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

Written By

Weidong Chen

Submitted: June 26th, 2018 Reviewed: September 16th, 2018 Published: July 24th, 2019

DOI: 10.5772/intechopen.81542

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The computation of the two-dimensional Fourier transform by the sampling points creates an ill-posed problem. In this chapter, we will cover this problem for the band-limited signals in the noisy case. We will present a regularized algorithm based on the two-dimensional Shannon Sampling Theorem, the two-dimensional Fourier series, and the regularization method. First, we prove the convergence property of the regularized solution according to the maximum norm. Then an error estimation is given according to the L2-norm. The convergence property of the regularized Fourier series is given in theory, and some examples are given to compare the numerical results of the regularized Fourier series with the numerical results of the Fourier series.


  • Fourier transform
  • band-limited signal
  • ill-posedness
  • regularizationAMS subject classifications: 65T40
  • 65R20
  • 65R30
  • 65R32

1. Introduction

The two-dimensional Fourier transform is widely applied in many fields [1, 2, 3, 4, 5, 6, 7, 8, 9]. In this chapter, the ill-posedness of the problem for computing two-dimensional Fourier transform is analyzed on a pair of spaces by the theory and examples in detail. A two-dimensional regularized Fourier series is presented with the proof of the convergence property and some experimental results.

First, we describe the band-limited signals.

Definition. For two positive Ω1, Ω2R, a function fL2R2 is said to be band-limited if


Here f̂ is the Fourier transform of:


We will consider the problem of computing f̂ω1ω2 from ft1t2.

For band-limited signals, we have the following sampling theorem [4, 10, 11]. For the two-dimensional band-limited function above, we have


where H1π/Ω1 and H2π/Ω2.

Calculating the Fourier transform of ft1t2 by the formula (2), we have the formula which is same as the Fourier series


where PΩω1ω2=1Ω1Ω1×Ω2Ω2 1, ω2) is the characteristic function of Ω1Ω1×Ω2Ω2.

In many practical problems, the samples fn1H1n2H2 are noisy:


where ηn1H1n2H2 is the noise


and fTL2 is the exact band-limited signal.

The noise in the two-dimensional case is discussed in [5, 6], and the Tikhonov regularization method is used. However, there is too much computation in the Tikhonov regularization method since the solution of an Euler equation is required.

The ill-posedness in the one-dimensional case is considered in [12, 13]. The regularized Fourier series


in [12] is given based on the regularized Fourier transform


in [14]. The regularized Fourier transform was found by finding the minimizer of the Tikhonov’s smoothing functional.

In this chapter, we will find a reliable algorithm for this ill-posed problem using a two-dimensional regularized Fourier series. In Section 2, the ill-posedness is discussed in the two-dimensional case. In Section 3, the regularized Fourier series and the proof of the convergence property are given. The bias and variance of regularized Fourier series are given in Section 4. The algorithm and the experimental results of numerical examples are given in Section 5. Finally, the conclusion is given in Section 6.


2. The ill-posedness

We will first study the ill-posedness of the problem (3) in the noisy case (4). The concept of ill-posed problems was introduced in [15]. Here we borrow the following definition from it.

Definition 2.1 Assume A:DU is an operator in which D and U are metric spaces with distances ρD and ρU, respectively. The problem


of determining a solution z in the space D from the “initial data” u in the space U is said to be well-posed on the pair of metric spaces DU in the sense of Hadamard if the following three conditions are satisfied:

  1. For every element uU, there exists a solution z in the space D; in other words, the mapping A is surjective.

  2. The solution is unique; in other words, the mapping A is injective.

  3. The problem is stable in the spaces DU:>0,δ>0, such that


In other words, the inverse mapping A1 is uniformly continuous. Problems that violate any of the three conditions are said to be ill-posed.

In this section, we discuss the ill-posedness of Af̂=f on the pair of Banach spaces (L2Ω1Ω1×Ω2Ω2,l(Z2)), where f̂ω1ω2 is given by the Fourier series in Eq. (3).

The operator A in Eq. (6) is defined by the following formula:


where =fn1H1n2H2:n1Zn2Z.

As usual, l is the space an:nZ2 of bounded sequences. The norm of l is defined by



  1. The existence condition is not satisfied.

  2. The uniqueness condition is satisfied.

  3. The stability condition is not satisfied. The proof is similar to the proof in [10].


3. The regularized Fourier series

Based on the one-dimensional regularized Fourier series in [12], we construct the two-dimensional regularized Fourier series:


where fn1H1n2H2 is given in (4). We will give the convergence property of the regularized Fourier series in this section.

Lemma 3.1


where a1+2πα2πα.



Lemma 3.2 For any band-limited function gt1t2 and ω1ω2Ω1Ω1×Ω2Ω2


Proof. By the sampling theorem


By Lemma 3.1 and the FOIL method, Eq. (10) is true.

Lemma 3.3 For each arbitrarily small c>0 and ωΩ+cΩc,


Proof. By the inequality a+b22a2+b2,


Lemma 3.4 For each arbitrarily small c>0 and ω1ω2Ω1+cΩ1c×Ω2+cΩ2c,


for α+0 and g that is Ω-band-limited.

Proof. By the Cauchy inequality,


where n1=n2=gn1H1n2H22 is bounded by Parseval equality, and


By Lemma 3.3, Eq. (12) is true.

Lemma 3.5








Lemma 3.6 For each arbitrarily small c>0 and ω1ω2Ω1+cΩ1c×Ω2+cΩ2c,


for α+0 and g that is Ω-band-limited.

Proof. By Cauchy inequality,


where n1=n2=gn1H1n2H22 is bounded by the Parseval equality, and


By Lemma 3.3 and Lemma 3.5 Eq. (14) is true.

Lemma 3.7


for δ+0 and α+0, where η and δ are given in (4) and (5) in Section 1.







For the same reason,


So Eq. (15) is true.

Theorem 3.1 Suppose fTL1R2L2R2 is band-limited. For each arbitrarily small c>0, if we choose α=αδ such that αδ0 and δ/αδ0 as δ0, then f̂αω1ω2f̂Tω1ω2 uniformly in ω1ω2Ω1+cΩ1c×Ω2+cΩ2c as δ0.

Proof. By Lemma 3.2, Lemma 3.4 and Lemma 3.6, we have




This implies




For any ε>0, there exists M>0 such that








as α0.


4. Error analysis

In last section we have proved the convergence property of the regularized Fourier series under the condition fTL1R2 . In this section, we give the error analysis of the regularized Fourier series according to the L2-norm for the functions fTL2R2. The bound of the variance of the regularized Fourier series is presented.

By Lemma 3.5, we have next lemma.

Lemma 4.1


for δ+0 and α+0, where η and δ are given in Eq. (4) and Eq. (5) in Section 1.

Theorem 4.1 Suppose fTL2R2 is band-limited. If we choose α=αδ such that αδ0 and δ2/αδ0 as δ0, then f̂αω1ω2f̂Tω1ω2 in L2Ω1Ω1×Ω2Ω2 as δ0.









by Lemma 4.1 and


For every ε>0, there exists N>0 such that








as α0.


Theorem 4.2 Suppose fTL2R2 is band-limited. If the noise in Eq. (4) is white noise such that Eηn1H1n2H2=0 and Varηn1H1n2H2=σ2, then the bias f̂Tω1ω2Ef̂αω1ω20 in L2Ω1Ω1×Ω2Ω2 as α0 and


if ασ0 and σ2/ασ0 as σ0.

Proof. We can calculate




By the proof of Theorem 4.1, we can see that f̂Tω1ω2Ef̂αω1ω20 in L2Ω1Ω1×Ω2Ω2 as α0 and Varf̂αω1ω1=Oσ2+Oσ2/α.


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:


Example 1. Suppose




where Ω1=1 and Ω2=1.

We add the white noise that is uniformly distributed in 0.00050.0005 and choose N=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=0.001 is in Figure 3.

Figure 1.

The exact Fourier transform in Example 2.

Figure 2.

The numerical results by Fourier series in Example 2.

Figure 3.

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


6. Conclusion

The problem of computing the two-dimensional Fourier transform is highly ill-posed. Noise can give rise to large errors if the Fourier series formula is used. The regularized two-dimensional Fourier series is presented. The convergence property is proved and tested by some examples. The convergence property and numerical results show that the regularized two-dimensional Fourier series is excellent in computation in noisy cases. The algorithm will be useful in image processing and multi-dimensional signal processing. The method will be of interest to: engineers who want higher precision in the gauging and design of function generators and analyzers; the electronic or electrical rectification industry; and also to the mathematics community for computing methods and the improvement of mathematics programs on signals and systems, for example, Simulink; and others since many problems in engineering involve noise.



The author would like to express appreciation to Sarah Lanand for her help in formatting the document to meet submission guidelines.


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Written By

Weidong Chen

Submitted: June 26th, 2018 Reviewed: September 16th, 2018 Published: July 24th, 2019