A comparison of digital beacon receiver frequency estimators
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Two algorithms for estimating the frequency and power of the carriers of 20 GHz and 30 GHz satellite signals are compared. Both algorithms operate on a prefiltered sequence generated by lowpass filtering followed by signal decimation for the purpose of sampling rate reduction. The lowpass filtering is accomplished via the overlap-add method of FIR filtering using the FFT. Carrier frequency prediction and tracking is accomplished with a Kalman predictor, for which the frequency drift process is modeled via polynomial extrapolation. The Kalman predictor operates on frequency measurements provided by one of two frequency estimators.
One of the frequency estimation algorithms, a refinement of the DFT-automatic frequency control technique, uses the Chirp-Transform algorithm in its aim for the maximum likelihood estimate of frequency and power. The averaged periodogram is computed from the prefiltered sequence and is used to measure the frequency of the drifting frequency signal as well as its power. One of the disadvantages of this algorithm is the bias present in the estimation of power. The bias can be removed only with knowledge of the noise power. The algorithm has the advantage of being almost exclusively a convolution and therefore is accomplished with minimal computation via the FFT.
An alternative parametric approach to frequency estimation is also investigated. In this approach the weighted least-squares modified Yule-Walker method of autoregressive model estimation is used on the prefiltered sequence to yield frequency estimates. Power estimation is accomplished next via modal decomposition of the estimated correlation sequence. The advantage of this approach is that for slowly varying frequency drift paths (24 hour cycle) the frequency estimates exhibit MSE approximately 3 dB less than the Chirp-Transform algorithm over a wide range of SNR. There are two disadvantages to the parametric algorithm. First the parametric algorithm estimates power with MSE approximately 2 dB greater than the nonparametric algorithm. Secondly the algorithm is more complicated than the nonparametric Chirp-Transform algorithm because it requires matrix inversions and the determination of the roots of a polynomial.
For the digital beacon receiver problem investigated here both algorithms perform similarly in two important respects. First both algorithms can lock onto a carrier signal whose frequency is drifting at the rate of 5 Hertz per second in a noise environment corresponding to a 15 dB/Hz SNR. Secondly both algorithms can make unbiased frequency estimates of the carrier signal allowing the receiver to track the carrier at 7 dB/Hz SNR. Both algorithms attain the Cramer-Rao bound for estimation of constant frequency sinusoids. For a simulated satellite signal with maximum frequency drift of 5 Hertz per second the Kalman frequency predictor is able to reduce the problem to nearly that of the constant frequency case so that the resulting performance corresponds to the Cramer-Rao bound for estimation of constant frequency sinusoids.
Where computational considerations are critical the nonparametric algorithm is preferred. In fact, unless the superior accuracy of the frequency prediction afforded by the parametric algorithm is paramount, the nonparametric algorithm is to be chosen.
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