Recovering the original physical signal from the statically distorted one by limited frequency range

Abstract

Abstract Is it possible to recover the original physical signal s(t) (t denotes time) distorted by the physical converter with the static nonlinear conversion function z = f(s), which is monotonic in some range s ∈ [ s min , s max ] ? Answer is yes, if the output signal f(s(t)) has an unlimited spectrum range. Then s(t) = f−1(z(t)), where f−1(z(t)) denotes the inverse function that returns only one real root s ( t ) ∈ [ s min , s max ] corresponding to z(t) (other roots are not in our interest). Is it possible to recover the original signal s(t) ∈ [ s min , s max ] , if the spectrum range of the output signal f(s(t)) is limited to the spectrum range of the original signal s(t)? It is known that nonlinear distortion of the signal produces a plenty of high harmonics. For the first look, the initial information about the original signal is spread in these multiple harmonics, and, if we want to extract this information, we should use whole signal only. This article shows that the initial information does not disappear by filtering out the higher harmonics, and how to recover the original signal from the statically distorted one, when the frequency range of the output signal is limited to the spectrum range of the original signal s(t). In other words, if orthogonal frequency components of s(t) are considered as independent states of the information space, the information space dimension or the number of the independent states does not change in the process of the static nonlinear distortion. Usually, conventional measurement methods use linear properties of the physical converters. However, ideal linear converters are not available, and linear methods are strongly influenced either by nonlinear distortions or the low signal-to-noise ratio, if the level of the signal is artificially decreased to diminish the nonlinear distortions. Nonlinear converters are the way to solve this dilemma. They permit to significantly increase the level of the signal, and to diminish noise simultaneously. The result of this research is a new wide spectrum measurement method of s(t), when converter is nonlinear, and the frequency range of the output signal f(s) is limited to the frequency range of the original signal s(t). The original signal s(t) is expressed as Fourier series. The spectral components of the converter output f(s(t)) are determined by using Fourier transform, and limited to the spectral range of the original signal s(t). The obtained system of nonlinear equations is solved using the least squares technique that permits to find s(t) in the presence of noise. This method allows one to eliminate the influence of nonlinear distortions, and consequently to eliminate harmonic distortion and intermodulation effects. This method enables the use of the converters that are normally not suitable due to the high nonlinearity.