Introduction to Synchronization Schemes in OFDM Systems

1. Introduction to Synchronization Schemes in OFDM Systems 2018/08/06

2. Outline • Introduction. • Impact of timing offset (TO). • Impact of carrier frequency offset (CFO). • TO estimation algorithm • CFO estimation algorithm • ML Estimation of Time and Frequency Offset in OFDM • Robust Frequency and Timing Synchronization for OFDM

3. Introduction (1/2) • Synchronization issue: • Symbol timing offset (TO): • Due to unknown transmission time. • Carrier frequency offset (CFO): • Oscillator mismatch. • Doppler effect. • Sampling clock offset (SCO): • Mismatch between ADC and DAC. • Phase noise: • Introduced by local oscillators used for up/down-conversion.

4. Introduction (2/2) • There are two categories in general: • Data-Aided methods. • The drawback of the data-aided scheme is the leakage of the bandwidth efficiency due to redundancy overhead. • Non-Data-Aided methods. • Non-data-aided methods or called blind relying on the cyclo-stationarity, and virtual sub-carriers, etc.. • The blind estimation requires a large amount of computational complexity, therefore, it may be not be available in short-burst wireless communication.

5. Discussion of symbol boundary

6. Impact of timing offset • Timing offset: where and are the integer part and fractional part of timing offset, respectively. • For fractional part of timing offset : • For integer part of timing offset : , Appendix-A

7. Impact of Carrier frequency offset (1/4)

8. Impact of Carrier frequency offset (2/4) Faded signal attenuated and rotated by CFO Inter-Carrier Interference (ICI)

9. Impact of Carrier frequency offset (3/4) • A simple representation of received signal with only fractional CFO is given by where is the ICI coefficient, Appendix-B

10. Impact of Carrier frequency offset (4/4) • Fractional CFO • Phase shift in time domain. • Induce the magnitude attenuation and ICI. • Loss of orthogonality. • Integer CFO • Phase shift in time domain. • No effect on the orthogonality. • Index shift.

11. 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.

12. Introduction • We present and evaluate the joint maximum likelihood (ML) estimation of the time and carrier-frequency offset in OFDM systems. • Our novel algorithm exploits the cyclic prefix preceding the OFDM symbols, thus reducing the need for pilots. • In the following analysis, we assume that the channel is nondispersive and that the transmitted signal s(k) is only affected by complex additive white Gaussian noise (AWGN) n(k).

13. System Model

14. System Model • Consider two uncertainties in the receiver of this OFDM symbol: the uncertainty in the arrival time of the OFDM symbol and the uncertainty in carrier frequency. • The first uncertainty is modeled as a delay in the channel impulse response , where is the integer-valued unknown arrival time of a symbol. • The latter is modeled as a complex multiplicative distortion of the received data in the time domain , where denotes the difference in the transmitter and receiver oscillators as a fraction of the intercarrier spacing.

15. ML Estimation • Assume that we observe 2N+L consecutive samples of r(k), as shown in Fig. 2, and that these samples contain one complete (N+L)-sample OFDM symbol.

16. ML Estimation • Define the index sets and • Collect the observed samples in the (2N+L) ×1-vector • Notice that the samples in the cyclic prefix and their copies r(k), are pairwise correlated, i.e., while the remaining samples r(k), are mutually uncorrelated.

17. ML Estimation • Using the correlation properties of the observations r, the log-likelihood function can be written as • Under the assumption that r is a jointly Gaussian vector and omit some factor, we can show that : (1)

18. ML Estimation • The joint complex gaussian PDF can be expressed as . • Note that utilize AB*+A*B=2Re[AB*], we can easily show and .

19. ML Estimation • The maximization of the log-likelihood function can be performed in two steps: • The maximum with respect to the frequency offset is obtained when the cosine term in (1) equals one. • This yields the ML estimation ofwhichis • We assume that an acquisition, or rough estimate, of the frequency offset has been performed and that 0, thus u=0. • Since the cosine tern equals to one, the log-likelihood function of θ becomes and the joint ML estimation of θ and ε becomes

20. Simulation

21. T. M. Schmidl and D. C. Cox, “Robust Frequency and Timing Synchronization for OFDM,” IEEE Transaction on Communications, vol. 45, no. 12, Dec. 1997.

22. TO Estimation Algorithm (1/10) • Schmidl’s Method [1]: First of all, we could design a training symbol which contains a PN sequence on the odd frequencies.

23. TO Estimation Algorithm (2/10) • Due to the property of IDFT, the resulting time domain training symbol would have a repetition form as shown below: • After sampling, the complex samples are denoted as rm. • Ex: Let the multipath channel L = [h0h1], the received sample r0 and rN/2 can be expressed by: (w/o CFO)

24. TO Estimation Algorithm (3/10) • Received signal without CFO: • Received signal with CFO: Extra phase rotation due to CFO Phase difference, which contains the information about CFO

25. TO Estimation Algorithm (4/10) • With CFO, if the conjugate of a sample from the first half is multiplied by the corresponding sample from the second half ( T/2 seconds later), there will be an extra phase difference caused by the CFO, as shown below.

26. TO Estimation Algorithm (5/10) • Let there be N/2complex samples in one-half of the training symbol (excluding the cyclic prefix), and let the sum of the pairs of products be : • Note that d is a time index corresponding to the first sample in a window of N samples. • The received energy for the second half-symbol is defined by • A timing metric can be defined as

27. TO Estimation Algorithm (6/10) • Delay correlator:

28. TO Estimation Algorithm (7/10) • If d’ is the correct symbol timing offset: • If d’ falls behind the correct symbol timing offset 1 sample:

29. TO Estimation Algorithm (8/10)

30. TO Estimation Algorithm (9/10) • Drawback: Plateau effect. Since CP is the copy of the last few samples, these two observation windows result in the same correlation(without noise).

31. Fractional CFO estimation • Fractional CFO estimation can be accomplished when the symbol boundary is detected. where L is the distance between two identical block, R is the block size. • For example: L = N; R = Ncp. L = N/2; R = N/2. N cp N N cp N

32. Estimation Range • The acquisition range of is , which depends on the repetition interval. • For example: • 802.16e-2005 (L=N/4): • DVB-T (L=N): • Note that , is the subcarrier spacing.

33. Appendix-AImpact of TO: derivation

34. Appendix-BImpact of CFO: derivation P. 16

35. HW • Number of subcarrier N = 128, length of CP NCP =16. • Channel: AWGN. • Pilot : ZC sequence (loaded on the subcarriers 0,2,4,…,N-2) • Time offset = 3 samples. • CFO

36. HW • Exercise 1: Timing offset estimation. • Plot the timing metric where • Exercise 2: Carrier frequency offset estimation. • Given a CFO estimator • Plot the mean square error of CFO estimation versus SNR.