This letter introduces a generalized version of Kay's estimator for the frequency of a single complex sinusoid in complex additive white Gaussian noise. The Kay estimator is a maximum-likelihood (ML) estimator at high signal-to-noise ratio (SNR) based on differential phase measurements with a delay of one symbol interval. In this letter, the corresponding ML estimator with an arbitrary delay in the differential phase measurements is derived. The proposed estimator reduces the variance at low SNR, compared with Kay's original estimator. For certain delay values, explicit expressions for the window function and the corresponding high SNR variance of the proposed generalized Kay (GK) estimator are presented. Furthermore, for some delay values, the window function is nearly uniform and the implementation complexity is reduced, compared with the original Kay estimator. For a delay value of two, we show that the variance at asymptotically high SNR approaches the Cramer-Rao bound as the sequence length tends to infinity. We also explore the effect of exchanging the order of summation and phase extraction for reduced-complexity reasons. The resulting generalized weighted linear predictor estimator and the GK estimator are compared with both autocorrelation-based and periodogram-based estimators in terms of computational complexity, estimation range, and performance at both low and high SNRs.

This article introduces a generalized version of Kay's (1989) estimator for the frequency of a single complex sinusoid in complex additive white Gaussian noise (AWGN). Kay developed a maximum likelihood (ML) estimator based on differential phase measurements with a delay of one symbol interval. In this paper the corresponding ML estimator with an arbitrary delay in the differential phase measurements is derived. The proposed estimator reduces the variance at low SNR compared with Kay's original estimator. The penalty for the reduced variance is a reduced estimation range. With sufficient delay the window function in the generalized Kay estimator is nearly uniform and the implementation complexity is reduced. Performance comparisons with an estimator based on differential phase measurements with a uniform window function is given. The proposed estimator is also compared with the Fitz (1991) estimator.

This paper presents pulses of finite duration for use in orthogonal frequency division multiplexing (OFDM) using offset QAM modulation. In such a system the pulses will overlap in both frequency and time without intersymbol or interchannel interference. The pulses are generated by an optimization procedure which minimizes the out-of-band energy under the constraints of zero intersymbol and interchannel interference. Low out-of-band energy is important to achieve high bandwidth efficiency, especially when the number of channels is small. The optimization procedure leads to an integral equation which is solved by expanding the pulse in truncated prolate spheroidal wave functions. Examples of optimized pulses and their spectrum are presented for lengths of 2 and 4 symbol intervals.