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.