only. The searching process is to sort all the signals which belong to the same constellation and to add all the weights which belong to the same signal. The detection scheme after sorting and adding all the weights is shown as
arg max ( | 1: )
Approach III : Find the expectation value from the marginal posteriori distribution s$k ( ) ( )
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
Y = H Y = H HS + H N (3.8), Since we assumeHmn(1)≈Hmn(2), for 4 transmitting antennas, the equivalent channel matrix