Chapter 4 Data Detection in MIMO-OFDM System Based on GPDA Detector
4.1 GPDA-MCPDA Detector
4.1.1 Markov Chain Monte Carlo Method
4.2.1.1 Real Sphere Decoding
Fig. 4.3 A sphere of radius and centered at
In communication system, the SD algorithm is used to solve the ML problem as follows:
d
r
.2
ˆML arg min
a X∈ i
=
a r - Ha (4.27) The computational complexity of above exhaustive search
m
method is really high. Therefore, the SD algorithm is brought up to avoid the exhaustive search and search only over the possible a which lie in a certain sphere of radius d around the given vector
r
, thereby reducing the search space and, hence, the required co putations (see Fig. 4.3).It is clear that the closest point inside the sphere will also be the closest point for the who
ny points,
2) oints are inside the sphere? If this requires testing the distance of
The SD algorithm does not really address the first question, but usually uses ZF solution to be le point. However, close scrutiny of this basic ideal leads to two key questions.
1) How do you choose radius d? Clearly, if radius is too large, we obtain too ma
and the search remains exponential in size, whereas if radius is too small, we obtain no points inside the sphere.
How can we tell which p
each point form
r
, then there is no point in SD, as we will still need an exhaustive search.the radius of the sphere. However it does propose an efficient way to answer the second. The basic observation is the following. Although it is difficult to determine the points inside a general NT-dimensional sphere, it is trivial to do so in the one-dimensional case. The reason is that a one dimensional sphere reduces to the endpoints of an interval, and so the desired points will be the integer values that lie in this interval. We can use this observation to go from dimension k to dimension k +1. Suppose that we have determined allk-dimensional points that lie in sphere of radius Then, for any such k -dimensional point, the set of admissible values of the
( k + 1)
th d ensional coordinate th lie in the higher dimensional sphere of the same radius ms an interval.The above means that we can determine all
a d.
im at
d for
points in a sphere of dimension NT and
radius d by successively determining all points in spheres of lower dimensions 1, , NT and the same radius d . Such an algorithm for determining the points in an NT-di
sphere essentially constructs a tree where the branches in the kth level of the tree correspond to the points inside the sphere of radius d and dimension k (see Fig. 4.4).
Moreover, the complexity of such an algorithm will depend on the size of the tree, i.e., on the number of points visited by the algorithm in different dimensions.
2, mensional
at hand. To this Fig. 4.4 Search tree of sphere decoding.
With this brief discussion, we can now be mo e specific about the problemr
end, we shall assume that NR≥NT. Note that, the point Ha lies inside a sphere of radius d centered at
r
if and only if2 2
d ≥ r - Ha (4.28)
In order to break the problem into the sub-problems described above, it is useful to consider the QR factorization of the matrix H
R T T
orthonormal columns of Q, respectively. The condition (4.28) can be written as
2
R. he upper triangular property of R
comes in handy. The rig nd side of t e above inequality can be expanded as
T T T T necessary condition for Ha lie inside the sphere
T T T T
NT where denotes rounding to the nearest larger element in
poi
eans sufficient. For every satisfying (4.33), defining
⎡ ⎤⋅
⎢ ⎥ the set of numbers that spans
the point. Similarly, ⋅⎢ ⎥⎣ ⎦ denotes rounding to the nearest smaller element in the set of
numbers that spans the nt.
Of Course, (4.33) is by no m condition can be found by looking at the first two terms in (4. −
belonging to the interval
One can continue in a similar fashion for
NT 2
a − and so on until
.2.1.2 Complex Sphere Decoding
ies on a real system where a is chosen from a
1, thereby obtaining all a
points belonging to (4.28).
4
The SD algorithm described above appl
real point, but in communication systems we face to deal with complex systems, because of
the modulation scheme we used is QPSK. In this case, equation (4.28) becomes as follows:
2 2
d ≥ r - Ha (4.35)
where
{ } { }
y problems of the SD algorithm are mentioned. The second
block diagram of the GPDA-SD detector is shown in Fig. 4.5. As the diagram solved via SD which we introduced in Section 4.2.1.1.
4
In the Section 4.2.1.1, two ke
one can be solved by the algorithm of SD, but we don’t know how to choose the radius in the first problem. In this case, we can use the distance calculated by GPDA as the radius of sphere.
The
indicates, it can be divided into three types. First, when SNR is below M dB, which SNR is quite low, we can use the GPDA detector directly to achieve near-optimal solution. Second, when SNR is between M dB and N dB, we can use the GPDA-SD detector to assist the GPDA detector to reach better performance. Finally, when SNR is over N dB, the complexity of the SD algorithm (using the ZF solution to be the radius of the sphere) may be lower than that of the GPDA-SD detector, hence using the SD algorithm directly will be a better choice.
Fig. 4.5 Block diagram of the GPDA-SD detector. Number of subcarrier 64 Length of cyclic prefix 16
Channel Rayleigh Fading
Path 2 Relative power (dB) (0,0)
Number of iterations (GPDA) 2 (if no mention)
Modulation
16QAM (M=16, N=30) 64QAM (M=21, N=35) Table 4.1 Simulation parameters for MIMO-OFDM system.
this Section, we compare the BER performance between MCMC and MCPDA, and then
In
we show the influence on the number of iterations and the number of parallel for the MCPDA detector. After that, we compare the BER performance of proposed two detectors
with the optimal ML detector, the SD algorithm, and the GPDA detector. We also compare the complexity of proposed two detectors with that of the SD algorithm. Finally, we will show the effect of channel estimation error for different detection scheme. Note that, our simulations are based on real-valued signal model (i.e. equation (3.21)).
Fig. 4.6 and Fig. 4.7 show the BER performance for the MCMC detector and the MCP
mance for the MCPDA detector with different iterations and
s the BER performance for parallel MCPDA detector with different comb
crea
DA detector with different initial point, respectively, where obtained by running 5 times iteration. As we can observe in Fig. 4.6, the initial point dominates the BER performance seriously, but in the Fig. 4.7, it does not absolutely dominate the BER performance. Thus, the MCPDA detector solved the initial point problem which in the MCMC detector. Moreover, if the initial point does not affect BER performance too much, why do not we choose an initial point which has lower complexity?
Fig. 4.8 shows the BER perfor
q=16. The result shows that the MCPDA detector only needs few times iteration to be converged. In other words, with the number of iteration increasing, the BER performance of the MCPDA will get little better. Thus, we must increase number of parallel to reach better BER performance.
Fig. 4.9 show
inative iterations and q=16. The result shows that when the parallel MCPDA detector with the number of parallel in sing, the BER performance of the parallel MCPDA can be
better. But, how many parallels we need? That is another question which is worth us pondering.
Fig. 4.10 shows the BER performance for different detection schemes with q=16. As we can observe form Fig. 4.10, at BER=10−3, the GPDA detector and the ML detection have about 5 dB disparity. The ML detector and the SD algorithm have the same performance; the GPDA-SD detector and the GPDA-MCPDA detector are really close to the ML detector. In this simulation, the SD algorithm use the distance calculated by ZF as the radius of sphere and GPDA-MCPDA is obtained by running 20 parallel randomly initialized MCPDA; each MCPDA has depth of 2. In Fig. 4.11, it shows the BER performance of the aforementioned detectors for the case q=64.
Fig. 4.12 shows the com lexity for different detection schemes with p q=16. The result shows that the complexity of the GPDA-SD detector and that of the GPDA-MCPDA are both lower than that of the SD algorithm, especially at low SNR regions. Nevertheless, at the high SNR regions, the complexity of the GPDA-SD detector is significantly less than that of the GPDA-MCPDA. I In addition, when the SNR exceeds the N dB, the complexity of the SD algorithm may continue to decrease and less than that of the GPDA-SD detector, thus we will switch the GPDA-SD detector to the SD algorithm. In Fig. 4.13, it shows the complexity of the aforementioned detectors for the case q=64.
Fig. 4.14 shows the channel estimation error for different detection schemes with
q=1
channe
6, at SNR=25 dB. The result shows that the GPDA detector is not too sensitive to l, whereas the ML detector, the GPDA-MCPDA detector and the GPDA-SD detector are sensitive to channel. Note that, E[|H| ]2 =E[|H| ]2 + ΔE[| H| ]2 where E H[| | ]2 is estimated channel composed of real channel E H[| | ]2 and estimation error E[|ΔH|2].
Fig. 4.6 The BER performance for MCMC detector with different initial point.
Fig. 4.7 The BER performance for MCPDA detector with different initial point.
Fig. 4.8 The BER performance for MCPDA detector with different iterations.
Fig. 4.9 The BER performance for parallel MCPDA with different combinative iterations
=16. Fig. 4.10 The BER performance for different detection scheme with q
=64. Fig. 4.11 The BER performance for different detection scheme with q
=16. Fig. 4.12 The complexity for different detection scheme with q
=64. Fig. 4.13 The complexity for different detection scheme with q
Fig. 4.14 The channel estimation error for different detection scheme.
Chapter 5
valence between the SSIC Algorithm
In this Chapter, we prove the Soft Successive Interference Cancellation (SSIC) algorithm
The SSIC Algorithm
In ccessive interference cancellation algorithm is a simple idea, which has been widely used in m
orm the left by
On the Equi
and the PDA Algorithm
and the PDA algorithm are equivalent. The simulation results demonstrate that the SSIC algorithm and the PDA algorithm have exactly the same BER performance.
5.1
communication systems, the su
any fields. Unfortunately, if some symbols were not reliably detected, it would make the BER performance worse. Therefore, instead of
“cancelling by signal subtraction” (which would cause error propagation in case of incorrect decoding), we resort to a “soft” mean value, which can deal with reliability information for symbols rather than with hard decisions only, i.e. SSIC [17] algorithm.
To obtain the system model of the SSIC algorithm, we multiply (2.1) f HH
to obtain
= +
y Ga n
(5.1)where y H r= H , G H H= H , and n H v= H .
The model of the SSIC algorithm is obtained by multiplying (5.1) from the left by G−1
lement is 1 and the other element are 0. In order to co
ed signal vector
z
can be expressed as follows:(5.3)
where is the interferences from undetected symbols, is the interference due to the and ei is a column vector
whose ith e obtain mputational efficiency
and to compare with the PDA algorithm under the fair condition, we choose (5.2) as the system model for the SSIC algorithm.
After interference-cancelled receiv
decision error of previously detected symbols, and we define D to be a set of symbols which have been detected. Thus, to estimate the associated probabilities for an element ai, we treat all other elements a j ij( ≠ as multivariate random variables, and from equation )
(5.3), we define
as the effective noise on ai, and approximate it as a Gaussian noise with matched mean and
covariance:
2 -1
, ( ) 1
Fig. 5.1 Block diagram of the basic SSIC procedure.
The basic procedure for the SSIC detector is as follows:
1. Based on the matrix in (5.1), we obtain the optimal detection sequence proposed for the V-BLAST OSIC in [9] and denote the sequence as
G
by approximate the in (5.3) as Gaussian distributed, and set the results equal to the corresponding elements of .
z
pk
6. If i N< T , let i= +i 1 and back to step 4. Otherwise, carry on step 7.
7. If ∀i, pi has converged, go to step 8. Otherwise, let iter=iter+1 and return to step 3.
8. For j=1,...,NT, make a decision ˆa for j a via j
ˆj j( ), arg max{ j( )}
a = x l l= m p m (5.13) The block diagram of the basic SSIC procedure is shown in Fig. 5.1.
5.2 The Equivalence of the SSIC Algorithm and the PDA Algorithm
In Section 5.1, we can calculate P(z ai =x mi( )) by equation (5.9) as follows:
( i i( )) exp( ( i i( ) i)H i1( i i( ) i
P z a =x m ∝ − −z ex m −B ∑− z e− x m −B )
Substituting equation (5.3) in to equation (5.9), we can get
, ,
Substituting equation (5.5) in to equation (5.14), we can obtain
, ,
where can be observed in equation (3.6) and equation (5.6).
Therefore, using equation (3.9) and equation (5.9) can obtain the same posterior probability . In other words, we confirm that the SSIC algorithm and the PDA algorithm are
[ ] [ ]
i Cov i Cov i
∑ = B = N =Ωi
i( ) P m
equivalent.
5.3 Simulation Results
Perfect channel information Perfect noise variance estimation Number of subcarrier 64 Length of cyclic prefix 16
Channel Rayleigh Fading
Path 2 Relative power (dB) (0,0)
Modulation 16QAM
Table 5.1 Simulation parameters for SSIC algorithm and PDA algorithm.
In this Section, we show the BER performance for the SSIC algorithm and the PDA algorithm. Furthermore, we will compare the complexity for the SSIC algorithm and the PDA algorithm. The simulation parameters are shown in Table 5.1.
Fig. 5.2 shows the BER performance for the SSIC algorithm and the PDA algorithm. As we can observe form Fig. 5.2, the SSIC algorithm has same the BER performance with the PDA algorithm.
Fig. 5.3 shows the complexity for the SSIC algorithm and the PDA algorithm. The result
shows that the complexity of the SSIC algorithm is slightly more than that of the PDA algorithm. Since the SSIC algorithm needs to subtract the soft information form y .
Fig. 5.2 The BER performance for the SSIC algorithm and the PDA algorithm.
Fig. 5.3 The complexity for the SSIC algorithm and the PDA algorithm.
Chapter 6 Conclusion
In this thesis, we establish the equivalence of the SSIC algorithm and the PDA algorithm.
Furthermore, we proposed two detectors, GPDA-MCPDA and GPDA-SD, to reduce the complexity of the SD algorithm for near-optimal detection in a MIMO-OFDM spatial multiplexing system with higher order QAM constellations (16QAM/64QAM). These two methods exploit the GPDA detector which performs well at the low SNR regions. At the high SNR regions, the first proposed detector is combining GPDA and MCMC, which incorporates the concept of PDA to calculate the covariance then construct a Markov Chain to make it converge to the target distribution; the second one is based on the SD algorithm using the GPDA detector solution to be the radius of the sphere. Simulation results demonstrate that both detectors can achieve near-optimal performance with lower complexity as compare with the SD algorithm, especially at the low SNR regions.
Bibliography
[1] G. Stuber et al., “Broadband MIMO-OFDM wireless communications,” Proceedings of the IEEE, vol. 92, no. 2, pp. 271-294, Feb. 2004.
[2] A. v. Zelst and T. Schenk, “Implementation of a MIMO OFDM-based wireless LAN system,” IEEE Trans. on Sign. Proc., vol. 52, no. 2, pp. 483-494, Feb. 2004.
[3] B. Hassibi and H. Vikalo, “On sphere decoding algorithm. I. Expected complexity,”
IEEE Trans. on Sign. Proc., vol. 53, no. 8, pp. 2806-2818, Aug. 2005.
[4] J. Luo, K. et al., “Near-optimal multiuser detection in synchronous CDMA using probabilistic data association,” IEEE Commun. Lett., vol. 5, pp. 361-363, Sept. 2001.
[5] D. Pham et al., “A generalized probabilistic data association detector for multiple antenna systems,” IEEE Commun. Lett., vol. 8, pp. 205-207, Apr. 2004.
[6] C.P. Robert and G. Casella, Monte Carlo statistical Methods, Springer-Verlag, New York, 1999.
[7] P. W. Wolniansky et al., “V-Blast: An architecture for realizing very high data rates over the rich-scattering channel,” in Proc. Int. Symp. Signals, Systems and Electronics, pp.
295-300, Sept. 1998.
[8] O. Edfors et al., “An introduction to orthogonal frequency-division multiplexing,”
Division of Signal Processing, Lulea University of Technol., Sweden, Research Report 1996:16, Sept. 1996.
[9] G. D. Golden et al., “Detection algorithm and initial laboratory results using the V-BLAST space–time communication architecture,” Electron. Lett., vol. 35, no. 1, pp.
14-16, Jan. 1999.
[10] W. W. Hager, “Updating the Inverse of a Matrix,” SIAM Review, vol. 31, no. 2, pp.
221-239, June 1989.
[11] S. Liu and Z. Tian, “Near-optimal soft decision equalization for frequency selective
MIMO channels,” IEEE Trans. on Sign. Proc., vol. 52, no. 3, pp. 721-733, Mar. 2004.
[12] X. Mao et al., “Markov Chain Monte Carlo MIMO Detection Methods for High Signal-to-Noise Ratio Regimes,” IEEE Globecom 2007 Wireless Communications , Washington DC, Nov. 26-30, 2007.
[13] B. Farhang-Boroujeny et al., “Markov chain Monte Carlo algorithms for CDMA and MIMO communication systems,” IEEE Trans. on Sign. Proc., vol. 54, no. 5, pp.
1896-1909, May 2006.
[14] http://icl.cs.utk.edu/papi/; http://www.mathworks.com/.
[15] G. Casella and E. I. George, “Explaining the Gibbs sampler,” The American Statistician, vol. 46, no. 3, pp. 167–174, Aug. 1992.
[16] U. Fincke and M. Pohst, “Improved methods for calculating vectors of short length in a lattice, including a complexity analysis,” Mathematics of Computation, vol. 44, pp.
463-471, Apr. 1985
[17] L. Song et al., “Successive Interference Cancellation Schemes for Time-Reversal Space-Time Block Codes,” IEEE Trans. on Veh. Technol., vol. 57, no. 1, pp. 642-648, Jan. 2008.