# Research Article Sign Data Derivative Recovery

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2 ISRN Applied Mathematics ad oise, X : Ω R, f v X are recorded for multiple iteratios of the oise. The average of the measuremet i.e., the expectatio E recovers the sigal E f v X f v.. If the oise is chose to be uiform, where ρ x is the desity fuctio such that ρ x a, a x a 0, else,. the the variace, E f v X E f v X, reduces to Var f v X 3 a..3 As reported by O Brie et al., it was empirically discovered that the average of the sigs of sigal plus oise recovers the sigal if the sigal-to-oise ratio is less tha or equal to oe. This ca be show mathematically 3 usig the sigum fuctio 4, sg x,x > 0, sg x,x <0, sg 0 0, E sg f v X sg f v x ρ x dx f f ρ x dx ρ x dx..4 Because ρ x is eve ad equals f f ρ x dx.5 E sg f v X f v a, f a, a..6 The variace is E sg f v X E sg f v X, reducig to Var sg f v X f v..7 a Cosequetly, the error is miimal whe the sigal-to-oise ratio is ear uity. The advatage of retaiig oly the sigs of sigal plus oise is the requiremet of approximately bit to record the iformatio as opposed to requirig 6 to 0 bits to record full amplitude data. The goal of this paper is to examie the recovery of derivatives from sig data i uiform oise. The issue is that recovery of sigal from sig data ca be exteded to recovery of derivatives of the sigal through the use of fiite differeces ad that recovery is costraied by the size of the variace. I this paper, we first examie sig data derivatives for both

3 ISRN Applied Mathematics 3 the discrete ad cotiuous case. We follow with a derivatio of variace. We coclude our aalysis with a computatioal test, which lists the true variace versus the variace estimate derived statistically for a test fuctio for selected step sizes.. Sig Data Derivatives Let the sigal f v be a th order differetiable fuctio. Based o sigal recovery from sig data, it ca be show that derivatives of the sigal are also recoverable. Usig the liearity of the expectatio value, Δ E v Δv sg f v X Δ v Δv E sg f v X,. where Δ v is the th order fiite differece operator with respect to the variable v 5. Ithis case, a ouit step size, Δv,isused e.g., 6. I detail, we ca write Δ v Δv sg f v X Δv i sg f v i Δv X i, i 0 i. where the otatio i represets the biomial coefficiet!/i! i! ad where X i X 0, X,...are idepedet represetatios of the radom variable, X. Substitutig from.6 ito. yields Δ E v Δv sg f v X Δ vf v a Δv..3 I the limit of ifiitesimal step size, this becomes a cotiuous derivative Δ lim E v Δv 0 Δv sg f v X d f v a dv.4 or d E dv sg f v X d f v a dv..5 Equatio.4 presets a alterative solutio to direct itegratio. For example, usig the rule, f x δ x dx f/ x δ x dx, 7, the itegral d 3 E dv sg f v X 3 d δ df 3 6 dδ du dv du df dv d f dv δ d3 f ρ x dx dv 3.6

4 4 ISRN Applied Mathematics loses all terms with derivatives of the delta fuctioal, reducig to ρ f d3 f dv 3. f x.7 I geeral, d E dv sg f v X ρ f d f dv d f f x a dv..8 It follows that the oise is restricted such that a f. 3. The Variace of Sig Data Derivatives Lettig S Δ v/ Δv sg f v X, compute the variace, E S E S.From.3,it follows that E S Δ vf/a Δv. E S ca be foud by iductively geeralizig from : E S b0 sg f Δv 4 0 X 0 b sg f X b sg f X Δv 4 b 0 b b b 0b f0 f f0 f b 0 b f f b b, 3. where f i f v i Δv, f k f v k Δv,adb i i i. These results geeralize to Var S Δv i 0 i Δv i / k i k fi f k Δ v f i k a Δv. 3. Sice f is differetiable, Δ vf/ Δv d f/dv <εad, thus, Δ vf/ Δv is fiite. Based o defiitio, Var S > 0. Cosequetly, lim Δv 0 Var S. Similarly, lim Var S, 0 < Δv <. The variace of a discrete sig derivative approaches ifiity with decreasig step size ad icreasig order. I additio, sice lim Δv 0 S d /dv sg f v X, Var d /dv sg f v X, so i the case of the cotiuous derivatives.5 the variace is ifiite. Use 3. to fid the variace of the first discrete sig derivative by lettig : Var S Δv f f

5 ISRN Applied Mathematics 5 Table : True variace, Var S, versus the variace estimate, Var N S, for the fuctio f si v, with the umber of iteratios N 000, a, ad v 3. Δv Var S Var N S Table : True variace, Var S, versus the variace estimate, Var N S, for the fuctio f si v,withthe umber of iteratios N 000, a, ad v 3. Δv Var S Var N S The variace of the secod discrete sig derivative is similarly computed as Var S 6 a f Δv 4 0 4f f Computatioal Tests These results ca be tested computatioally. Variace ca be estimated for N iteratios with Var N S N N S m E S, m 4. where the idex m desigates the sample umber. Cosider the test fuctio f si v. Usig the first-order sig data derivative, compare Var S to Var N S, ad usig the secod-order sig data derivative, compare Var S to Var N S for N 000, a, ad v 3. The results are show i Tables ad. We illustrate the chage i variace i Figure, which shows three curves, each cosistig of N 000 iteratios. The first curve i blue shows the sig data recovery of the fuctio f si v or E S 0 for a adδv 0.5. The secod curve i gree shows the sig data recovery E S, which approximates f for a adδv 0.5. The third curve i red shows the sig data recovery E S, which approximates f for a adδv 0.5.

6 6 ISRN Applied Mathematics.5 ES ES ES Figure : The expectatio value curves for S / Δv i 0 i i sg f v i Δv X i or E S for 0,,, f v si v, a, ad Δv 0.5. The umber of iteratios i the expectatio values is N 000. E S 0 correspods to the blue curve ad approximates f, E S correspods to the gree curve ad approximates f,ade S correspods to the red curve ad approximates f. 5. Coclusios Recovery of sigal from the sigs of sigal plus oise icurs a variace, which oly depeds o the oise amplitude, while recovery of discrete derivatives from the sigs of sigal plus oise i.e., sig data icurs a variace which grows ifiite for ifiitesimal step size ad ifiite order. The applicatio problem is that sig data ca be used i the seismic idustry i processes which may differetiate the data. I such cases, if the step size or order of the fiite differece is ot costraied, the process will icur large variace ad covergece of the process will be miimized. While methods for smoothig oisy data for fiite differece calculatios are kow, sig data requires oisy data. I this paper, we have characterized the problem by explicitly evaluatig the variace of discrete sig data derivatives. Appedix Clarificatio of E S E S b0 sg f Δv 4 0 X 0 b sg f X b sg f X Δv 4 b 0 sg f 0 X 0 b0 sg f 0 X 0 b sg f X b sg f X b sg f X b0 sg f 0 X 0 b sg f X b sg f X b sg f X. A.

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