THE computation of Fourier transforms (see Definition

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1 IAENG International Journal of Applied Mathematis, 48:, IJAM_48 6 The Computation of Fourier Transform via Gauss-Regularized Fourier Series Qingyue Zhang Abstrat In this paper, we onstrut a new regularized Fourier series, that is the Gauss-regularized Fourier series. Moreover, we find this regularized Fourier series an be used to ompute the Fourier transforms of bandlimited signals. When the regularization parameter tends to zero, we prove that this regularized Fourier series is uniform and L -onvergene. Numerial results demonstrate the superiority of the new method over some previous methods. Index Terms Gauss-regularization, ill-posedness, bandlimited signal, Fourier transform. I. INTRODUCTION THE omputation of Fourier transforms see Definition.) of signals is a highly ill-posed problem [. It is not reliable to ompute the Fourier transforms of signals by their definitions in pratie. Therefore, for omputing Fourier transforms of signals, many regularization methods were raised [ 5. In partiular, in [, Chen onstruted a polynomial-regularized Fourier series ˆf p ω) = h fkh) + π + πkh) e ikhω χ [ Ω,Ω ω), where >, h = π Ω, fkh) = f Ωkh) + ηkh), {ηkh)} is the noise ηkh) and f Ω L R) is the exat bandlimited signal see Definition.). Moreover, he gave that this polynomial-regularized Fourier series is more effetive in ontrolling the noise than Fourier series when we use it to ompute the Fourier transforms of bandlimited signals. When the regularization parameter tends to zero, the uniform and L -onvergene of this regularized Fourier series are proved by Chen. In this paper, we onstrut a new regularized Fourier series, that is the Gauss-regularized Fourier series ˆf ω) = h fkh)e 4π kh) e ikhω χ [ Ω,Ω ω). Moreover, we find this regularized Fourier series an be used to ompute the Fourier transforms of bandlimited signals. When the regularization parameter tends to zero, we prove the uniform and L -onvergene of this regularized Fourier series. Numerial results demonstrate the Gauss-regularized Fourier series has the better performane in ontrolling the noise than the polynomial-regularized Fourier series when we used it to ompute the Fourier transforms of bandlimited signals. Next, we review some definitions, notations and basi results. Manusript reeived July 8, 7; revised Otober, 7. This work was supported by National Natural Siene Foundation of China No. 4435). The authors are with the Shool of Sienes, Tianjin University of Tehnology, Tianjin, China jzhangqingyue@63.om). Definition.: For f L R), the Fourier transform of f is defined by ˆfω) = ft)e itω dt. ) Definition.: Let f L R), if there exists a positive Ω suh that ˆfω) = for ω > Ω, then f is said to be Ω-bandlimited. By Shannon sampling theorem [6 3, the Ω-bandlimited signal fx) an be exatly reonstruted from its samples fkh) and ft) = sinωt kh)) fkh) Ωt kh)), ) where h = π Ω and the series ) onverges both uniformly on R and in L R). By ) and the time invariane of the bandlimited signals, it follows that h fkh) = f L, 3) For the detail throughout see [3). Taking the Fourier transform on both sides of ), we have ˆfω) = h fkh)e ikhω χ [ Ω,Ω ω). For any f, g L R), define their onvolution f g)x) = fx y)gy)dy. R This paper is organized as follows: In setion, we give the onvergene property of Gauss-regularized Fourier series, that is the uniform and L -onvergene. Numerial results are presented in setion 3. Setion 4 onludes the paper with some outlook. Finally, in setion 5, we provide proofs of Theorem. and Theorem.. II. THE UNIFORM AND L -CONVERGENCE OF ˆf In this setion, we will give the onvergene property of Gauss-regularized Fourier series, that is the uniform and L -onvergene. The following theorem gives the uniform onvergene of Gauss-regularized Fourier series. Theorem.: Assume that f Ω L R) L R) is Ω- bandlimited. For eah fixed < < Ω, if the regularization parameter = ) satisfies ) and ) when, then ˆf onverges to ˆf Ω uniformly on [ Ω +, Ω when. Proof: See setion 5. Next, we give L -onvergene of Gauss-regularized Fourier series. Theorem.: Assume that f Ω L R) is Ω-bandlimited. If the regularization parameter = ) satisfies ) Advane online publiation: February 8)

2 IAENG International Journal of Applied Mathematis, 48:, IJAM_48 6 and when, then ˆf onverges to ˆf Ω in ) L [ Ω, Ω) when. Proof: See setion III. NUMERICAL RESULTS In this setion, we give some numerial results to show that the Gauss-regularized Fourier series is more effetive in ontrolling the noise than the polynomial-regularized Fourier series when we used it to ompute the Fourier transforms of bandlimited signals. For a large N, we use the next formulas in pratial omputation and ˆf ω) = h ˆf p ω) = h N fkh)e 4π kh) e ikhω χ [ Ω,Ω ω), 4) N ˆfω) = h fkh)e ikhω + π + πkh) χ [ Ω,Ωω) 5) N fkh)e ikhω χ [ Ω,Ω ω). 6) Here fkh) = f Ω kh) + ηkh) and {ηkh)} is the noise ηkh). Example 3.: Comparison between different regularization methods. Let f Ω = os t πt. It is easy to see that ˆf Ω ω) = ω )χ [, ω). We hoose the noise that is uniformly distributed in [.5,.5 and N =. Figure shows the result using the Gaussregularized Fourier series with =., the result using the polynomial-regularized Fourier series with =. and the result using the Fourier series. Figure presents the result using the Gauss-regularized Fourier series with =., the result using the polynomial-regularized Fourier series with =. and the result using the Fourier series. By Figure and Figure, the Gaussregularized Fourier series has good performane at least in some ases when we used it to ompute the Fourier transforms of bandlimited signals. IV. CONCLUSION Computing the Fourier transforms of bandlimited signals by Fourier series is an ill-posed problem. As a result, many regularization methods were raised. In this paper, we present a new regularized Fourier series, that is the Gauss-regularized Fourier series. Moreover, we prove the uniform and L - onvergene of this regularized Fourier series when the regularization parameter tends to zero. Numerial results show that the Gauss-regularized Fourier series is more effetive in ontrolling the noise than the known-regularized Fourier series the polynomial-regularized Fourier series) when we used it to ompute the Fourier transforms of bandlimited signals. Studying other regularization methods is the goal of future work a) b) ) Fig.. Comparison of three algorithms with =.. a) is the result using Gauss-regularized Fourier series formula 4)). b) is the result using polynomial-regularized Fourier series formula 5)). ) is the result using Fourier series formula 6)). V. PROOFS OF THEOREM. AND THEOREM. A. Proof of Theorem. To prove Theorem., we need the following lemma. Lemma 5.: Suppose that g L R) is Ω-bandlimited. Then for eah fixed < < Ω and for any Ω + < ω < Ω, we have h gkh)e 4π kh) e ikhω χ [ Ω,Ω ω) gt)e 4π t e itω dt g L 3hπ e /8π + 4h g L π e dt) t. Proof: For Ω+ < ω < Ω, by Shannon s sampling Advane online publiation: February 8)

3 IAENG International Journal of Applied Mathematis, 48:, IJAM_ = h 4π Note that e 4π kh) = [5), we have gkh)e ikhω e 4π kh) 4π e t /6π e itkh dt ) e t /6π e itkh dt see h gkh)e 4π kh) e ikhω χ [ Ω,Ω ω) d) e) f) Fig.. Comparison of three algorithms with =.. d) is the result using Gauss-regularized Fourier series formula 4)). e) is the result using polynomial-regularized Fourier series formula 5)). f) is the result using Fourier series formula 6)). formula ), we obtain that = gkh) = h = h Therefore gt)e 4π t e itω dt sinωt kh)) e 4π t e itω dt Ωt kh) gkh) π e iωkh χ [ Ω,Ω ω) π e ω /6π gkh)e ikhω 4π e t /6π e itkh dt. h gkh)e 4π kh) e ikhω χ [ Ω,Ω ω) gt)e 4π t e itω dt gt)e 4π t e itω dt = h gkh) e ikhω 4π e t /6π e itkh dt π ) + e t /6π e itkh dt. Therefore, for ω [ Ω +, Ω, by Cauhy-Shwarz s inequality and the inequality a + b a + b ) h gkh)e 4π kh) e ikhω χ [ Ω,Ω ω) gt)e 4π t e itω dt ) h 8π 3 gkh) e t /6π e itkh dt + h ) 8π 3 gkh) e t /6π e itkh dt h 8π 3 gkh) e t /6π e itkh dt + e t /6π ) e itkh dt. Sine,k =,k = e t /6π e itkh dt e t /6π e itkh dt e itkh ikh e t /6π,k e t /6π te itkh ikh8π dt e i)kh e ) /6π = ikh,k e t /6π te itkh e ) /8π kh) ikh8π dt,k + e t /6π te itkh ikh8π dt Advane online publiation: February 8)

4 IAENG International Journal of Applied Mathematis, 48:, IJAM_48 6,k + kh8π ) [ e ) /8π kh) [ e ) /8π = kh),k + ) kh) e t /6π = [ e ) /8π kh),k = then we have,k,k,k,k 4e ) /8π kh) 4e /8π kh), ) e t /6π tdt + e ) /8π kh) e t /6π e itkh dt 4e /8π ) kh) + e t /6π dt 4e /8π kh) + e t /6π dt). Similarly, we have e t /6π e itkh dt Ω+ω,k 4e /8π kh) + e t /6π dt). Therefore, for ω [ Ω +, Ω, h gkh)e 4π kh) e ikhω χ [ Ω,Ω ω) gt)e 4π t e itω dt h 8π 3 gkh) 8e /8π kh),k ) + = h 8π 3 g L e t /6π dt,k + e t /6π dt = g L 3hπ e /8π + h g L 4π 3 8e /8π kh) ) = g L 3hπ e /8π + 4h g L π ) e t /6π dt e dt) t, the first equality holds by 3). Proof of Theorem.: Sine for any ω [ Ω+, Ω ˆf ω) ˆf Ω ω) = h f Ω kh) + ηkh))e 4π kh) e ikhω ˆf Ω ω) = h +h f Ω kh)e 4π kh) e ikhω ηkh)e 4π kh) e ikhω therewith, by Lemma 5. where f Ω t)e itω dt, ˆf ω) ˆf Ω ω) f Ω t)e 4π t e itω dt + h ηkh)e 4π kh) e ikhω + f Ω L e /6π 3hπ + h f Ω L = I + II + f Ω L e /6π 3hπ + h f Ω L I = e t dt e t dt, f Ω t) e 4π t ) e itω dt f Ω t)e itω dt and II = h ηkh)e 4π kh) e ikhω. We treat I first: Sine f Ω t) e 4π t ) fω t) and f Ω t) L R), by Lebesgue s dominated onvergene theorem see [6) I f Ω t) e 4π t ) dt, as. We now treat II: Sine II h h = h + h h + h = h + h ηkh) e 4π kh) e 4π kh) k= k e 4π kh) k= k e 4π th) dt e 4π th) dt = h + h e 4π th) dt, Advane online publiation: February 8)

5 IAENG International Journal of Applied Mathematis, 48:, IJAM_48 6 then II onverges to uniformly on R if the regularization parameter = ) is hosen suh that when ). Finally, sine f Ω L 3hπ e /6π + h f Ω L when, the proof is finished. B. Proof of Theorem. Sine ˆf ω) ˆf Ω ω) = h ηkh)e 4π kh) e ikhω χ [ Ω,Ω ω) e t dt h f Ω kh) e 4π kh) ) e ikhω χ [ Ω,Ω ω), using the inequality a + b a + b ), we obtain ˆf ω) ˆf Ω ω) L h ηkh)e 4π kh) e ikhω χ [ Ω,Ω ω) +h f Ω kh) e 4π kh) ) e ikhω χ [ Ω,Ω ω) L = 8Ω h ηkh)e 4π kh) +8Ω h f Ω kh) e 4π kh) ). L REFERENCES [ N. Tikhonov and Y. Arsenin, Solution of ill-posed problems, New York: Winston/Wiley, 977. [ W. Chen, Computation of Fourier transforms for noisy bandlimited signals, SIAM J. Numer. Anal., vol. 49, no., pp. 4,. [3 H. Kim, B. Yang and B. Lee, Iterative Fourier transform algorithm with regularization for the optimal design of diffrative optial elements, J. Opt. So. Amer. A, vol., no., pp , 4. [4 H. Song, M. Yi and Y. Pan, Numerial solutions of frational partial differential equations by using Legendre wavelets, Engineering Letters, vol. 4, no. 3, pp , 6. [5 H. Zhuang, M. Yang, Z. Cui and Q. Zheng, A method for stati hand gesture reognition based on non-negative matrix fatorization and ompressive sensing, IAENG International Journal of Computer Siene, vol. 44, no., pp. 5 59, 7. [6 Z. Xia, C. Yuan, X. Sun, D. Sun and R. Lv, Combining wavelet transform and LBP related features for fingerprint liveness detetion, IAENG International Journal of Computer Siene, vol. 43, no. 3, pp. 9 98, 6. [7 A. Aldroubi and K. Gröhenig, Nonuniform sampling and reonstrution in shift-invariant spaes, SIAM Rev., vol. 43, no. 4, pp ,. [8 A. Aldroubi, J. Davis and I. Krishtal, reonstrution of signals in evolutionary system via spatiotemporal trade-off, J. Fourier Anal. Appl., vol., no., pp. 3, 5. [9 J. Benedetto, and G. Ferreira, Modern sampling theory: mathematis and appliations, Boston: Birkhäser,. [ H. Huo and W. Sun, Average sampling theorem, Chinese Si. Bull., vol. 45, no. 9, pp. 43 4, 5. [ X. Zhou and W. Sun, On the sampling for wavelet subspaes, J. Fourier Anal. Appl., vol. 5, no. 4, pp , 999. [ Q. Zhang L. Wang and W. Sun, Signal denoising with average sampling, Digit. Signal Proess, vol., no., pp. 6 3,. [3 L. Butzer, A survey of the Whittaker-Shannon sampling theorem and some of its extensions, J. Math. Res. Exposition, vol. 3, no., pp. 85, 983. [4 D. Urynbassarova, Z. Li and R. Tao, The Wigner-Ville distribution in the linear anonial transform domain, IAENG International Journal of Applied Mathematis, vol. 46, no. 4, pp , 6. [5 M. Stein and G. Weiss, Introdution to Fourier analysis on Eulidean spaes, Prineton: Prineton Univ. Press, 97. [6 W. Rudin, Real and omplex analysis, New York: MGraw-Hill, 987. It follows similar lines of the treatment of II in the proof of Theorem. that 8Ω h ηkh)e 4π kh), if the regularization parameter = ) satisfies when. By 3), we have 8Ω h ) f Ω kh) e 4π kh) ) 8Ω h f Ω kh) = 8Ω h f L <. Using Lebesgue s dominated onvergene theorem 8Ω h f Ω kh) e 4π kh) ) when. Hene we obtain ˆf ω) ˆf Ω ω) L if the regularization parameter = ) is hosen suh that ) and when. This finishes the ) proof. Advane online publiation: February 8)