AD-03 Datasheet, PDF (1/4 Pages) National Semiconductor (TI) – Effects of Aperture Time and Jitter in a Sampled Data System

N
Application Note AD-03
Effects of Aperture Time and Jitter
in a Sampled Data System
Mark Sauerwald
May 1994
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:
+∞
∫ ^{fs(n) =}
f (t) δ (t − nτ) dt
−∞
where δ(t-nτ) is a dirac delta distribution, f(t) is the input
signal and f_S(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:
+∞
∫ ^{fs =}
f (t) ψ (t − nτ) dt
−∞
A shorthand way of expressing this is f_S = < 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(t₀) – f(t₀+ζ). If
we consider the Taylor expansion of f(t) about t₀:
f (t) ≈ f ₍t₀₎ + f'₍t − 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.
SNR vs. Input Frequency
120
100
80
60
40
1k
10k
100k
1M
10M
Input Frequency (Hz)
1PS
5PS
10PS
15PS
20PS
25PS
100M
Figure 1: SNR vs. Input Frequency for Various
Values of Aperture Jitter
© 1994 National Semiconductor Corporation
Printed in the U.S.A.
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