Sign Data Derivative Recovery

Given only the signs of signal plus noise added repetitively or sign data, signal amplitudes can be recovered with minimal variance. However, discrete derivatives of the signal are recovered from sign data with a variance which approaches infinity with decreasing step size and increasing order. For industries such as the seismic industry, which exploits amplitude recovery from sign data, these results place constraints on processing, which includes differentiation of the data.While methods for smoothing noisy data for finite difference calculations are known, sign data requires noisy data. In this paper, we derive the expectation values of continuous and discrete sign data derivatives and we explicitly characterize the variance of discrete sign data derivatives.


Introduction
Sign-bit recording systems discard all information on the detailed motion of the geophone and ask only whether its output is positive or negative, whether it is going up or coming down.In a sign-bit system, therefore, the signal waveform is converted into a square wave.All amplitude information is lost 1 .
It is well known that, for a range of signal-to-noise ratios between about 0.1 and 1, the final result of sign-bit recording, after stacking, correlating, and other processing, looks no less good, to the eye, than the result from full-fidelity recording.This is considered to be as intriguing as it is surprising 1 .Alternatively, what we present in this paper is evidence that the processing of sign-bit data i.e., sign data can be limited for certain cases relative to the processing of the full-bandwidth data.
Model signal appears as a one-dimensional function, f v , and noise as a random variable, X.In industries like the seismic industry, measurements of signal, f v : R → R 2 ISRN Applied Mathematics and noise, X : Ω → R, f v X are recorded for multiple iterations of the noise.The average of the measurement i.e., the expectation E recovers the signal If the noise is chosen to be uniform, where ρ x is the density function such that As reported by O'Brien et al. 2 , it was empirically discovered that the average of the signs of signal plus noise recovers the signal if the signal-to-noise ratio is less than or equal to one.This can be shown mathematically 3 using the signum function 4 , sgn x 1, x > 0, sgn x −1, x < 0, sgn 0 0, Because ρ x is even and equals The variance is Consequently, the error is minimal when the signal-to-noise ratio is near unity.The advantage of retaining only the signs of signal plus noise is the requirement of approximately 1 bit to record the information as opposed to requiring 16 to 20 bits to record full amplitude data 2 .
The goal of this paper is to examine the recovery of derivatives from sign data in uniform noise.The issue is that recovery of signal from sign data can be extended to recovery of derivatives of the signal through the use of finite differences and that recovery is constrained by the size of the variance.In this paper, we first examine sign data derivatives for both the discrete and continuous case.We follow with a derivation of variance.We conclude our analysis with a computational test, which lists the true variance versus the variance estimate derived statistically for a test function for selected step sizes.

Sign Data Derivatives
Let the signal f v be an nth order differentiable function.Based on signal recovery from sign data, it can be shown that derivatives of the signal are also recoverable.Using the linearity of the expectation value, where Δ n v is the nth order finite difference operator with respect to the variable v 5 .In this case, a nonunit step size, Δv, is used e.g., 6 .
In detail, we can write where the notation n i represents the binomial coefficient n!/i! n − i ! and where X i X 0 , X 1 , . . .are independent representations of the random variable, X.
Substituting from 1.6 into 2.1 yields In the limit of infinitesimal step size, this becomes a continuous derivative lim Equation 2.4 presents an alternative solution to direct integration.For example, using the rule, f x δ n x dx − ∂f/∂x δ n−1 x dx, 7 , the integral loses all terms with derivatives of the delta functional, reducing to In general, It follows that the noise is restricted such that a ≥ |f|.

The Variance of Sign Data Derivatives
n can be found by inductively generalizing from n 2: The variance of a discrete sign derivative approaches infinity with decreasing step size and increasing order.In addition, since lim Δv → 0 S n d n /dv n sgn f v X , Var d n /dv n sgn f v X ∞, so in the case of the continuous derivatives 2.5 the variance is infinite.Use 3.2 to find the variance of the first discrete sign derivative by letting n 1: The variance of the second discrete sign derivative n 2 is similarly computed as Var S 2 1 3.4

Computational Tests
These results can be tested computationally.Variance can be estimated for N iterations with where the index m designates the sample number.Consider the test function f sin v .Using the first-order sign data derivative n 1 , compare Var S 1 to Var N S 1 , and using the second-order sign data derivative n 2 , compare Var S 2 to Var N S 2 for N 1000, a 1, and v 3. The results are shown in Tables 1 and 2.
We illustrate the change in variance in Figure 1, which shows three curves, each consisting of N 1000 iterations.The first curve in blue shows the sign data recovery of the function f sin v or E S 0 for a 1 and Δv 0.5.The second curve in green shows the sign data recovery E S 1 , which approximates f for a 1 and Δv 0.5.The third curve in red shows the sign data recovery E S 2 , which approximates f for a 1 and Δv 0.5.

Conclusions
Recovery of signal from the signs of signal plus noise incurs a variance, which only depends on the noise amplitude, while recovery of discrete derivatives from the signs of signal plus noise i.e., sign data incurs a variance which grows infinite for infinitesimal step size and infinite order.
The application problem is that sign data can be used in the seismic industry in processes which may differentiate the data.In such cases, if the step size or order of the finite difference is not constrained, the process will incur large variance and convergence of the process will be minimized.While methods for smoothing noisy data for finite difference calculations are known, sign data requires noisy data.In this paper, we have characterized the problem by explicitly evaluating the variance of discrete sign data derivatives.

Appendix
A.1 This simply reduces to A.2 In order to compute A.2 , we must compute an integral of the form The probability densities are both uniform: A.6

Figure 1 :
Figure 1: The expectation value curves for S n 1/ Δv nn i 0 −1 i n i sgn f v n − i Δv X i or E S n for n 0, 1, 2, f vsin v , a 1, and Δv 0.5.The number of iterations in the expectation values is N 1000.E S 0 corresponds to the blue curve and approximates f, E S 1 corresponds to the green curve and approximates f , and E S 2 corresponds to the red curve and approximates f .

Table 1 :
True variance, Var S 1 , versus the variance estimate, Var N S 1 , for the function f sin v , with the number of iterations N 1000, a 1, and v 3.

Table 2 :
True variance, Var S 2 , versus the variance estimate, Var N S 2 , for the function f sin v , with the number of iterations N 1000, a 1, and v 3.