Elias Nemer

1. Carrier Estimation For QAM Receivers Using Higher Order Cumulants Elias Nemer enemer@ieee.org New algorithmic approach for estimating the frequency and phase offset of a QAM carrier in AWGN conditions Using HOC

2. Outline • Context and Motivation • Problem statement • Carrier Estimation with HOC • Frequency offset • Phase offset • Performance analysis • Simulation Data • Analysis • Conclusion

3. Problem Context K samples / symbol In Rx Nyq LPF PLL FFE FBE Qn Slicer Rx Nyq LPF Equalizer Carrier Estimation 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. Problem Context • Non-decision directed recovery 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

5. Motivation...Why HOS ? • Inherent suppression of Gaussian and symmetric processes • - Symmetric process --> third cumulant = zero • - Gaussian process -->All cumulants > order 2 are = zero y(n) = x(n) + g(n) --> HOS (y) = HOS (x) Linear ? HOS = 0 HOS != 0 ? Non-linear ! Gaussian H(f) • Detection of Non-linearity and phase coupling C 4 - - - - - - - - - - - - - - - - - - - - 2 < < inf • Peculiar boundaries 2  C  2 -2 -0.75 0 3 inf Normalized kurt Gaussian Impulse process rand binary Laplacian 2 sinusoids Unif. phase

6. Carrier Estimation K samples / symbol In Rx Nyq LPF Qn Rx Nyq LPF Carrier Estimation Assuming: 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

7. Carrier Estimation Freq offset A demodulated M-PSK signal can be represented as : 2nd order Freq offset: The freq offset may be estimated from the autocorrelation of Zn 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]

8. Carrier Estimation • Generalized M-QAM signal at the receiver output : Freq offset Freq offset: 4th Order • Consider the 1D slice of the 4th order cumulant • Where the sub-terms are :

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

10. Carrier Estimation • Consider the 1D slice of the 3rd 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) : • If the frequency offset is zero (or known), then the phase may be estimated as:

11. Other Factors Effect of Nyquist Filter and Channel The Cumulants at the output of a channel / filter may be written in terms of that at the input and the coefficients : W Y h(k) Relations between Normalized Cumulants

12. Method SNR Df=200 Hz 2 kHz 10 kHz 2nd order 40 dB 199 Hz 2000 Hz 10000 Hz 4th order 200 2007 10026 2nd order 20 190 2002 9999 4th order 198 2009 10024 2nd order 15 205 1982 9992 4th order 202 1990 10016 2nd order 8 221 1954 9992 4th order 198 1972 10016 Fc = 5 MHz Fsym = 160 ksym/s Block size : 100 symbols Performance Results Freq offset … but not at high freq offsets 4th order yields better accuracy at small freq offsets

13. Complex sinusoid Analysis Gaussian noise When computing the HOC of Zn, using time averages, added terms occur due to : • 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 4th stat - Function of underlying process (noise) energy - May be reduced by increasing segment length

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

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

16. Analysis Variance : Case of Gaussian Noise - 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 4th order moment at par with the 2nd order.

17. Conclusion • New estimators for the carrier phase and frequency offset were developed based on newly established expressions of the 3rd and 4th-order cumulants of the demodulated QAM signal. • The higher order estimator is more robust to noise for small values of frequency offsets, though it is not the case for larger ones. • 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. • more noise-robust methods for (blind) carrier estimation may be developed based on higher order statistics.