Signal Extraction for Nonstationary Time Series with Diverse Sampling Rules

Throughout, we shall assume that d>0, since the d=0 case is essentially handled in Kailath et al., (2000). In order to prove the theorem, it suffices to show that the error process e(t)=sˆ(t)−s(t) is orthogonal to the underlying process {y(h)}. By (19), it suffices to show that {e(t)} is orthogonal to {w(t)} and the initial values y^∗(0). So we begin by analyzing the error process produced by the proposed weighting kernel ψ=F−1[g]. We first note the following interesting property of ψ. The moments of ψ

for k<d exist by the smoothness assumptions on g, and are easily shown to equal zero if 0<k<2d (i. e., for d≤k<2d, the moments are zero so long as they exist – their existence is not guaranteed by the assumptions of the theorem). Moreover, the integral of ψ is equal to 1 if d>0. These properties ensure (when d>0) that the filter Ψ(L) passes polynomials of degree less than d. This is because Ψ(L)tj=tj for j<d. We first note that representation (19) also extends to the signal: s(t)=∑j=0d−1tjjs(j)(0)+[Idu](t). Then the error process is

Since Ψ(L) passes polynomials, (ψ∗s)(t)−s(t)=∫(ψ(x)−Δ0(x))[Idu](t−x)dx, where Δ0 is the Dirac delta function. Note that any filter that does not pass polynomials cannot be MSE optimal, since the variance of the error process will grow unboundedly with time. So we have

which is orthogonal to y^∗(0) by Assumption A. Due to the representation (19), it is sufficient to show that the error process is uncorrelated with [I^dw](t). For any real h

which uses the fact that [Idw](t)=[Idu](t)+n(t)−∑j=0d−1tjjn(j)(0). Now we have

If f_u is integrable, we can write Ru(h)=12π∫fu(λ)eiλhdλ. If Ru∝Δ0 instead, then fu∝1; we can still use the above Fourier representation of R_u in (35), because the various integrals will take care of the non-integrability of f_u automatically. Since ∫0xeiλydy=(1−eiλx)/(iλ), we obtain that (35) is equal to

When integrated against ψ(x)−Δ0(x), we use the moments property of ψ to obtain

This uses Ψ(e−iλ)−1=−λ2dfn(λ)/fw(λ), which is not integrable if fn∝1; yet f_uf_n/f_w will be integrable under the conditions of the theorem. As for the noise term in (34), we first note that n^(j)(t) exists for each j<d since w(t) exists by assumption; this existence is interpreted in the sense of Generalized Random Processes (Hannan 1970). In particular

This Fourier representation is valid even when fn∝1, since Ψ(e−iλ) is integrable by assumption. Similarly,

where the derivatives are interpreted in the sense of distributions – i. e., when this quantity is integrated against a suitably smooth test function, the derivatives are passed over via integration by parts:

Since λjΨ(e−iλ) for j<d is integrable by assumption, we have ψ(j)(x)=12π∫(iλ)jΨ(e−iλ)eiλxdλ, and the second term in (34) becomes

This cancels with the first term of (34), which shows that Ψ(L) is MSE optimal. Using similar techniques, the error spectral density is obtained as well. □

First we show that the difference between the two filters has no bias. Letting a(x)=xp for integer p and aτ=a(δτ),

follows from binomial expansion. Matching coefficients, we see that a necessary and sufficient condition for similar polynomial treatment is

for j=0,1,2,⋯,p. In the case of handling I(d) processes, we would impose this condition with p=d. But (36) can be compactly expressed in frequency domain as (26) using Fourier Transforms. Using the representation in eq. [5] of MT, any discrete filter that satisfies this condition ensures that the discretization error Ψ(L)y(δτ+δc)−Ψδ(B)yτ only involves stochastic portions. Next, using induction and results of Hannan (1970), we can represent an I(d) CT process (with square integrable differentiated process w) as

Typically the initialization values y(j)(0) are random variables assumed to be independent of the differentiated process w, to which the orthogonal increments process dZ pertains. Stock-sampling the above representation is fairly clear, but note that flow-sampling will result in the factor (1−e−iλδ)/(1−e−iλδ)(iλ)(iλ) multiplying the complex exponential. In either the stock or flow case we can apply the discrete-lag and continuous-lag filters and cancel out the deterministic terms, leaving the stated expressions for the discretization error. The expression (27) for the discretization MSE then follows at once. This integral (27) is not guaranteed to be finite, unless there is a suitable degree of decay in f_w or the other integrands (clearly λ−2d assists integrability). Also note that (26) ensures that a suitable number of zeroes occur in the integrand at frequency zero, to offset the explosive behavior of λ−2d at λ=0. □

We first establish the decomposition of the discretization MSE. Using the notation [f]δ(λ)=δ−1∑h=−∞∞f(λ+2πh/λ+2πhδδ) for the fold of the function f (cf., MT), the stock ODF is given by the formula uδ=[gecfwm2d]δ/uδ=[gecfwm2d]δ[fwm2d]δ[fwm2d]δ, where mj(λ)=λ−j. Then (28) follows from [uδfwm2d]δ=[gecfwm2d]δ, which holds due to a property of folds, implying that the cross-terms are zero. Also the total PADF discretization MSE can be rewritten as 1πδ∫0πK(λ/λδδ)dλ, where

which is convenient for numerical computation. The integral expression for (27) is easily approximated via a Riemann sum. The ODF discretization MSE can be computed using the formula (also discussed in MT)

This too can be rewritten as 1πδ∫0πH(λ/λδδ)dλ, where

This can also be computed via a Riemann approximation; of course, computation of these quantities require a knowledge of the true spectrum f_w, and thus is a theoretical exercise. Hence the PADF discretization MSE equals

which decomposes the error in terms of the ODF discretization MSE and the extra MSE due to using a sub-optimal discretization.