1. Jump to first page enemer@ieee.org New algorithmic approach for estimating the frequency and phase offset of a QAM carrier in AWGN conditions Using HOC

2. Jump to first page E. Nemer - 2 Context and Motivation Problem statement Carrier Estimation with HOC Frequency offset Phase offset Performance analysis Simulation Data Analysis Conclusion

3. Jump to first page E. Nemer - 3 LPF InIn QnQn Rx Nyq Rx Nyq LPF Carrier Estimation K samples / symbol FFE PLL FBE Slicer Equalizer Often done in 2 stages : First : blind (no reliance on apriori symbol knowledge) does not require a proper timing estimation uses the output of the RxNyq (oversampled) Second : decision-directed can only correct for small offset uses typically 1 sample / symbol

4. Jump to first page E. Nemer - 4 ML-based schemes Feed-forward, feedback Criteria in designing algorithms Performance degradation in low SNR conditions Complexity issues : simplified ML approaches are often used Issues related to acquisition time, hang up are key in burst receivers Non-decision directed recovery

5. Jump to first page E. Nemer - 5 - Symmetric process--> third cumulant = zero - Gaussian process --> All cumulants > order 2 are = zero y(n) = x(n) + g(n) --> HOS (y) = HOS (x) Inherent suppression of Gaussian and symmetric processes Detection of Non-linearity and phase coupling Peculiar boundaries C 4 C 2  2 -------------------  - 2 < < inf rand binary 2 sinusoids Unif. phase Gaussian Laplacian Impulse process  inf Normalized kurt H(f) Gaussian Linear ? HOS = 0 HOS != 0 ? Non-linear !

6. Jump to first page E. Nemer - 6 Symbol rate (but not timing) is known Channel is equalized Statistics of the sent symbols are known (but not exact values) Estimate the frequency offset Assuming: LPF InIn QnQn Rx Nyq Rx Nyq LPF Carrier Estimation K samples / symbol

7. Jump to first page E. Nemer - 7 A demodulated M-PSK signal can be represented as : Freq offset: The freq offset may be estimated from the autocorrelation of Z n at lag L, assumed to be a multiple of K, the symbol time In-phase and quadrature components of the noise are small (zero) at the epok [4] Freq offset

8. Jump to first page E. Nemer - 8 Generalized M-QAM signal at the receiver output : Freq offset: Consider the 1D slice of the 4 th order cumulant Where the sub-terms are : Freq offset

9. Jump to first page E. Nemer - 9 Assume the diagonal symbols are used (S1, S3) Then higher order moments can be expressed in terms of the signal energy The (normalized) 4 th -order cumulant becomes : From which, the frequency offset may be deduced Freq offset

10. Jump to first page E. Nemer - 10 If the frequency offset is zero (or known), then the phase may be estimated as: Consider the 1D slice of the 3 rd order cumulant, defined as : Phase offset and can be shown to be (for diagonal symbols) : Consider the sum of the real and imaginary parts : and the normalized sum (by the energy) :

11. Jump to first page E. Nemer - 11 Relations between Normalized Cumulants h(k) W Y The Cumulants at the output of a channel / filter may be written in terms of that at the input and the coefficients : Effect of Nyquist Filter and Channel

12. Jump to first page E. Nemer - 12 MethodSNRDf=200 Hz2 kHz10 kHz 2 nd order40 dB 199 Hz2000 Hz10000 Hz 4 th order200200710026 2 nd order2019020029999 4 th order198200910024 2 nd order1520519829992 4 th order202199010016 2 nd order822119549992 4 th order198197210016 4 th order yields better accuracy at small freq offsets … but not at high freq offsets Freq offset Fc = 5 MHz Fsym = 160 ksym/s Block size : 100 symbols

13. Jump to first page E. Nemer - 13 Bias (sine wave) Bias (Gaussian noise) Variance Gaussian noise - Occurs when frame length not an integer number of periods - Time estimator is only asymptotically unbiased. - Need to define a new unbiased estimator of 4 th stat - Function of underlying process (noise) energy - May be reduced by increasing segment length Complex sinusoid Gaussian noise When computing the HOC of Zn, using time averages, added terms occur due to :

14. Jump to first page E. Nemer - 14 - Occur when the frame length not an integer number of periods - Can be reduced by computing HOC over several segments and averaging them. - More bias terms are present in the 4 th order moment than the 2 nd order 2 nd and 4 th order moments of a sine wave Bias : Case of a Sine wave T: sample time w : frequency  : Phase

15. Jump to first page E. Nemer - 15 - Time-average based estimator is only asymptotically unbiased. - Need to define a new unbiased estimator of 4 th stat : Biased ! Define a (new) unbiased estimator : Valid only for white Gaussian noise Bias : Case of Gaussian Noise

16. Jump to first page E. Nemer - 16 - Function of underlying process (noise) energy - May be reduced by increasing segment length The segment size N has to be significantly increased (by 48 ) in order to bring the variance of the 4 th order moment at par with the 2 nd order. Variance : Case of Gaussian Noise

17. Jump to first page E. Nemer - 17 F New estimators for the carrier phase and frequency offset were developed based on newly established expressions of the 3 rd and 4 th - order cumulants of the demodulated QAM signal. F The higher order estimator is more robust to noise for small values of frequency offsets, though it is not the case for larger ones. F Clearly the improvement depends on the ability to find better ways to compute the HOS in a way to reduce the large bias and variance, when using a finite data set. F more noise-robust methods for (blind) carrier estimation may be developed based on higher order statistics.