ELIMINATING THE OFDM CYCLIC PREFIX J. D. Bakker Information and Communication Theory Group Faculty of Information Technology and Systems, Delft University of Technology P.O. Box 5031, 2600 GA Delft, The Netherlands Tel./Fax: +31 15 278 3635 / 1843 E-Mail:
[email protected]
Abstract – A novel method is presented to estimate the frame time and carrier frequency offset in orthogonal frequency multiplexing (OFDM) systems [1] using a variable number of time-domain pilot signals. The performance and computational intensity of the proposed system is compared to a traditional OFDM system using redundant information from the cyclic prefix [2]. Keywords – OFDM, modulation, cyclic prefix, synchronization I. INTRODUCTION OFDM systems have recently gained increased interest. They are used in Digital Audio Broadcasting (DAB) and the 802.11a standard for local area networks. OFDM systems operate by modulating the data onto multiple orthogonal subcarriers. The advantage of OFDM systems for high data rate communications is that the symbol length is relatively long compared to the channel delay spread, thus reducing intersymbol interference (ISI). To combat ISI even further most current OFDM systems use a cyclic prefix, where a portion of the end of each symbol is prepended to the symbol. When this cyclic prefix is longer than the channel delay spread, all ISI is absorbed into the cyclic prefix. A synchronization system can use this added redundancy to estimate the frame time and carrier frequency offset at the receiver. In typical OFDM systems the cyclic prefix amounts to 5%-30% of the total symbol length, depending on symbol duration and the expected channel impulse response. The proposed system replaces the cyclic prefix with a binary sequence that is time multiplexed with the OFDM signal. The sequence is used as a pilot for time and frequency synchronization; an equalizer is used to reduce ISI in the received signal. This paper shows that this added binary sequence enables superior synchronization performance than the cyclic prefix at equal overhead. Additional advantages of the added binary sequence are that it can be used for user identification, and that the amount of added redundancy can be dynamically adapted. In Section II an overview is given of the existing cyclic prefixbased synchronization. Section III describes the new
0-7803-7589-0/02/$17.00 ©2002 IEEE
synchronization method and its advantages. Section IV shows simulation results. Conclusions are offered in Section V. II. CYCLIC PREFIX-BASED SYNCHRONIZATION A common method for determining time and frequency offsets in an OFDM system is described in [2]. This method uses the cyclic prefix normally found in OFDM systems (see Figure 1). Considering a discrete transmitted signal s(k) and a received signal r(k) which is the transmitted signal subject to Additive White Gaussian Noise (AWGN), represented by n(k). Furthermore, the receiver adds an uncertainty in the timing, τ, and frequency δf of the signal, giving:
r ( k ) = s( k − τ )e
j 2 πδfk N
+ n( k )
(1)
where k is the sample number and N is the number of samples in the OFDM symbol. This simplified representation assumes
time → symbol n-1
prefix n
prefix n+1
symbol n
symbol n+1
moving window *
– Figure 1: Simplified operation of OFDM synchronization based on the cyclic prefix –
PIMRC 2002
non-dispersive channel that models the delay element, τ, of the timing offset by a channel response of the form
h( k ) = δ ( k − τ )
(2)
Hence the unknown arrival time of the symbol is now modelled as a delay of τ, while the unknown carrier oscillator frequency offset is modelled as a complex rotation of the time domain signal. Important to the analysis of the signal is the assumption that the individual samples of s(k) can be considered to be statistically independent. This is a reasonable assumption if we consider that the signal s(k) has been generated by the IDFT of the data symbols xk which we assume to be statistically independent. This means s(k) is a linear combination of identically distributed random variables which will approximate a complex gaussian process by the central limit theorem.
III. BINARY SEQUENCE-BASED SYNCHRONIZATION Instead of using the existing OFDM cyclic prefix for synchronization, it is also possible to insert dedicated synchronization patterns in the transmitted waveform. Several such systems have been proposed, including systems based on a dedicated preamble [3,4] and embedded spread spectrum sequences [5]. All of these systems keep the cyclic prefix, however. Method The scheme proposed in this paper completely removes the cyclic prefix and replaces some or all of them with timedomain binary sequences. These sequences, p(k), are system parameters and thus known by both the transmitter and the receiver. The sequences are defined as
±1 1 ≤ k ≤ L p( k ) = 0 otherwise
The inclusion of the guard adds redundancy to the signal rendering the process non-white allowing information to be extracted about the time and carrier frequency offset in r(k). This fact is used in [2] where a simultaneous Maximum Likelihood estimation of both the carrier frequency and time offset is derived. The repeated signal fragment is exploited by recognising the pairwise correlation properties of the samples in the guard and the samples in the signal fragment they were copied from:
σ + σ E[r ( k )r ( k + m)] = σ e 0 ∗
2 2 s n 2 − j 2 πδf s
m=0 m=N otherwise
(3)
where σs2 = E[s(k)2] and σn2 = E[n(k)2] are the signal and the noise power, respectively. Any samples that fall outside the cyclic prefix will be uncorrelated. Essentially this means that the correlation function of the received signal with a version of itself delayed by N, the length of a symbol, will peak when correlation is taken within the guard and the copied signal fragment. This allows us to detect the start of a symbol by finding the location of the peak in the correlation function, while the phase angle of the correlation, which is proportional to δf, allows us to detect the frequency offset of the carriers in the receiver. The correlation of the guard showing the location of a symbol by peak detection. Frequency offset can be determined from the phase angle of the peak. Robust synchronization is achieved by maximizing the log-likelihood of the pdf of the samples in r(k), and we can obtain a useful algorithm in the estimation of the synchronization parameters τ and δf.
(4)
Any sequence can be used for p(k), although sequences with a low autocorrelation for offsets other than zero and a flat frequency spectrum are optimal. Note that maximum length pseudonoise (PN) sequences generally only possess the latter quality. In the simplest implementation p(k) directly replaces the cyclic prefix, so packets are constructed by alternately sending p(k) and OFDM symbols, as shown in Figure 2. In the receiver, the procedure is as follows. First, the receiver calculates the cross-correlation between the received signal and the original
time → symbol n-1
symbol n
p(k)
p(k) (original)
p(k)
symbol n+1
*
– Figure 2: OFDM synchronization based on correlation with a time-domain binary sequence–
p(k). Using (2) for the AWGN case, this yields a repeating peak: j 2 πδfk +ϕ 0 N
= δ ((k mod N ) − τ )e
+ n ′( k )
j 2 πδfk +ϕ 0 N
+ n ′( k )
Cyclic prefix (15 samples) 15-bit binary code 7-bit binary code
0.0001
(5)
where f0 is the initial (constant) phase offset of the received signal, and n’(k) includes both the channel noise and the crosscorrelation between p(k) and the OFDM symbols. As the OFDM symbols are independent of p(k), the crosscorrelation produces a Gaussian signal. The arguments of the peaks in (5) produce an estimate of δf, from the location of the peaks follows τ. In a multipath environment (5) should be deconvolved with the autocorrelation of p(k) (which can be precomputed); this may be done with the FFT engine present in each OFDM system. The method works best in systems that transmit bursts of OFDM symbols. In that case the estimates of h(k) from (5) can be averaged, which significantly reduces the contribution of n’(k). An example is given in [6]: an OFDM system which transmits bursts of 32 symbols of 64 carriers each.
RMS frequency error
h( k mod N )e
0.001
1e-05
1e-06 0
5
10
15
20
SNR (dB)
– Figure 3: Simulated RMS frequency error –
p(k) being a binary code makes it very attractive to implement the correlator (see Figure 2) in hardware, as a correlator with one input limited to binary values can be implemented in adders rather than (expensive) multipliers. This technique combines well with the power-saving measures suggested in [4].
Advantages One of the main advantages of the method proposed here is that it separates synchronization and data. The synchronization process does not depend on the exact modulation used for the data. This increases the modularity of the entire system, and allows IP to be reused between systems. An added advantage is that the synchronization process produces a channel estimate. This channel estimate can be used for subsequent equalization or soft error decoding. If equalization is used, it is possible to reduce the number of synchronization blocks per transmitted burst.In such a setup, OFDM symbols are transmitted back-to-back, with only a few synchronization words in between. In the system described in [6] only nine p(k) words are used: one at the start of the burst, and one after each fourth OFDM symbol. As all synchronization words are identical, the blocks of four OFDM symbols plus one p(k) word can be seen as cyclically convolved with h(k), which simplifies equalization to a cyclic deconvolution over this block. As the synchronization waveform is binary, it has a peak-toaverage power ratio (PAR) of 0dB. OFDM systems normally have a PAR of 10log(number of carriers), although methods exist to decrease the PAR [7]. The low PAR for p(k) means that more power is available for synchronization, especially as many commercial RF amplifiers are more efficient when they produce more output power (see for example [8]).
IV. SIMULATION RESULTS The performance of the new algorithm was evaluated with a Monte-Carlo simulation. The results, shown in Picture 3, are for an OFDM system with eight OFDM symbols in a burst over an AWGN channel. Each point represents over 35000 simulated bursts. V. CONCLUSIONS A robust synchronization method for OFDM has been proposed. The performance is better than the current cyclicprefix based systems. The method enhances flexibility and reduces system overhead, by replacing the cyclic prefix that is normally used in an OFDM system. Future work includes more simulations to fully evaluate the perfoemance of the method in fading environments. REFERENCES [1] Chimini, L.J., “Analysis and simulation of a digital mobile channel using orthogonal frequency division multiplexing”. IEEE Transactions on Communication, vol. COM-33, no. 7, pp. 665-675, July 1986. [2] Van de Beek, J.J., M. Sandell, P.O. Börjesson, “ML estimation of time and frequency offset in OFDM
systems”, IEEE Transactions on Signal Processing, vol. 45, no. 7, July 1997. [3] Schmidl, T.,, D. Cox, “Robust frequency and timing synchronization for OFDM”, IEEE Transactions on Communications, vol. 45, no. 12, pp. 1613-1621, December 1997. [4] Tufvesson, F., M. Faulkner, O. Edfors, “Time and frequency synchronization for OFDM using PNsequence preambles”, Proceedings of IEEE Vehicular Technology Conference, Amsterdam, The Netherlands, September 19-22, 1999, pp. 2203-2207. [5] Tufvesson, F., M. Faulkner, P. Hoeher, O. Edfors, “OFDM Time and Frequency Synchronization by Spread Spectrum Pilot Technique”, 8th IEEE Communication Theory Mini Conference in conjunction to ICC'99, Vancouver, Canada, June 7-10, 1999, pp. 115-119.
[6] Bakker, J.D., F.C. Schoute, R. Prasad, “An Air Interface for High Bandwidth Cellular Digital Communications on Microwave Frequencies”. Proceedings of IEEE Vehicular Technology Conference VTC '98, Ottawa, May 1998. [7] A. D. S. Jayalath and C. Tellambura, “Reducing the Peakto-Average Power Ratio of Orthogonal Frequency Division Multiplexing Signal through Bit or Symbol Interleaving,” Elec. Letts. Vol. 36, pp. 1161-1163, June 2000. [8] http://www.rfmd.com/DataBooks/db97/2172.pdf
