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

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:

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

Then the error generated on each sample is approxi-

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

()

−

()

−∞

+∞

∫

δτ

ff

t

t

n

dt

()

−

()

−∞

+∞

∫

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