A comparative study of the optimal and interpolation methods for restoration a stationary continuous signal from discrete values

In this paper, we consider the task of restoration of a continuous signal by its discrete values. We compare the optimal and interpolation restoration techniques for the continuous-discrete observation model. The expression for the root-mean-square error of the restoration of a stationary continuous signal from its discrete values is derived. For the optimal restoration case, the signal is reconstructed by the means of an optimal restoration filter. For the interpolation restoration, the signal is reconstructed with the use of linear interpolation. The results of a comparison of the root-mean-square errors of the optimal restoration and interpolation procedures with the same parameters of the input signal, additive noise, and dynamic distortion are presented.


Introduction
The task of continuous signal restoration [1 -4] by its discrete values is a common task of signal processing. There are several ways to get the continuous signal estimation by its discrete values. The simplest solution is to select the interpolation function [4,5] and restore the continuous signal with it. An alternative solution [1, 6 -10] is to set an optimal restoration filter that minimizes the root-meansquare error (RMSE).
The aim of this article is to find out how the optimal restoration method of a stationary continuous signal outperforms the interpolation one. If the RMSE of the optimal recovery method is not much smaller than the error of the interpolation method, then the use of optimal recovery is not practical, since it is more computationally complex and requires information about the signal degradation.
The article is organized in the following way. First, expressions are derived for the RMSE of the optimal recovery method for the general case and for the case of a continuous-discrete linear observation model. The following section presents the derivation of the expression for the RMSE of the interpolation recovery method. Then, the experimental research of the two methods is described. It contains the parameters of the original signal, the degrading system and the additive noise for which experiments were conducted and obtained results.

General case
In this paper, we use one-dimensional signals for the sake of simplicity, but the results can be extended to large dimensions.
Let ( ) y n be the observed degraded discrete observation model-independent signal. The recovery task is to obtain ˆ( ) opt x t the estimation of the original continuous signal ( ) x t .
First let us denote continuous coordinates via discrete ones: where t is the real variable of time defined over the whole numerical axis, n is the integer argument of the resulted sequences, given on the whole numerical axis, T is the period of continuous signal sampling.
Signal recovery performed by linear and invariant to a shift (LIS) system for a fixed moment of time looks as: is the impulse response of the restoring filter for time τ .
During optimal recovery, the RMSE 2 opt τ ε for timeτ is minimized: Let us take the derivative of (3). The resulting sequence convolution is a variation of the Wiener-Hopf equation [1]: where ( ) Φ с xy is the original and observed signal relative energy spectrum,

( )
G Ω τ is the frequency response of optimal recovery filter, is the observed signal energy spectrum.
Let us derive the optimal filter frequency response from (5): Let us now consider the RMSE expression (3). Having (4) and (6), its spectral representation will look as: We can use (7) to calculate the RMSE value for a certain moment of time: The resulting RMSE value 2 ε is the following:

Discrete-continuous observation model
Let us specify the obtained expressions for the continuous-discrete observation model case. This observation model assumes that the original signal is subject to degradations in the continuous domain, and then the signal is sampled. As a result, only samples of a degraded discrete signal are available for observation. In this case, the distorting system is considered to be a LIS system. Having that, the observed signal can be written in the following way: where ( ) Ω с H is the frequency response of degradation LIS system, ( ) is the energy spectrum of the additive noise.
Having (17) and (18), the energy spectrum of optimal recovery RMSE looks as:

Stationary continuous signal interpolation by discrete samples
Let us consider an interpolation method for continuous signal recovery from its discrete values. This method does not take into account the distortion to which the signal was subject to. Let us return to the continuous domain, specifying a distorted signal in the form of a delta functions grid: Let us describe (16) in more detail: where ( ) is the spectrum of the additive noise, is the interpolation system frequency response.
Let us consider the energy spectrum of (17): RMSE value can be calculated in the following way:

Experimental study of optimal and interpolation recovery methods
An experimental study was conducted for both described recovery methods. The purpose of the experimental research is to determine whether the optimal method RMSE is significantly lower than the interpolation one. During the study, the RMSE values were obtained for both recovery methods with the same parameters of the signal, the degradation system, and the additive noise. The results were obtained for several values of the noise variance of the original signal, its correlation coefficient and noise variance of the degradation LIS system. A bi-exponentially correlated signal was taken as the original continuous signal. Its energy spectrum can be written as follows: where β is the correlation coefficient.
The impulse response of the distorting system was defined as the Gauss function, which is the traditional model of a distorting LIS system for real optical system. Pulse and frequency responses of the used distorting system are presented below: ( ) where 2 h σ is the degradation system variance.
We also used additive white noise: Experimental studies were carried out with the use of linear interpolation.     As can be seen from the dependencies, the optimal recovery method gives 1.2-1.63 times less RMSE than the interpolation recovery method (with the examined signal, noise and degradation parameters). This allows us to conclude that the use of the optimal recovery method is much more efficient than the use of interpolation one. Better recovery quality of optimal recovery justifies computational complexity and the need of degradation parameter estimation.

Conclusions
In this paper, the expressions for the RMSE of the continuous stationary signal recovery from its discrete values using two methods interpolation and optimal were derived.