L1-Norm Convergence Properties of Correlogram Spectral Estimates

This paper establishes the following results concerning the estimation of the power spectrum of a single, deterministic, infinitely long signal. a) If ŝx is the signal's power spectral density, correlogram spectral estimates obtained from increasingly longer signal segments tend to ŝx * ŵ/2π in the L-norm, where ŵ is the Fourier transform of the window used to generate the estimates. b) The L-norm of ŝx − ŝx * ŵ /2π can be made arbitrarily small by an appropriate choice of window. It is further shown that the accuracy of the spectral estimates generated by a given window is related to a newly introduced function, termed the windowing error kernel and that this function yields bounds on the asymptotic error of the estimates. As an example, correlogram spectral estimates are used to analyze spectral regrowth in an amplifier. Copyright Notice This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder. 4354 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 9, SEPTEMBER 2007 L -Norm Convergence Properties of Correlogram Spectral Estimates Giorgio Casinovi, Senior Member, IEEE Abstract—This paper establishes the following results concerning the estimation of the power spectrum of a single, deterministic, infinitely long signal. a) If ^ is the signal’s power spectral density, correlogram spectral estimates obtained from increasingly longer signal segments tend to ^ ^ 2 in the -norm, where ^ is the Fourier transform of the window used to generate the estimates. b) The -norm of ^ ^ ^ 2 can be made arbitrarily small by an appropriate choice of window. It is further shown that the accuracy of the spectral estimates generated by a given window is related to a newly introduced function, termed the windowing error kernel and that this function yields bounds on the asymptotic error of the estimates. As an example, correlogram spectral estimates are used to analyze spectral regrowth in an amplifier.This paper establishes the following results concerning the estimation of the power spectrum of a single, deterministic, infinitely long signal. a) If ^ is the signal’s power spectral density, correlogram spectral estimates obtained from increasingly longer signal segments tend to ^ ^ 2 in the -norm, where ^ is the Fourier transform of the window used to generate the estimates. b) The -norm of ^ ^ ^ 2 can be made arbitrarily small by an appropriate choice of window. It is further shown that the accuracy of the spectral estimates generated by a given window is related to a newly introduced function, termed the windowing error kernel and that this function yields bounds on the asymptotic error of the estimates. As an example, correlogram spectral estimates are used to analyze spectral regrowth in an amplifier.