COMPUTATION OF FOURIER TRANSFORMS FOR NOISY B FOR NOISY BANDLIMITED SIGNALS

Transcription

1 COMPUTATION OF FOURIER TRANSFORMS FOR NOISY BANDLIMITED SIGNALS October 22, 2011

2 I. Introduction Definition of Fourier transform: F [f ](ω) := ˆf (ω) := + f (t)e iωt dt, ω R (1)

3 I. Introduction Definition of Fourier transform: F [f ](ω) := ˆf (ω) := + f (t)e iωt dt, ω R (1) Definition of band-limited function: A function f L 2 (R) is said to be Ω-band-limited if ˆf (ω) = 0, ω / [ Ω, Ω]. Here ˆf (ω) is the Fourier transform of f.

4 The inversion formula: f (t) = 1 Ω ˆf (ω)e iωt dω, a.e. t R (2) 2π Ω

5 The inversion formula: f (t) = 1 Ω ˆf (ω)e iωt dω, a.e. t R (2) 2π Ω The problem: We will consider the problem of computing ˆf (ω) from f (t) by solving the integral equation: 1 Ω ˆf (ω)e iωt dω = f (t). (3) 2π Ω

6 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) = where H := π/ω. n= sin Ω(t nh) f (nh) (4) Ω(t nh)

7 Fourier Serious Calculating the Fourier transform of f (t) by the formula (4), we have the formula which is same as the Fourier series [4] p.3 ˆf (ω) = H n= f (nh)e inhω P[ Ω, Ω] (5) where P[ Ω, Ω] is the characteristic function of [ Ω, Ω].

8 The problem in practice In many practical problems, the samples {f (nh)} are noisy: f (nh) = f T (nh)+η(nh) (6) where {η(nh)} is the noise η(nh) δ (7) and f T L 2 is the exact band-limited signal.

9 A brief review of some previous results in this problem 1. In [7] and [8], some spectral estimators are given. However the ill-posedness is not discussed (Definition of ill-posed is in the next slide).

10 A brief review of some previous results in this problem 1. In [7] and [8], some spectral estimators are given. However the ill-posedness is not discussed (Definition of ill-posed is in the next slide). 2. To overcome the ill-posedness, regularization techniques are used for the problem of computing ˆf in [9] and [10]. Those regularization techniques need the process of calculus of variations.

11 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 initial data u in the space U is well-posed on the pair of metric spaces (D, U) in the sense of Hadamard if we have: 1. Existence: A is onto. 2. Uniqueness: A is Stability: A 1 is continuous.

12 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 initial data u in the space U is well-posed on the pair of metric spaces (D, U) in the sense of Hadamard if we have: 1. Existence: A is onto. 2. Uniqueness: A is Stability: A 1 is continuous. Remark: Problems that violate any of the three conditions are ill-posed.

13 The Definition of the Operator A The operator A on last page is defined by the following formula: Aˆf := {..., f ( nh),...f ( H), f (0), f (H),...f (nh),...}. (8) We discuss the ill-posedness of the problem on the pair of spaces of (L 2, l ) where L 2 = {F : F L 2 [ Ω, Ω]} and F (ω) L 2 = Ω Ω F (ω) 2 dω. l is the space {a(n) : n Z} of bounded sequences and a l = sup a(n). n Z

14 The problem is ill-posed on the pair of spaces (L 2, l ). 1. Existence: The existence condition is not satisfied. We can choose the sampling {f (nh)} to be the samples of a signal whose frequency distribution is not in [ Ω, Ω]. This is equivalent to the fact A(L 2 ) l where A(L 2 ) is the range of A.

15 The problem is ill-posed on the pair of spaces (L 2, l ). 1. Existence: The existence condition is not satisfied. We can choose the sampling {f (nh)} to be the samples of a signal whose frequency distribution is not in [ Ω, Ω]. This is equivalent to the fact A(L 2 ) l where A(L 2 ) is the range of A. 2. Uniqueness: The uniqueness condition is satisfied, since f (nh) 0 implies f (t) 0. So A is injective.

16 The problem is ill-posed on the pair of spaces (L 2, l ). 1. Existence: The existence condition is not satisfied. We can choose the sampling {f (nh)} to be the samples of a signal whose frequency distribution is not in [ Ω, Ω]. This is equivalent to the fact A(L 2 ) l where A(L 2 ) is the range of A. 2. Uniqueness: The uniqueness condition is satisfied, since f (nh) 0 implies f (t) 0. So A is injective. 3. Stability: The stability condition is not satisfied. In other words, A 1 is not continuous from A(L 2 ) to L 2. Proof of Instability By Parseval equality, ˆη 2 = 2πH n= η(nh) 2, it is easy to see the stability condition is not satisfied. So, this is a highly ill-posed problem.

17 III. Regularization method Definition of regularizing operator: Assume Az e = u e ( e for exact). An operator R(, α) : U D, depending on a parameter α, is called a regularizing operator for the equation Az = u in a neighborhood of u e if there exists a function α = α(δ) of δ such that if lim δ 0 u δ u e = 0 and z α = R(u δ, α(δ)) then lim z α z e = 0. δ 0 The approximate solution z α = R(u, α) to the exact solution z e obtained by the method of regularization is called a regularized solution.

18 Regularized Fourier Transform Since this is a highly ill-posed problem [1], the formula (1) is not reliable in practice. In [2], a regularized Fourier transform is presented: F α [f ] = where α > 0 is the regularization parameter. f (t)e iωt dt 1 + 2πα + 2παt 2, ( )

19 Regularized Fourier Transform Since this is a highly ill-posed problem [1], the formula (1) is not reliable in practice. In [2], a regularized Fourier transform is presented: F α [f ] = where α > 0 is the regularization parameter. f (t)e iωt dt 1 + 2πα + 2παt 2, ( ) Remark: This formula needs the information of f (t), t (, ). We are considering computation of ˆf from f (nh).

20 The Regularized Fourier Series Based on the regularized Fourier transform (3) and the Fourier series (5), we construct the regularized Fourier Series: ˆf α (ω) = H n= where f (nh) is given in (6). f (nh) 1 + 2πα + 2πα(nH) 2 einhω P[ Ω, Ω] (9)

21 Lemma 1. η(nh) 1 + 2πα + 2πα(nH) 2 n= 2 = O(δ 2 ) + O(δ 2 / α) where η and δ are given in (6) and (7) in section I.

22 The Convergence Property I Theorem 1. If we choose α = α(δ) such that α(δ) 0 and δ 2 / α(δ) 0 as δ 0, then ˆf α (ω) ˆf T (ω) in L 2 [ Ω, Ω] as δ 0.

23 The Convergence Property I Theorem 1. If we choose α = α(δ) such that α(δ) 0 and δ 2 / α(δ) 0 as δ 0, then ˆf α (ω) ˆf T (ω) in L 2 [ Ω, Ω] as δ 0. Remark: According this theorem, α should be chosen by the error of the sampling. We can choose α = kδ µ where k > 0 and 0 < µ < 4. Then ˆf α (ω) ˆf T (ω) in [ Ω, Ω] as δ 0.

24 The Convergence Property II Theorem 2. If the noise in (6) is white noise such that E[η(nH)] = 0 and Var[η(nH)] = σ 2, then the bias ˆf T (ω) E[ˆf α (ω)] 0 in L 2 [ Ω, Ω] as α 0 and Var[ˆf α (ω)] = O(σ 2 ) + O(σ 2 / α).

25 IV. Experimental Results In practical computation, we first filter the noise out of the band [ Ω, Ω] by the convolution f (t) = f δ (t) sin(ωt) πt where f δ (t) is the signal with noise w(t): f δ (t) = f T (t) + w(t).

26 The Algorithm We choose a large integer N and use the next formula in computation: ˆf α (ω) = H N n= N f (nh) 1 + 2πα + 2πα(nH) 2 einhω P[ Ω, Ω] (10) where f (nh) is the sampling data after filtering the noise out of the band [ Ω, Ω]. The regularization parameter α can be chosen according to the noise level in Theorem 1.

27 Example 1. Suppose Then f T (t) = 1 cos t πt 2. ˆf T (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 the regularized Fourier series with α = are in figure 1. The vertical axes are the values of the Fourier series and regularized Fourier series respectively. The result of the regularized Fourier series with α = and α = are in figure 2.

28 Discussion on the result From the numerical results in Figure 1 and Figure 2, we can see that bias is smaller if α is reduced from to This means that ˆf T (ω) E[ˆf (ω)] is close to zero which is the result of Theorem 2. However if α is too small such as the case α = in Figure 2, the error of the regularized Fourier series is large. This is due to the second term O(σ/ α) in the estimation of Var[ˆf (ω)] in Theorem 2.

29 The Tikhonov regularization method I In example 2, we will compare the regularized Fourier series with the Tikhonov regularization method used in [1], [9] and [10]. We introduce the Tikhonov regularization method next. First we write equation (2) by the finite difference method Aˆf = f where and A = ( ) 1 2π eiω j t k h, h = Ω/M (2N+1) (2M+1) ˆf = (ˆf M,..., ˆf 0,..., ˆf M ) f = (f N,..., f 0,..., f N ).

30 The Tikhonov regularization method II Define the smoothing functional M α [ˆf, f ] = Aˆf f 2 + α ˆf 2. Here the norms of ˆf and f are defined by ˆf 2 = M j= M ˆf j 2, f 2 = N k= N f k 2. We can find the approximate Fourier transform by minimizing M α [ˆf, f ]. And the minimizer is the solution of the Euler equation ( A T A + αi ) ˆf = A T f.

31 Example 2. The signals are the same as the signals in Example 1. We choose M=400. The results of the regularized Fourier series and the Tikhonov regularized solution with α = are in Figure 3, 4 and 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 is more accurate, but it is not as good as the regularized Fourier series.

32 References I A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-Posed Problems. Winston/Wiley, W. Chen, An Efficient Method for An Ill-posed Problem Band-limited Extrapolation by Regularization. IEEE Trans. on Signal Processing, vol 54, pp , C. E. Shannon, A mathematical theory of communication. The Bell System Technical Journal, vol. 27, July J. R. Higgins, Sampling Theory in Fourier and Signal Analysis Foundations. Oxford University Press Inc., New York, 1996.

33 References II R. Bhatia, Fourier Series. The Mathematical Association of American, A. Steiner, Plancherel s Theorem and the Shannon Series Derived Simultaneously. The American Mathematical Monthly, vol. 87, no. 3, Mar. 1980, pp A. R. Nematollahi, Spectral Estimation of Stationary Time Series: Recent Developments. J. Statist. Res. Iran 2, 2005, pp

34 References III S. V. Narasimhan and M. Harish, Spectral estimation based on discrete cosine transform and modified group delay. Signal Processing 86, 2004, pp H. Kim, B. Yang and B. Lee, Iterative Fourier transform algorithm with regularization for the optimal design of diffractive optical elements. J. Opt. Soc. Am. A Vol. 21, No. 12, pp , I. V. Lyuboshenko and A. M. Akhmetshin, Regularization Of The Problem Of Image Restoration From Its Noisy Fourier Transform Phase. International Conference on Image Processing, Volume 1, 1996 pp