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AD-03 데이터시트(PDF) 1 Page - National Semiconductor (TI) |
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AD-03 데이터시트(HTML) 1 Page - National Semiconductor (TI) |
1 / 4 page Application Note AD-03 Effects of Aperture Time and Jitter in a Sampled Data System Mark Sauerwald May 1994 N © 1994 National Semiconductor Corporation http://www.national.com Printed in the U.S.A. The Nyquist sampling theorem states that band limited signals may be represented without error by sampled data. Among the effects which place additional restric- tions on the signal, beyond those which are given by the Nyquist theorem, are aperture effects. Aperture effects such as aperture jitter, and aperture time, contribute to a signal degradation which is frequency dependent. In applications for high performance A/D converters, the aperture effects may be the dominant source of noise in the digitization process. The purpose of this paper is to define the terms relating to aperture effects, to develop a mathematical framework to represent these effects, and to predict the errors that will be introduced into the sampled signal as a result of aperture effects. Introduction and Terminology The result of a sampling event is ideally the value of the input function at the sampling instant. This can be expressed as: where δ(t-nτ) is a dirac delta distribution, f(t) is the input signal and fS(n) is the sampled function. In practice the sampling event is somewhat different, the instantaneous sampling is replaced by an integration over a small period of time, and there is uncertainty in the actual instant at which the sample is taken. These two non- idealities: the finite sampling time and the uncertainty of the sampling instant can be accounted for by replacing the delta distribution by another distribution Ψ(t). Ψ(t) will be an ordinary, continuous function of time which includes a random variable: ζ to denote the uncertainty in sampling instant. ζ will not be explicitly written but it is assumed that Ψ(t) is Ψ(t+ζ). ζ represents the uncertainty in the sampling instant or aperture jitter and is a random variable with a mean value of 0. The RMS value of ζ is what will be specified as aperture jitter. The result of a sampling operation may now be written as: A shorthand way of expressing this is fS = < f,Ψ >. We are now in a position to evaluate the effects of aperture non-idealities in the sampling process on the sampled signal. One key is that since we are dealing with a random process, the result of a single sample is meaningless, we must consider the effects upon the signal, or the collection of the individual samples to have meaningful results. The Effects of Sampling Instant Uncertainty (Aperture Jitter) To consider Aperture Jitter we will assume that the sampling function Ψ is δ(t+ζ), with ζ representing the uncertainty in sampling instant. The error generated in sampling is now < f, δ > – < f, δ(t+ζ) > or f(t 0) – f(t0+ζ). If we consider the Taylor expansion of f(t) about t0: Then the error generated on each sample is approxi- mately f'(t) • ζ. This is intuitively comfortable since it states that the error is proportional to the slew rate multiplied by the aperture jitter. When a signal is sampled, the expected error that will be generated will be the RMS value of the slew rate multi- plied by the RMS value of ζ (the RMS value of ζ is what is commonly specified as Aperture Jitter). Hence for a full scale sinusoidal input signal of 10MHz, the RMS slew rate is 0.02 full scale ranges/ns. If the Aperture Jitter is 10ps then the expected error would be 0.02 * 0.01ns or 213ppm. This corresponds to a signal to noise ratio of 73dB or approximately the same size error as the quantization error from an ideal 12 bit converter. In Figure 1 the Signal-to-Noise Ratio is plotted vs. the Input frequency for various different values of aperture jitter. Figure 1: SNR vs. Input Frequency for Various Values of Aperture Jitter fn f t t n dt s () = () − () −∞ +∞ ∫ δτ ff t t n dt s = () − () −∞ +∞ ∫ ψτ ft ft f' t t 00 () ≈ ()+− () SNR vs. Input Frequency Input Frequency (Hz) 120 100 40 1k 10k 100k 80 60 1PS 5PS 20PS 10PS 15PS 1M 10M 100M 25PS |
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