# Linear Correlator

A linear correlator is a digital correlator with a linear spacing in the channel layout. It finds application for DLS, DWS, FCS and other light scattering technologies. All these techniques require the unnormalized photon rate autocorrelation function, which in a first approximation reads:

\begin{gather*}G(\tau_k)=\frac{1}{N}\sum_{i=1}^{N}n(t_i)n(t_i-\tau_k)\end{gather*}

where the time axis has been discretized over \(N\) samples and \(\tau_k\) is the time lag between the photon count rates \(n(t_i)\) and \(n(t_j)\). Computation of the previous quantity can be carried out through a digital correlator, whose simplest schematic is showed in the figure below. The basic operations performed by a digital correlator for \(N\) samples are:

- Count photoelectron pulses upon a sampling time \(\Delta t\).
- For each correlation channel \(k\), delay the samples by \(\Delta t\), such that \(\tau_k = k \Delta t\).
- For each channel, multiply the delayed samples by the current one.
- For each channel, sum the products.

This scheme, known as the linear correlation scheme, is characterized by a subsequent set of \(k\) equally spaced time operations for a fixed sampling time \(\Delta t\). The extension of the lag time range is evidently determined by the maximum number of channels \(k\) available. The main limitation of this system lies in the finite implementable number of channels. In fact, to obtain high measurement accuracy, a very short sampling time, usually of the order of 10 ns is required. Under these constraints, a total measurement duration of even 1 s would require a dynamic range of 10^{8 }s and a corresponding number of channels of 10^{8}, which is technically unfeasible.

*Linear correlator architecture*

In addition, the inherent discretization of the time axis, and the resulting averaging of the photoelectron signal, sampled over \(\Delta t\), yields distortions of a decaying correlation function, increasingly evident when the lag time \(\tau_k\) is considerably larger than \(\Delta t\). For normalization of the numerical correlation function please check the mutli-\(\tau\) correlator page.