Some cross correlation properties for distorted signals

For a nondecreasing distortion characteristic
\phi(\cdot)
and a given signal
x(\cdot)
, the "cross correlation" function defined by
R_{\phi} (\tau) \triangleq \int_{-\infty}^{\infty} \phi[x(t)]x(t - \tau) dt
is shown to satisfy the inequality
R_{\phi}(\tau) \leq R_{\phi}(0)
, for all
\tau
, generalizing an earlier result of Richardson that required
\phi(\cdot)
to be continuous and strictly increasing. The methods of the paper also show that, under weak conditions, \begin{equation} R_{\phi,\psi}(\tau) \triangleq \int_{-\infty}^{\infty} \phi[x(t)]\psi[x(t - \tau)] dt \leq R_{\phi,\psi}(0) \end{equation} when
\psi
is strictly increasing and
\phi
is nondecreasing. In the case of hounded signals (e.g., periodic functions), the appropriate cross correlation function is \begin{equation} \mathcal{R}_{\phi,\psi}(\tau} \triangleq \lim_{T \rightarrow \infty} (2T)^{-l} \int_{-T}^T \phi[x(t)]\psi[x(t - \tau)] dt. \end{equation} For this case it is shown that
\mathcal{R}_{\phi,\psi} (\tau) \leq \mathcal{R}_{\phi,\psi}(0)
for any nondecreasing (or nonincreasing) distortion functions
\phi
and
\psi
. The result is then applied to generalize an inequality on correlation functions for periodic signals due to Prosser. Noise signals are treated and inequalities of a similar nature are obtained for ensemble-average cross correlation functions under suitable hypotheses on the statistical properties of the noise. Inequalities of this type are the basis of a well-known method of estimating the unknown time delay of an observed signal. The extension to nondecreasing discontinuous distortion functions allows the use of hard limiting or quantization to facilitate the cross correlation calculation.