The Regularized Low Pass Filter
Weidong Chen

Abstract

In this paper the low pass filter is discussed in the noisy case. And a regularized low pass filter is presented. The convergence property of the regularized low pass filtering algorithm is proved in theory and tested by numerical results.

Share and Cite:

Chen, W. (2014) The Regularized Low Pass Filter. Journal of Signal and Information Processing, 5, 14-16. doi: 10.4236/jsip.2014.51003.

First, we describe the band-limited signals. The details can be seen in [6].

Definition: A function is said to be band-limited if

Here is the Fourier transform of:

(1)

We then have the inversion formula:

In many practical problems, the signal is noisy:

(2)

where is the noise

(3)

and is the exact band-limited signal.

In this paper, we will consider the problem low pass filtering:

(4)

If the signal is noisy however, the filter is not reliable. We will give an example to show that the noise can become very large after the low pass filtering process. So this filter is not reliable in the noisy case. And a regularized low pass filtering algorithm will be presented.

In section 2 we give the property of the low pass filter. A regularized filtering algorithm and the proof of its convergence are in section 3. The numerical results of some examples are given in section 4. Finally, the conclusion is given in section 5.

2. The Property of the Low Pass Filter

In this section, we discuss the property of the low pass filter.

Example. Assume the noise is where is a given point in the time domain and is close to zero. Then the noise signal after the filtering is

We can see that. However, the noise at after the filtering is

Also at any point,

So the error after the filtering becomes.

Remark. This is only an example for analysis. In the section of numerical results we will show that the low pass filter (4) is not very effective for white noise.

3. The Regularized Filtering Algorithm

First, we consider the regularized Fourier transform [7]:

where is the regularization parameter. Here is the minimizer of a smoothing functional. We have proved converges to the exact Fourier transform as the error of approaches to zero. In [7], we have successfully used the regularized Fourier transform in extrapolation. So the weight function

is helpful to solve ill-posed problems.

Based on the regularized Fourier transform we present the regularized filtering formula:

(5)

where is given in (2).

The convergence property of this regularized filtering formula is given in the theorem below.

Theorem 3.1. For, if and as, then according to the maximum norm as.

Proof.

where

For each, there exists such that

Then

where

as.

4. Numerical Results

In this section, we give some examples to show that the regularized filtering algorithm (5) is more effective in reducing the noise than the convolution (4).

Suppose the exact signal in example 1 and 2 is

Then construct

where.

Example 1. We consider the noise

where, , and. This noise is used in the analysis of the stability in Section 2.

The result of (4) and the result of the regularized filtering algorithm with are in figure 1.

Example 2. We consider the noise to be white noise that is Gauss distribution whose variance is 0.01. The result of (4) and the result of the regularized sampling algorithm with are in figure 2.

5. Conclusion

The filter of convolution with sinc function is not stable. For some noises the results of the filtering are not reliable.

Regularized filtering algorithm is more effective in reducing the noise.

Figure 1. The numerical results of example 1.

Figure 2. The numerical results of example 2.

Acknowledgements

I would like thank University of Georgia for supporting my post doctoral work.

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] D. Georgakopoulos and W. Q. Yang, “Circuit Noise Reduction by Analogue Lowpass Filtering and Data Averaging,” Electronics Letters, Vol. 37 No. 19, 2001, pp. 1147-1148. http://dx.doi.org/10.1049/el:20010782 [2] P. Daripa and W. Hua, “A Numerical Study of an Ill-Posed Boussinesq Equation Arising in Water Waves and Nonlinear Lattices: Filtering and Regularization Techniques,” Applied Mathematics and Computation, Vol. 101, No. 2-3, 1999, pp. 159-207. http://dx.doi.org/10.1016/S0096-3003(98)10070-X [3] O. Scherzer and J. Weickert, “Relations between Regularization and Diffusion Filtering,” Journal of Mathematical Imaging and Vision, Vol. 12, No. 1, 2000, pp. 43-63. http://dx.doi.org/10.1023/A:1008344608808 [4] T. Daboczi and I. Kollarl, “Multiparameter Optimization of Inverse Filtering Algorithms,” IEEE Transactions On Instrumentation and Measurement, Vol. 45, No. 2, 1996, pp. 417-421. http://dx.doi.org/10.1109/19.492758 [5] D. Tschumperle and R. Deriche, “Vector-Valued Image Regularization with PDEs: A Common Framework for Different Applications,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 27, No. 4, 2005, pp. 506-517. http://dx.doi.org/10.1109/TPAMI.2005.87 [6] W. Chen, “An Efficient Method for An Ill-Posed Problem —Band-limited Extrapolation by Regularization,” IEEE Transaction on Signal Processing, Vol. 54, No. 12, 2006, pp. 4611-4618. http://dx.doi.org/10.1109/TSP.2006.881255 [7] W. Chen, “Band-Limited Extrapolation by Regularization,” LAP Lambert Academic Publishing, 2010.