International Journal of Scientific & Engineering Research, Volume 6, Issue 4, April-2015 519

ISSN 2229-5518

ANALYASIS OF MIMO-OFDM SYSTEM IN CARRIER FREQUENCY OFFSET

USING PARTICLE FILTER

K.CHAITHANYA NIKLESH1 P.MALARVEZHI2

1SRM University, Department of Electronics & Communication Engineering Kattankulathur, Chennai, India chaithanyaniklesh@gmail.com

2SRM University, Department of Electronics & Communication Engineering Kattankulathur, Chennai, India malarvizhi@ktr.srmuniv.ac.in

Multiplexing (OFDM), sequential Monte Carlo, Particle Filter.

—————————— ——————————

The combination of orthogonal frequency division multiplexing (OFDM) and multiple antennas at both the transmitter and the receiver, referred to as multiple input multiple-output (MIMO). In the MIMO-OFDM transmission system the carrier frequency offset (CFO) induced by the local oscillator and the Doppler shift results in the interference of sub-carriers and further incurs an error floor effect. It is required to estimate and compensate for the CFO at the receiver side. Moreover, the unknown channel parameters mixed in the transmission will deteriorate the acquisition condition in practice .In contrast to frequency synchronization schemes of single-input single-output (SISO)-OFDM systems this problem is extended to multi- dimensional parameter acquisition in MIMO systems. Hence, it incurs much difficulty for the coherent detection. Various methods have been investigated to estimate multiple time-invariant frequency offset and

channel parameters, such as maximum-likelihood

estimation correlation-based method, expectation maximization (EM) and iterative methods. They all use the Cramer Rao bound as the precision reference. In addition, an extended Kalman filter (EKF) is designed to track time-variant parameters.

Recently, a sequential Monte Carlo approach in the Bayesian framework has been widely investigated in the wireless communication systems. These particle filtering algorithms estimate the state information with a posterior probability. Moreover, they can achieve theoretical optima in the presence of nonlinear and non-Gaussian models. In comparison with other nonlinear estimation methods such as EKF and unscented Kalman filter (UKF), a particle filter will yield more accurate results along with the flexibility of the algorithm design .However, in terms of this high-dimensional parameter estimation, it is very hard to get appropriate particles with small distribution variance on each dimension by the general sequential importance sampling (SIS) method.

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When the transmit/receive antennas in a

MIMO system share one oscillator, the difference of

CFOs among different antenna pairs can be

time-domain vector by an 𝑁-point inverse discrete

Fourier transform (IDFT):

𝑝 𝑝

negligible. Under such a scenario, the particle filter is

investigated to marginalize out the unknown time- variant Rayleigh channel from the sampling space. The channel is estimated by the Kalman filter on the assumption that its response model is known. The particle weights of CFO are evaluated according to the estimation error. Simulation results demonstrate the algorithm can estimate the CFO and the channel jointly with high accuracy, as well as its robustness to small variations of the channel model.

𝑑𝑘 (𝑛) = 𝑤𝐻 𝑑𝑘 (𝑛) (1)

Where is the IDFT matrix. To prevent inter-

symbol interference (ISI), a cyclic prefix (CP) of 𝑁𝑔

symbols is appended in front of each IDFT output

block. The resulting vector of length is digital-to- analog converted by a pulse-shaping filter with a

finite support on [0, 𝑇𝑑] where with 1/(𝑁𝑇𝑠) being

the subcarrier spacing. Finally, the analog signal from

the pulse-shaping filter is transmitted from the 𝑝-th

antenna over the channel.

That is, be the normalized carrier frequency offset with respect to (w.r.t.) the carrier spacing

1/(𝑁𝑇𝑠) between transmit antennas of the 𝑘-th user and the 𝑞-th receive antenna of the BS. In the

presence of CFOs, the received vector signal after

removing the guard interval becomes

𝑞 𝑝

𝑝.𝑘

𝑟𝑞 (𝑛) = ∑𝑘

∆(∈𝑘 (𝑛) ∑𝑚𝑡

𝐷𝑘 (𝑛)ℎ𝑘

(𝑛) +

𝑣𝑞(𝑛) 0

Figure1 Block diagram for OFDM SIGNAL

𝑘 𝑝 (𝑛)ℎ𝑝.𝑘 (𝑛) + 𝑣𝑞 (𝑛)

GENERATION

= ∑𝑘=1 𝐷𝑘 𝑘

(2)

Consider an uplink OFDM system with 𝑁 subcarriers and 𝐾 active users. The base station (BS) and each active user are equipped with 𝑀𝑟 and 𝑀𝑡 antennas, respectively. Each user is assigned 𝑁𝑘

exclusive subcarriers, we denote the index set of

carriers assigned to the 𝑘-th user as where 1 ≤ 𝑖𝑙 ≤ 𝑁

for 𝑙 = 1, 2, 𝑁𝑘. The proposed algorithms are

applicable to any carrier assignment scheme (CAS).

Note that even though we are using orthogonal subcarriers, self-interference and MAI are inevitable

Recall that the CP length equals the channel length plus timing offset. Under such an , the timing errors do not explicitly appear in the received signal model. Thus, we have suppressed the timing errors in (2). Several approaches have been proposed to model the time-varying channels and frequency offsets in mobile environments. Since we assume a normalized

Doppler spread 𝑓𝐷𝑇𝑑 ≪ 1,we adopt the following

first-order autoregressive (AR) parametric model

widely used to characterize the time-varying frequency offset and channel responses.

𝑞 (𝑛) = 𝛼𝑞 ∈𝑞 (𝑛 − 1)

due to CFOs Denote by the data symbols transmitted

∈𝑘

𝑘 𝐸 𝑘

by the 𝑘-th user from the 𝑝- th transmit antenna over the 𝑛-th OFDMA block. For convenience, we assume

that the data symbols are taken from the same

+ 𝑤𝑞 (𝑛)

𝑞 (𝑛) = 𝛼𝑞 ℎ𝑞 (𝑛 − 1) + 𝑤𝑞 (𝑛) (3)

complex-valued finite alphabet and independently

identically distributed. The 𝑖-th entry of is non-zero

if and only if .next, is converted to the corresponding

ℎ𝑘

𝑘 𝐸 𝑘

𝑘 ℎ

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Furthermore, we assume that Note that we assume independent fading across transmit antennas and multi-paths. The time variation in CFO in (4) arises from (a) local oscillator instability due to temperature/voltage variations and (b) changes in relative platform Doppler velocity. In the next section, we propose a pilot-aided parallel Schmidt extended particle filter approach to estimate based on

the received signal (𝑛), assuming exact knowledge of

and in the BS.

The incoming input binary streams are first mapped into constellation points according to any of the digital modulation schemes such as QPSK/QAM. In QPSK (Quadrature Phase Shift Keying) modulation, the incoming binary bits are combined in the form of two bits and are mapped into constellation point. The N constellation points are modulated using N sub- carriers whose carrier frequencies are orthogonal in nature. The modulation is similar to taking inverse discrete/fast fourier transform (IDFT/IFFT) operation. The output of N point (IFFT) block is the OFDM signal. Now the N OFDM signal samples are combined and then transmitted i.e., the parallel samples are now

Ec is the energy of subcarrier, All the preamble based frequency offset estimation methods given in literature aims at accuracy and increasing the range of frequency offset estimation. The importance of frequency offset estimation in various high speed broadband wireless applications can be understood

Particle filtering is a sequential sampling method built on the Bayesian paradigm. From the Bayesian theory, at sample k , the posterior distribution p(x0:k | r0:k) is the main entity of interest. However, due to the nonlinearity of the measurement equation, its analytical expression is not tractable. Alternatively, particle filtering can be applied to approximate this PDF by stochastic samples generated using a sequential importance sampling strategy. Particle filtering is an extension of the sequential methodology. It consists in recursively estimating the required posterior density function p(x0:k | r0:k) by a set of M random samples with associated weights, denoted by

𝑀

converted into serial sequence and then it is

𝑥0

𝑗 �𝑤(𝑗) (6)

transmitted. The OFDM baseband signal at the transmitter is expressed as in

𝑝̂ �

𝛾0:𝑘

� = � 𝛿�𝑥0:𝑘 − 𝑥0:𝑘 𝑘

𝐽=1

1/√𝑛 ∑𝑁−1 𝑥(𝑘)𝑒𝑗2𝜋𝑛𝑘/𝑛

𝑥(𝑛) =

Where is drawn from the importance function is the

Dirac delta function and is the normalized importance

where

𝑘=0 . 0<n<N-1 (4)

weight associated with the j-th particle. The weights are updated according to the

(𝑚) ,𝛾 �𝑝�𝛾 𝑥 (𝑚) ,𝛾 �

n -time domain sample index

𝑤(𝑚) 𝛼

𝑝�𝛾 𝑘 𝑥0:𝑘−1

(𝑚) 𝑚

0:𝑘−1

𝑤𝑚 (7)

X (k) -modulated QPSK data symbol on the kth

subcarrier

𝜋(𝑥𝑘

/𝑥0:𝑘−1 𝛾0:𝑘)

N -total number of subcarriers and x (n) -OFDM signal.

In order to maintain a signal to noise ratio (SNR) of

20 decibels or greater for the OFDM carriers, offset is limited to 4% or less than the inter carrier spacing which is simulated in the lower bound for the SNR at the output of the DFT for the OFDM carriers in a channel with AWGN and frequency offset is derived as and is given by

𝑆𝑁𝑅 ≥ �𝐸𝑐 � �𝑠𝑖𝑛𝜋𝑣 � /{10.594(𝐸𝑐 )}( {𝑠𝑖𝑛𝜋𝑣)2 } (5)

After a few iterations, particle filtering is known to suffer from degeneracy problems. So we integrate a resampling step to select particles for new generations in proportion to the importance weightsThis is due to the fact that the particle filter is then only used to estimate the nonlinear states, while the remaining conditional linear-Gaussian states are estimated using the closed-form particle filter. In our case, conditionally on the nonlinear state variables the DSS model contains a linear substructure on h,

subject to Gaussian noise. Using the Bayes' theorem,

𝑁0

𝜋𝑣 2

𝑁0

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the posterior density function of interest can be

written as

𝐸𝑝 [𝐵(𝑥𝜖𝐴)] = � 𝑝(𝑥). 𝐵(𝑥𝜖𝐴)𝑑𝑥

𝑝 �𝑥0:𝑘� = 𝑝( ℎ

) (8)

𝑝 (𝑥)

∫

𝜋(𝑥)

. 𝜋(𝑥). 𝐵(𝑥𝜀𝐴)𝑑𝑥 (13)

𝛾0:𝑘

(𝜃0:𝑘,𝜀0:𝑘,𝛾0:𝑘)(𝜃0:𝑘,𝜀0:𝑘,𝛾0:𝑘)

where _ is a distribution for which we require that

Where is analytically tractable and can be obtained via a particle filter associated with each particle. Indeed, the j-th PDF is a multi dimensional gaussian

𝑝(𝑥) > 0, 𝜋(𝑥) > 0

Thus, we can define a weight w(x) a

𝑝(𝑥)

probability density function. The mean and the

covariance can be obtained using the particle

𝑤(𝑥) =

𝜋(𝑥)

(14)

filtering equations given by the time update equations

This weight w is used to account for the differences between p and the _. This leads to

(𝑗)

𝑘

−1

𝑘

(𝑗)

𝑘

−1

𝑘

∑ 𝑗𝑘

𝑘−1=

∑ 𝑗𝑘

−1

𝑘

(9)

Ep [B(xϵA)] = � p(x). w(x). B(xϵA)dx

A particle filter is a nonparametric implementation of the Bayes filter and is frequently used to estimate the state of a dynamic system. The key idea is to represent a posterior by a set of hypotheses. Each hypothesis represents one potential

state the system might be in. The state hypotheses are

= Eπ [w(x)B(xϵA)] (15)

Let us consider again the sample-based

representations and suppose the sample are drawn from. By counting all the particles that fall into the region A, we can compute the integral of over A by the sum over samples

represented by a set S of N weighted random samples

[∫ π(x)dx ≈ 1/n ∑n

B(s ∈ A)

i=1 ] (16)

𝑆 = {< 𝑆[𝑖] , 𝑊 [𝑖] > 𝑖 = 1, … … 𝑁} (10)

where s[i] is the state vector of the i-th sample and

w[i] the corresponding importance

weight. The weight is a non-zero value and the sum over all weights is 1. The sample

set represents the distribution

𝑁

3.1 CFO Estimation on SNR:

The increment of SNR estimation accuracy of channel tends to be higher .It can also be observed that the number of the particles is another direction Factor of the estimation accuracy

𝑝(𝑥) = ∑𝑖=1 𝑤𝑖 𝛿𝑠 (𝑖)(𝑥)

(11)

:

We are faced with the problem of computing the

expectation that x ∈ A, where A

is a region. In general, the expectation Ep[f(x)] of a

function f is defined as

𝐸𝑝 [𝑓(𝑥)] = � 𝑝(𝑥). 𝑓(𝑥)𝑑𝑥 (12)

Let B be a function which returns 1 if its

argument is true and 0 otherwise. We can

express the expectation that x ∈ A by

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0.18

0.16

0.14

CFO estimation SNR

snr 0dB snr 6dB snr 12dB

0.25

0.2

CFO estimation NP

Np=20

Np=50

Np=100

0.12

0.1

0.15

0.08

0.06

0.1

0.04

0.02

0.05

0

0 5 10 15 20 25 30 35 40 45 50

Calculation steps

Figure2: CFO Estimation on SNR

3.2 INFERENCE:

0

0 5 10 15 20 25 30 35 40 45 50

Calculation steps

FIGURE 3: CFO Estimation on number of Particles

3.4 INFERENCE:

For SNR(db) | At Calculation steps | Estimation error |

0 | 13.2 | 0.1667 |

6 | 13.2 | 0.07143 |

12 | 13.2 | 0.0625 |

Table1: Comparison for CFO Estimation SNR

3.3 CFO Estimation on number Of Particles:

As the of particles increase the estimation error will be lower. In addition the particle with large weights have large number than that of the particle with lower weight.

Table2: Comparison for CFO Estimation NP

Particle filtering algorithm for frequency synchronization in MIMO-OFDM systems. It is based on the sequential Monte Carlo approaches in the Bayesian framework, and simplified to a one- dimensional particle sampling in order to improve the computation efficiency. Numerical simulation shows that the algorithm can obtain high accuracy of CFO and estimation performance. Additionally, it can simultaneously estimate time-varying CFO and channel. It also has some robustness to the variation

of the channel model.

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[1] ZHANG Jing J Shanghai Univ (Engl Ed), 2009,

13(6): 438–443 Particle filter for joint frequency offset and channel estimation in MIMO-OFDM systems

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648–656.

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469–473.

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[8] S. Shahbazpanahi, A. B. Gershman, and G. B. Giannakis, “Joint blind channel and carrier

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1 K.CHAITHANYA NIKLESH is pursuing his master’s degree in communication systems from department of electronics and communication, SRM University, Kattankulathur. His research interests include wireless communication, multiple-input- multiple-output.(MIMO).

2 P.MALARVEZHI is working as assistant professor in department of electronics and communication, SRM University, Kattankulathur

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