# Filtering the period and amplitude of a sine wave corrupted by noise

Let $$W$$ be a standard Brownian motion and $$\mathcal F_t$$ its natural filtration.

Suppose $$\theta, A$$ are positive $$L^1$$ random variables independent of $$\mathcal F_t$$.

Let $$Y_t$$ be the process

$$Y_t := A \sin \, (\theta t) + W_t$$

and denote by $$\mathcal Y_t$$ its natural filtration.

Question: Is it true that $$\mathbb E[A| \mathcal Y_t] \to A$$, and $$\mathbb E[\theta| \mathcal Y_t] \to \theta$$ almost surely?

By the Levy zero-one law, the question is equivalent to deciding whether $$A$$ and $$\theta$$ are measurable with respect to $$\mathcal F_\infty$$, the common refinement of all the $$\mathcal F_t$$. The answer is positive.

For $$\alpha>0,\beta\ge 0$$, consider $$Z_n=Z_n(\alpha,\beta):=Y_{\beta+(2n+1)\alpha }-Y_{\beta+(2n-1)\alpha }=U_n+X_n$$ where, for $$n=0,1,2,\ldots$$, the sequence $$U_n:=A\sin\Bigl(\theta (\beta+(2n+1)\alpha)\Bigr)-A\sin\Bigl(\theta (\beta+(2n-1)\alpha)\Bigr)=2A\sin(\alpha \theta) \cdot \cos(\theta\beta+2n\theta\alpha)$$ is an almost periodic sequence obtained as a function of the (zero entropy) rotation by angle $$2\alpha\theta$$, and $$X_n:=W_{\beta+(2n+1)\alpha }-W_{\beta+(2n-1)\alpha }$$ is a sequence of i.i.d. Gaussian variables with mean 0 and variance $$2\alpha$$.

In [1], H. Furstenberg defined disjointness of dynamical systems and proved in Theorem I.2 that i.i.d. processes and zero-entropy processes are disjoint. In Theorem 1.5 of the same paper, he showed that if $$(U_n)_{n \ge 1}$$ and $$(X_n)_{n \ge 1}$$ are sequences of integrable real random variables which define two disjoint stationary processes, then the sum sequence $$(U_n+X_n)_{n \ge 1}$$ determines $$(U_n)_{n \ge 1}$$. A generalization is in Theorem 7 of [2].

In particular, from the sequence $$\{Z_n(\alpha,\beta)\}_{n \ge 1}$$ we can obtain $$(U_n)_{n \ge 1}$$ =$$(U_n(\alpha,\beta))_{n \ge 1}$$ for every $$\alpha$$ and a.e. $$\beta$$, and using continuity, for all $$\beta$$. Then $$\theta=\pi/(4\alpha_*)$$, where $$\alpha_*=\min\{\alpha>0: U_1(\alpha,0)=0\}$$ and then $$A$$ is easy to determine.

Theorem 1.6 in [3] gives another proof.

For an elementary direct argument, define the deterministic function $$f_0(\alpha)=\lim_n \frac{1}{n}\sum_{k=1}^n Z_n(\alpha,0)^2$$ and observe that the smallest positive $$\alpha$$ where $$f_0(\alpha)\alpha^{-2}$$ is minimized is $$\alpha_{min}=\pi/\theta$$. Moreover, $$f_0(\alpha_{min}/2)=4A^2+\alpha_{min}$$ which yields $$A$$.

[1] H. Furstenberg (1967), Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. systems theory 1, pp. 1-49. https://mathweb.ucsd.edu/~asalehig/F_Disjointness.pdf

[2] Furstenberg, Hillel, Yuval Peres, and Benjamin Weiss. "Perfect filtering and double disjointness." In Annales de l'IHP Probabilités et statistiques, vol. 31, no. 3, pp. 453-465. 1995.http://www.numdam.org/article/AIHPB_1995__31_3_453_0.pdf

[3] Lev, Nir, Ron Peled, and Yuval Peres. "Separating signal from noise." Proceedings of the London Mathematical Society 110, no. 4 (2015): 883-931. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.749.2820&rep=rep1&type=pdf

• Amazing as usual. I like the elementary proof too, taking advantage of the zero one law to make $f$ deterministic. I should take a look at the papers [2] and [3] of yours! Stochastic filtering is really fun. Jan 30 at 6:28
• Hm, slight thing - should the normalising factor $\alpha^{-2}$ in $f_0 (\alpha) \alpha^{-2}$ be $\alpha^{-1}$ instead? Since the squared fluctuations in $W$ are of order $\alpha$. Jan 30 at 9:57
• Yes, corrected the normalization. Jan 30 at 16:28