Vol 2 Issue 6

1. International Journal of Research in Advent Technology, Vol.2, No.6, June 2014
E-ISSN: 2321-9637
233
Design and Implementation of SD for MIMO System
Using FPGA
Rashmi Pahal1, Dinesh Kumar Verma2Ajay sharma3
Electronics and communication Department1,2,3
P.D.M college of Engg1,2, DKOP Labs Pvt. Ltd.3
Rashmipahal04@gmail.com1, erdineshverma@gmail.com2,ajay@dkoplabs.com3
Abstract-This paper has shown the use of Newton Iterative Method for matrix inversion, it reduces the complexity
of calculating the unconstrained solution in Sphere Decoding for Multiple input Multiple output system. The paper
purposes the initialization procedure for Newton Iterative method and Q-Cholskey method for decomposition of
matrix to upper triangular matrix. This paper has shown the result of minimum Euclidian distance for a 2×2 MIMO
with 4-QAM.
Index Term- FPGA, Newton Iterative Method, MIMO, QAM
1. INTRODUCTION
MIMO system is a wireless system employing
multiple transmit and receive antennas[1].As the
capacity increase with the minimum number of
transmit and receive antenna[2], in the last years
significant interest in large MIMO
schemes[3][4][5].Hence low complexity detection
technique for such system are crucial for practical
application. A high performance detection method
Sphere Decoding (SD) [6] and its variation. Sphere
Decoding demands the inversion of channel matrix,
so an approximate inversion method is used to get the
approximate inverse[7][12]. The Sphere Decoding
algorithm was first introduced in[8] as a method for
finding lattice vectors of short length, and its
complexity is polynomial in dimension of the
lattice[9]. Sphere decoding was first applied to
communication problems in a paper on lattice code
decoding[10]. The use of newton-iterative method [7]
of inversion of matrix for SD target MIMO systems,
and focuses on minimum Euclidean Distance Vector.
This technique can be interested also to other MIMO
detection scheme that require inversion of large
matrices.
In this paper, we introduce a SD for measurement of
minimum Euclidean distance vector, Newton
algorithm for the calculation of matrix inversion. In
section II, we review the system Model and decoding
algorithm for Sphere Decoder. In section III, we
show our stimulation result for considered MIMO
system. Section IV show the conclusion of SD
Algorithm.
2. SYSTEM MODEL AND DECODING
ALGORITHM
Consider a multiple antenna system with nT transmit
antennas and nR receive antennas over a flat fading
Rayleigh channel. Such a system is represented by
̂=
+ , … (1)
The case in which nT = nR. The baseband signal
vector transmitted is denoted as = [1,2,……,nT],
where each component of the vector is independently
matrix whose elements mij represent the complex
transfer functions from the jth transmit drawn from a
complex constellation such as QAM or PSK, our
work is on 4-QAM. M is nT×nR channel antenna to
the ith receive antenna. Each element of M is an
independently identically distributed (i.i.d) zero mean
circular complex Gaussian random variable of unit
variance. Noise vector, is an i.i.d. complex
Gaussian random variable of variance N0 that is
independent of M and u.
To obtain a lattice representation of this multiple
antenna system, we begin by transforming the
complex matrix equation into the real matrix
equation. This lattice representation is given as:
r = [

2. {̂ᵀ} {̂ᵀ}]T …(2)
M =

3. {
ᵀ} −{
ᵀ}
{
ᵀ}

4. {
ᵀ}
…(3)
u = [

5. {ᵀ} {ᵀ}]T …(4)
w = [

6. { ᵀ} { ᵀ}]T …(5)

7. International Journal of Research in Advent Technology, Vol.2, No.6, June 2014
E-ISSN: 2321-9637
234
Since the elements of M are i.i.d. complex Gaussian,
the rank of the matrix M is almost always 2nT, so we
can think of the 2nT columns of M as basis vectors
{mi} of a lattice lying in a 2nR-dimensional space.
The vector acts as the “coordinates” of a lattice
point.
Receiver is familiar with the channel matrix, so
maximum likelihood (ML) detection is achieved by
searching for a possible transmitted that generate
the smallest Euclidean distance
dML =arg mind ǁr – Mu ǁ2 …. (6)
An efficient method of solving (6) is provide by
SD[6][11] that requires an equivalent from of (1). It
is easily shown that (7) is equivalent to
dML =arg mind ǁ R(u-) ǁ 2 …(7)
where, R is the upper triangle matrix obtained from
the Q-Cholesky factorization, and the unconstrained
solution is given by =(CHC)-1 CHr[6].
For an invertible channel matrix C.
=M-1 r …(8)
This form is the basic of scheme implemented
in[17][18]. The implemented scheme [17] is
attractive for large MIMO systems.
We term the process of calculating the unconstrained
solution and upper triangular matrix R as
preprocessing and determining the ML solution. The
SD algorithm employs depth first tree searching[16]
for finding the ML data vector. The unconstrained
solution
is calculated during the preprocessing, a
task that can contribute significantly to complexity if
no of antennas at any of the end is large. We use
Newton iterative approach for obtaining the matrix
inverse, as it is massively parallelizable with good
numerical stability and requires O(log2n) parallel
time units to achieve an accuracy of 2-O(log2n) when
N is large[15]. Other method used for approximate
matrix inverse Jacobi iteration [13][14] converges
linearly, while Newton Iterative method converges
Quadratically.
The Newton algorithm for inverting M is given
by[12][7]
XK+1=XK(2I–MXK) …(9)
Theorem 2 in [12] shows the initialization X0 = α0MT
, With α0 positive and sufficiently small. Initial
choice of α0 is crucial in the convergence process for
calculating the inverse of matrix.
The residual matrix, EK is a measure of deviation of
computed inverse from the actual inverse of given
matrix A. The Residual matrix EK after k iteration is
given as
EK=I–M M+ …(10)
It is easy to show that in this method we have XK+1 =
XK
2 revealing quadratical convergence. By increasing
the number of iteration , accuracy of generalized
inverse matrix also increases. Q-Cholesky algorithm
is followed for decomposition of matrix to upper
triangular matrix[11].
3. SIMULATION RESULTS
The objective of the computer stimulation is to
investigate the effect of Newton iteration in
calculation of Unconstrained solutions on the overall
MIMO system. We have done all of our work on
Xilinx ISE tool for analysis of unconstrained solution
obtained by SD by use of Newton iterative method
for matrix inversion.
We analyze the result of Newton algorithm and
observe that the result obtained at 4th iteration are not
accurate while result obtained at 7th iteration are
converging to zero. So we use the result of 7th
iteration for SD algorithm. The analysis of Newton
result is by comparing the residue matrix at 4th and
7th iteration. We implement the SD for 2 × 2 MIMO
with 4 QAM. Considering the 4 QAM with average
symbol energy ,Es of 42, so the possible lattice
coordinates are S4 = [-3,-1,1,3].
Fig 1 The stimulation of Newton Iterative method for
matrix Inversion.

8. International Journal of Research in Advent Technology, Vol.2, No.6, June 2014
E-ISSN: 2321-9637
235
Fig 2 The stimulation of Q-cholesky
Fig. 3 The stimulation result of Sphere Decoding
algorithm.
4. CONCLUSION
In stimulation nT =2 transmit antennas and nR =2
receiving antennas, the average symbol energy was
42. The complexity face while doing the work is,
selection of initial choice of sphere radius. As if
radius chosen in small then it will leave some of the
lattice points, while if radius chosen is large then it
will search the points which are not present in
constellation, thus making the system divergent.
Second problem we faces is the usage of divider and
square Root in SD algorithm, as it is very difficult for
Verilog to code the divider and Square Root. Our
thesis has shown the stimulation only not the
synthesis. This work has a future scope that one can
use the IP CORE processor for Divider and Square
Root and can synthesis the SD.
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9. International Journal of Research in Advent Technology, Vol.2, No.6, June 2014
E-ISSN: 2321-9637
236
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