RECTANGULAR POWER SPECTRAL DENSITIES OF ORTHOGONAL FUNCTIONS

摘要

In this application, a set of orthogonal functions is introduced whose power spectral densities are all rectangular shape. To find the orthogonal function set, it was considered that their spectrums (Fourier transforms of the functions) are either real-valued or imaginary-valued, which are corresponding to even and odd real-valued time domain signals, respectively. The time domain functions are all considered real-valued because they are actually physical signals. The shape of the power spectral densities of the signals are rectangular thus, the Haar orthogonal function set can be employed in the frequency domain to decompose them to several orthogonal functions. Based on the inverse Fourier transform of the Haar orthogonal functions, the time domain functions with rectangular power spectral densities can be determined. This is equivalent to finding the time-domain functions by taking the inverse Fourier transform of the frequency domain Walsh functions. The obtained functions are sampled and truncated to generate finite-length discrete signals. Truncation destroys the orthogonality of the signals. The Singular Value Decomposition method is used to restore the orthogonality of the truncated discrete signals.