COMPUTATION OF FOURIER TRANSFORMS FOR … COMPUTATION OF FOURIER TRANSFORMS FOR NOISY BANDLIMITED...

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COMPUTATION OF FOURIER TRANSFORMSFOR NOISY BANDLIMITED SIGNALS

Weidong Chen, Georgia State University

October 22, 2011

Weidong Chen, Georgia State University COMPUTATION OF FOURIER TRANSFORMS FOR NOISY BANDLIMITED SIGNALS

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I. Introduction

Definition of Fourier transform:

F [f ](ω) := f (ω) :=

∫ +∞

−∞f (t)e iωtdt, ω ∈ R (1)

Definition of band-limited function:A function f ∈ L2(R) is said to be Ω-band-limited if

f (ω) = 0, ∀ω /∈ [−Ω,Ω].

Here f (ω) is the Fourier transform of f .


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The inversion formula:

f (t) =1

2π

∫ Ω

−Ωf (ω)e−iωtdω, a.e. t ∈ R (2)

The problem:

We will consider the problem of computing f (ω) from f (t) bysolving the integral equation:

1

2π

∫ Ω

−Ωf (ω)e−iωtdω = f (t). (3)


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Sampling Theorem

For band-limited signals, we have the following sampling theorem[3].

Shannon Sampling Theorem. The Ω-band-limited signal f (t)can be exactly reconstructed from its samples f (nH), and

f (t) =∞∑

n=−∞

sin Ω(t − nH)

Ω(t − nH)f (nH) (4)

where H := π/Ω.


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Fourier Serious

Calculating the Fourier transform of f (t) by the formula (4), wehave the formula which is same as the Fourier series [4] p.3

f (ω) = H∞∑

n=−∞f (nH)e inHωP[−Ω, Ω] (5)

where P[−Ω,Ω] is the characteristic function of [−Ω,Ω].


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The problem in practice

In many practical problems, the samples f (nH) are noisy:

f (nH) = fT (nH)+η(nH) (6)

where η(nH) is the noise

|η(nH)| ≤ δ (7)

and fT ∈ L2 is the exact band-limited signal.


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A brief review of some previous results in this problem

1. In [7] and [8], some spectral estimators are given. Howeverthe ill-posedness is not discussed (Definition of ”ill-posed” isin the next slide).

2. To overcome the ill-posedness, regularization techniques areused for the problem of computing f in [9] and [10]. Thoseregularization techniques need the process of calculus ofvariations.


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II. The Ill-posedness of the Problem

Definition: Assume A : D → U is linear. The problem

Az = u

of determining the solution z in the space D from the “initialdata” u in the space U is well-posed on the pair of metricspaces (D,U) in the sense of Hadamard if we have:

1. Existence: A is onto.2. Uniqueness: A is 1-1.3. Stability: A−1 is continuous.

Remark:Problems that violate any of the three conditions are ill-posed.


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The Definition of the Operator A

The operator A on last page is defined by the following formula:

Af := ..., f (−nH), ...f (−H), f (0), f (H), ...f (nH), .... (8)

We discuss the ill-posedness of the problem on the pair of spacesof (L2, l∞) where L2 = F : F ∈ L2[−Ω, Ω] and

||F (ω)||L2 =

∫ Ω

−Ω|F (ω)|2dω.

l∞ is the space a(n) : n ∈ Z of bounded sequences and

||a||l∞ = supn∈Z

|a(n)|.


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The problem is ill-posed on the pair of spaces(L2, l∞).

1. Existence: The existence condition is not satisfied.We can choose the sampling f (nH) to be the samples of asignal whose frequency distribution is not in [−Ω,Ω]. This isequivalent to the fact A(L2) 6= l∞ where A(L2) is the rangeof A.

2. Uniqueness: The uniqueness condition is satisfied, sincef (nH) ≡ 0 implies f (t) ≡ 0. So A is injective.

3. Stability: The stability condition is not satisfied. In otherwords, A−1 is not continuous from A(L2) to L2.

Proof of Instability

By Parseval equality, ||η||2 = 2πH∑∞

n=−∞ |η(nH)|2, it is easy tosee the stability condition is not satisfied. So, this is a highlyill-posed problem.


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III. Regularization method

Definition of regularizing operator:

AssumeAze = ue (”e” for exact).

An operator R(∗, α) : U → D, depending on a parameter α, iscalled a regularizing operator for the equation Az = u in aneighborhood of ue if there exists a function α = α(δ) of δ suchthat if

limδ→0

||uδ − ue || = 0 and zα = R(uδ, α(δ))

thenlimδ→0

||zα − ze || = 0.

The approximate solution zα = R(u, α) to the exact solution ze

obtained by the method of regularization is called a regularizedsolution.


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Regularized Fourier Transform

Since this is a highly ill-posed problem [1], the formula (1) is notreliable in practice. In [2], a regularized Fourier transform ispresented:

Fα[f ] =

∫ ∞

−∞

f (t)e iωtdt

1 + 2πα + 2παt2, (∗)

where α > 0 is the regularization parameter.

Remark:This formula needs the information of f (t), t ∈ (−∞,∞). We areconsidering computation of f from f (nH).


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The Regularized Fourier Series

Based on the regularized Fourier transform (3) and the Fourierseries (5), we construct the regularized Fourier Series:

fα(ω) = H∞∑

n=−∞

f (nH)

1 + 2πα + 2πα(nH)2e inHωP[−Ω,Ω] (9)

where f (nH) is given in (6).


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Lemma 1.

∞∑n=−∞

∣∣∣∣ η(nH)

1 + 2πα + 2πα(nH)2

∣∣∣∣2 = O(δ2) + O(δ2/√

α)

where η and δ are given in (6) and (7) in section I.


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The Convergence Property I

Theorem 1. If we choose α = α(δ) such that α(δ) → 0 andδ2/

√α(δ) → 0 as δ → 0, then fα(ω) → fT (ω) in L2[−Ω,Ω] as

δ → 0.

Remark:According this theorem, α should be chosen by the error of thesampling. We can choose α = kδµ where k > 0 and 0 < µ < 4.Then fα(ω) → fT (ω) in [−Ω, Ω] as δ → 0.


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The Convergence Property II

Theorem 2. If the noise in (6) is white noise such thatE [η(nH)] = 0 and Var [η(nH)] = σ2, then the biasfT (ω)− E [fα(ω)] → 0 in L2[−Ω,Ω] as α → 0 and

Var [fα(ω)] = O(σ2) + O(σ2/√

α).


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IV. Experimental Results

In practical computation, we first filter the noise out of the band[−Ω,Ω] by the convolution

f (t) = fδ(t) ∗sin(Ωt)

πt

where fδ(t) is the signal with noise w(t):

fδ(t) = fT (t) + w(t).


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The Algorithm

We choose a large integer N and use the next formula incomputation:

fα(ω) = HN∑

n=−N

f (nH)

1 + 2πα + 2πα(nH)2e inHωP[−Ω,Ω] (10)

where f (nH) is the sampling data after filtering the noise out ofthe band [−Ω,Ω]. The regularization parameter α can be chosenaccording to the noise level in Theorem 1.


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Example 1.

Suppose

fT (t) =1− cos t

πt2.

ThenfT (ω) = (1− |t|)P[−Ω, Ω]

where Ω = 1.We add the white noise that is uniformly distributed in[−0.025, 0.025] and choose N = 500 after the filtering process.The result with the Fourier series and the result with theregularized Fourier series with α = 0.005 are in figure 1. Thevertical axes are the values of the Fourier series and regularizedFourier series respectively. The result of the regularized Fourierseries with α = 0.001 and α = 0.00001 are in figure 2.


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Discussion on the result

From the numerical results in Figure 1 and Figure 2, we can seethat bias is smaller if α is reduced from 0.005 to 0.001. Thismeans that fT (ω)− E [f (ω)] is close to zero which is the result ofTheorem 2. However if α is too small such as the caseα = 0.000001 in Figure 2, the error of the regularized Fourierseries is large. This is due to the second term O(σ/

√α) in the

estimation of Var [f (ω)] in Theorem 2.


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The Tikhonov regularization method I

In example 2, we will compare the regularized Fourier series withthe Tikhonov regularization method used in [1], [9] and [10]. Weintroduce the Tikhonov regularization method next.First we write equation (2) by the finite difference method

Af = f

where

A =

(1

2πe iωj tk h

)(2N+1)×(2M+1)

, h = Ω/M

andf = (f−M , ..., f0, ..., fM)

f = (f−N , ..., f0, ..., fN).


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The Tikhonov regularization method II

Define the smoothing functional

Mα[f , f ] = ||Af − f ||2 + α||f ||2.

Here the norms of f and f are defined by

||f ||2 =M∑

j=−M

|fj |2, ||f ||2 =N∑

k=−N

|fk |2.

We can find the approximate Fourier transform by minimizingMα[f , f ]. And the minimizer is the solution of the Euler equation(

ATA + αI)

f = AT f .


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Example 2.

The signals are the same as the signals in Example 1. We chooseM=400. The results of the regularized Fourier series and theTikhonov regularized solution with α = 0.001 are in Figure 3, 4and 5 for the cases H = π, H = π/10 and H = π/50.We can see that if H is smaller the Tikhonov regularized solution ismore accurate, but it is not as good as the regularized Fourierseries.


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References I

A. N. Tikhonov and V. Y. Arsenin,Solution of Ill-Posed Problems.Winston/Wiley, 1977.

W. Chen,An Efficient Method for An Ill-posed Problem—Band-limitedExtrapolation by Regularization.IEEE Trans. on Signal Processing, vol 54, pp.4611-4618, 2006.

C. E. Shannon,A mathematical theory of communication.The Bell System Technical Journal, vol. 27, July 1948.

J. R. Higgins,Sampling Theory in Fourier and Signal Analysis Foundations.Oxford University Press Inc., New York, 1996.


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References II

R. Bhatia,Fourier Series.The Mathematical Association of American, 2005.

A. Steiner,Plancherel’s Theorem and the Shannon Series DerivedSimultaneously.The American Mathematical Monthly, vol. 87, no. 3, Mar.1980, pp. 193-197.

A. R. Nematollahi,Spectral Estimation of Stationary Time Series: RecentDevelopments.J. Statist. Res. Iran 2, 2005, pp.107-127..


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References III

S. V. Narasimhan and M. Harish,Spectral estimation based on discrete cosine transform andmodified group delay.Signal Processing 86, 2004, pp.279-305..

H. Kim, B. Yang and B. Lee,Iterative Fourier transform algorithm with regularization forthe optimal design of diffractive optical elements.J. Opt. Soc. Am. A Vol. 21, No. 12, pp. 2353-2356, 2004.

I. V. Lyuboshenko and A. M. Akhmetshin,Regularization Of The Problem Of Image Restoration From ItsNoisy Fourier Transform Phase.International Conference on Image Processing, Volume 1, 1996pp. 793 - 796.


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