By Susskind Alfred K
Those notes from certain in depth summer time courses on Analog-Digital conversion held at MIT from 1956-1957 specialise in difficulties created while electronic gear is associated with actual structures. A "language" challenge arises, for the language of the information-processing apparatus is electronic and the language of conversation within the remainder of the approach is sort of continuously within the kind of electric signs or mechanical displacements analogous the the actual parameters concerned. hence, there's a want for units to accomplish the language translation. units taht practice analog-to-digital conversion are referred to as coders, and units that practice digital-to-analog conversion are referred to as decoders.The subject material is split itno 3 components. the 1st half relates to structures facets of electronic details processing that impression the requirements for analog-to-digital and digital-to-analog conversion units. within the moment half, a close engineering research and overview of various conversion units is gifted. The 3rd half is dedicated to a case examine in line with improvement paintings performed on the Servomechanisms Laboratory of the MIT division of electric Engineering.
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Where j = v:r � W e will demonstrate this fact for the fir st moment of x(t) . i. e •• the aver a ge value of x(t). If w (x) i s a nice enough function. X d - ' (X x w (x) cra ( e J w (x) (-jx)e )dx - ' /Xx J dx -00 W e showed in the last s ection that x(t) = s o that s ince e - ' aO J = e 0 1 00 xw (x)dx. -00 = 1 . we have the de sired r e sult 2 -44 0l = 0 = f oo 0 xw (x} e dx - 00 f = oo xw (x}dx = x (t) . -00 4 . Quantization as A r e a Sampling We will now inve stigate the effe cts of using di s c r ete rather than c ontinuous measureme nts.
The z-Transform Since the Laplace transform is merely a modified Fourier transform with a real a -axis and an imaginary j Cal-frequency axis, we would expect the frequency part of the Laplace transform to experience the same frequency folding, due to sampling, as the Fourier transform. In other words, we would expect to find that the s-plane is folded accordian fashion for a sampled function. This fact is most easily illustrated through an example. Figure 2 - 1 5 shows an exponential function of the form e- a t before and after sampling.
By the use of suitable tech niques, the Fourier transform can be made to surmount these difficulties, but a much more painless and conventional approach is to use the Laplace transform instead. The Laplace transform differs from the Fourier transform in that a "squeezing function" of the t form e- a , for a suitable a, is introduced such that the integral fOO o f(t) e- at e- jwt dt 2-15 is finite. Quite clearly cr will depend upon f (t ) , since some f (t ) 's need no squeezing at all. Notice that since e-crt stretches instead of squeezes if t is negative, the integral goes only from zero to in finity.