1. 2015-暑期訓練課 Introduction to Synchronization Schemes in OFDM Systems
2015/7/21
王森弘

2. Outline Introduction. Signal model. 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

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. Outline Introduction. Signal model. 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

6. Signal Model (1/3) : the frequency domain data at the k-th subcarrier.
After the IDFT operation:
Channel is assumed to be quasi-stationary. Therefore, its discrete-time impulse response can be expressed as:
Received signal in time-domain:

7. Signal Model (2/3)
where F is the normalized Fourrier matrix, and C is a circulant matrix:
Thus, would be a diagonal matrix.

8. Signal Model (3/3)
In a time-invariant channel, the channel matrix H would be a diagonal matrix.
In a fast fading channel, i.e., the channel coherent time < OFDM symbol period, the Doppler effect must be taken into consideration. Hence, the channel matrix H would not be a diagonal matrix, resulting ICI.
ICI

9. Outline Introduction. Signal model. 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

10. Discussion of symbol boundary

11. Impact of 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

12. Impact of timing offset
Once upon the boundary of DFT window is not located in the ISI-free region, it will induce some extra ISI.
The phase shift and caused by integer and fractional part of timing offset,
Both are depend on frequency index k .
Not Differentiable from the phase of Hk .
Phase Shift can be resolved by Differential Encoding /Decoding , or carrier recovery with pilots.

13. Outline Introduction. Signal model. 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

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

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

16. 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

17. 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
No effect on the orthogonality.
Index shift.

18. Outline Introduction. Signal model. 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

19. 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.

20. 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 = [h0 h1], the received sample r0 and rN/2 can be expressed by: (w/o CFO)
Appendix-C

21. 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

22. 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.

23. TO Estimation Algorithm (5/10)
Let there be N/2 complex 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

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

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

26. TO Estimation Algorithm (8/10)

27. 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).

28. TO Estimation Algorithm (10/10)
Advantage: Low computational complexity. The product P(d) can be implemented with the iterative formula:
The method also called delay correlator. 扣掉 相加

29. Other TO estimation algorithms (1/4)
Minn’s Method [2]: B -B

30. Other TO estimation algorithms (2/4)
Park’s Method [3]:
where D is the symmetric version of C. C D C* D*

31. Other TO estimation algorithms (3/4)
Ren’s Method [4]:
where S is the PN sequence weighted of the original preamble.
A*S

32. Other TO estimation algorithms (4/4)

33. Outline Introduction. Signal model. 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

34. 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

35. 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.

36. Maximum CFO Maximum CFO in 802.16e WiMAX OFDM mode
Oscillator deviation: 8ppm
Highest carrier frequency : 10.68GHz
Maximum CFO: 16 ppm x GHz = 171 kHz
171 kHz subcarrier spacing subcarrier spacing.
Since , estimation for subcarrier spacing is required.
CFO = 1.7 fs = ( ) fs; CFO = 2.7 fs = (4 – 1.3) fs

37. Integer CFO estimation algorithm (1/2)
Time domain correlation
Match filters with coefficient of conjugated preamble waveform modulated by different integer CFO.
Example: CFO is 4.2 fs

38. Integer CFO estimation algorithm (2/2)
Frequency domain autocorrelation
In DVB-T, take advantage of continual pilot subcarriers.
Assume similar CFR in two consecutive symbols.

39. J. -J. van de Beek, M. Sandell, and P. O
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 , Jul

40. 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).

41. System Model

42. 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.

43. System Model
Now, consider the transmitted signal s(k). This is the IDFT of the data symbols xk, which we assume are independent.
Hence, s(k) is a linear combination of independent, identically distributed random variables. If the number of subcarriers is sufficiently large, we know from the central limit theorem that s(k) approximates a complex Gaussian process whose real and imaginary parts are independent.
This process, however, is not white since the appearance of a cyclic prefix yields a correlation between some pairs of samples that are spaced N samples apart.
Hence, r(k) is not a white process either, but because of its probabilistic structure, it contains information about the time offset and carrier frequency offset .

44. 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.

45. 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.

46. 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)

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

48. 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 of which is
We assume that an acquisition, or rough estimate, of the frequency offset has been performed and that , 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

49. Simulation

50. Appendix-A Impact of TO: derivation

51. Appendix-B Impact of CFO: derivation

52. Appendix-C
Transmitted signal CP
channel
P. 20

53. Reference
[1] T. M. Schmidl and D. C. Cox, “Robust Frequency and Timing Synchronization for OFDM,” IEEE Transaction on Communications, vol. 45, no. 12, Dec
[2] H. Minn, V. K. Bhargava, and K. B. Letaief, “A Robust Timing and Frequency Synchronization for OFDM Systems,” IEEE Transaction on Wireless Communications, vol. 2, no. 4, Jul
[3] B. Park, H. Cheon, and C. Kang, “A Novel Timing Estimation Method for OFDM Systems,” IEEE Transaction on Communications Letters, vol. 7, no. 5, May 2003.
[4] G. Ren, Y. Chang, and H. Zhang, “Synchronization Method Based on a New Constant Envelop Preamble for OFDM Systems,” IEEE Transaction on Braodcasting, vol. 51, no. 1, Mar
[5] H. Minn, V. K. Bhargava, and K. B. Letaief, “On Timing Offset Estimation for OFDM Systems,” IEEE Transaction on communications Letters, vol. 4, no. 7, Jul
[6] 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 , Jul
[7] J. Lee, H. Lou, and D. Toumpakaris, “Maximum Likelihood Estimation of Time and Frequency Offset for OFDM Systems,” Electronics Letters, vol. 40,  no. 22, pp , Oct

54. 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

55. 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.