Carrier Frequency Offset Estimator Based onUnique Word Cross-correlation

Arturs Aboltins, Member, IEEE,

Abstract—Many carrier frequency offset estimation algorithmsfor multicarrier communication systems utilize autocorrelationof repeating sequences. Limited frequency acquisition range duehigh temporal distance between the repeating parts do not allowto use these algorithms in communication systems with uniqueword prefixed block-wise transmission. This contribution is ad-dressed to solve fine carrier frequency offset estimation problemin such communication systems by proposing simple algorithmsthat employ cross-correlation between training sequence andreceived time domain samples. Numerous computer simulationsand theoretical analysis confirm applicability of the proposedmethods to the multicarrier communication systems.

Keywords—correlation, frequency estimation, synchronization

I. INTRODUCTION

In order to provide transmission in the required frequency

range, communication system usually include units responsible

for the translation of the baseband spectrum to the required

carrier frequency and back. Discrepancies between transmitter

an receiver carrier frequencies causes shift of the frequency

spectrum of the baseband signal on the frequency axis. This

effect is called carrier frequency offset (CFO).

Demand for the high data transmission rates in modern

communication systems require to employ advanced modu-

lation schemes. Multicarrier transmission schemes, such as

orthogonal frequency division multiplexing (OFDM) [1] have

gained particularly large interest during two last decades due

its ability to cope with severe multipath propagation and

multiple access interference in an efficient manner. However,

utilization of unitary transforms [2], such as well-known

discrete Fourier transform (DFT) in OFDM or less famil-

iar Generalized unitary rotation (GUR) [3] in Generalized

Orthogonal Nonsinusoidal Division Multiplexing [4] makes

communication systems extremely sensitive to the CFO. Data

transmission in these communication systems is impossible

without appropriate automatic frequency offset compensation

mechanisms.

Frequency synchronization starts with CFO estimation and

there are plenty of methods for that purpose. Meyr et.al in

their book [5] classify frequency synchronizers into ones that

work independently of timing information and ones that are

located after chip sampler and, therefore, are dependent on

timing information. Timing dependent CFO estimation enables

higher CFO estimation range, however precise timing must be

established before CFO estimation.

Timing synchronization solution for GUR based commu-

nication system [6] employs insertion of unique word (UW)

A. Aboltins is with the Institute of radio electronics, Riga TechnicalUniversity, Latvia, e-mail: aboltins@rtu.lv

prefix [7] before each useful time domain sample block.

This method enables high precision timing estimation at the

moderate level of complexity. However, attempts to employ

widely known basic CFO estimation methods (see section III)

have shown, that they suffer from the limited frequency offset

estimation range.

Usually estimation CFO is performed in two stages. Large,

integer frequency offsets, which are multiples of the chip

rate, can be estimated using coarse CFO estimation methods.

In contrast, fractional frequency offsets are estimated using

completely different methods. This contribution is devoted to

the fractional CFO estimation in communication systems with

UW-prefixed block-wise transmission.

II. SIGNAL MODEL

In order to provide high-accuracy block boundary detection

(timing synchronization), each block of N complex data

samples is prefixed by the UW, i.e. block of M complex

training samples. Resulting block of D = M + N samples

x(k) is being shaped and up-converted to the carrier frequency

fct. If in the output of the transmitter baseband we have signal

x(t), then translation of this signal the carrier frequency fct(up-conversion) can be described as:

s(t) = x(t)ej2πfctt (1)

If the stream of up-converted blocks is transmitted into

ideal communication media, then the received passband signal

r(t) is equal to the transmitted passband signal s(t). Down-

conversion process can be described as multiplication with

complex exponent having opposite argument sign:

y(t) = r(t)e−j2πfcrt , (2)

where r(t) is received passband signal and fcr is carrier

frequency of the receiver. If transmitter and receiver carrier

frequencies are equal, received baseband signal is also equal

to the transmitted baseband signal. However, as a rule carrier

frequencies are generated by different oscillators and they are

not equal exactly, therefore causing the CFO:

y(t) = x(t)ej2π(fct−fcr)t (3)

Carrier frequency offset projection to the baseband can be

described as multiplication of received baseband signal by the

complex exponent with frequency equal to the CFO:

y(k) = x(k)ej2πεk (4)

where ε = (fct − fcr)τ is normalized CFO, τ is sample time

and k is time domain sample index.

486

III. PROBLEM DEFINITION

Our objective is to design such CFO estimator, which is

able to detect ε < 0.01 in stream of data blocks, where each

block is prefixed by the same UW.

In accordance with (4), CFO causes rotation of the received

time domain samples in in-phase, quadrature (IQ) plane. If

coherent modulation schemes, for example, QPSK, are used,

such rotation causes periodical communication failures. More-

over, CFO causes increasing timing offset, which uncorrected

will lead to total failure of the communication. On other

hand, if N data samples come from the unitary transform,

which is not based on complex exponents, then CFO leads to

intercarrier interference.

Many publications related to the fractional CFO estimation

in digital communications [5], and especially in multicarrier

systems [8] employ repeating parts of time domain signal,

located at the beginnings of the blocks or frames.

If we have an ideal communication channel with no disper-

sion and noise, then received time domain baseband signal

is described by (4). Multiplication of two equal samples

x(k) = x(k+D) having temporal distance between each other

D samples gives:

y(k)y∗(k +D) = |x(k)|2ej2πεD (5)

CFO can be expressed directly from (5) as follows:

ε =arg(y(k)y∗(k +D))

2πD(6)

For higher estimation accuracy, series of multiplications

can be averaged over block with length M and it leads to

autocorrelation (AC) based CFO estimator. Since all phase

increments are expected to be the same, complex exponent

can be brought out of the sum

M−1∑k=0

y(k)y∗(k +D) = e−j2πDεM−1∑k=0

|y(k)|2 (7)

from where CFO can be expressed as:

ε =1

2πDarg

(M−1∑k=0

y(k)y∗(k +D)

)(8)

In accordance with [5] (page 488) this is so called D-spaced

data-aided (DA) CFO estimator. Since argument of (8) can not

exceed π, detection range of mentioned estimator is limited by

the condition

εmax =1

2D(9)

In one of the most influential work [8] in area of frequency

synchronization for OFDM, authors propose to use training

block with two equal halves. Proposed algorithm operates

near Cramer-Rao lower bound, however it suffers from the

same limitations as algorithm (8). Morelli and Mengali in [9]

improved estimator [8] by utilization more than two repeating

training sequences, but this enhancement do not allow to

extend frequency acquisition range.

Maximum likelihood CFO estimation algorithm [10] based

on cross product between data samples and cyclic prefix

(CP) samples [1] for OFDM communication systems is also

UW � (.)∗

�

yk buffer1:M

������ �

�

�

�

select2:M

select1:M-1

�

�����+-

� ∑ ������� �ε

Fig. 1: CFO estimator based on averaging of cross correlation phaseincrement

based on equation (8). This methods exploits feature that CP

is identical to the tail of the data block. Unfortunately CP

is efficient only in DFT based communication systems [2].

Moreover, acquisition range of this algorithm is limited to

εmaxCP = 12N .

IV. ESTIMATOR ALGORITHMS

Proposed data-directed (DD) CFO estimation algorithms are

based on the cross-correlation (XC) between known training

sequence stored in the receiver and received samples. In other

words, proposed CFO estimator is based on a matched filter.

CFO estimators are derived from data-directed algorithms

proposed by Meyr et al in [5].

A. Average increment based method

If there is no any channel impairments, except CFO, re-

ceived baseband signal is described by (4). Multiplication

of respective transmitted and received samples yields the

following result:

y(k)x(k)∗ = |x(k)|2ej2πεk (10)

From the equation (10) could be seen that phase of

y(k)x(k)∗ grows linearly with the time with the speed which

is proportional to the CFO. Using this observation, we can

derive XC based CFO estimator. Let’s take two sample pairs

with indexes k and k + D, where D is integer offset. If

multiplication of second sample pair is described by the similar

expression

y(k +D)x(k +D)∗ = |x(k +D)|2ej2πε(k+D), (11)

then subtracting arguments of equations (10) and (11) results

in

arg[y(k +D)x(k +D)∗]− arg[y(k)x(k)∗] = 2πεD ,(12)

Now CFO can be expressed from (12) as follows:

ε =arg[y(k +D)x(k +D)∗]− arg[y(k)x(k)∗]

2πD(13)

Averaging offsets within a block of adjacent samples (D =1) allows to obtain expression, which can be used in real CFO

estimator:

ε =1

2(M − 1)π

M−2∑k=0

(α(k + 1)− α(k)) (14)

where

α(k) = arg[y(k)x(k)∗] (15)

Structure of resulting CFO estimator is shown in Fig. 1

Noticeably, that detection range of the proposed estimator

is still limited by the condition (9). However, due the fact that

487

UW � (.)∗

�

yk buffer1:M

������ �

�

�∑

�

�����+-

������� � ∑ �ε

Fig. 2: CFO estimator based on linear regression of correlation phaseincrement

D = 1 in this estimator, εAVGmax = 1/2. Similarly as in the

case with AC based estimator (8), accuracy of given estimator

can be increased at the cost of the detection range by taking

D > 1. However, collection of block of M phase increments

will require D times more time.

B. Least squares based method

Expression (14) represents simplified line fit within a block

of samples. However, simple averaging between phase incre-

ments is not optimal in sense of mean squared error (MSE).

Least squares (LS) estimator based on simple linear regression

[11], might improve the performance of the device. Replacing

averaging with LS estimator yields an expression:

ε =1

2π

M−1∑k=0

(α(k)− α)(k − k)∑M−1m=0 (k − k)2

(16)

Due to symmetry of multiplier (k − k) in equation (16),

mean of angles α can be replaced by the sum, therefore saving

one division to M :

ε =1

2π

M−1∑k=0

(α(k)−∑M−1m=0 α(m))(k − k)∑M−1

m=0 (k − k)2(17)

Block diagram of the obtained CFO estimator is depicted

in Fig. 2.

Since DLS = 1, frequency acquisition range of the proposed

estimator (17) remains εLSmax = 1/2.

C. Method for CAZAC sequences

Equation (14) can be changed if we use training samples

from constant amplitude zero autocorrelation (CAZAC) se-

quences. Such training samples can be obtained, for example,

using Chu-Zadoff algorithm [12]. Using this method, (non-

linear) complex argument operation can be exchanged with

the sum. In this case equation (14) can be rewritten as:

ε =1

2(M − 1)π

(arg

M−2∑k=0

z(k + 1)− argM−2∑k=0

z(k)

)(18)

where

z(k) = y(k)x(k)∗ (19)

In this variant of XC based CFO estimator, argument is

taken from the result of XC between training block pattern

and the received samples. The largest advantage of estimator

(18) is that it can be used in conjunction with XC based

timing offset estimator [6]. Resulting CFO estimation scheme

is shown in Fig. 3.

If sequences with good correlation properties are used in

the role of UW, magnitude of either ”timing” signal can be

used for block timing offset estimation.

UW � (.)∗

�

yk buffer1:M

�����×

�

�

�

select2:M

select1:M-1

�

�

∑

∑

�

�

�

�

�

�

timing1

timing2

�

�

�

�����+-

������� �ε

Fig. 3: CFO estimator optimized to CAZAC sequences

0 5 10 15 2010

−8

10−6

10−4

SNR, dB

MS

E

AC, ε=0.001

AC, ε=0.01

AVG, ε=0.001

AVG, ε=0.01

LS, ε=0.001

LS, ε=0.01

CZ, ε=0.001

CZ, ε=0.01

Fig. 4: Performance of CFO estimators in communication systemwith AWGN channel.

For estimator (18) frequency acquisition range of is also

εCZmax = 1/2.

V. SIMULATION RESULTS

Performance of the proposed XC based CFO estimators

has been verified using computer simulations. The transmitted

sequence contained 100 blocks, each consisting of N = 64random data chips and M = 64 UW chips produced using

Chu-Zadoff algorithm [12]. Each model has been tested with

two different normalized CFOs ε1 = 0.001 and ε2 = 0.01. In

accordance with (9), estimation limits of our AC based estima-

tor (8) was ε < 12(64+64) which is equal to εACmax = 0.0039,

whereas estimation limits of all XC based estimators was

εXCmax = 0.5.

In the first series of simulations signal with CFO was been

sent over additive white Gaussian noise (AWGN) communica-

tion channel. MSE obtained during the simulations is depicted

in Fig. 4. Plots with legend AC of auto-correlation based

estimator (8) are given for the refernce. Cross correlation based

algorithms (14), (17) and (18) correspond to the abbreviations

”AVG”, ”LS” and ”CZ”.

From simulation results it could be seen that at the small

CFO (ε = 0.001) MSE of AC based estimator is order

of magnitude smaller than XC based ones. In contrast, at

relatively high signal to noise ratio (SNR), performance of

XC based estimator (17) starts to surpass performance of AC

based one. Between proposed XC based CFO estimators, LS

based solution (17) demonstrates the best results, therefore

confirming excellence of this signal processing technique.

488

0 5 10 15 2010

−8

10−6

10−4

10−2

SNR, dB

MS

E

AC, ε=0.001

AC, ε=0.01

AVG, ε=0.001

AVG, ε=0.01

LS, ε=0.001

LS, ε=0.01

CZ, ε=0.001

CZ, ε=0.01

Fig. 5: Performance of CFO estimators in communication systemwith frequency selective channel.

0 0.002 0.004 0.006 0.008 0.01

10−9

10−7

10−5

Normalized CFO

MS

E

AC

AVG

LS

CZ

Fig. 6: Dependence of estimator accuracy from frequency offset

In the second series of simulations, the same set of signals

has been sent over dispersive channel having 4-tap long

impulse response. Simulation results are depicted in Fig. 5.

Performance of XC based estimators was degraded severely,

whereas MSE of AC based estimator remained almost un-

changed. Static convolution in communication channel affects

both samples of AC based estimator in equal manner, therefore

its influence is compensated. It is suggested, that in time-

variant communication channels AC based estimator would

perform much worse, whereas performance of XC based

estimator would not change significantly.

In Fig. 6 dependence of MSE of various estimators from

CFO is depicted. Turns out that accuracy of AC based estima-

tion is proportional to the frequency offset, whereas XC based

estimators have almost constant MSE over whole frequency

offset range. Fig. 6 confirms limitations of the AC based

estimator in sense of frequency acquisition range.

VI. CONCLUSIONS

Theoretical results and numerous simulations confirmed

that proposed CFO estimators provide sufficient accuracy for

utilization in multicarrier communication systems. It must be

taken into account, that sufficiently large training sequences

must be used in order to achieve sufficient accuracy. For

example, 64 sample sequence provides MSE = 10−8 at

SNR = 20dB. Moreover, performance of proposed XC based

algorithm surpasses performance of auto-correlation based

algorithms in non-dispersive communication channels with

high SNR.

Performance of the proposed algorithms in communication

systems with dispersive channels is acceptable, however its

performance was inferior to that of the AC based methods.

One of the largest advantages of proposed methods is large

(εXCmax = 0.5) fractional CFO estimation range.

Accuracy of XC based estimators can be increased by mea-

suring phase increment between non-adjacent sample pairs.

However increase of the accuracy will occur at the cost of

decreased frequency acquisition range.

ACKNOWLEDGMENT

The author would like to thank prof. Peteris Misans for

valuable advises regarding linear regression. This work was

partly supported by the National Program of the Ministry of

Education and Science of Republic of Latvia.

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