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frame and carrier recovery in two symbols. We formulate these methods with a common notation and this leads us to address an oversampled scheme.
A NEW TIME-FREQUENCY SYNCHRONIZATION SCHEME FOR OFDM-TDMA SYSTEMS Jose M. Paez-Borrallo, Santiago Zazo, M. Julia Fernandez-Getino

ETS Ingenieros de Telecomunicacion - Universidad Politkcnica de Madrid. Ciudad Universitaria s h , Madrid 28040, Spain Phone: 34-91-3367280; Fax: 34-91- 3367350; e-mail: [email protected] Abstract - Bursty data transmission over OFDM systems requires a rapid time-fiequency synchronization using a desirable short-time training sequence. Several sophisticated methods provide frame and carrier recovery in two symbols. We formulate these methods with a common notation and this leads us to address an oversampled scheme which allows synchronization in only one symbol. In our proposal, the limit of the acquisition range for the carrier fiequency offset can be extended to *N the subcarrier spacing instead of fl the subcarrier spacing as previous methods. Discussions about its robustness against channel and noise are included and supported by several representative computer simulations.

Several methods for time-frequency acquisition have been developed in the recent years (see [2] and references therein). Usually, symbol-timing search is performed first by using correlation methods; in practice, time searching must be evaluated accurately in order to set optimal conditions for fiequency acquisition. In our approach, we will propose a two steps procedure which obtain a very accurate time estimation. Frequency offset compensation in burst operation mode is addressed, in most of the methods, on the phase comparative between two identical halves in one training symbol; these schemes provide the correction within a fiactional part of the carrier spacing and includes an extra symbol to correct the remaining frequency offset as an integer number of the subcarriers frequency separation [2,3].

I. INTRODUCTION

Our proposal is suitable for burst mode applications where using more that one training symbol is not recommended because the loss in the system throughput. Our approach formulates the basic equations of methods dealing with the duplication of the first symbol in two identical halves with matrix notation and the concept of Decimation in Frequency DFT Algorithms [4]. This formulation improves the knowledge about the performance of these methods and it also allows a new synchronization scheme by means of the concept of Decimation in Time DFT Algorithms. Our final scheme extends the acquisition range over the transmission bandwidth by just oversampling the received sequence by a factor of two.

Orthogonal Frequency-Division Multiplexing (OFDM) systems is a large interesting alternative to single carrier schemes using either a continuous stream of data as in a broadcast application (DVB, DAB) or a burst operation mode as in a Wireless Local Area Network over a fiequency selective channel. The main advantage of the OFDM transmission technique is the fact that there is no intersymbol interference (ISI) at all, if a long enough Cyclic Prefix ,CP, is included. This effectively simulates a channel performing cyclic convolution which implies orthogonality over dispersive channels. Synchronization of the symbol timing and carrier frequency offset is probably the most difficult task in these systems because they must be performed very accurately or there will be loss of orthogonality between the subsymbols placed at the different fiequencies. It is well known that OFDM systems are very sensitive to carrier frequency o&ets since they can only tolerate which are a small fiaction of the subcarrier spacing without a large degradation in system performance [I].

0-7803-5565-2/99/$10.00 0 1999 IEEE 2408

The behavior of this method and, in particular, its robustness against additive white Gaussian noise and channel characteristics are also discussed and supported by several representative computer simulations.

11. FORMULATION OF THE OFDM

SYNCHRONIZATION SYSTEM Let us consider the following block diagram in Fig. 1 which remarks the main topics related to the

generation, transmission, and synchronization of a standard N-OFDM system in the absence of noise where N means the number of subcaniers:

Time synchronization intends to find, with maximum accuracy, the unknown symbol time location (r), that is, to determine the starting sample of the synchronization symbol and therefore a time reference for every CP, which must be removed, and for every OFDM symbol which must be demodulated. The standard fiequency ofiet acquisition is performed by comparing the phase between the two halves of an special training symbol in which the fist half is identical to the second half in the time domain. It is well known that the replication of one sequence in the time domain is related with the linear interpolation with zeros in the DFT domain. In this application, it means that only the odd or even carriers are active,meanwhile, the remainder are idle.

I ? . ^ I Fig. 1. Synchronization scheme in OFDM systems where X means the transmitted symbol by all subcarriers, r is the received time domain symbol and Y represents the received symbol by each subcarrier (observe that we have assumed bold letters for the time domain representation and capital bold letters for the fiequency domain representation). Let us now describe the operators defined in figure 1 : a) Input vector X can be decomposed in two parts: first part is the S (Nxl) vector which represents the OFDM symbol to be transmitted; fiom the point of view of frequency synchronization, this symbol has a particular pattern which will be discussed later in the sequel; second part is a zero-padding vector 0 of length N(R-I) which allows a simple formulation of the time oversampling technique (where R means the oversampling factor). (1)

b) Matrix H is a diagonal matrix which represents the attenuation complex hctor per subcarrier, as the channel transfer function H ( o ) at the corresponding subcarrier frequency: diag(H)=[H(O) I(-&) ... H( 2a(RN-l) RN

)]

c) Matrix T (T')represents the DFT (IDFT) operator in matrix form. Usually, in the analysis, we will include a subscript remarking its dimension for clarity purposes. d) Matrix E is also a diagonal matrix which represents the fiequency shifiing effect as a linear increasing phase offset in the time domain; this matrix ispzkametrized by E as the relative frequency offset of the channel (the ratio of the actual fiequency offset to the intercanier spacing). (3) diag(E)= 1 eJRK ..-

c

.*=E

If matrix EzI (identity matrix), interference (ICI) appears.

intercarrier 2409

For convenience, let us define the following notation: given an arbitrary vector z =[z(l), 2(2), ..., z(ZV)]~, (superscript means Hermitian and N is even) we derive: z,=[z( l), 2(2), ..., Z(N/2)IH, zb=[z(N/2+1), z(N/2+2), ...,Z(N)lH, (4) z,=[z( l), 2(3), ..., z(N-1)IH, ze=Cz(2), 2(4), @91H, -*e,

where basically subscript a means the first half of the given vector and subscript b, means the second half Also, subscript o means the odd components of the original vector and subscript e means the even components. This definition can also be derived for diagonal matrices by direct application to the main diagonal vector.

111. TIME SYNCHRONIZATION

ALGORITHM. The duplicate structure of the synchronization symbol proposed in [2], suitable for fiequency offset correction, also applies properly for time synchronization. However, this timing metric which basically correlates both symbol halves in order to find a maximum value, in fhct reaches a plateau which has a length related to the CP length and the channels duration. This lack of accuracy is not acceptable in many systems, in particular if the fiequency recovery is severely degraded. From our point of view, we perform the time synchronization in two steps: a first step that correlates both halves which is identical to the method described in [2]. However the goal of this approach is to carry out a first step to obtain a narrower range for potential fine adjustment, and a second step performs a correlation of samples within certain window length with the external reference (training symbol) which allows the proper synchronization.

I

t Windowing

Second step

Fig.2. Improved Time Synchronizationscheme The behavior of these two steps can be compared in the following example for the AWGN channel (SNR=lO dB).

(9)

The main results related to practical frequency synchronization methods [2, 31 can be formulated from the point of view of the decomposition of the synchronization symbol into two identical halves, and by comparison of the relative phase shifting between both halves. In figure 4, we show the received sequence at the synchronization symbol where both halves ro, rb, should be identical. r rb

Fig.4. Duplication of the training symbol. Assuming that the CP length is long enough, no IS1 appears and therefore, the mulplicative channel model applies (see Fig. 1); r = ET-'= (5) By using the concept of decimation in frequency DFT alporithm, each half can be expressed as follows (in terms of both even and odd carrier symbols). 1 2

r, =-E,(T,&~ H,X, +w-~T,&, H,x,) 1

H,X, -w-'T& H,x,)

(8)

b) Odd carriers: E^ = ungZe(r," rb) / a - 1

IV. FREQUENCY SYNCH. ALGORITHM

2

1 b) Odd carriers: r," r, =-(-XplHo12 4 Xo) e'='

This notation also shows the equivalence between the correlatation in the time [2] or in the frequency domain [3] because the DFT is an orthogonal matrix. In both cases, ungZe operator retrieves the offset information: a) Even carriers: E^ = ungZe(r," r, )/ a

Fig. 3 Timing synchronizationbehavior

r, =--E,(T&,

This formulation supports the result given in [2] whose metric is summarized now for convenience (in the absence of noise): 1 4 Xe) e'*' a) Even carriers: r," rb =-(XflHel'

(6)

where W is a known diagonal frequency shifting matrix whose main diagonal is: 2410

The performance of this estimator is analyzed in detail in [2] showing a very robust behavior against the additive Gaussian noise and also it is independent of the channel frequency response. However, it is obvious that the limit of the acquisition range is fl the subcarrier spacing. In most of cases, this range must be increased by using a second symbol.

V. INCREASING THE ACQUISITION RANGE BY OVERSAMPLING In section IV, it is addressed the main ideas and matrix formulation that describe a standard method for offset correction. In this section, we propose a new method which enables an unambiguous acquisition in the range M. The main idea of our proposal lies on the fact that those methods using the idea shown in previous section are based on the comparison between two identical halves which allows the recovery of the fkequency offset. Let us regard that oversampling do really introduce a sequence replication in this case between adjacent samples (instead of halves): the main fact to be remarked is that phase comparison between odd and even samples in the time domain training symbols will also provide the ofEet information. More indeed, it is intuitive that the comparison between samples much closer in the time domain will increase significantly the acquisition range.

7 First balf

1 2

re = -EeT;' (HaXa + H,,X,)

I aamplcsqmtion -NcedJ0&tfSdraagc

NIzW4lpkSreperai -1cszl a&ctrange

Second haIf

FbthaIf

ssondhdf

Fig.5. Replication of the synchronization symbol

Let us consider a generic case where the fiequency offset is greater than one intercarrier spacing, that is: &=&k+&,=k+Ef

Imposing Xb=O in equation (16) because the second half of subcarrier symbols are idle, and realizing that:

(10)

.ZZ&

E,=e

I-

2N

E,

4is the fractional part of the frequency o s e t and Ek=kis the integer part (k-intercarrier spacing where E Z). Therefore the shifting matrix E can be expressed as the product of these two diagonal matrices: E = EkE (1 1)

the correlation between odd and even samples yields: .277&

k

,

Let us suppose that method described in (8) and (9) performs optimally, and therefore, we are able to estimate with very high accuracy the fiactional fkquency offset: E, =E, (13) Going back to the frequency domain let's observe the effect of the ambiguous determination of the integer intercarrier spacing (see fig. 1): Y = T~;ET-*HX TE,T-IHX = I,HX (14)

r,Hro =e'=

X;H,HW-IH,X,

(18)

Expression in eq.( 18) shows two important features of this offset measure: on one hand it is able to estimate the frequency o s e t and the acquisition range is increased to iN. On the other hand, it is channel dependent: let us observe that factor: X ~ H ;w -IH,x, (19) should be pure real or imaginary in order to provide an accurate measure; otherwise an unknown bias is always introduced. Recalling that frequency ofiet may be decomposed as a fiactional part and an unknown number of intercarrier spacing (lo), shifting matrix E, can be expressed as the product of these two diagonal matrices: = (20) we can propose the following metric: r,HAro

(21)

where I k represents a matrix as the k-circulant identity matrix.

where matrix A is defined as follows: A = ~~&T;BT~&$

(22)

The result of (14) lead us to a cyclic rotation of vector HX, that is no IC1 appears but there is a cyclic rotation between the original and the actual location of the intercarrier symbols. Let us develop analytically this formulation in order to propose a channel independent metric for offset acquisition.

is performed by the estimation of the fractional part of the ikequency o W t by method described in equation (9). The structure of matrix B will be developed in the sequel. After some algebra, (21) yields:

The time oversampling problem (oversampling factor R=2) can be formulated in an appropriate form-by the concept of decimation in time DFT algorithm in terms of first (&) and second (X,) halves in the fiequency domain (observe that &=O because time oversampling is obtained by zero padding in the frequency domain): X, = TNxe+ WTNx, (15) x, = TNX,- W N X , Therefore, the received sequence can be expressed in the following way: 241 1

2m

r:Aro

=X,"~T,E~Ee,T~BT~E$EeT~W-'HaX e'=a

(23)

In terms of the k-circulant identity matrix , (23) becomes:

Let us remark some ideas about the definition of B: it should allow the proper identification of parameter k, as the integer intercarrier offset: if &O, that is the frequency o&et is limited to the intercarrier spacing, the choice of B=W is obvious; in a generic case

where k is unknown, this last choice is not optimum at all; However, if we consider matrix B, where s is a shifting parameter of the s-circulant identity matrix, that is: B, =I,WI," (25) observe that when s=k,

I,HB,I,w-'

=I

and in a generic case: IfB,vIkw-' =

-I

(k-s)x(k-s) O

-'1.

Let us observe that

,274-k)

(27)

Equation (28) states an important conclusion: the angle of this metric is blind to the true shifting value k, but this information is included into the modulus: observe that the condition of maximum modulus is obtained when s=k; therefore, the evaluation of the modulus of metric described by (28) for arbitrary shifting values of s E (0:N-1) will provide the integer intercarrierspacing (k) at its maximum condition. 29) k = mux(lr:A,r,

I)

VI. COMPUTER SIMULATIONS In order to support our theoretical derivations we have developed several simulations of the fiequency offset correction algorithm as is described in equations (9) and (29) for a 32-OFDM system. Our proposal is decomposed in two steps; first step is devoted to the fiactional frequency offset by application of metric described in (9). Afterwards, asecond step uses this result in order to apply the metric discussed in section V (equation 29) to obtain the integer intercarrier spacing. The global result is the addition of results obtained by both steps.

& - w .'5s

*.n*I*4yvT*I"

Our scenario considers a AWGN channel with several signal to noise ratios (SNR): low S N R (10 dB) and high S N R (20 dl3). The first set of results is performed by the deterministic sweeping of the nominal frequency oflket in the range -15