5. The algorithm and experimental results

In this section, we give the algorithm and an example to show that the regularized Fourier series is more effective in controlling noise than the Fourier series.

In practical computation, we choose a large integer N and use the next formula in computation:

^<sup>f</sup> <sup>α</sup>ð Þ¼ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup>

Figure 3.

57

Figure 2.

H1H<sup>2</sup> ∑ N n1¼�N

∑ N n2¼�N

The numerical results by Fourier series in Example 2.

DOI: http://dx.doi.org/10.5772/intechopen.81542

Computation of Two-Dimensional Fourier Transforms for Noisy Band-Limited Signals

Example 1. Suppose

f nð Þ <sup>1</sup>H1; <sup>n</sup>2H<sup>2</sup> <sup>e</sup>in1H1ω1þin2H2ω<sup>2</sup>

<sup>2</sup> h i <sup>P</sup>Ωð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> :

<sup>2</sup> h i <sup>1</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup>

1 � cost<sup>1</sup> πt 2 1

1 � cost<sup>2</sup> πt 2 2

:

1 þ 2πα þ 2παð Þ n1H<sup>1</sup>

The numerical results by the regularized Fourier series in Example 2.

f <sup>T</sup>ð Þ¼ t1; t<sup>2</sup>

Figure 1. The exact Fourier transform in Example 2.

Computation of Two-Dimensional Fourier Transforms for Noisy Band-Limited Signals DOI: http://dx.doi.org/10.5772/intechopen.81542

Figure 2. The numerical results by Fourier series in Example 2.

if α σð Þ! 0 and <sup>σ</sup>2=α σð Þ! 0 as <sup>σ</sup> ! 0.

� � � 2 <sup>L</sup><sup>2</sup> <sup>¼</sup> <sup>H</sup><sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>2H<sup>2</sup>

1 þ 2πα þ 2παð Þ n1H<sup>1</sup> <sup>2</sup> h i

> ∑ ∞ n2¼�∞

½ �� � �Ω1; <sup>Ω</sup><sup>1</sup> ½ � <sup>Ω</sup>2; <sup>Ω</sup><sup>2</sup> as <sup>α</sup> ! 0 and Var ^<sup>f</sup> <sup>α</sup>ð Þ ½ � �ω1; <sup>ω</sup><sup>1</sup>

1H<sup>2</sup>

<sup>2</sup> � ∑ ∞ n1¼�∞

<sup>2</sup> <sup>þ</sup> <sup>2</sup>πα <sup>þ</sup> <sup>2</sup>παð Þ <sup>n</sup>1H<sup>1</sup> <sup>2</sup> � �

1 þ 2πα þ 2παð Þ n1H<sup>1</sup> <sup>2</sup> h i<sup>2</sup>

In this section, we give the algorithm and an example to show that the regularized Fourier series is more effective in controlling noise than the Fourier series. In practical computation, we choose a large integer N and use the next formula

By the proof of Theorem 4.1, we can see that ^<sup>f</sup> <sup>T</sup>ð Þ� <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> <sup>E</sup> ^<sup>f</sup> <sup>α</sup>ð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup>

∑ ∞ n2¼�∞

1 þ 2πα þ 2παð Þ n2H<sup>2</sup> <sup>2</sup> h i

h i

σ2

2πα þ 2παð Þ n2H<sup>2</sup> <sup>2</sup> � �

1 þ 2πα þ 2παð Þ n2H<sup>2</sup> <sup>2</sup> h i<sup>2</sup> :

h i

<sup>¼</sup> <sup>O</sup> <sup>σ</sup><sup>2</sup> ð Þþ <sup>O</sup> <sup>σ</sup><sup>2</sup> ð Þ <sup>=</sup><sup>α</sup> .

! 0 in

Proof. We can calculate

Recent Advances in Integral Equations

4πα þ 2παð Þ n1H<sup>1</sup>

� f <sup>T</sup>ð Þ n1H1; n2H<sup>2</sup>

� �

�

�

and

L2

Var ^<sup>f</sup> <sup>α</sup>ð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> h i

in computation:

Figure 1.

56

The exact Fourier transform in Example 2.

^<sup>f</sup> <sup>T</sup>ð Þ� <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> <sup>E</sup> ^<sup>f</sup> <sup>α</sup>ð Þ <sup>ω</sup>1; <sup>ω</sup><sup>2</sup> � h i �

> � 2

¼ ∑ ∞ n1¼�∞

5. The algorithm and experimental results

Figure 3. The numerical results by the regularized Fourier series in Example 2.

$$\begin{aligned} \hat{f}\_a(\mathbf{u}\_1, \mathbf{u}\_2) &= \\ H\_1 H\_2 \sum\_{n\_1=-N}^N \sum\_{n\_2=-N}^N \frac{f(n\_1 H\_1, n\_2 H\_2) e^{in\_1 H\_1 \mathbf{u}\_1 + in\_2 H\_2 \mathbf{u}\_2}}{\left[1 + 2\pi a + 2\pi a (n\_1 H\_1)^2\right] \left[1 + 2\pi a + 2\pi a (n\_2 H\_2)^2\right]} P\_\Omega(\mathbf{u}\_1, \mathbf{u}\_2) \dots \end{aligned}$$

Example 1. Suppose

$$f\_{\,\,T}(t\_1, t\_2) = \frac{1 - \cos t\_1}{\pi t\_1^2} \frac{1 - \cos t\_2}{\pi t\_2^2}.$$

Then

$$
\hat{f}\_T(\mathfrak{o}\_1, \mathfrak{o}\_2) = (\mathfrak{1} - |\mathfrak{o}\_1|)(\mathfrak{1} - |\mathfrak{o}\_2|) P\_\mathfrak{\mathcal{Q}}(\mathfrak{o}\_1, \mathfrak{o}\_2),
$$

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Article ID 1403181, 17 pages

where Ω<sup>1</sup> ¼ 1 and Ω<sup>2</sup> ¼ 1.

We add the white noise that is uniformly distributed in ½ � �0:0005; 0:0005 and choose N ¼ 20. The exact Fourier transform is in Figure 1. The result of the Fourier series is in Figure 2. The result of the regularized Fourier series with α ¼ 0:001 is in Figure 3.
