Chapter 2 MIMO-OFDM System
2.4 Detection Schemes on the Receiving Side
The complexity of the Maximum Likelihood (ML) detector grows exponentially with the number of transmitting antennas and the size of the signal constellation. This motivates the use of simpler suboptimum detectors in practical applications. Among those are:
Zero Forcing (ZF) detectors, which invert the channel matrix. The ZF receiver has a very small complexity that does not depend on the modulation. However, it does not completely exploit the system diversity and suffers from bad performance at low SNR.
Minimum Mean Square Error (MMSE) detectors, which reduce the combined effect of interference between the parallel channels and the additive noise. The MMSE receiver slightly improves the performance of the ZF receiver, but it requires knowledge of the SNR. Besides, it does not completely exploit the channel diversity either.
V-BLAST Ordered Successive Interference Cancellation (OSIC) [9], which exploits the timing synchronism inherent in the system model. Furthermore, linear nulling (i.e., ZF) or
MMSE is used to perform the detection. In other words, SIC is based on the subtraction of interference of already detected element of a from the received signal vector r. This results in a modified received vector in which effectively fewer interferers are present.
SD [3] algorithm, which reduces the number of symbol values used in the ML detector.
Note that this type of detectors may preserve optimality while reducing implementation complexity.
Thus, if signal conditions are excellent, the data rate will be more than twice depending on the number of antennas used in both the transmitter and receiver. In that case, the channel matrix is better conditioned and the performance degradation of suboptimal detectors is reduced.
Chapter 3
Probabilistic Data Association Based Detectors
3.1 Probabilistic Data Association Detector
The Probabilistic Data Association (PDA) detector [4] is a highly successful approach to target tracking in the case that measurements are unlabeled and may be spurious. It is based on two approximations. Firstly, the PDA detector only looks at one transmitted symbol at a time, treating the received symbols as statistically independent. The second approximation is the Gaussian approximation (“Gaussian forcing”) of the probability density function (PDF) of the interference and noise. This is a bold and to some extent unjustifiable step, but it is difficult to argue with good performance and low complexity.
Now, to obtain the system model of the PDA detector, we multiply (2.1) form the left by HH
to obtain
other element are 0. In order to obtain computational efficiency, we choose (3.2) as the system model for the PDA algorithm.
3.1.1 Basic Algorithm
In the reformulated MIMO-OFDM spatial multiplexing system model (3.2), we treat the element of a as independent multivariate random variable where the ith element, , is a member of possible set: associate a vector whose mth element,
Xi ai
pi p mi( ), is the current estimate of the posterior that ai =xi(m). Since direct evaluation of Pr a
(
i =xi( )m y)
is computationally prohibitive, the PDA algorithm attempts to estimate by using “Gaussian forcing” idea to approximate( )
Pr ai = x mi( ) y p,{ j}∀ ≠j i , which will serve as the update value for p mi( ).
An important factor in the performance of the PDA algorithm is the order in which the probability vectors
{ }
pi ∀i are updated. In this case, we use V-BLAST OSIC method [9] toorder the update sequence according to SNR in descending order, which can detect the first signal that belongs to the highest order of SNR. It is useful to provide the reliable symbols from the high SNR, and we can use these reliable symbols to detect the other symbols to make the BER performance better.
To estimate the associated probabilities for an element , we treat all other elements as multivariate random variables, and from (3.2), we define
ai
as the effective noise on , and approximate it as a Gaussian noise with matched mean and
covariance:
probability P mi( ) is then given as
Fig. 3.1 Block diagram of the basic PDA procedure.
The basic procedure for the PDA detector is as follows:
1. Based on the matrix in (3.1), we obtain the optimal detection sequence proposed for the V-BLAST OSIC in [9] and denote the sequence as
G
corresponding elements of . The block diagram of the basic PDA procedure is shown in Fig. 3.1.
3.1.2 Computational Refinements
A. Speed-Up–Matrix arithmetic
As noted in [4], although the computation in step 4 is no longer exponential, to calculate the inverse of for each element directly is still expensive. Further simplifications can be evaluated by applying the Sherman-Morrison-Woodbury formula [10] twice consecutively.
Ωi
1 1
In our simulations, we have observed that the algorithm generally converges within 2 to 4 iterations for SNR < 14 dB, and within 1 to 2 iterations for SNR > 14 dB. However, the overall complexity can be high if one or two elements of exhibit slow convergence. To reduce the complexity in these instances, successive skip is applied each iteration. Note that, the successive skip method is different from the successive cancellation method which mentioned in [4]. The advantage of the successive skip method is implemented easily and lower complexity than the successive cancellation method. The main idea of the successive skip tactic is that the posterior probability of some elements is high enough to let us believe it within the process of converges. So, if we have this element in the next iteration, we can simply skip it.
After zth iteration, we define to be the set of elements that satisfy
max{ ( )} )P mi ∀m ≥ ∀ε, i (3.17)
where ε = −1 (0.2σ2) is a small positive number. At z+1th iteration, we will skip the elements that belong to V .
3.2 Generalized PDA Detector
The Generalized PDA (GPDA) detector [5] is present for the special case of
square/rectangular (sqr/rect) QAM. In the case of sqr/rect q-QAM, the GPDA algorithm differs from the PDA approach of [11] by reducing the number of probabilities associated with each transmit symbol. As an apparent consequence of reducing the number of probabilities for sqr/rect QAM, the GPDA shows an improved error probability over the PDA approach used in [11]. A further advantage of GPDA is that it offers a reduced computational cost over that of [11] for the case when the number of receive antennas is greater than the number of transmit antennas.
To obtain the system model for sqr/rect QAM version of the GPDA detector, we begin
by transforming (2.1) into the real-valued vector equation
= + modeled as i.i.d. complex Gaussian, will almost always have full rank and consequently the symmetric matrix will be positive definite with probability nearly one.
= T
G H H n H v= T
H
H
G
The model for sqrt/rect QAM version of the GPDA detector is obtained by multiplying imaginary parts of the QAM symbols respectively. Thus equation (3.21) can be solved via PDA which we introduced in Section 3.1. Note that, when we calculate equation (3.9) in the GPDA algorithm, it should be modeling by the real Gaussian distribution rather than the complex Gaussian distribution and the noise variance should be
Sℜ ℑ
2 2
σ rather than σ . 2
3.3 Simulation Results
Perfect Channel State Information (CSI) Perfect noise variance estimation
Number of subcarrier 64 Length of cyclic prefix 16
Channel Rayleigh Fading
Path 2
Relative power (dB) (0,0)
Modulation QPSK, 16QAM, 64QAM
Table 3.1 Simulation parameters for comparing PDA with GPDA.
In this Section, we use several computer simulation examples to show the performance and the computational cost of the PDA detector and the GPDA detector. We also compare the PDA detector and the GPDA detector with the V-BLAST ZF OSIC [9] and the optimal ML detector in the examples. The simulation parameters are shown in Table 3.1.
In Fig. 3.2, we compare the BER performance for the GPDA detector and the PDA detector with . The result shows that the BER performance of the GPDA detector has about 1 dB improvements as compared with the PDA detector. Since the dimension of the GPDA detector is two times as the dimension of the PDA detector making the PDF of the interference and noise closer to Gaussian distribution. In Fig. 3.3, it shows the performance of the aforementioned detectors for the case
q=4
q=16.
In Fig. 3.4, we compare the effect of sorting for the GPDA detector and the PDA detector with . The result shows that the BER performance of the PDA and the GPDA detectors has approximately more 2 dB than that of the unsorted PDA and the unsorted GPDA detectors. Thus, we can identify that sorting is an important factor for the GPDA and the PDA detectors.
q=4
In Fig. 3.5, we compare the number of iterations for the GPDA detector and the PDA
detector with . The result shows that both the GPDA detector and the PDA detector just
need 2-4 iterations to converge. In Fig. 3.6, it shows the BER performance of the aforementioned detectors for the case
q=4
q =16.
In Fig. 3.7, we compare the complexity for the GPDA detector and the PDA detector with . For the case of complexity of the system, the measurement was calculated using FLOPS function in MATLAB [14], which counts the approximated floating point operations
that the algorithm needs to complete decoding in one block of transmitted symbols. The result shows that there is a great gap of the original complexity between the GPDA detector and the PDA detector. After using the matrix speed-up (speed-up I) tactic, the gap between the GPDA detector and the PDA detector has reduced. If we use matrix speed-up tactic and successive skip tactic simultaneously (speed-up II), the complexity may reduce once more. Overall, the complexity of the GPDA detector is slight more than that of the PDA detector for
q=4
q=4.
In Fig. 3.8, we compare the complexity for the GPDA detector and the PDA detector with . As the figure suggests that, after using speed-up tactic, the complexity of the GPDA detector will be significantly less than that of the PDA detector.
q=16
In Fig. 3.9, we compare the complexity for the GPDA detector and the PDA detector with different modulation order, at SNR=0 dB. We can observe that with greater modulation order, the gap between the GPDA detector and the PDA detector gets wider.
Fig. 3.10 shows the noise variance estimation error for the GPDA detector and the PDA
detector with , at SNR=25 dB. The result shows that the GPDA detector and the PDA detector are almost free from the impact of the noise variance estimated error. Note that,
q =16
E[
2 2
[ ] [ ]
Eσ =Eσ + Δσ2] where E[ ]σ2 is estimated noise variance composed of real noise
variance E[σ2] and estimation error E[Δσ2].
After above comparisons, the GPDA detector shows improved BER performance over the PDA detector by reducing the number of probabilities associated with each transmit symbol. Moreover, the complexity of the GPDA detector is much less than that of the PDA detector, especially in high order modulation. Hence, that is the reason why we choose the GPDA detector rather than the PDA detector.
Fig. 3.2 The BER performance for PDA and GPDA with q=4.
Fig. 3.3 The BER performance for PDA and GPDA with q=16.
Fig. 3.4 The effect of sorting for PDA and GPDA with q=4.
Fig. 3.5 The number of iterations for PDA and GPDA with q=4.
Fig. 3.6 The number of iterations for PDA and GPDA with q=16.
Fig. 3.7 The complexity for PDA and GPDA with q=4.
Fig. 3.8 The complexity for PDA and GPDA with q=16.
Fig. 3.9 The complexity for PDA and GPDA with different modulation order.
Fig. 3.10 The noise variance estimation error for GPDA and PDA.
Chapter 4
Data Detection in MIMO-OFDM System Based on GPDA Detector
4.1 GPDA-MCPDA Detector
The basic idea of the GPDA-MCPDA detector is using the GPDA detector at the low SNR regions, and using parallel MCPDA method to generate numbers of random samples at
the high SNR regions. After generating samples, we will pick up a sample from the final iteration of parallels, and the sample which has minimum distance (i.e. arg min 2
a X∈ i r - Ha ).
Thus, we can get a solution from the GPDA-MCPDA detector.
4.1.1 Markov Chain Monte Carlo Method
A major limitation towards more widespread implementation of Bayesian approaches is that obtaining the posterior distribution often requires the integration of high-dimensional functions. This can be computationally very difficult, but several approaches short of direct integration have been proposed. The MCMC methods [6], which attempt to simulate direct draws from some complex distribution of interest. MCMC approaches are so-named because one uses the previous to randomly generate the next sample value, generating a Markov Chain (as the transition probabilities between sample values are only a function of the most recent
sample value).
The realization in the early 1990’s that one particular MCMC method, the Gibbs sampler, is widely applied to a broad class of Bayesian problems has sparked a major increase in the application of Bayesian analysis.
4.1.1.1 Monte Carlo Integration
The original Monte Carlo approach was a method developed by physicists to use random number generation to compute integrals. Suppose we wish to compute a complex integral
b ( )
This is referred to as Monte Carlo integration.
Monte Carlo integration can be used to approximate posterior (or marginal posterior) distributions required for a Bayesian analysis. Consider the integral ( )I y =
∫
f y x p x d( | ) ( ) x , which we approximate by1
It was observed in the preceding Section 4.1.1.1 that the integral can be approximate by Monte Carlo integration. However, not every density p x( ) can be drawn directly. Now, we suppose the density q x( ) roughly approximates the density p x( ), then
This forms the basis for the method of importance sampling, with
1
4.1.1.3 Introduction to Markov Chains
Before introducing the Gibbs sampler, a few introductory comments on Markov Chains are in order. Let denote the value of a random variable at time , and let the state space refer to the range of possible
Xt t
X values. The random variable is a Markov process if the
transition probabilities between different values in the state space depend only on the random variable’s current state, i.e.,
1 0 1
Pr(Xt+ =s Xj| =sk, ,Xt =si)=Pr(Xt+ =s Xj| t =si) (4.7)
Thus for a Markov random variable the only information about the past needed to predict the future is the current stage of the random variable, knowledge of the values of earlier states do not change the transition probability. A Markov chain refers to a sequence of random variables generated by a Markov process. A particular chain is defined most critically by its transition probabilities, vector of the state space probabilities at step . We start the chain by specifying a starting vector . Often all the elements of are zero except for a single element of 1, corresponding to the process starting in that particular state. As the chain progresses, the probability values get spread out over the possible state space.
t
π ( ) t
t
)
π (0) π (0
The probability that the chain has state value si at time (or step) is given by the Chapman-Kolomogrov equation, which sum over the probability of being in a particular state at the current step and the transition probability from that state into state
t+1
si,
1
Successive iteration of the Chapman-Kolomogrov equation describes the evolution of the chain.
We can more compactly write the Chapman-Kolomogrov equation in matrix form as follows. Define the probability transition matrix P as the matrix whose i j, th element is
( , )
P i j , the probability of moving from state to state i j , P i( → j) . The
Chapman-Kolomogrov equation becomes
( 1) t + = ( ) t
π π P
(4.11)Using the matrix form, we immediately see how to quickly iterate the Chapman-Kolomogrov
equation, as
( )t = (t−1) =( (t−2) ) = (t−2) 2
π π P π P P π P (4.12)
Continuing in this fashion show that
( )t = (0) t
π π P (4.13)
Defining the n-step transition probability p as the probability that the process is in state i j( ),n j given that it started in state , n step ago, i.e., i
Finally, a Markov chain is said to be irreducible if there exists a positive integer such that
( ) ,n 0
pi j > for all . That is, all states communicate with each other, as one can always go from any state to any other state (although it may take more than one step). Likewise, a chain is said to be aperiodic when the number of steps required to move between two states (say
i j,
x
and ) is not required to be multiple of some integer. Put another way, the chain is not forced into some cycle of fixed length between certain states.
y
π*
* *
A Markov chain may reach a stationary distribution , where the vector of probabilities of being in any particular given state is independent of the initial condition. The
stationary distribution satisfies
= π π P
*
(4.15) The condition for a stationary distribution is that the chain is irreducible and aperiodic. When a chain is periodic, it can cycle in a deterministic fashion between states and hence never settles down to a stationary distribution.
A sufficient condition for a unique stationary distribution is that the detailed balance
equation holds,
( ) *j ( ) k
P j→k π =P k→ j π (4.16)
If equation (4.16) holds for all , the Markov chain is said to be reversible, and hence equation (4.16) is also called the reversibility condition. Note that this condition implies
, as the
with the last step following since rows sum to one.
4.1.1.4 Gibbs Sampler
One problem with applying Monte Carlo integration is in obtaining samples from some complex probability distribution. Attempts to solve this problem are the roots of MCMC methods. In particular, they trace to attempts by mathematical physicists to integrate very complex functions by random sampling, and resulting Metropolis-Hastings sampling [6]. The Gibbs sampler [6] [15] (introduced in the context of image processing by Geman 1984), is a special case of Metropolis-Hastings sampling wherein the random value is always accepted (i.e. α =1). The task remains to specify how to construct a Markov Chain with values converged to the target distribution. The key to the Gibbs sampler is that we only consider the univariate conditional distributions (the distribution when all of the random variables but one is assigned fixed value). Such conditional distributions are far easier to simulate than complex joint distributions and usually have simpler forms. Thus, we simulate n random variables sequentially from the n univariate conditions rather than generating a single n-dimensional vector in a single pass using the full joint distribution.
To introduce the Gibbs sampler, consider a bivariate random variable ( , )x y , and suppose we want to compute one or both marginal, p x( ) and p y( ). The idea behind the sampler is that it is far easier to consider a sequence of conditional distributions, p x y( | ) and
( | )
p y x , than to obtain the marginal by integral of the joint density p x y( , ), e.g.,
( ) ( , )
p x =
∫
p x y dy. The sampler starts with some initial value y0 for y and obtains x0 by generating a random variable from the conditional distribution, p x y( | = y0). We use x0 to generate a new value of from the sampler. To do so, we draw from the conditional distribution based on the valuey1 Repeating this process for k times, then it generates a Gibbs sequence of length k, where a subset of points ( ,x y for j j) 1≤ ≤j m k< is taken as our simulated draws from the full joint
distribution. The first m times of the process, called burn-in period, can make the Markov Chain converge to the distribution that near its stationary one.
When more than two variables are involved, the sampler is extended in the obvious fashion. In particular, the value of the kth variable is drawn from the distribution
where denotes a vector containing all of the variables but k. Thus, during the ith iteration of the sample, to obtain the value of i( )k
For example, if there are four variab y z)
|
Now, we consider the equation (3.2) and we use Gibbs sampler to generate samples of
generate initial samples randomly
sample from ( | , , , , )
i( )m
The first Nb iterations of the loop, called burn-in period, are to let the Markov Chain converge to near its stationary distribution. During the next Ns iterations, the Gibbs sampler generate the Ns samples, i.e.,
t sample can be istribution should be converged
after
las the solution of the distribution. Since the d
b s
N +N times iteration.
There are two problems of Gibbs sampler:
1) How do you choose an initial point? A poor choice of initial point will greatly increase the required burn-in time, and an area of much current research is whether an optimal initial point can be found.
2) How many iterates are needed? This question does not have exact answer, the majority
2) How many iterates are needed? This question does not have exact answer, the majority