Polynomial Approximation of Signals Corrupted By Noise Dorina Isar and Alexandru Isar * * IEEE Member Electronics and Communications Faculty, 2 Bd. V. Parvan, 300223, Timisoara, Romania, isar@etc.utt.ro Abstract — The polynomial approximation is a very useful procedure for the optimization of the design of electronic equipment. This procedure is more complex when the function to be processed is perturbed by noise. Exploiting the connection between wavelets and polynomials and the well known abilities of the Discrete Wavelet Transform to denoise, in this paper is proposed a new method for the accurate polynomial approximation of a function, despite the noise that perturbs it. This method can be used to enhance the estimation of any polynomial function perturbed by noise. Examples, containing applications of the proposed method, are also presented. The good simulation results obtained recommend the use of the proposed method in any difficult estimation problem. Index Terms — polynomial approximation, Discrete Wavelet Transform, denoising, estimation, instantaneous frequency. I. Introduction The polynomial approximation is a very important procedure for the optimization of the design of electronic equipment. For example the approximation of the function describing the input output relation of a given system with a first degree polynomial permits the linearization of that system. The approximation of a curve with a second degree polynomial permits its representation with lines and circles. The list of examples for the application of the polynomial approximation can be easy continued. There is a strong connection between the wavelets theory and the polynomial theory. Every polynomial of degree r belongs to the generating space of a multiresolution analysis. So all the details of the discrete wavelet transform, DWT, of a polynomial of degree r, computed using a mother wavelets with a number of vanishing moments greater than r, are equals with zero. Hence computing the DWT of a signal, x[n], with the aid of a mother wavelets with r+1 vanishing moments, eliminating the details in the DWT domain and computing the inverse discrete wavelet transform, IDWT, the polynomial approximation of the considered signal with a polynomial of degree r , x r [n], is obtained. So, the information contained in the signal x r [n] can be recovered using a very small number of coefficients of the DWT of the signal x[n]. This number will be noted in the following with Q and will correspond to the number of samples of the considered signal divided by 4. The small value of Q is the reason why the DWT is used in compression applications. The approximation already described can be done even if the signal to be processed, x[n], is perturbed by noise, following a denoising procedure. In this case the signal to be processed is of the form x i [n] = x[n] + n i [n], where n i [n] is a stationary noise with zero mean. To estimate the signal x[n], Donoho, proposed the following method [1]: 1. The DWT of the signal x i [n] is computed, obtaining the signal y i [n]. 2. A non-linear filtering procedure, called soft thresholding, is applied in the wavelet transform domain: [] [] { } [] ( ) [] ⎩ ⎨ ⎧ ≥ - = not if 0 th n y , th n y n y sgn n y i i i 0 (1) where th is a threshold. 3. Taking the inverse DWT, IDWT, the denoised version, x o [n] is obtained. If the treshold th from relation (1) is well selected then all the noise from the DWT details coefficients is suppressed. A significant part of the noise from the DWT approximation coefficients can be also suppressed if the nonlinear filter in (1) is used to process these coefficients when the mean of the signal x[n] equals zero. If the signal x[n] is a polynomial then the suppression of all its DWT detail coefficients do not introduce any distortion. So, only the filtering of the DWT approximation coefficients affects the distortion level of the reconstruction realized using the IDWT. There are few recent papers dealing with this denoising method, [2] – [4], but the extraction of the polynomial part of the signal x[n] is not analyzed. The success of the Donoho's denoising method is due to two properties of the DWT. This transform concentrates the energy of the signal x[n] and spreads the noise energy. The DWT has a whitening effect, [5]. So the noise spreading effect is obvious. But the DWT conserves the energy. This is the reason why the noise and the useful components of the analyzed signal are separated in the DWT domain. The performances of the denoising method depends on the quality of this separation. The parameters of the Donoho's denoising method are: the mother wavelets, ψ(t), the number of DWT's iterations, IT and the value of the threshold, th. The case of a polynomial input signal x[n] of degree P was not analyzed yet and is very interesting due to the strong connection