Complexity Reduction in ML Decoding For MIMO Systems

(1)

Complexity Reduction in ML Decoding For MIMO

Systems

Ramya Jothikumar, Nakkeeran Rangaswamy

Abstract—In this paper, we propose a combined Breadth first tree search ML (Maximum Likelihood)-ZF (Zero Forcing) method of detection for Spatial Multiplexed MIMO (Multiple Input Multiple Output) systems with reduced complexity. The detection of real and imaginary parts of QAM (Quadrature Amplitude Modulation) modulated symbol is carried out in successive level of tree which makes parallel processing possible. Reduction in complexity compared to conventional ML for a 2x2 system is 80% and for a 4x4 system is 83%.

Keywords–ML, ZF, Spatial Multiplexing, MIMO

I. I

NTRODUCTION

Higher data rates with the available spectrum are achieved by improving the spectral efficiency of the wireless system [1]. Spatial Multiplexing MIMO (Multiple Input Multiple Output) communication systems increase spectral efficiency, throughput and capacity at the cost of increased computational complexity in the receiver. The main task is designing an efficient detector for MIMO system. In order to reduce the complexity, linear detection methods like zero forcing (ZF) and minimum mean square equalization has been proposed, the drawback is, they show a poor bit-error-rate (BER) performance. Ordered successive interference cancellation decoders such as vertical bell laboratories layered space time (V_BLAST) show slightly better performance, but suffer from error propagation [2]. Studies state that, Maximum Likelihood (ML) detection is the optimal detection technique but with more computational complexity, as the number of antennas and signal constellation size increases. In order to overcome the problem, the Sphere Decoder (SD) [3] has been introduced for MIMO detection, where it searches within a sphere of radius R for ML solution. If the initial radius is too large then excess burden of computation is required and if the radius is small then it cannot guarantee for optimal BER performance. However, the complexity is also a variable.

Further, K-best lattice decoder approach provides a fixed complexity but it is considerably higher than the complexity of the SD in order to guarantee a quasi-ML performance [4]. In such cases, ML detection is preferred where the complexity is fixed and the performance is optimum. In this letter, we propose to combine ML of breadth first tree search decoding with ZF equalization for a shuffled channel structure which reduces the complexity and requirements for hardware implementation of decoder without any degradation in the performance of MIMO system. The combination of breadth first ML with linear ZF detection to reduce the complexity of hardware realization is analyzed for the very first time in literature.

II. M

ODELING OF

M

IMO

S

YSTEMS

In a MIMO system, with NT transmit antennas and NR receive antennas, the received signal can be represented by

Y H X n (1)

Where X =(X1, X2,…….., XNT) T

denotes the vector of transmitted symbol. n = (n1, n2…… nNT)Tis the vector of independent and identically distributed (i.i.d) Gaussian noise and Y = (Y1, Y2…….YNT)Tis the vector of received symbols. H denotes the NT x NRchannel matrix where hij represent the complex transfer function from the jth transmit antenna to the ithreceive antenna. For numerical simulations, the entries of H are modeled as i.i.d, Rayleigh fading. Hence, H can be represented as

1,1 1,2 1,

2,

2,1 2,2

,1 ,2 ,

T T

R R R T

N

N N N N

H H H

H

H H

H

H H H

 

 

  

 

 

 

  



(2)

The values for Xj ,j=1……. NT are chosen from the complex constellation Ω with c bits per symbol, i.e.

|Ω|=2c, where the set of all possible transmitted vector symbol is denoted by ΩNT .Assume that all the components of H are known at the receiver and NT= NRis assumed in sequel. As very large scale integration (VLSI) or digital signal processors (DSP) [5] do not support complex-valued operations in practice the complex values of received vectors and channel matrix are decomposed from the N-dimensional problem in to a 2N-dimensional real-valued problem [5], which can be written as

( ) ( ) ( ) ( ) ( )

R Y R H I H R X R n I Y I H R H I X I n



  _    _ 

       

       

(3)

With the new matrices having larger dimensions

M=2NTand N=2NR, where R(Y) and I(Y) indicate the real and imaginary part of Y respectively. The complexity is reduced since the real-valued signal provides a simpler Partial Euclidean Distance (PED) calculation and the number of visited nodes is less than in complex-valued signal model [6]. The complex valued system of equations can be decomposed in to a system of equations with only real valued numbers as follows

Y  H X  n (4)

Each i

X ,i=1…….NRin X is chosen from the set of

real numbersΩ, which in the case of 4-QAM modulation isΩ= {-1, 1} and for 16 QAM modulation isΩ= {-3, -1, 1, 3}.The maximum likelihood which minimizes the value

2

(2)

H  Q R , Y  H X 2 can be rewritten in recursive manner as follows

2

2 1

( )

ˆ H

M

j i j

i M j i

T X Y H X

Q Y R X

Y R X

 

 





 



(5)

Where ˆ H

Y  Q Y , rijdenotes the (i, j) th

element of R

and i

X is the jthelement ofX , the norm is computed for

‘M’ antenna’s. Assuming ( 1 )

1( ) 0

M M

T _ X   for first

iteration the PED are given by

2

( ) ( 1 ) ( )

1

( i ) ( i ) ( i )

i i i

T X  T  X   e X (6)

Where Ti(X ( )i )is the PED of the ith element and

i=1….M is the layer number in the tree search and the

branch cost can be obtained as follows 2 2

( )

1

ˆ

( )

M i

i i i j j

j

e X Y R X



   

(7) Equation (5) can be represented by a tree structure with 2N levels and each node in the tree contains Ω child

nodes. Thus the ML solution can be achieved by finding the path with the smallest path metric in the tree constructed by (5) through (7) as cited from [7].

III. P

ROPOSED

A

LGORITHM

The zero forcing detection method is one of the simplest linear detection methods to decode the received signal. However, it suffer from error propagation and results in poor BER performance. The performance of the ZF is limited by the higher order antenna index that is decoded first. Though the ML detection gives superior performance in terms of BER, the complexity grows exponentially. ML detection requires exhaustive exponentially growing search among the entire candidate that become complex when the number of antennas and modulation order increases. For example in a 4x4 MIMO system with 16 QAM modulations, 65,536 candidate vector symbols have to be taken in to account for each received vector. To address this problem and to reduce complexity ML of breadth first tree search with reduced node is used for detection in higher indexed antenna which is decoded first and for remaining, detection is

done with ZF. Find 1

S  H  Y and calculate

2

d  Y  R S and set d as root node. Starting at the root node the partial Euclidean distance (PED) from the root node to the child nodes were computed at the Nth layer. The search is extended with minimum valued node to the (N -1)thlayer. Only the undiscarded node at each level is traced. The undiscarded node here refers to the node with minimum PED. The decision for X is made at their corresponding level. This procedure is repeated until a vector corresponding to the minimum distance at the first layer is found. The number of nodes being searched gets reduced as the approach reaches (N-1)thlayer till first layer giving a path for reduction in complexity. Clearly

this can be illustrated with a diagramatic representation as in Fig.1.

Fig.1. Example of breadth first tree search algorithm (l =4 layers)

The proposed scheme uses a channel shuffling strategy in which the searching order of the QAM modulated signal is varied. The shuffled form of channel which is given in [8], is cited in Eq. (6).

1,1 1,1 1, 1,

,1 ,1 , ,

( ) ( ) ( ) ( )

N N N N N N N N N N N N N N N N

R H I H R H I H

I H R H I H R H

H

R H I H R H I H

I H R H I H R H

 

 

 



 

 

 

 



 



(6) The order of detection of symbols changes from

 ( 1) , , ( ) ( 1) , , ( )

T

N N

X  R X   R X I X   I X (7)

To

 ( 1) ( 1) , , ( ) ( )

T N N

X  R X I X   R X I X (8)

(3)

operations required to perform a task. If more layers are detected with ZF beyond the subscribed one there exist a considerable amount of performance degradation and if less layers are detected with ZF, the complexity of the system increases. The optimum scenario between these two is the proposed one, here Zero forcing is merged with the Modified ML with reduced nodes without any degradation in performance and thus complexity is reduced to a greater extend. This could be illustrated with an example.

Example: Let us consider a MIMO system with NT=NR=2 which employs 4-QAM modulation. Construct a tree with 2N=4levels, accompanied by breadth first ML detection, applying real value decomposition and QR decomposition to the channel the relationship between input and output is given below as cited from [8],

2

1

1 1 1 2 1 3 1 4 1

2

2 2 2 2 3 2 4 2

3 3 3 4 3

3

4 4 4

4

ˆ

ˆ ₀

ˆ

ˆ 0 0

0 0 0

ˆ

Y _R _R _R _R _X

Y R R R X

Y R X

R R X

Y

R X

Y  

 

 

  

 

   

   

  

 

(9) Obtain S from the ZF algorithm and calculate d. Let l=N and d=d+1;

STEP1:

4

2

4 1 4 4 4 4 4

4 1 4

ˆ

, ( )

ˆ _{a r g m i n} ₍ ₎

X

X l e t f X Y R X

X f X

 

   





(10) STEP2:

3

2 2

3 2 3 4 4 4 4 3 3 3 3 3 4 4

3 2 3

ˆ _ˆ ˆ _ˆ

, ( )

ˆ _{a rg m in} ₍ ₎

X

X le t f X Y R X Y R X R X

X f X



      



 

(11) STEP 3:

2

2 2

2 3 2 4 44 4 3 33 3 34 4

2

2 22 2 23 3 24 4

2 3 2

ˆ _ˆ ˆ _ˆ _ˆ

, ( )

ˆ _ˆ _ˆ

ˆ _{arg min} ₍ ₎

X

X let f X Y R X Y R X R X

Y R X R X R X

X f X



      

   



 



(12) STEP 4:

Already the ZF solution is found as

1

S  H  Y _,

instead of finding the last layer solution with ML, zero forcing is used thereby reducing the complexity. The solution for the last layer is given as

1

1 1

S  H  Y

(13) Therefore the complexity for breadth first ML with reduced node is 2NΩ, where that of proposed algorithm is

(2N-1)Ω for 2x2 method and (2N-4)Ω for 4x4 method.

IV. S

IMULATION

R

ESULTS

Figs.2, 3, 4, 5, 6, 7 gives the complexity of 4 QAM, 16 QAM, and 64 QAM, for 2x2 and 4x4 MIMO systems.

Where the complexity refers to the number of real multiplication needed to decode the transmitted complex symbols. In the proposed algorithm, X1and X2for N=2, and for N=4, X1, X2, X3, X4were calculated by applying zero forcing, where the computational burden is higher. We denote the Modified ML with reduced nodes with ZF as proposed algorithm.

The reduction in complexity of Modified ML with reduced nodes with conventional SD is 51% for 2x2 and 60% for 4x4 MIMO systems. Since ML MIMO detection technique is a NP hard analysis, here the comparison for complexity is made with Sphere Decoder (SD) which is considered to be the less complex and give similar optimum performance as ML. So it is referred here as conventional ML in this paper as per the literature. The proposed algorithm reduces the overall complexity for 2x2 by 80% and for 4x4 by 83% with that of the conventional SD.

The complexity reduction is quite high in 4x4, since as we go down the tree structures the number of multiplication become higher and for those levels ZF is used, which reduce complexity to a considerable amount.

Fig.2. Complexity of 2x2 MIMO for 4QAM

(4)

Fig.8 shows the performance of Modified ML with reduced nodes in which order of detection is considered as in Equation (7) and in shuffled Modified ML the order is taken as per Equation (8).

Fig.6 Complexity of 4x4 MIMO for 16QAM

From simulation it is observed the performance remains same for both the ordering. The shuffled Modified ML is worthy to implement parallel, with reduced requirement of hardware.

Fig.8. Performance of Modified ML for reduced nodes with and without shuffled channel matrix

V. C

ONCLUSION

In this paper, the proposed algorithm reduces the complexity compared to conventional detection algorithm. Breadth first search Modified ML with reduced number of nodes and zero forcing at selected layers in addition to shuffled channel matrix paved the way for overall complexity reduction. Since complexity is fixed at each layer this feature can be utilized to enhance the performance, by adopting the parallel processing while implementation.

R

EFERENCES

[1] P.W.Wolniansky, G.J.Foschini, G.D. Golden, and

R.A.Valenzuela, “V-BLAST: An architecture for realizing very

high data rates over the rich-scattering wireless channel,” Proc.

URSI Int. Symp. Signals, Systems and Electronics, Pisa, Italy,

pp.295–300, Sep.1998.

[2] Andreas Burg, Moritz Borgmann, Markus Wenk, Martin

Zellweger, Wolfgang Fitchner and Helmut Bolcskei, “VLSI

Implementation of MIMO detection using the sphere decoding

algorithm,”IEEE Journal of Solid- State circuits., vol. 40, no. 7, pp. 1566-1576, July. 2005

[3] B.Hassibi and H.Vikalo, “On the sphere decoding algorithm I.

Expected Complexity, “IEEE Transactions on Signal

Processing, vol.53, no.8, pp.2806–2818, Aug. 2005

[4] Z.Guo and P.Nilsson, “Algorithm and implementation of the K

-best sphere decoding for MIMO detection,”IEEE Journal of

Selected Areas of Communication., vol. 24, no. 3, pp. 491–503, Mar. 2006

[5] Janne, janhunen, ollisilven, Markkujuntti, “Programmable

processor implementations of K-best list sphere detector for

MIMO receiver,“Elsevier Journal of signal processing., vol.

90, pp. 313-323, July. 2009

[6] Santra Roger, Alberto Gonzalez, vicenc almenar, Antonio

M.vidal”Practical aspects of preprocessing techniques for K

-Best tree search MIMO detectors”, Elsevier Journal of Computers and Electrical Engineering, vol. 37, pp. 451-460, May 2011

[7] Chung-An Shen, Ahmed M. Eltawil, Khaled N. Salama, and

Sudip Mondal “A Best-First Soft/Hard Decision Tree Searching

MIMO Decoder for a 4x 4 64-QAM System,” IEEE

Transactions on Very Large Scale Integration (VLSI) Systems,

(5)

[8] L.Azzam and E.Ayanoglu, “Reduced complexity sphere decoding via a reordered lattice representation,” IEEE

Transactions on Communication, vol. 57, no. 9, pp. 2564–2569,

Sep. 2009 .

A

UTHOR

’

S

P

ROFILE

Ramya Jothikumar

received Bachelor of Engineering in Electronics and Communication Engineering (ECE) in 2000 and Master of Engineering in Communication systems in 2004 from Madras University and Anna University, India respectively. She is pursuing her Ph.D. programme in the Department of ECE, Pondicherry Engineering College. She is currently working as Associate Professor in the Department of ECE at Sri Manakula Vinayagar Engineering College, Puducherry, India. Her research interests include wireless communication, MIMO and Digital Communication.

Complexity Reduction in ML Decoding For MIMO Systems