Comparative Performance Study of LS and MMSE Channel Estimation over Time Varying Channel in OFDM System

Comparative Performance Study of LS and MMSE Channel Estimation over Time Varying Channel in OFDM System

Berthe Souleymane, S. I. M. M. Raton Mondol School of Communication and Information Engineering, Chongqing University of Posts and Telecommunications Chongqing, 400065, P.R. China

Zhou Wei

School of Optoelectronic Engineering Chongqing University of Posts and Telecommunications

Chongqing, 400065, P.R. China

Abstract- Orthogonal Frequency Division Multiplexing (OFDM) has been recently applied widely in wireless communication systems, due to high data rate, transmission capability with high bandwidth efficiency and its robustness to multipath delay. Channel estimation is an essential problem at the receiver, where the wireless channel is usually frequency selective and time varying. The estimation of channel at pilot subcarriers is based on Least Square (LS), Minimum Mean Square Error (MMSE) while interpolation is done using linear interpolation. For performance evaluation of LS and MMSE channel estimators in OFDM system, the previous works use the block type pilot arrangement, where the pilot tones are transmitted into all subcarriers but at specific period which is suitable for frequency selective fading. In order to keep track of the time-varying channel characteristic, the pilot tones symbols must be placed as the coherence time which causes data reduction. Aiming at this disadvantage, we propose to use the comb-type pilot arrangement where the pilot tones are transmitted at each time to track the rapid variation of the channel. The Clarke model is used to perform the time varying channel and 16-QAM as the modulation scheme. The performance of the algorithms is measured in terms of Mean Square Error (MSE) and Symbol Error Rate (SER). Simulation results reveal that MMSE estimator provides better performance to track the time-varying channel .

Keywords: OFDM System; LS Estimator; MMSE Estimator; comb-type pilot arrangement; Block-type pilot arrangement

1. INTRODUCTION

   Orthogonal Frequency Division Multiplexing OFDM) has received a broad attention in wireless communication system due to its resistance against multipath fading and high spectral efficiency. For these reasons, it has been accepted by many of the future generation system such as LTE, IEEE802.11 and WIMAX [1].

   At high data rates, channel distortion to data is very significant; therefore channel estimation becomes essential before the demodulation of OFDM signals [2].

   In OFDM system, channel estimation methods can be

   In pilot based channel estimation, pilot tones that are known at priori by the receiver are multiplexed along with the data for channel estimation [4].

   The pilot-based channel estimation can be performed by either inserting pilot tones into all subcarriers of the OFDM symbol with a specific period (block type pilot arrangement) or inserting pilot tones into each OFDM symbols with a specific period of frequency(comb-type pilot arrangement).

   By now, many channel estimation algorithms are used such as LS estimator and MMSE estimator. The previous performance comparison between the two channel estimation estimators are reported previously [5-7]. All use the block- type pilot arrangement, which is suitable for frequency selective fading. In practice, with the mobility between the transmitter and the receiver, the wireless channel is time- varying channel. In order to keep track of the time-varying channel characteristic, it might incur too much pilot tones which causes data reduction.

   In this paper, our contribution is to compare the performances of LS and MMSE over time-varying channel using comb-type pilot arrangement. Monte Carlo matlab simulation is used to measure the performance in term of mean square error (MSE) and symbol error rate (SER).

   This paper is organized as follows. In section II overview of pilot based OFDM system model is explained. Section III is divided into two subsections: III.A described LS estimator,

   III.B presented MMSE estimator. Section IV presented our proposed method. Section V explained linear interpolation. Simulation results are discussed in section VI. Conclusion of the work is presented in section VII.

2. OFDM SYSTEM

   OFDM system model block diagram in Fig1 can be described as follow: at beginning the binary is mapped according to modulation followed by serial to parallel conversion. After that, pilot sub-carriers are multiplexed with

   divided into two classes: blind channel estimation and pilot

   data sub-carriers, which give

   X (k ) samples. Then IFFT is

   based channel estimation. The blind channel estimation is carried out by using the statistical properties of the received signals [3].

   performed on X (k ) sample that transform frequency domain samples X (k ) into time domain x(n) , which can be shown as

   NFFT 1

   x(n) IFFT{X (k)}

   n0

   j 2 kn

   X (k)e NFFT

   (1)

   where H (k ) FFTh(n)and W (k ) FFTw(n).

   W (k ) and H (k ) are the Fast Fourier Transform of w(n) and

   Mapper

   Mapper

   binary

   Channel

   Channel

   Adding Pilot

   S/P

   S/P

   IFFT

   IFFT

   P/S

   P/S

   X (k )

   x(n)

   Adding Cyclic Prefix

   xg (n)

   h(n) respectively.

   After that, the received pilots sub-carriers Y (k ) are obtained

   p

   Demapper

   Demapper

   P/S

   P/S

   Y (k)

   FFT

   FFT

   y(n)

   yg (n)

   w(n)

   S/P

   S/P

   +

   from Y (k ) and then H (k ) is calculated from the information

   binary

   Channel & Data Estimation

   Remove Cyclic Prefix

   carried by H p (k ) .The transmitted data samples can be:

   Y (k )

   Figure 1 OFDM System

   X (k )

   H (k )

   (8)

   Where NFFT represents the number of FFT points. To combat

   where H (k )

   is the estimated channel transfer function and

   inter-symbol interference guard band interval denoted by NG

   are added in each OFDM symbol and samples becomes xg (n) , which can be expressed as:

   Y (k ) is the received signal. Finally de-mapping takes place, where binary data are obtained at the receiver output.

3. CHANNEL ESTIMATION

   x( N

   x( N

   • n)

   • n)

   n N

   n N

   , N

   , N

   , …, 1

   , …, 1

   xg (n) FFT G G1

   A. LS ESTIMATOR

   A. LS ESTIMATOR

   (2)

   x(n)

   n 0, 1, 2, , NFFT 1

   Let X P

   be the diagonal matrix of pilots given as

   After passing through the time variant channel, the received signal yg (n) becomes

   X P diag{X P (1) X P (2)

   one OFDM symbols,

   X P ( NP )} , N P is the number of pilots in

   hP is the impulse response of the

   yg (n) xg (n) h(n) w(n)

   (3)

   channel at pilot sub-carrier in one OFDM symbol, and Wp is

   Where w(n) represents additive white Gaussian noise. The

   the AWGN channel noise at pilot sub-carrier. If there is no

   impulse response of the time variant channel expressed using Clarkes model as:

   h(n)

   can be

   Inter Symbol Interference (ISI), the signal received is written as

   M

   M

   2 j[2 f cos( ) ]

   YP XP FhP Wp

   (9)

   h(n) e d r r

   (4)

   M r 1

   Where YP the vector of output signal is after demodulation as

   Where M represents the number of propagation paths,

   Y [Y (1) Y (2) Y ( N

   )]T

   , T is transpoe, F the Fourier

   ~ U[ , ) and ~ U[ , ) are the random phase

   P p p p p

   r r Transform matrix. . The purpose of LS algorithm is to

   and angle of arrival of the

   rth

   multipath component

   minimize the cost function K without noise.

   respectively, and fd is the maximum Doppler frequency due 2

   to the relative motion between transmitter and receiver. After serial to parallel conversion, removal of guard interval takes

   K YP X P FhP

   (10)

   place from yg (n)

   Let H P,LS is the estimate impulse response of the channel

   y(n) yg (n NG )

   n 0,1, 2, , NFFT 1

   (5)

   H P,LS

   1

   X

   X

   P YP

   (11)

   Then y(n) is send to FFT block for the following operation

   LS has low computational complexity as compared to other channel estimators, but its major drawback is high MSE value.

   1 NFFT 1

   • j 2 kn

   (6)

   Y (k) FFT{y(n)} N

   y(n)e NFFT

   , n 1, 2, , N

   FFT

   FFT n0

   B. MMSE ESTIMATOR

   suppose that there is no ISI, because channel impulse response length is smaller than the guard band interval.The output of the FFT block is Y (k ) , which can be represented as

   If the channel and AWGN are not correlated, MMSE estimate of H is given by [8]

   H R R1

   Y (12)

   Y (k ) X (k )H (k ) W (k ) , k 1, 2, , NFFT

   (7)

   MMSE

   HPYP

   YPYP P

   Where

   R HPYP

   E{H Y H } R

   X H HPHP P

   Aiming at this disadvantage, we propose to multiplex the pilot tones with data at each time but at specific

   P P

   P P

   R E{YY H } X R

   X H 2 I

   subcarriers(Figure3). The channel is estimated at pilot

   YPYP

   P P HPHP P

   w Np

   subcarriers using the LS and MMSE estimators. With

   are the cross covariance matrix between

   H P and YP , and

   interpolation techniques the estimation of channel at data subcarrier can be obtained using channel estimation at pilot

   auto-covariance matrix of YP

   respectively R is auto-

   H H

   H H

   P P

   subcarriers. The Np pilot signals are uniformly inserted into

   covariance matrix of H . 2 is the noise-variance. If R

   the subcarriers

   X (k ) according to the following equation.

   P w HPHP

   w

   w

   and 2 are known to the receiver, CIR could be calculated by

   X (k ) X (iL l )

   l 0, 1, 2, , L 1

   MMSE estimator as below:

   X (k )

   p

   p

   l 0

   (14)

   H R R 1 Y

   Data l 1, 2, , L 1

   MMSE HPYP YPYP P

   R X H ( X R X H 2 I

   )1 X

   H 1

   (13)

   Where L = number of subcarriers (N)/number of pilots ( N p ) ,

   HPHP P P HPHP P w Np

   P P,LS

   i = pilot carrier index.

   R HPHP

   (R HPHP

   • 2 ( X H X

   ))1

   H P,LS

   w P P

   w P P

   A major drawback of the MMSE estimator is its high computational complexity, especially the matrix inversions is needed each time the data change.

4. PROPOSED METHOD

   A block type pilot arrangement is depicted in Figure2. In this time type, OFDM symbols with pilot at all subcarriers are transmitted periodically for channel estimation. Using these pilots, a time-domain interpolation is performed to estimate the channel along the time axis. Let st denote the period of pilot symbols in time. In order to keep track of the time- varying channel characteristic, the pilot symbols must be placed as frequently as the coherence time is. As the coherence time is given in an inverse form of the Doppler

   Figure 3 Comb-type pilot arrangement

5. LINEAR INTERPOLATION

   In this method two consecutive pilots are used to estimate channel at data sub-carriers k, iL k (i 1)L which are located in between the pilots, as showed below [9-10]:

   H (k ) H (iL l ) (15)

   frequency

   fDoppler

   in the channel, the pilot symbol period l

   must satisfy the following inequality:

   H (k ) ( H (i 1) H (i)) H (i) , i 0, 1, 2,

   L

   , Np 1

   (16)

   st

   1

   fDoppler

6. SIMULATION AND RESULTS

   This section discusses the results of the simulation using MATLAB that were performed based on the channel

   For fast fading, the block type is not suitable to track the rapid variation of the channel characteristic.

   Figure 2 Block -type pilot arrangement

   estimators discussed in the section III.

   For the simulation of basic OFDM system, we used the following parameters as shown in Table.

   Parameters	Specification
   No. of FFT points	128
   Length of Cyclic Prefix	16
   Total no. of Subcarriers	128
   Total no. of Symbol	144
   Pilot Spacing	2
   Number of Pilots	64
   Pilot Arrangement	Comb Type
   Signal Constellation	16QAM
   Interpolation	linear
   Channel Model	Clarke model
   Number of Channel Taps	5
   Channel Estimation Techniques	LS,MMSE
   Doppler Frequency	80HZ

   Parameters	Specification
   No. of FFT points	128
   Length of Cyclic Prefix	16
   Total no. of Subcarriers	128
   Total no. of Symbol	144
   Pilot Spacing	2
   Number of Pilots	64
   Pilot Arrangement	Comb Type
   Signal Constellation	16QAM
   Interpolation	linear
   Channel Model	Clarke model
   Number of Channel Taps	5
   Channel Estimation Techniques	LS,MMSE
   Doppler Frequency	80HZ

   Table: Simulation parameters

   Figure 4 LS and the MMSE channel estimators for OFDM system based on the parameter of Mean square error

   SNR	0	5	10	15	20	25
   MSE_LS	0.779	0.247	0.079	0.024	0.008	0.002
   MSE_MMSE	0.128	0.049	0.017	0.006	0.002	8e-04

   Figure 5 LS and the MMSE channel estimators for OFDM system based on the parameter of Symbol Error Rate

   SNR	0	5	10	15	20	25
   SER_LS	0.744	0.566	0.375	0.178	0.046	0.006
   SER_MMSE	0.635	0.447	0.248	0.087	0.012	7e-04

   Fig.4 shows the mean square error (MSE) of channel estimation at different SNR in dB. As SNR increases mean square error decreases for both LS and MMSE. Fig.5 shows the symbol error rate (SER) at different SNR in dB. As SNR increases symbol error rate decreases for both cases For a given SNR, MMSE estimator performs better than LS estimator. The complexity of MMSE estimator is larger than LS estimator but gives better performance in comparison to LS.

7. CONCLUSION

This paper highlights the comparative performance study between LS and MMSE estimators over time varying channel using comb-type pilot arrangement. The channel estimation is one of the fundamental issues of OFDM system design. The transmitted signal under goes many effects such reflection, refraction and diffraction. Also due to the mobility, the channel response can change rapidly over time. At the receiver these channel effects must be canceled to recover the original signal. From the present simulation based comparative study, it can be concluded that MMSE estimator provides better performance to track the time-varying channel.

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