Block Synchronization Using a Unique Word for a Generalized UnitaryRotation Based Communication System

Arturs Aboltins

Institute of Radio Electronics, Riga Technical University, Azenes 12, LV1048, Riga, Latvia, E-mail: aboltins@rtu.lv

ABSTRACT: Generalized Unitary rotation (GUR) provides amethod for construction of unitary transformations in para-metric multicarrier communication systems. Such commu-nication systems require special timing offset estimation andcorrection schemes, since a unitary transform in those sys-tems is dynamic. Moreover, they demand higher timing ac-curacy than traditional time domain and frequency domainapproaches. This contribution proposes block timing offsetestimation and a correction scheme suitable for GUR basedparametric communication systems with block-wise transmis-sion.

1. IntroductionMulticarrier (MC) communications are one of the most

attractive technologies for providing high speed digital

transmission. Multiple carriers provide an additional di-

mension for the mitigation of distortions caused by the

propagation media and multiple access interference. MC

systems with sinusoidal basis functions (BFs), such as or-

thogonal frequency division multiplexing (OFDM), are one

of the most widely used MC technologies. During last

years filter bank based solutions [1] have gained a large

attention in scientific community. Filter banks utilize ad-

vanced waveforms, for instance wavelets, for obtaining

higher spectral density and reduced inter-carrier interfer-

ence.

Factorization of unitary transformation matrices, which

are used for production of the waveforms, would allow to

decrease the complexity of communication systems. More-

over, parametric communication systems, where the uni-

tary transform is dynamic and parametrized, can be built.

Along with other factorization methods, using of rotation

angles [2], GUR [3] provides an excellent possibility to

build such communication systems. Papers of Oka [4] and

other authors (see references in [5]) as well as our first ex-

periment [5] have confirmed the feasibility of the proposed

concept. However, the absence of equalization and syn-

chronization methods in these communication systems is

the main obstacle in the development of practical solutions.

The purpose of given contribution, in conjunction with

our contribution devoted to equalization [6], is to fill gaps

fromcoder� QPSK

mapper��X IGUR ��x +pad � P/S � ↑ 4 � shap

ing �

channel

�to de-coder� QPSK

demapper��Y GUR ��y -pad � S/P

�

sync �

�

Figure 1: Communication system baseband model

in synchronization and equalization methods for paramet-

ric multicarrier communication systems. The paper is or-

ganized as follows. Section 1 describes the structure of the

communication system and signal transformations. Section

2 provides a description of algorithms for timing offset es-

timation and correction. Section 3 gives the validation of

the theoretical results using simulations. Finally we make

conclusions and propose future work in Section 4.

2. Materials and Methods2.1. System model

In an MC communication system, a time domain signal

(1) is the sum of N orthogonal carriers modulated by Nparallel data streams. Such signal can be obtained using

unitary transformation which converts blocks of samples�X , mapped from binary data, to modulated signal�x:

�x = Φ−1�X (1)

Before transmission, time domain blocks �x undergo

padding, parallel-to-serial (P/S) conversion, upsampling,

pulse shaping and upconversion to passband.

At the receiver side, the signal is being downconverted

to baseband and sampled. Afterwards, the received signal

must undergo serial-to-parallel (S/P) conversion. If there is

no inter-block interference, the signal at the receiver base-

band input is:

�y = H�x+�w, (2)

where H is channel matrix, which in case of static chan-

nel is a convolution matrix, and �w is additive noise. Af-

ter removal of padding, the received time domain signal

149

2012 13th Biennial Baltic Electronics Conference (BEC2012) Tallinn, Estonia, October 3-5, 2012

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is converted back into transform domain by means of for-

ward unitary transform and the received useful symbols are

referred to as:�Y = Φ�y, (3)

The diagram of communication system baseband model is

depicted in figure 1.

2.2. Generalized unitary rotationThe GUR transform matrix is defined as follows:

Φ =1

∏p=log2(N)

Bp(φ,γ,ψ), (4)

In this equation φ, γ and ψ are angles which can take any

value ∈ [0;2π], and the stairs-like orthogonal generalized

rotation matrix (SOGRM) Bp is defined as:

Bp =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

τ11,p 0 0 . . . 0 0

0 0 τ12,p . . . 0 0

. . . . . .. . . . . .

0 0 0 0 . . . τ1N/2,pτ21,p 0 0 . . . 0 0

0 0 τ21,p . . . 0 0

. . . . . .. . . . . .

0 0 0 0 . . . τ2N/2,p

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(5)

This matrix contains only mutually independent plane rota-

tions, thus it is necessary to factorize log2(N) such matrices

in order to obtain the matrix (4), which rotates all planes si-

multaneously. The elements of each SOGRM are obtained

from unitary four-element single-plane rotation matrices.

One variant of such matrix could be:

[τ1q,pτ2q,p

]=

[ ∓sinφq,pe− jψq,p cosφq,pe jγq,p

cosφq,pe− jγq,p ±sinφq,pe jψq,p

](6)

Totally is is possible to construct 64 variants of single-plane

rotation matrix.

2.3. Statement of the problemReceived serial stream of samples must undergo serial-

to-parallel (S/P) conversion before forward transformation.

Separation of the stream must be done at correct positions,

otherwise demodulation of data symbols (3) by means of a

unitary transform unit based on factorized matrix (4) will

be impossible. This paper is addressed to the development

of mechanism for block boundary detection and S/P con-

version in the receiver of communication system.

2.4. Related workThe overview of synchronization and equalization meth-

ods for OFDM is given in our overview paper [7]. Most of

the existing timing estimators for OFDM are based on au-

tocorrelation (AC) of cyclic prefix (CP) [8]. Unfortunately

CP is inefficient [9] in systems based on non-sinusoidal

BFs, and CP based methods are not directly applicable.

On the other hand, one of the classical methods for time

synchronization is a unique word (UW) (see subsection

2.5). UWs traditionally are used for the synchronization in

single carrier (SC) communication systems [10]. Recently

UW based padding has gained interest in context of MC

communications [11].

2.5. Timing offset estimationIn order to obtain the correct time offset of blocks, a va-

riety of estimation methods can be used. All of them can be

divided into training sequence based estimation methods,

and blind methods. Although training sequences reduce

efficiency of the communication system, they significantly

simplify synchronizer design.

In our contribution we will prepend each block by UW

- fixed combination of samples. The use of UW allows

to choose best sequences for continuous time offset esti-

mation. Moreover, UW can be reused for continuous fre-

quency synchronization and channel estimation [6], when

the receiver is in the tracking mode.

For time synchronization codes with good periodic cor-

relation properties and low peak to average power ratio

(PAPR) must be used. Zadoff-Chu (ZC) codes [12] have

a sharp autocorrelation function and a constant amplitude.

Due to specific features ZC sequences are called constant

amplitude zero autocorrelation (CAZAC) sequences. They

are widely used in modern communication systems, such

as Long Term Evolution (LTE), where ZC codes are used

for cell search. ZC codes are generated using equations:

a(k) =

{e j M

K πk2 if K is odd

e j MK πk(k+1) if K is even

(7)

where k = 0,1, . . . ,K − 1 is time domain index and M is

integer coprime with K.

There are several ways how to estimate time instants of

the beginning of the blocks:

• autocorrelation (AC) between neighboring UWs,

• cross-correlation (XC) with UW sequences.

In this paper both approaches are efficiently combined

in a single low-complexity solution.

Cross-correlation based estimator If equal UW se-

quences are placed in each block, a direct, low complex-

ity, XC based timing offset estimator can be designed. If

the transmitted sequence is known and the channel is cor-

rupted by additive white Gaussian noise (AWGN), then in

accordance with Meyr et al.[10], XC is the optimal data-

aided ML estimator. If we denote received samples as yand known UW samples as a, then correlation window op-

eration will be described by the following equation:

cxc(k) =L

∑m=1

y(k+m)a(m)∗ (8)

150

UW � (·)∗ �����× � ∑ � | · |

�

yk � buffer �

�

� | · | � ∑�

����÷ �vxc

> 0.7 �v−xc

Figure 2: Cross-correlation based block timing estimator

where L is total length of UW padding and (·)∗ denotes

complex conjugation. In order to provide normalization, it

is necessary to divide the obtained result by average incom-

ing signal magnitude:

vxc(k) =cac(k)

y(k)=

∣∣∑Lm=1 y(k+m)a(m)∗

∣∣∑L

m=1 |y(k+m)| (9)

The time delay estimate can be found by taking the argu-

ment of cross-correlator output samples vxc with magnitude

larger than a certain threshold vxc0:

θxc = argθ{vxc(θ)}

∣∣∣∣vxc>vxc0

(10)

The diagram of unit providing XC based timing estimation

is given in figure 2.

Autocorrelation based estimator AC based time offset

estimation can be used in systems with a cyclic structure of

time domain blocks, such as systems with CP or systems

with repeating UW. Most of AC based block timing esti-

mators described in the literature are based on the idea of

autocorrelation between repeating parts:

vac(k) =k+L−1

∑m=k

y(m)y∗(m+N), k ∈ {0, ...,θ, ...N +L−1}(11)

Such approach is also suitable for UW based communica-

tion systems, where the estimate for the block delay can be

found using:

θac = argmaxθ

{vac(θ)} (12)

However, the authors of one of the most influential studies

[8] for OFDM suggested using AC in conjunction with the

additional statistic obtained by taking into account proba-

bility distributions of samples and applying maximum like-

lihood (ML) estimation. Unfortunately this method is suit-

able only to CP based systems, because in UW based sys-

tems the design of ML estimator depends on statistical

properties of the training sequences. We are currently re-

searching this topic.

The structure of block timing offset estimator based on

AC is depicted in figure 3. Its operation is based on equa-

tions (11) and (12).

2.6. Timing offset correctionIf ZC sequences (7) are used, the XC algorithm based

estimator (9) outputs a sharp, 1 sample long peak at the end

z−N � (·)∗ �

�����× � moving

sum� | · | �vac

yk �

�

Figure 3: Autocorrelation based block timing estimator

yk �

� XCEstimator

�v−xc S/P � find

offset�θxc

� ACEstimator

�vac S/P � argmax �θac com

bine

r

θ

�

�

�

� NCO � PIDcontroller

�

block clock

Figure 4: Block synchronizer

of each UW (see figure 5). Although this pulse can be used

directly for keying serial-to-parallel (S/P) converter in the

receiver front-end, a connection of the estimator output to

control system, which mitigates false detections, provides

better results.

In contrast, AC based estimator (11) outputs a slowly

varying signal (see figure 5) and it is not suitable for direct

keying of S/P conversion block. Maximums of vac peaks at

the output of the autocorrelator must be tracked in order to

obtain correct block synchronization metrics. The result of

this tracking operation is block timing offset, which can be

used for controlling S/P clock.

For adjusting a numerically controlled oscillator (NCO),

which produces S/P clock signal, a proportional-integral-

derivative (PID) controller can be used. Figure 4 depicts

the structure of obtained device. It resembles data aided

(DA) feedback synchronizer from [10]. Synchronizer uses

a combined output from both - XC and AC based estima-

tors. In our model we utilize a combiner, which selects the

estimate with minimum magnitude.

3. Simulation resultsThe performance of the above-described methods has

been evaluated using simulations. For this purpose a model

of baseband communication system without equalization

was created (see figure 1). Time synchronizer (unit "sync"

0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

1

time, ms

v

ac

vxc

Figure 5: Estimator outputs

151

0 5 10 15 2010

−3

10−2

10−1

100

SNR, dB

BE

R

AC

XC

Composite

Figure 6: Performance of communication system using various

estimators

in figure 1) was implemented in accordance with the design

depicted in figure 4. 106 bits were transmitted using 64

subcarriers produced by 30◦ CRAOT [13] through channel

with AWGN . Output signals of estimators are presented in

figure 5.

After careful tuning of PID controller, the system has

demonstrated stable, but slow convergence to synchronized

state within approximately 20 blocks. Slow and precise be-

havior of feedback synchronizer makes it suitable for track-

ing mode operation. The bit error rate (BER) achieved in

synchronized state is presented in figure 6.

The communication system demonstrates high sensitiv-

ity to block timing offsets, a delay in 1 chip leads to un-

recoverable loss of data. The accuracy of time offset cor-

rector is determined by the accuracy of XC based estima-

tor, whereas the AC based estimator provides robustness

against false detections. However, false detections and jit-

ter limit BER at approximately 10−3.

4. Conclusions and future workThe proposed solution provides an efficient low-

complexity block synchronization solution for communica-

tion systems with dynamic transform and block-wise trans-

mission. It allows to start building real GUR based commu-

nication system prototypes. A more intelligent combining

of estimator outputs would lead to decreased false detec-

tions and lower BER. Utilization of ML estimation [8] in

AC timing estimator would increase synchronization accu-

racy.

Communication systems based on simplest GUR trans-

forms, like CRAOT, are extremely sensitive to timing off-

set, because transforms are time-variant. Simulated GUR

based communication system requires timing error toler-

ance less than 1 chip, while, for instance, OFDM timing

tolerance is equal ± the length of CP. Therefore, the re-

search on synthesis of wavelet-like time-invariant trans-

forms using GUR is necessary.

The accuracy of XC based estimator is limited to 1 chip,

therefore fine estimation and tuning within 1 chip bound-

aries would decrease jitter and improve performance. Clas-

sical time-domain methods described in the literature [10]

can be used for this purpose.

As an interesting alternative to UW padding, a su-

perimposed pseudo-noise (PN) sequence synchronization

scheme [14] can be used for timing offset estimation.

References

[1] B. Farhang-Boroujeny, “OFDM Versus Filter Bank Multi-

carrier,” Signal Processing Magazine, IEEE, vol. 28, no. 3,

pp. 92–112, 2011.

[2] P. Dita, “Factorization of Unitary Matrices,” Journal de

Physique (main title), vol. A 36, p. 12, 2001.

[3] G. Valters and P. Misans, “FPGA implementation of El-

ementary Generalized Unitary Rotation,” 2009 Norchip,

no. 2, pp. 1–4, Nov. 2009.

[4] I. Oka andM. Fossorier, “A General Orthogonal Modulation

Model for Soft-ware Rados,” IEEE Transactions on Com-

munications, vol. 54, no. 1, pp. 7–12, 2006.

[5] P. Misans and G. Valters, “Initial FPGA design for gen-

eralized Orthogonal Nonsinusoidal Division Multiplexing,”

2009 Norchip, pp. 1–5, Nov. 2009.

[6] A. Aboltins and P. Misans, “Singular Value Decomposition

Based Phi Domain Equalization For Multi-Carrier Commu-

nication System,” Electronics and Electrical Engineering,

vol. 9, no. 125, 2012.

[7] A. Aboltins and D. Klavins, “Synchronization and correc-

tion of channel parameters for an OFDM-based communi-

cation system,” Automatic Control and Computer Sciences,

vol. 44, no. 3, pp. 160–170, Jul. 2010.

[8] J.-J. van de Beek, M. Sandell, and P. O. Borjesson, “ML

estimation of time and frequency offset in OFDM systems,”

IEEE Transactions on Signal Processing, vol. 45, no. 7, pp.

1800–1805, Jul. 1997.

[9] A. Aboltins, “Comparison of Orthogonal Transforms for

OFDM Communication System,” Electronics and Electrical

Engineering, vol. 5, no. 5, pp. 77–80, 2011.

[10] H. Meyr, M. Moeneclaey, and S. A. Fechtel, Digital Com-

munication Receivers: Synchronization, Channel Estima-

tion, and Signal Processing. Wiley, 1998.

[11] H. Witschnig, T. Mayer, A. Springer, A. Koppler, L. Maurer,

M. Huemer, and R. Weigel, “A different look on cyclic pre-

fix for SC/FDE,” The 13th IEEE International Symposium

on Personal, Indoor andMobile Radio Communications, pp.

824–828, 2002.

[12] Chu, “Polyphase codes with good periodic correlation prop-

erties,” IEEE Transactions on Information Theory, no. 6, pp.

531–532, 1972.

[13] P. Misans, M. Terauds, A. Aboltins, and G. Valters, “MAT-

LAB/SIMULINK Implementation of Phi-TransformsâASA

New Toolbox Only or the Rival of Wavelet Toolbox for

the Next Decade?” in Nordic MATLAB User Conference,

2008, pp. 1–8.

[14] W.-W. Hu and C.-P. Li, “Super-Imposed Training Scheme

for Timing and Frequency Synchronization in OFDM Sys-

tems,” 2007 IEEE 65th Vehicular Technology Conference -

VTC2007-Spring, pp. 1718–1722, Apr. 2007.

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