# MIT 2 693 - The Sampling Process (18 pages)

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**View the full content.**## The Sampling Process

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The Sampling Process Geophysical processes are continuous processes in time but are not sampled continuously and never for infinitely long time spans Processes are almost always 1 sampled at discrete points in time usually equally spaced for 2 a finite length of time and 3 recorded in digital form Digital recording has the advantage of high resolution and ease in the subsequent analysis by the use of a computer Understanding how to sample the environment properly so the statistics of the process can be accurately estimated involves knowledge of 1 The discrete sampling process This includes the effects of quantization in time or the sample interval t and the quantization of the parameter being measured x 2 The response of our instrument to the environment 3 Gating sampling for only a finite length of time The relationship between the discrete digitized sample set what we have to work with and the original continuous function is covered by certain sampling theorems The Sampler Define the sampler III Shah as III t t n n Eq 47 which is an infinite set of unit spaced impulses or delta functions By suitable limiting process to take care of sharp peaks and infinite length Bracewell shows that 1 III t III f Eq 48 Then some properties of the sampler given n as an integer and a as a scalar are 2 III t n III t 3 III t 1 2 III t 1 2 4 III t III t 5 III at 1 a t n a n 6 Eq 50 n III t dt 1 n 7 III t x t x t n n Eq 51 This says that the convolution of x t with the sampler produces an infinite sum of shifted versions of x t The general results of this is a complex mess If however x t is gated so that it 24 is time limited to t 1 2 and x t 0 otherwise the convolution produces an infinite replication of x t in each interval t 8 III t x t x n t n n Eq 52 This is the sampling operation The continuous function x t is changed into a set of pulses The product function is determined by its values at t n and can be represented by a series of numbers xt as the discrete representation of x t

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