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

Saed SAMADI; Kaveh MOLLAIYAN; Akinori NISHIHARA

doi:10.1587/transfun.E92.A.722

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

Saed SAMADI, Kaveh MOLLAIYAN, Akinori NISHIHARA

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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 M π/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.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E92-A No.3 pp.722-732

Publication Date: 2009/03/01

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Online ISSN: 1745-1337

DOI: 10.1587/transfun.E92.A.722

Type of Manuscript: Special Section PAPER (Special Section on Latest Advances in Fundamental Theories of Signal Processing)

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Saed SAMADI, Kaveh MOLLAIYAN, Akinori NISHIHARA, "Discrete Wirtinger-Type Inequalities for Gauging the Power of Sinusoids Buried in Noise" in IEICE TRANSACTIONS on Fundamentals, vol. E92-A, no. 3, pp. 722-732, March 2009, doi: 10.1587/transfun.E92.A.722.
Abstract: 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 M π/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.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E92.A.722/_p

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@ARTICLE{e92-a_3_722,
author={Saed SAMADI, Kaveh MOLLAIYAN, Akinori NISHIHARA, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Discrete Wirtinger-Type Inequalities for Gauging the Power of Sinusoids Buried in Noise},
year={2009},
volume={E92-A},
number={3},
pages={722-732},
abstract={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 M π/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.},
keywords={},
doi={10.1587/transfun.E92.A.722},
ISSN={1745-1337},
month={March},}

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TY - JOUR
TI - Discrete Wirtinger-Type Inequalities for Gauging the Power of Sinusoids Buried in Noise
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 722
EP - 732
AU - Saed SAMADI
AU - Kaveh MOLLAIYAN
AU - Akinori NISHIHARA
PY - 2009
DO - 10.1587/transfun.E92.A.722
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E92-A
IS - 3
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - March 2009
AB - 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 M π/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.
ER -