Robust Blind Carrier Frequency Offset Estimation Algorithm ... · PDF fileRobust Blind Carrier...

Robust Blind Carrier Frequency Offset EstimationAlgorithm for OFDM Systems

Arunprakash Jayaprakash1 • G Ramachandra Reddy1

Published online: 3 September 2016� Springer Science+Business Media New York 2016

Abstract In this paper, a joint covariance power fitting and phase based blind estimation

method for carrier frequency offset (CFO) is proposed for orthogonal frequency division

multiplexing (OFDM) systems using constant modulus constellations. Based on the

assumption that the channel varies slowly within two adjacent OFDM symbols, the

influence of the channel on corresponding covariance values and phase of the two OFDM

symbols will be the same. Utilizing this, a robust method is formulated based on covari-

ance power fitting criterion and phase information between two nearby OFDM symbols.

The mean square error and bit-error-rate performance of the proposed estimation method is

compared with that of the prominent conventional estimation schemes under noisy mul-

tipath channels with high delay spreads and Doppler spreads. Based on Monte Carlo

simulations, it is shown that the proposed method is robust under different channel con-

ditions and provides more accurate CFO estimates.

Keywords Blind estimation � Carrier frequency offset � Frequency selective � Orthogonalfrequency division multiplexing (OFDM)

1 Introduction

Orthogonal frequency division multiplexing (OFDM) has been the popular modulation

technique for high data-rate wireless transmission over multipath frequency selective

channels [1]. It is employed in different wireless standards like digital audio [2] and video

broadcasting [3], wireless local area network standards IEEE 802.11a/g [4], 4G LTE [5]

etc. The popularity of OFDM is because of its high spectral efficiency due to the mutually

orthogonal sub-carriers. Even if the subcarriers are closely spaced without any guard

& Arunprakash [email protected]

1 School of Electronics Engineering, VIT Univeristy, Vellore, Tamil Nadu 632014, India

123

Wireless Pers Commun (2017) 94:777–791DOI 10.1007/s11277-016-3650-9

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bands, there will not be any inter subcarrier interference. But any mismatch between the

transmitted and received frequencies distorts the orthogonality of the subcarriers. This loss

of orthogonality happens under doubly dispersive channels in which the impulse response

of the channel is time variant and undergoes time spreading. A relative velocity between

the transmitter and receiver introduces a Doppler shift in the transmitted frequency and the

data is received at a different subcarrier frequency. The difference between the transmitted

and received frequency, known as carrier frequency offset (CFO), distorts the orthogo-

nality between the subcarriers. The CFO can also be induced due to a dissimilarity in local

oscillator frequencies of the transmitter and the receiver. These result in the interference of

one subcarrier with one or more of its neighboring subcarriers which is known as inter

carrier interference (ICI). Therefore, accurate estimation of CFO has been an active area of

research.

A variety of blind CFO estimation schemes for constant magnitude modulations are

studied in the literature. A blind CFO estimation scheme for OFDM is proposed in [6],

where the cost function is formulated based on a kurtosis type criterion. Another CFO

estimation scheme is presented in [7] where CFO is found out by utilizing the variance of

the interference which occurs owing to the loss in orthogonality between the subcarriers. A

non-supervised CFO estimation method called PDE-F method is proposed in [8]. In this

method, the channel effect upon two adjacent subcarriers in the same OFDM symbol is

assumed to be non-varying. Hence, CFO is estimated by minimizing a cost function

formulated based on the power difference between received samples corresponding to two

the adjacent subcarriers. The accuracy of estimation by PDE-F scheme depends on the time

dispersion of the channel impulse response and its performance gets degraded under highly

frequency selective channels. In [9], a robust non-supervised CFO estimation procedure

called PDE-T method is proposed, under the assumption that the channel can be highly

time dipsersive and frequency selective but remains almost invariant at consecutive symbol

time instants. Here, the objective function is formulated upon the sum of power difference

between demodulated symbols on similar subcarrier indices of two adjacent OFDM

symbols. In [10], CFO is evaluated by maximizing an objective function formulated as the

product of the magnitude of the demodulated data on similar subcarrier indices of nearby

OFDM symbols. In [11], CFO is estimated by utilizing out-of band elements of covariance

matrix which is generated using a single OFDM symbol. The covariance based technique

requires information about the length of the channel for accurate formulation of the cost

function. Moreover, the number of out-of-band elements of the covariance matrix reduces

with an increase in the length of the channel, which affects the estimation accuracy of the

method. An accurate iterative CFO estimation scheme based on modified Viterbi and

Viterbi (VAV) algorithm is proposed in [12]. In this method, CFO is estimated from the

difference in phase information between consecutive OFDM symbols which are raised to

the power of modulation order of the modulated symbols. The angle based technique

exhibits good performance in terms of estimation accuracy than the conventional methods,

but the range of fractional CFO values which can be precisely estimated decreases with an

increase in the modulation order. A modified transmission scheme using two different

constellation sizes within the OFDM symbol is also proposed in [12] to improve the

estimation range of iterative VAV scheme, but at the cost of reduced data throughput.

In the present study, we present a new non-supervised CFO estimation technique for

OFDM system using constant magnitude constellations. The CFO is estimated utilizing the

power invariance of covariance fitting values as well as the phase information in the

demodulated OFDM symbols so that the range of CFO estimation is improved at the same

time providing better CFO estimates than the conventional counterparts. The performance

778 A. Jayaprakash, G. R. Reddy

123

of the proposed covariance fitting method is analyzed and compared with other conven-

tional methods in terms of mean square error (MSE) and bit error rate (BER) for doubly

dispersive channel conditions.

The paper is structured as follows. Section 2 is devoted to the system model for OFDM

symbol affected with CFO. The formulation of the proposed method is presented in

Sect. 3. The numerical results and analysis are provided in Sect. 4. Finally, the paper is

concluded in Sect. 5.

In this paper, vectors are represented with lower case boldface letters, while matrices

are denoted by upper case boldface letters. Scalar quantities are indicated with normal

letters. The notations j � j and ð�Þ� denote respectively the modulus and complex conjugate

operations. The notation k � k represents Eucledian norm of a vector. The notations Ref�gand Imf�g represent the real part and imaginary part of a variable respectively.

2 System Model for OFDM Symbol Affected with CFO

Consider a fully loaded OFDM system consisting of N complex data symbols modulated

onto N orthogonal subcarriers. The complex source symbols are uniformly drawn from a

constant modulus constellation like M-ary phase shift keying (PSK). Consider a complex

data vector bm ¼ bmð0Þ; bmð1Þ; . . .; bmðN � 1Þ½ �T which is used for the generation of mth

OFDM symbol block. The mth OFDM symbol xm ¼ xmð0Þ; xmð1Þ; . . .; xmðN � 1Þ½ �T is

obtained by [9]

xmðnÞ ¼1ffiffiffiffi

Np

X

N�1

k¼0

bmðkÞej2pknN; n ¼ 0; 1; . . .;N � 1 ð1Þ

If F represents the inverse discrete Fourier transform matrix, xm ¼ Fbm. In order to avoid

inter symbol interference (ISI) between two successive OFDM symbols, a guard time

called cyclic prefix is included at the beginning of each symbol, which is filled with the last

Np values of xm, where Np corresponds to a time interval greater than the delay spread of

the channel. Due to the presence of cyclic prefix the transmitted data at each subcarrier can

be recovered by dividing the received data with the frequency response of the channel at

that subcarrier.

If the channel frequency response matrix corresponding to the mth symbol index is

represented as Hm, the mth time domain OFDM symbol affected by channel is obtained as

FHmbm. Hm is a diagonal matrix which consists of the Fourier transform of the time

domain channel taps evaluated at the subcarriers. Let e represent the CFO value normalized

by subcarrier spacing. The received OFDM symbol affected by CFO and channel under the

assumption of noise free transmission is given in [9]

ym ¼ ej2pemðNþNpÞ

N HeFHmbm ð2Þ

where He is a diagonal matrix given by

He ¼ diag 1; e2peN�1; e

2peN�2; . . .; e

2peN�ðN�1Þ

h iT� �

The scalar term, which is the first exponential term in (2) is dependent on m, which

represents the common phase shift. Let dm ¼ dmð0Þ; dmð1Þ; . . .; dmðN � 1Þ½ �T¼ FHmbm

Robust Blind Carrier Frequency Offset... 779

123

represents the channel affected OFDM symbol and the effect of CFO is denoted by

Um ¼ ej2pemðNþNpÞ

N He. Hence the received symbol reduces to

ym ¼ Umdm ð3Þ

dm ¼ U�1m ym ð4Þ

If the estimated CFO value is represented as e, the OFDM symbol after the correction of

CFO is [9]

dm ¼ e�j2pemðNþNpÞ

N H�eym ð5Þ

3 Proposed CFO Estimation Algorithm for OFDM Systems

Different CFO estimators presented in [9–13] for constant modulus constellation are based

upon the assumption that the channel parameters are invariant between the adjacent OFDM

symbols. The basic principle behind these estimators is that for constant modulus con-

stellations, the magnitude of the demodulated symbols will be same as the magnitude of

the channel’s frequency response if the CFO is compensated accurately [9]. i.e.

FHdm�

�

�

� ¼ Hmj j bmj j ¼ Hmj j. Owing to the invariance of the channel between two nearby

OFDM symbols, FHdm�

�

�

� ¼ FHdmþ1

�

�

�

�. This happens because, the magnitude of the data

symbols bmj j ¼ bmþ1j j ¼ 1. This is true for all subcarriers, if and only if, CFO is perfectly

estimated and corrected [9]. However, the present study utilizes the invariance of the

covariance matrix values and the phase information of two adjacent OFDM symbols.

Let ðeÞ be a trial value of CFO used for compensation and the corrected OFDM signal

be represented as a vector dm;e ¼ dm;eð0Þ; dm;eð1Þ; . . .; dm;eðN � 1Þ� �T

given by [9]

dm;e ¼ e�j2pemðNþNpÞ

N H�eym ð6Þ

A matrix, Dm;e is formed by stacking circularly shifted dm;e column vectors. i.e.,

Dm;e ¼

dm;eð0Þ dm;eð1Þ . . . dm;eðN � 2Þ dm;eðN � 1Þdm;eð1Þ dm;eð2Þ . . . dm;eðN � 1Þ dm;eð0Þ

..

. ... . .

. ... ..

.

dm;eðN � 2Þ dm;eðN � 1Þ . . . dm;eðN � 4Þ dm;eðN � 3Þdm;eðN � 1Þ dm;eð0Þ . . . dm;eðN � 3Þ dm;eðN � 2Þ

2

6

6

6

6

6

6

6

6

4

3

7

7

7

7

7

7

7

7

5

ð7Þ

The covariance matrix corresponding to m-th OFDM symbol is obtained from Dm;e as [11]

Rm;e ¼1

NDm;eD

Hm;e ð8Þ

In [11], it is established that, the covariance matrix obtained by the cyclic shift method

depends solely on the channel properties for constant magnitude constellations, if CFO

compensation is perfect. Therefore, two adjacent OFDM symbols should correspond to a

single covariance matrix since the channel characteristics remain unaltered between them.

Moreover, the covariance matrix is Hermitian Toeplitz which can be represnted as [11]


123

Rm;e ¼ Toeplitz rm;e�

ð9Þ

where rm;e ¼ rm;eð0Þ; rm;eð1Þ; . . .; rm;eðN � 2Þ; rm;eðN � 1Þ� �T

represents the first column of

the covariance matrix. Since the elements in the Rm;e are repeating, the first column vector

rm;e is sufficient for the formulation of cost function. From (7) and (8), rm;e is obtained as

rm;e ¼1

NDm;ed

�m;e ð10Þ

If the channel characteristics remains unchanged between L consecutive OFDM symbols,

the objective function to be minimized is formulated as

JðeÞ ¼X

L�1

m¼1

rm;e � rmþ1;e

2 ð11Þ

where rm;e and rmþ1;e correspond to the first column vectors of the covariance matrices of

the mth and (m?1)th OFDM symbol respectively.

The estimate of actual CFO e, represented as be, is obtained by minimizing the objective

function J eð Þ represented in (11).

be ¼ arg mine2 �0:5;0:5ð Þ

J eð Þ ð12Þ

The objective function given by (11) follows the shape of a sinusoid which is similar to that

in [9] and hence the optimum value of CFO can be obtained by a three point substitution

method instead of going for complex exhaustive search methods. The cost function can be

represented as [11]

J eð Þ ¼ A cos 2p e� eð Þð Þ � A ð13Þ

Here A is independent of CFO. Three different values of e ¼ 0; 0:25 and -0.25 are

substituted in (13). The optimum value of e is obtained as [11]

be ¼

1

2ptan�1 b=að Þ for a� 0

1

2ptan�1 b=að Þ þ 1

2for a\0 and b� 0

1

2ptan�1 b=að Þ � 1

2for a\0 and b� 0

8

>

>

>

>

>

<

>

>

>

>

>

:

ð14Þ

where a ¼ J 0:25ð Þ þ J �0:25ð Þð Þ=2ð Þ � J 0ð Þf g and b ¼ J 0:25ð Þ � J �0:25ð Þð Þ=2ð Þf g.The problem formulation was carried out in the absence of noise. Due to the presence of

additive white Gaussian noise, there will be a small error in estimation. The residual error

de ¼ e� be will be very small. When CFO is corrected using (5), the vector containing the

channel affected demodulated source symbols €dm ¼ €dmð0Þ; €dmð1Þ; . . .; €dmðN � 1Þ� �T

is

obtained by correcting the CFO using the estimated CFO be as in (5) and applying inverse

FFT which is represented in [9] as

€dm ¼ FHe�j2pbemðNþNpÞ

N H�eym ð15Þ

Since be ¼ e� de, (15) is reduced to


123

€dm ¼ FHej2pðdeÞm NþNpð Þ

N HdeFHmbm ð16Þ

Similarly, the compensated vector for ðmþ 1Þ-th symbol instant is given as

€dmþ1 ¼ FHe�j2pbe mþ1ð ÞðNþNpÞ

N H�eymþ1 ð17Þ

€dmþ1 ¼ FHej2pðdeÞ mþ1ð Þ NþNpð Þ

N HdeFHmþ1bmþ1 ð18Þ

From (16) and (18), it may be inferred that the elements €dmðnÞ and €dmþ1ðnÞ, correspondingto the same subcarriers in the vectors €dm and €dmþ1 have similar phase except the scalar

term involving the residual CFO value and the phase of source symbols drawn from M-ary

PSK constellations. If the elements in both the vectors are raised to the power of M, the

angle of the M-PSK modulated symbols reduces to zero and the phase difference between

the elements in both the vectors solely depend on the residual CFO. Here, since the residual

CFO is small, even though the symbols are raised to the power of M, phase ambiguity is

not observed at higher M values. Therefore the CFO, de can be accurately estimated in a

single step as

bde ¼N

2pM N þ NPð Þ angleX

L�1

m¼1

X

N�1

n¼0

dmþ1ðnÞd�mðnÞ� M

( )

ð19Þ

The proposed CFO estimation algorithm is summarized in Table 1.

4 Simulation Results and Analysis

An OFDM system with number of subcarriers, N = 64, using PSK modulation, cyclic

prefix length, Np ¼ 16, subcarrier spacing of 8.33 kHz and subcarriers data rate which

equals to 16.68 kb/s is considered in the simulation. The number of OFDM symbols over

which the channel remains invariant is taken as L = 2. The frequency selective channel

models which are used in [8, 9] are utilized for the performance comparison of various

estimators under different delay spread and Doppler spread conditions. Three channel

types with five paths having Jakes Doppler spectrum are used for simulations. Channel-1

has average path gains of [0.35, 0.25, 0.18, 0.13, 0.09] and path delays of [0, 1, 2, 3, 4]

samples. Channel-2 has a higher delay spread than Channel-1 with path delays of [0, 1, 2,

Table 1 Proposed algorithm

Step Operation

I Generate rm;e and rmþ 1;e from the received OFDM symbols for three values of e ¼ 0; 0:25 and

�0:25 using (6), (7) and (10)

II Evaluate JðeÞ for the three values of e using (11) and the covariance vectors obtained in Step I

III Obtain an initial estimate of CFO, be using (14)

IV Substitute the value of be in (15)and (17) and obtain the vectors €dm and €dmþ 1

V Calculate the residual CFO estimate bde using (19) and update be ¼ be þ bde


123

6, 11] and average path gains of [0.34, 0.28, 0.23, 0.11, 0.04]. Channel-3 has a much higher

delay spread with path delays of [0, 4, 8, 12] samples and uniform average path gains of

[0.25, 0.25, 0.25, 0.25]. The proposed covariance and phase based method is compared

with the estimators in [9] and [11] in terms of MSE and BER for various channel con-

ditions under the effect of AWGN noise using Monte Carlo simulations. The MSE per-

formance is obtained through K = 10000 estimations of CFO and averaging them. i.e.

MSE ¼ 1K

PKk¼ 1 e� bekð Þ2

n o

where e is the true value of CFO and bek is the estimated value

of CFO for kth iteration. The CFO value is kept constant over L = 2 OFDM symbols and

changes randomly from one pair of OFDM symbols to the other. The PDE-T based [9]

method and covariance based method [11] are the state-of-the-art blind carrier frequency

offset frequency estimation methods for constant modulus constellations. PDE-T method is

proved to be better than the other power difference based estimators [9]. The VAV

algorithm based method [12] is another blind CFO estimation method but, unlike the other

power difference based estimators, the range of CFO estimation depends on the constel-

lation size, computational complexity is comparatively larger and is iterative. The

covariance based method is proved to be better than PDE-T based method with similar

computational complexity, which is investigated in [11] under certain channel conditions.

But its performance can vary based on the length of the channel. Hence, the two prominent

blind CFO estimation methods are considered as benchmark for comparison. In order to

analyze the robustness of each algorithm under different frequency selective conditions,

the various estimators are analyzed under the three Rayleigh fading channels with zero

Doppler shift. The MSE versus signal to noise ratio (SNR) variation and the BER versus

SNR comparison for Channel-1 are respectively presented in Figs. 1 and 2. The source

symbols are drawn from 4-PSK constellations. The MSE versus SNR comparison for the

considered estimators under Channel-3 is illustrated in Fig. 3. From Figs. 1 and 3, it can be

inferred that the proposed method attains an MSE of 10�6 at 17 dB where as the same is

attained by the conventional covariance based method and PDE-T method at SNR levls of

24 and 28 dB respectively. Similarly, a BER of 10�2 is attained by the proposed method at

20 dB where as the conventional methods attain it at 26 and 30 dB.

In order to analyse performance under time varying Doppler channels, channels with

different maximum Doppler frequencies and delay spreads are considered for simulations.

0 5 10 15 20 25 30 35 4010−10

10−8

10−6

10−4

10−2

SNR(dB)

MSE

PDE−Tcovariance methodproposed method

Fig. 1 MSE versus SNRcomparison for 4-PSK, Channel-1 and zero Doppler frequency


123

The Doppler frequencies are normalized with the subcarrier bandwidths. The MSE versus

SNR plots for maximum Doppler frequencies of 50 and 200 Hz for moderately frequency

selective Channel-2 is shown in Fig. 4. The frequencies 50 and 200 Hz correspond to

vehicle speeds of nearly 24.5 and 98 km/h respectively. The MSE versus SNR plots for

maximum Doppler frequencies of 50 and 200 Hz for highly frequency selective Channel-3

is shown in Fig. 5. From Figs. 4 and 5, it can be seen that the even though the MSE versus

SNR graphs converge at very high SNRs for the three estimators, the proposed method

attains a significantly improved performance at low SNRs. At Doppler shifts of 50 Hz, the

MSE of the proposed method is 10�5 at 11 dB where as the other methods attain the same

MSE level at SNRs beyond 17 and 19 dB. The SNR(dB) at which target MSE (T-MSE) is

achieved by various estimators for Doppler frequencies (fD) and Channel-3 is summarized

in Table 2.The proposed estimator also shows better performance than the other techniques

at high Doppler shift of 200 Hz. The BER versus SNR plots for maximum Doppler

frequency of 200 Hz for Channel-1 and Channel-2 are given in Fig. 6.The performance

0 5 10 15 20 25 30 35 4010−5

10−4

10−3

10−2

10−1

100

SNR(dB)

BER


Fig. 2 BER versus SNRcomparison for 4-PSK, Channel-1 and zero Doppler frequency

0 5 10 15 20 25 30 35 4010−10

10−8

10−6

10−4

10−2

SNR(dB)

MSE


Fig. 3 MSE versus SNRcomparison for 4-PSK, Channel-3 and zero Doppler frequency


123

improvement in accuracy of CFO estimation is reflected in the BER performance. Even at

high Doppler shift, the BER achieved by the proposed technique at SNR of 20dB is

attained by the conventional estimators at 26 and 30 dB. In order to illustrate the per-

formance comparison of the proposed technique for higher order modulations, the BER

versus SNR plots for maximum Doppler frequencies of 300 Hz using 4-PSK and 8-PSK for

0 5 10 15 20 25 30 35 4010−6

10−5

10−4

10−3

10−2

MSE

SNR(dB)

PDE−T (50 Hz)covariance method (50 Hz)proposed method (50 Hz)PDE−T (200 Hz)covariance method (200 Hz)proposed method (200 Hz)

Fig. 4 MSE versus SNRcomparison 4-PSK, Channel-2,Doppler frequency = 50 and200Hz

0 5 10 15 20 25 30 35 4010−6

10−5

10−4

10−3

10−2

MSE

SNR(dB)

PDE−T (50 Hz)covariance method (50 Hz)proposed method (50 Hz)PDE−T (200 Hz)covariance method (200 Hz)proposed method (200 Hz)

Fig. 5 MSE versus SNRcomparison 4-PSK, Channel-3,Doppler frequency = 50 and200Hz

Table 2 SNR(dB) at which tar-get MSE (T-MSE) is achievedvarious estimators for Dopplerfrequencies (fD) and Channel-3

fD ¼ 0,T-MSE

¼ 10�6

fD ¼ 50,T-MSE

¼ 10�5

fD ¼ 200,T-MSE

¼ 5� 10�5

Proposed method 17 11 12

Covariance method[11]

24 17 16

PDE-T [9] 28 19 20


123

moderately frequency selective Channel-2 is shown in Fig. 7. The BER versus SNR plots

for maximum Doppler frequencies of 300 Hz using 4-PSK and 8-PSK for highly frequency

selective Channel-3 is presented in Fig. 8. The BER versus SNR plots of various estimators

using 8-PSK and 16-PSK modulations, under maximum Doppler frequency of 300 Hz in

Channel-3 is illustrated in Fig. 9. The SNR(dB) at which target BER ¼ 10�2 is achieved

for various estimators and constellation sizes for Doppler shift = 300 Hz and Channel-3 is

summarized in Table 3. The proposed method maintains its superiority in performance for

higher order constellations under highly frequency selective channels and at high Doppler

shifts. The BER at a given SNR for various estimators and constellation sizes for Doppler

shift = 300 Hz and Channel-3 is consolidated in Table 4. In order to illustrate the per-

formance difference at a wide range of Doppler shifts, the MSE performance versus

maximum Doppler spreads, under Channel-3 and SNR = 15 dB for the various estimators

with 4-PSK modulation are depicted in Fig. 10. Similarly, the BER performance versus

maximum Doppler spreads, under highly frequency selective Channel-3 and

SNR = 20 dB for the various estimators with 4-PSK modulation are depicted in Fig. 11.

SNR(dB)0 5 10 15 20 25 30 35 40

BER

10-5

10-4

10-3

10-2

10-1

100

PDE-T (Channel-1)covariance method (Channel-1)proposed method (Channel-1)PDE-T (Channel-2)covariance method (Channel-2)proposed method (Channel-2)

Fig. 6 BER versus SNRcomparison 4-PSK, Channel-1and Channel-2, Dopplerfrequency = 200Hz

0 5 10 15 20 25 30 35 4010−5

10−4

10−3

10−2

10−1

100

BER

SNR(dB)

PDE−T (4−PSK)covariance method (4−PSK)proposed method (4−PSK)PDE−T (8−PSK)covariance method (8−PSK)proposed method (8−PSK)

Fig. 7 BER versus SNRcomparison 4-PSK and 8-PSK,Channel-2, Doppler frequency =300Hz


123

The proposed estimator is showing significantly improved BER and MSE performance

than the other estimators in both low and high Doppler spread conditions.

The initial cost function in (11) is formulated with the first columns of the covariance

matrices of two OFDM symbols which can be generated using FFT operation, for which

the complexity is of the order of N. In the case of PDE-T method in [9], the estimation is

0 5 10 15 20 25 30 35 4010−5

10−4

10−3

10−2

10−1

100

BER

SNR(dB)


Fig. 8 BER versus SNRcomparison 4-PSK and 8-PSK,Channel-3, Doppler frequency= 300Hz

0 5 10 15 20 25 30 35 4010−4

10−3

10−2

10−1

100

BER

SNR(dB)


Fig. 9 BER versus SNRcomparison 8-PSK and 16-PSK,Channel-3, Doppler frequency= 300Hz

Table 3 SNR(dB) at which target BER ¼ 10�2 is achieved for various estimators and constellation sizesfor Doppler shift =300 Hz and Channel-3

4-PSK 8-PSK 16-PSK

Proposed method 20 27 34

Covariance method [11] 26 32 38

PDE-T [9] 30 35 [40


123

performed using the demodulated OFDM symbol, obtained as the FFT of the received

OFDM symbol, which has similar complexity. The cost function in (11) consists of the

norm of difference of two vectors of length N. This calculation involves N complex

multiplications and 2N � 1 complex additions. Even though the phase difference based

Table 4 BER at a given SNR for various estimators and constellation sizes for Doppler shift =300 Hz andChannel-3

4-PSK, SNR = 24 dB 8-PSK, SNR = 32 dB 16-PSK, SNR = 40 dB

Proposed method 2.2 9 10-3 1 9 10-3 6.7 9 10-4

Covariance method [11] 3.2 9 10-2 9.8 9 10-3 3.3 9 10-3

PDE-T [9] 8.2 9 10-2 3.5 9 10-2 1.6 9 10-2

0 50 100 150 200 25010−6

10−5

10−4

Doppler frquency (Hz)

MSE


Fig. 10 MSE versus Dopplerspread comparison for 4-PSK,Channel-3 and SNR = 15 dB

0 50 100 150 200 250

10−2

10−1

100

Doppler frquency (Hz)

BER


Fig. 11 BER versus Dopplerspread comparison for 4-PSK,Channel-3 and SNR = 20 dB


123

estimation step involves power computations, accurate estimates are achieved using very

less number of subcarriers which is N. The number complex multiplications involved in

the conventional PDE-T method [9] and the covariance method [11] are of the order of N.

Therefore, even though the total number of multiplications required by the proposed

method is slightly higher than that required by conventional methods, it is still of the order

of N as well as exhibits better performance under all channel conditions.

5 Conclusion

In this paper, we have put forward an unsupervised CFO estimation scheme for OFDM

system with constant modulus constellations based on joint covariance power fitting and

phase criterion. The proposed method is robust and shows a substantial performance

improvement than the conventional prominent CFO estimation schemes under time

varying channels with low and high delay spreads. The effectiveness of the proposed

method to adapt to time varying channels with high frequency selectivity is due to the

careful formulation of the objective function based on covariance power fitting and phase

criteria. The performance of the proposed method is compared with the conventional

methods in terms of MSE and BER under time varying channels with different frequency

selectivity and Doppler shifts. The proposed method shows excellent performance

improvement than the conventional methods under all the considered channel conditions,

even at low SNRs. The effectiveness of the proposed estimation schemes for higher order

constellations has also been evaluated through numerical simulations. The effect of various

Doppler shifts on the performance of the various estimators is also analyzed under time

varying channels and the proposed estimator attains improved MSE and BER performance

under low and high Doppler shifts.

Appendix

The derivation of the cost function given by (13) is presented in this Appendix. Let the first

column of the covariance matrix Rn eð Þ be rm lð Þ ¼ rmð0Þ; rmð1Þ; . . .; rmðN � 1Þ½ �T . Rm eð Þisobtained using the matrix Dm eð Þ. From (6) and (8), the l-th element of the rm lð Þ vector isobtained as

rm lð Þ ¼X

N�l�1

p¼0

dm pþ lð Þdm�ðpÞ þ

X

N�1

p¼N�l

dm pþ l� Nð Þdm�pð Þ: ð20Þ

The relationship between bdm and dm is obtained from (2), (3) and (6) as

bdm ¼ e�j2pem NþNpð Þ

N H�ee

j2pem NþNpð ÞN Hedm

If dm ¼ dmð0Þ; dmð1Þ; . . .; dmðN � 1Þ½ �T , the ath element of the vector bdm is given by

dm að Þ ¼ ej2p e�eð Þm NþNpð Þ

N ej2p e�eð Þa

N dm að Þ: ð21Þ

Using (21) in (20),


123

rm lð Þ ¼ ej2p e�eð Þ lð Þ

N

X

N�l�1

p¼0

dm pþ lð Þd�m pð Þ þ ej2p e�eð Þ l�Nð Þ

N

X

N�1

p¼N�l

dn pþ l� Nð Þd�m pð Þ: ð22Þ

LetPN�l�1

p¼0 dm pþ lð Þd�m pð Þ be km;l andPN�1

p¼N�l dm pþ l� Nð Þd�m pð Þ be lm;l. Then

rm lð Þ ¼ ej2p e�eð Þl

N ðkm;l þ e�j2p e�eð Þlm;lÞ: ð23Þ

Similarly,

rmþ1 lð Þ ¼ ej2p e�eð Þl

N ðkmþ1;l þ e�j2p e�eð Þlmþ1;lÞ

and

rmþ1 lð Þ � rm lð Þð Þ ¼ ej2p e�eð Þl

N kmþ1;l � km;l�

þ e�j2p e�eð Þ lmþ1;l � lm;l�

� �

Let kmþ1;l � km;l�

=kl and lmþ1;l � lm;l�

=ll. Then rmþ1 lð Þ � rm lð Þk k2 can be minimized

to

rmþ1 lð Þ � rm lð Þk k2 ¼ klj j2 þ llj j2 þ 2Re kll�l

�

cos 2p e� eð Þ � 2Im k�l ll �

sin 2p e� eð Þ:ð24Þ

Since the channel characteristics remains invariant between two adjacent OFDM symbols,

in the absence of CFO and at high SNR, rmþ1 lð Þ ¼ rm lð Þ. Hence,

kmþ1;l þ lmþ1;l

�

¼ km;l þ lm;l�

. Therefore, kmþ1;l � km;l�

¼ � lmþ1;l � lm;l�

. i.e.

kl ¼ �ll. Hence the term Im k�l ll �

, associated with the sinusoidal term in (24) becomes

�Im klj j2n o

which is zero. Substituting kl ¼ �ll in (24), rmþ1 lð Þ � rm lð Þk k2 ¼ 2 klj j2�2 klj j2 cos 2p e� eð Þ. If, �2 klj j2 ¼ A, rmþ1 lð Þ � rm lð Þk k2 ¼ Acos 2p e� eð Þ � A. The con-

stant in the equation is independent of e� eð Þ and the cost function of (11) can be

approximated as J eð Þ ¼ A cos 2p e� eð Þð Þ � A.

References

1. van Nee, R., & Prasad, R. (2000). OFDM for wireless multimedia communications. Norwood, MA:Artech House, Inc.

2. ETSI Normalization Committee et al. (1995). Radio broadcasting systems, digital audio broadcasting(DAB) to mobile, portable and fixed receivers (Vol. 300, No. 401). Norme ETSI, Sophia-Antipolis,France, Doc. ETS, pp. 1995–1997.

3. Reimers, U. (1998). Digital video broadcasting. IEEE Communications Magazine, 36(6), 104–110.4. Li, Y. G., & Stuber, G. L. (2006). Orthogonal frequency division multiplexing for wireless communi-

cations. Berlin: Springer.5. Fazal, K., & Kaiser, S. (2008). Multi-carrier and spread spectrum systems: From OFDM and MC-

CDMA to LT. New York: Wiley.6. Yao, Y., & Giannakis, G. B. (2005). Blind carrier frequency offset estimation in SISO, MIMO, and

multiuser OFDM systems. IEEE Transactions on Communications, 53(1), 173–183.7. Al-Dweik, A. J. (2004). Robust non data-aided frequency offset estimation technique. In 15th IEEE

international symposium on personal, indoor and mobile radio communications, 2004. PIMRC 2004(Vol. 2), pp. 1365–1369

8. Zeng, X. N., & Ghrayeb, A. (2008). A blind carrier frequency offset estimation scheme for OFDMsystems with constant modulus signaling. IEEE Transactions on Communications, 56(7), 1032–1037.


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9. Al-Dweik, A., Hazmi, A., Younis, S., Sharif, B., & Tsimenidis, C. (2010). Carrier frequency offsetestimation for OFDM systems over mobile radio channels. IEEE Transactions on Vehicular Technol-ogy, 59(2), 974–979.

10. Lmai, S., Bourre, A., Laot, C., & Houcke, S. (2014). An efficient blind estimation of carrier frequencyoffset in OFDM systems. IEEE Transactions on Vehicular Technology, 63(4), 1945–1950.

11. Oh, J.-H., Kim, J.-G., & Lim, J.-T. (2011). Blind carrier frequency offset estimation for OFDM systemswith constant modulus constellations. IEEE Communications Letters, 15(9), 971–973.

12. Al-Dweik, A., Hazmi, A., Younis, S., Sharif, B., & Tsimenidis, C. (2010). Blind iterative frequencyoffset estimator for orthogonal frequency division multiplexing systems. IET Communications, 4(16),2008–2019.

13. Roman, T., Visuri, S., & Koivunen, V. (2006). Blind frequency synchronization in OFDM via diago-nality criterion. IEEE Transactions on Signal Processing, 54(8), 3125–3135.

Arunprakash Jayaprakash received his B.Tech degree in Electronicsand Communication Engineering from University of Kerala, Trivan-drum, India, in 2008 and his M.Tech degree from VIT University,Vellore, India in 2013. He worked as a Lecturer in Electronics andCommunication Engineering Department of University College ofEngineering, Kariavattom, Trivandrum from September 2009 to May2011. Currently he is working as adhoc faculty in Electronics andCommunication department at NIT Calicut. He is working towards hisPh.D. degree in Signal Processing and Communication Engineering atVIT University, Vellore, India. His research areas of interest includeDigital Signal Processing, Digital Image Processing and signal pro-cessing for Wireless Multicarrier Communication systems.

G. Ramachandra Reddy (SM IEEE’90) received the M.Sc. andM.Sc.(Tech.) degrees from the Birla Institute of Technology andScience, Pilani, India, in 1973 and 1975, respectively, and the Ph.D.degree from the Indian Institute of Technology, Madras, India, in 1987.In 1976, he joined the Department of Electrical and ElectronicsEngineering, College of Engineering, Sri Venkateswara University,Tirupati, India, as a Lecturer, and superannuated as professor in theyear 2010. He is currently working as senior professor in the School ofElectronics Engineering, VIT University, Vellore, India. FromFebruary 1989 to May 1991, he was with the Department of Electricaland Computer Engineering, Concordia University, Montreal, QC,Canada, as a Visiting Scientist. Dr. Ramachandra Reddy is a Fellow ofthe Institution of Electronics and Telecommunication Engineers and amember of the Indian Society for Technical Education.


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