Match Fltr Using PN

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The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC07)

MATCHED FILTER TIME AND FREQUENCY SYNCHRONIZATION METHOD

FOR OFDM SYSTEMS USING PN-SEQUENCE PREAMBLES

Henri Puska and Harri Saarnisaari

University of Oulu, Centre for Wireless Communications (CWC)

P.O.Box 4500, FI-90014 Oulu, FINLAND

ABSTRACT

Matched filter (MF) time and frequency synchronization

method for orthogonal frequency division multiplexing

(OFDM) systems is presented. The proposed method uses

the same pseudo-noise (PN)-preamble generation as was used

in [1], but signal processing in the receiver is different. The

performance of the proposed method is investigated using the-

oretical and simulated synchronization probabilities. The ob-

tained results indicate that the proposed synchronizer attains

the optimum theoretical performance.

I. INTRODUCTION

In orthogonal frequency division multiplexing (OFDM) sys-

tems, time and frequency synchronization are important issues

to be solved [2]. If these tasks are not performed with suffi-

cient accuracy, the orthogonality among the sub-carriers is lost,

and the communication system suffers from intersymbol inter-

ference (ISI) and intercarrier interference (ICI). Several tech-

niques have been proposed recently for OFDM timing and fre-

quency synchronization. Those suggested in [3], [4] propose

to use the correlation that exists between the samples of the

cyclic prefix and the corresponding portion of the OFDM sym-bol. Since received cyclic prefix is usually affected by ISI, the

performance of this estimator depends on the channel. To avoid

this problem in timing synchronization, the method in [1] uses

a training symbol with two identical halves in time domain.

However, the timing metric used in [1] suffers from broad au-

tocorrelation function (ACF), which may result large timing es-

timation error. This blurry ACF can be eliminated and, hence,

timing offset estimation error can be reduced by designing a

special training symbols which give more sharper timing met-

ric trajectory [5], [6].

It is well known that pseudo-noise (PN)-sequences have

good autocorrelation properties. Therefore, many authors have

proposed the use of time domain PN-sequence as preamble in

OFDM systems [7], [8]. The method used in [8] is also applied

in [9] but there the principle of frequency domain PN synchro-

nizer is also presented.

In this paper, we propose a method which uses the Schmidl

and Coxs [1] training symbol generation. Therein, two iden-

tical halves in time domain are generated by transmitting PN-

sequence on the even frequencies, while zeros are transmitted

on the odd frequencies. As a difference, instead of correlat-

ing two identical halves with each other in the receiver [1], our

This research was supported by the URANUS (Universal RAdio-link Plat-

form for EfficieNt User-centric AccesS) project of the EUs 6th frameworkprogramme (Project No. IST-27960 (URANUS))

proposal utilizes matched filter (MF) for time and frequency

synchronization.

The rest of the paper is organized as follows. The proposed

time and frequency synchronization methods are described in

Sections II and III, respectively. In Section IV simulated results

are compared with theoretical ones in additive white Gaussian

noise (AWGN) channel for verifying obtained results. The re-

sults are also compared to the Tufvessons method [8], where

time domain PN-preamble and slightly different timing metric

are used. Finally, conclusions are drawn in Section V.

II. PROPOSED TIME SYNCHRONIZATION METHOD

A block diagram of the typical OFDM system is shown in Fig.

1. Therein, input data bits are first encoded, interleaved and

mapped to data symbols, each of which modulates a different

subcarrier. In the proposed scheme, the transmitter transmits

every now and then a PN-preamble, whose length is L. This

PN-sequence is then zero padded such that zeros are placed

on the odd frequencies after serial-to-parallel (S/P) conversion.

Therefore, the total length of the PN-preamble is 2L, which is

equal to the number of subcarriers (Nc). Multicarrier modula-tion is performed by taking the inverse fast Fourier transform

(IFFT) of data symbols. The discrete time baseband OFDMsignal can be expressed as

s(n) =1Nc

Nc1k=0

D(k)ej2kn/Nc, (1)

where D(k) is a data symbol which modulates kth subcarrier.When the number of subcarriers is sufficiently large, s(n) ap-proximates a complex Gaussian process with zero mean and

variance 2s = E{|s(n)|2} [4]. An OFDM symbol has a use-ful symbol period Tu and preceding each symbol is a cyclicprefix of length Tcp, which is longer than the channel impulseresponse so that there will be no ISI [1]. The baseband signal

is digital-to-analog (D/A) transformed and up-converted to the

radio frequency (RF) and then transmitted through the channel.

At the receiver, discrete time baseband signal r(n) is ob-tained by down-converting and analog-to-digital (A/D) trans-

forming the continuous time RF-signal. In an AWGN channel,

without frequency offset, r(n) is given by

r(n) = s(n )ej + (n), (2)where is the propagation delay, is the carrier phase and(n) is AWG noise with zero mean and variance 2 =E{|(n)|2}. Cyclic prefix can not be removed until the re-ceiver has knowledge about frequency, phase, and symbol syn-

chronization (Fig. 1). In this section, we concentrate on timing

1-4244-1144-0/07/$25.00 c2007 IEEE


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RF RX

S/P P/S

P/SIFFTS/P

RF TX

0 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 01 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 1

Synchronization

Data

outDeinterleaving

+ decoding

Symbol

demapping

in

Data

Cyclix prefix

removal

Channel Cyclix prefix

addition

FFT

Coding +

interleaving

D/A

A/D

r(n)

PNpreamble

addition

Symbol

mapping

Figure 1: A block diagram of the OFDM system.

synchronization where the starting point of the OFDM symbol

is estimated. After cyclic prefix removal, multicarrier demodu-

lation is performed by taking the fast Fourier transform (FFT).

Then the received information symbols are demapped to bits

which are deinterleaved and decoded.

A. Time Domain MF Synchronizer

Fig. 2 shows the block diagram of the noncoherent MF timing

synchronizer implemented in time domain. The sampled signal

r(n), from A/D converter (Fig. 1), has signal-to-noise-ratio(SNR) = 2s/

2 .

PDI

2r(n) syncz(n)y(n)

threshold

Matched filter (MF)

Figure 2: Noncoherent MF time synchronization unit in time

domain.

The signal r(n) is then passed through MF whose impulseresponse h(n) is complex conjugated, time reversed and de-layed version of one half of the sPN(n), i.e., h(n) = s

PN(Ln). When the preamble signal sPN(n) is filtered using this MF,then two identical sharp ACFs occur as can be seen from Fig.

3. This is because the transmitted preamble consisted of two

identical halves. Post detection integration (PDI) can be used

to combine these two ACFs and time synchronization is made

from this combined signal shown in Fig. 4. As a reference,

timing metric of the Schmidl and Coxs method [1] is shownin Fig. 5. By comparing these figures, it can be seen that the

proposed scheme provides sharper timing metric.

Next, the theoretical synchronization probabilities for the

proposed method are presented. When sPN(n) and h(n) arenot synchronized, then the output of the MF (y(n)) is (if anideal ACF is assumed) complex valued Gaussian distributed

random variable with zero mean and variance equal to L2/2per I and Q branch. Therefore, the decision variable z(n) aftermagnitude squaring and PDI follows the central chi-square dis-

tribution with 4 degrees of freedom and the probability of false

alarm (PFA) is

PFA = expTH

L2

1k=0

1k! TH

L2

k, TH 0. (3)

0 50 100 150 200 2550

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Sample index

abs(MFoutput)

SNR = 0 dB

TCP

= 0

Nc

= 256

L=Tu/2

Angle separation ()

between thosespikes is proportional

to frequency offset (f)

L = 128

Figure 3: Magnitude squared output of the MF.

from which the detection threshold TH can be calculated inorder to obtain the desired PFA.

In the synchronous code phase position decision variable

z(n) is characterized by the noncentral chi-square distributionand the probability of detection (PD) is

PD = Q2

4L2s2

,

2THL2

. (4)

where Q() is the generalized Marcums Q-function [10].When the decision is made according to the maximum se-

lection, then the largest sample from L possible samples is se-lected from the set {z(n)} = [z(n) z(n + 1) z(n + L1)].The selected largest sample can be taken either from the cor-

rect (z(c)) or from the wrong code phase (z(w)). The prob-ability of selecting the correct sample from the set {z(n)} isdenoted by PMax = P(z(c) > z(w)i, i), where i refers toall of the L 1 possible false code phases. The probabilityPMax is obtained as [11]

PMax = P(z(w)i < z (c), i)

=

p(z(c))[P(z(w) < z (c))]L1 dz(c), (5)

where p(z(c)) is the probability density function (PDF) of therandom variable z(c). The probability that the maximum sam-ple is taken from the correct time instant and, in addition, it

also exceeds the threshold TH can be calculated as

PD,Max = PMaxPD, (6)

which is a useful probability when both maximum selection

and threshold comparison are utilized.

B. Frequency Domain MF Synchronizer

Fig. 6 illustrates the frequency domain implementation of

the equivalent time domain MF synchronizer structure (Fig.


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0 20 40 60 80 100 1270

0.05

0.1

0.15

0.2

0.25

Sample index

Timingmetricoftheproposedme

thod

SNR = 0 dB

Nc

= 256

L = 128

TCP

= 0

Figure 4: Noncoherently combined autocorrelation function.

2). As is well known, convolution in time domain corre-

sponds multiplication in frequency domain. Because OFDM

systems already have IFFT/FFT blocks for multicarrier modu-

lation and demodulation, it may be advantageous to use them

also in matched filtering. When the MF is implemented in

frequency domain, the incoming signal r(n), from A/D con-verter (Fig. 1), has to be transformed via FFT to the fre-

quency domain (R(f) = FFT[r(n)]). Impulse response h(n)of the MF has to be also transformed to frequency domain

(H(f) = FFT[h(n)]). Then R(f) and H(f) are multipliedand the result of this multiplication is IFFT transformed back

to time domain. Remaining part of Fig.6 is exactly the same

as was described in Fig. 2. It is explained, e.g., in [9], how

this filtering can be performed via the overlap-save method in

frequency domain.

III. FREQUENCY OFFSET ESTIMATION

A carrier frequency offset f causes a phase rotation of [1]

= f Tu (7)

between the two halves of the MF output signal y(n). Thephase difference can be estimated from the y(n) (Fig. 3)after timing synchronization by

= angle(y()y( + Tu2

)), (8)

where is the timing estimate. If|| < , then the frequencyoffset estimate is [1]f = /(Tu). (9)The Cramer-Rao bound (CRB) for the normalized frequency

offset can be expressed as [1]

Var[f Tu] = Var = 2

2L2s. (10)

0 50 100 150 200 2550

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Sample index

TimingmetricoftheS&C

algorithm

SNR = 0 dB

Nc

= 256

L = 128

TCP

= 0

Figure 5: Timing metric of the Schmidl and Coxs method [1].

S/P P/SIFFT PDIFFT

y(n) 2r(n) Multiply by

H(f)

z(n) sync

Threshold

Figure 6: Noncoherent MF time synchronization unit in fre-

quency domain.

IV. RESULTS

Timing synchronization performance of the proposed method

is first evaluated. Therefore, theoretical optimum synchroniza-

tion probabilities from (3), (4), (5) and (6) are calculated when

L = 128, Nc = 256, f = 0 and Tcp = 0. The channelis AWGN throughout this study. Obtained theoretical proba-

bilities are then verified by simulations. As a reference, these

results are also compared to those given by the Schmidl and

Coxs (S&C) [1] and the Tufvessons methods (Tufv.) [8].

Figs. 7 and 8 show PD performances as a function of SNR.Theoretical and simulated detection probabilities of the pro-

posed method are shown in Fig. 7. In addition, there are sim-

ulated PD results of the Tufvessons method. It can be seenthat simulated and theoretical results of the proposed schemecoincide very well. The performances of the proposed method

and the Tufvessons method are almost the same too.

The PD performances of the proposed method and the S&Cmethod are compared in Fig. 8. It can be seen that the S&C

method has quite large performance loss ( 10 dB). Despite ofa broad timing metric of the S&C method (Fig. 5), it achieves

probability 1 because PD just tells the probability that thethreshold is exceeded when a sample is taken from the correct

time instant. However, the S&C method tends to raise false

alarm rate due to broad timing metric.

Figs. 9 and 10 show PMax and PD,Max performances as afunction of SNR. Theoretical and simulated results of the pro-

posed method are shown in Fig. 9. Simulated results of the


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0.1

0.2

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0.7

0.8

0.9

1

SNR [dB]

PD

Theory, 101

Simul., 101

Simul., Tufv. 101

Theory, 102

Simul., 102

Simul., Tufv. 102

Theory, 103

Simul., 103

Simul., Tufv. 103

Theory, 104

Simul., 104

Simul., Tufv. 104

Nc

= 256

L = 128

TCP

= 0

f = 0

Figure 7: PD results with different PFA values.

Tufvessons method are shown also. It can be seen that sim-

ulated and theoretical results of the proposed scheme coincide

very well again. The performances of the proposed method and

the Tufvessons method are almost the same too.

PMax and PD,Max performances of the proposed methodand the S&C method are compared in Fig. 10. It is observed

that now there is a big difference in the performance. The S&C

method does not attain probability 1 because a maximum sam-

ple may be selected easily from the incorrect time instant.

Frequency synchronization performance of the proposed

method is evaluated in Fig. 11, where f is 0.1 subcarrierspacings. It can be seen that the proposed method achieves

theoretical CRB when Ec/N0 > 10 dB.

V. CONCLUSIONS

A Matched filter based time and frequency synchronization

method for OFDM systems was proposed, which provides

fast synchronization using PN-sequence preambles. The pro-

posed method transmits a known PN-preamble in the frequencydomain and it has two identical halves in the time domain.

Because MF in the receiver is matched to one half of this

time domain preamble, two identical autocorrelation functions

are achieved from the MF output. These two autocorrela-

tion functions are used for timing and frequency synchroniza-

tion. In timing synchronization, these autocorrelation functions

are combined noncoherently. In frequency synchronization, a

phase difference between those is estimated. The obtained re-

sults indicate that the proposed method attains the optimum

theoretical performance.

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0.1

0.2

0.3

0.4

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0.6

0.7

0.8

0.9

1

SNR [dB]

PD

Proposed method, 101

S&C 101


S&C 102

Proposed method, 10

3

S&C 103


S&C 104

Nc

= 256

L = 128

TCP

= 0

f = 0

Figure 8: PD results with different PFA values.

REFERENCES

[1] T. M. Schmidl and D. C. Cox, Robust Frequency and Timing Synchro-nization for OFDM, IEEE Transactions on Communications, vol. 45,no. 12, pp. 1613 - 1621, Dec. 1997.

[2] M. Speth, S. A. Fechtel, G. Fock and H. Meyr, Optimum ReceiverDesign for Wireless Broad-Band Systems Using OFDM Part - I, IEEETransactions on Wireless Communications, vol. 2, no. 3, pp. 424 - 430,May 2003.

[3] M. Sandell, J. J. van de Beek and P. O. Borjesson, Timing and Fre-

quency Synchronization in OFDM Systems Using the Cyclic Prefix, inProceedings of the IEEE International Symposium on Synchronization,1995, pp. 16-19.

[4] J. J. van de Beek, M. Sandell and P. O. Borjesson, ML Estimation ofTime and Frequency Offset in OFDM Systems, IEEE Transactions onSignal Processing, vol. 45, no. 7, pp. 1800 - 1805, July 1997.

[5] H. Minn, M. Zeng and V. K. Bhargava, On Timing Offset Estimationfor OFDM Systems, IEEE Communication Letters, vol. 4, no. 7, pp.242 - 244, July 2000.

[6] B. Park, H. Cheon, C. Kang and D. Hong, A Novel Timing Estima-tion Method for OFDM Systems, IEEE Communication Letters, vol. 7,no. 5, pp. 239 - 241, May 2003.

[7] J. L. Zhang, M. Z. Wang and W. L. Zhu, A Novel OFDM Frame Syn-chronization Scheme, in Proceedings of the IEEE International Con-

ference on Communications, Circuits and Systems and West Sino Expo-

sitions, 2002, pp. 119-123.[8] F. Tufvesson, O. Edfors and M. Faulkner, Time and Frequency Syn-

chronization for OFDM Using PN-sequence Preambles, in Proceed-ings of the IEEE Vehicular Technology Conference, 1999, vol. 4, pp.2203-2207.

[9] J. E. Kleider and S. Gifford, Synchronization for Broadband OFDMMobile Ad Hoc Networking: Simulation and Implementation, in Pro-ceedings of the IEEE International Conference on Acoustics, Speech,

and Signal Processing, 2002, vol. 4, pp. 3756-3759.

[10] J. G. Proakis, Digital Communications. McGraw-Hill, Inc., New York,USA, 1995.

[11] J. Iinatti, On the Threshold Setting Principles in Code Acquisition ofDS/SS Signals IEEE Journal on Selected Areas in Communications.vol. 18, no. 1, pp. 62-72, Jan. 2000.


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1

SNR [dB]

PMax

&PD,Max

PMax

, theory

PMax

, simul.

PMax

, simul., Tufv.

PD,Max

, theory, 101

PD,Max

, simul., 101

PD,Max

, simul., Tufv., 101

PD,Max

, theory, 102

PD,Max

, simul., 102

PD,Max


PD,Max

, theory, 103

PD,Max

, simul., 103

PD,Max


PD,Max

, theory, 104

PD,Max

, simul., 104

PD,Max


Nc

= 256

L = 128

TCP

= 0

f = 0

Figure 9: PMax and PD,Max results with different PFA values.

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0.1

0.2

0.3

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0.8

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SNR [dB]

PMax

&PD,Max

PMax

, Proposed method

PMax

, S&C

PD,Max

, Proposed method, 101

PD,Max

, S&C, 101

PD,Max


PD,Max

, S&C, 102

PD,Max


PD,Max

, S&C, 103

PD,Max


PD,Max

, S&C, 104

Nc

= 256

L = 128

TCP

= 0

f = 0

Figure 10: PMax and PD,Max results with different PFA val-ues.

20 15 10 5 0 5 1010

5

104

103

102

101

100

SNR [dB]

Frequencyerrorvariance

Simulated

CRB

Figure 11: Performance of the frequency offset estimator.

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