Signal Analysis: Estimation of Spectra

Abstract

In Chap. 5 we introduced stochastic processes. When a stochastic processes is observed, the result of a measurement can only be recorded for discrete times, e.g., t, t + Δt, t + 2Δt, .... The time step Δt is called the sampling time and 1/Δt the sampling frequency. A sampling frequency of, say, 1 kHz implies that a value is registered 1000 times a second, i.e., each millisecond. If we choose the unit of time such that Δt = 1, the sampling times are t, t + 1, ... or t = 1, ..., N. Hence, we will always write for a time series Y(t), t = 1, ..., N. The finite time series is the result of an observation, and therefore we should denote the values by small letters y(t), since they are to be considered as realizations of random variables. For later purposes, however, we will use the random variable Y(t) associated with each observation y(t).