Discrete Wirtinger-Type Inequalities for Gauging the Power of Sinusoids Buried in Noise

Two discrete-time Wirtinger-type inequalities relating the power of a finite-length signal to that of its circularly-convolved version are developed. The usual boundary conditions that accompany the existing Wirtinger-type inequalities are relaxed in the proposed inequalities and the equalizing sinusoidal signal is free to have an arbitrary phase angle. A measure of this sinusoidal signal's power, when corrupted with additive noise, is proposed. The application of the proposed measure, calculated as a ratio, in the evaluation of the power of a sinusoid of arbitrary phase with the angular frequency π/N, where N is the signal length, is thoroughly studied and analyzed under additive noise of arbitrary statistical characteristic. The ratio can be used to gauge the power of sinusoids of frequency π/N with a small amount of computation by referring to a ratio-versus-SNR curve and using it to make an estimation of the noise-corrupted sinusoid's SNR. The case of additive white noise is also analyzed. A sample permutation scheme followed by sign modulation is proposed for enlarging the class of target sinusoids to those with frequencies Mπ/N, where M and N are mutually prime positive integers. Tandem application of the proposed scheme and ratio offers a simple method to gauge the power of sinusoids buried in noise. The generalization of the inequalities to convolution kernels of higher orders as well as the simplification of the proposed inequalities have also been studied.