```Application Note AD-03
N
Effects of Aperture Time and Jitter
in a Sampled Data System
Mark Sauerwald
May 1994
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 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 nonidealities: 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 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
Printed in the U.S.A.
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 ( t 0 ) + f' ( t − t 0 )
Then the error generated on each sample is approximately 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 multiplied 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
SNR
The Nyquist sampling theorem states that band limited
signals may be represented without error by sampled
data. Among the effects which place additional restrictions 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.
80
1PS
60
40
1k
10k
100k
1M
10M
5PS
10PS
15PS
20PS
25PS
100M
Input Frequency (Hz)
Figure 1: SNR vs. Input Frequency for Various
Values of Aperture Jitter
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The Effects of Non-zero Aperture Time
For the analysis of aperture time we will assume ζ to be
0. This is the case of no aperture jitter. We will, however,
consider the effects of Ψ ≠ δ. The error (as compared to
an ideal sampling process) will then be < f, Ψ > – < f, d >
This is equivalent to < f, (Ψ – δ) >. To progress we must
now make some assumptions about the nature of Ψ. If
we consider the physical process of sampling that takes
place within a sample-and hold amplifier we can make an
educated guess as to the shape of Ψ. Consider the
sample-and hold circuit shown in Figure 2. When the
switch is closed, VOUT takes the form of the equation
below:
t
Vout ( t) =
∫
Vin
τ−t
RC
( τ) e
an input terminal. In this case, due to the symmetry of
the sampling process, we can expect Ψ to be an even
function:
2
t − t0 )
(
f ( t) ≈ f ( t 0 ) + f' ( t 0 ) ( t − t 0 ) + f" ( t 0 )
2
such as a gaussian, or a rectangular pulse then the
odd nature of (t-t0) will force < f'(t0) (t-t0),Ψ > = 0. Now in
order to determine the error generated by the non-zero
sampling time we must consider the next term in the
Taylor series expansion.
Now the expected error is:
fs ( t 0 ) error ≈
dτ
−∞
Vin
Vout
The RC time constant of this circuit will be proportional
to the aperture time. This implies that Ψ has the form
indicated below:
ψ
ψ = 0, t > 0
To get an idea of what effect this has on the sampling
process we will expand f as a Taylor series expansion
about the sampling instant: t. Once again we will
consider the first two terms of the expansion:
Now we can examine the value of < f, (Ψ – δ) >. < f, δ >
is by definition f(t0) and the other terms of the expansion
do not contribute to the result. Since Ψ has been
normalized, < f(t0),Ψ > is f(t0): the ideal result of the
sampling event. If we consider the next term: f'(t0) (t-t0)
this is a constant
t
1) Demodulate the signal and digitize the 100kHz base
band signal at a rate of 1MHz. This scheme results in
very low input slew rates and would be the preferred
method from an aperture error standpoint.
fs ( t 0 ) error = f' ( t 0 ) te RC dt
∫
−∞
slope going through 0 at the sampling point. This,
expanded out gives us an approximation of the error
generated by the non-zero sampling time. This implies
that the error generated is proportional to the slew rate of
the input, multiplied by the aperture time. This result is
very similar to the result for aperture jitter.
2) Digitize the modulated signal at a 1MHz rate, allowing
aliasing to perform the down conversion. In this method
the usual mixers are eliminated and the digitized signal
is identical to that obtained in method 1 above. The
problem is that the input slew rates seen by the
converter are over one thousand times greater than
those seen in the above example and aperture related
errors may dominate.
Let us now look at the shape of Ψ that might be expected
in a CCD or SAW device where the sample consists of
the charge deposited in a bin as the bin passes beneath
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2
Reducing the Input Slew Rate
In some cases, the input slew rate is higher than it needs
to be to recover the information content of the signal to
be digitized. An example would be if an input signal were
being digitally down converted through aliasing. As an
example if a 100kHz signal, modulated on a 101MHz
carrier is to be digitized, then there are several possible
approaches:
f ( t) ≈ f ( t 0 ) + f' ( t − t 0 )
0
< (t − t0 ) , Y >
How to Minimize Aperture Induced Errors
As we have seen in the preceding analysis the results of
both aperture time and aperture jitter is an error signal
which increases in amplitude as the slew rates at the
input terminal of the A/D increase. One set of strategies
that are used to reduce aperture errors therefore focus
on minimizing this input slew rate. In fact, from an
aperture error standpoint the only thing that is important
is that the slew rate be small at the instant that the
converter is sampling the input, rapid slewing between
samples does not contribute to aperture error. Another
tack that can be taken to minimize aperture related errors
is one which takes advantage of the fact that the noise
generated through aperture effects has a random
characteristic and therefore lends itself to reduction
through some standard signal processing techniques.
C
Figure 2: Simplified Sample-And-Hold Circuit
t
RC
=e ,t<0
2
If the input function f(t) is sinusoidal in nature, all of it’s
derivatives have approximately equal values so the
major difference between this case and the previous
case comes in the comparison of < t,Ψ > and 1/2 < t2,Ψ >.
Vin
R
f' ( t 0 )
2
3) Down convert to an I.F. of 1MHz then digitize at a rate
of 1MHz. In this method, some of the advantages of
digital down conversion can be retained - multiple
channels can be demodulated simultaneously - and
complexity is reduced from a system that down converts
to baseband. From an aperture standpoint we are still
seeing input slew rates an order of magnitude greater
than we did in example 1, but two orders of magnitude
less than those seen in example 2.
Another signal conditioning method that can be used
toreduce the noise both from aperture effects as well as
other sources is implemented in SD A/D converters. A
block diagram of a simple converter is shown in Figure 3.
4) Place a sample and hold circuit, sampling at a rate
of 1MHz, in front of the A/D in scenario 2. This results in
the A/D converter seeing a very low slew rate, but the
aperture errors are shifted to the sample-and-hold circuit.
In many cases a sample-and-hold has much lower
aperture errors associated with it than a comparable
speed A/D converter.
Figure 3
Digital
Out
Signal
In
+
H(s)
A/D
D/A
If we replace the SD A/D and D/A pair with a simple
additive noise where this noise represents the noise
contributed by the sampling process, both quantization
noise and aperture jitter related noise, then the block
diagram is modified to look like that in Figure 4. The
transfer function for the signal in this system is given by:
In all of the above scenarios, the digital output from the
converter is at a 1 MHz rate and the digital output would
be identical if the A/D converter were an ideal converter
with no aperture jitter. With real world converters, there
will be a vast difference in the signal to noise ratio in each
of these three scenarios.
STF =
H ( s)
1+ H (S)
Sampling & Quantization
Noise In
The solution involving placing a sample and hold circuit in
front of an A/D converter is especially interesting with many
high speed, high resolution, monolithic A/D converters.
Many of these devices have particularly poor aperture
performance and as a result even if the input frequencies
are reduced to sub-nyquist rates, the aperture error may
be the dominant error source. Use of a separate sample
and hold will allow for the burden of sampling the signal
to be shifted to the sample and hold, reducing the
sensitivity to aperture jitter in the A/D converter.
Signal
In
+
+
H(s)
Out
Figure 4
It can be seen from this that if H(s) is large for the
frequency range in which the signal is, then the signal
transfer function is near unity.
The noise however, sees a different transfer function:
Reducing Noise Through Signal Processing
A/D converter outputs contain noise that has as its
origins quantization noise, noise that comes from
aperture related effects and noise from other sources.
Often it is desirable to lump all of these together and just
treat them as noise and work towards reducing them.
One technique that can be used for this is simply to oversample the input then digitally low pass filter the output.
If we take the example above, where we have a 100MHz
carrier modulated with a 100kHz signal, if we sample at
a rate of 2MHz instead of a rate of 1 MHz, the noise is
distributed over a band that is twice as wide. If the
digital signal is then filtered, and the half of the band that
does not contain the signal of interest is thrown away, the
result is a 3dB improvement in the signal to noise level.
This could be carried on as far as the speed of the
converter permits with the cost being carried mainly in
the power and complexity of the digital filtering hardware.
Imagine sampling the signal at 1GHz, then filtering out all
but the lowest 500kHz band to obtain the equivalent of
1MHz sampling: we would be able to obtain a 30dB
improvement in SNR over what we would have sampling
at 1MHz.
NTF =
1
1+ H (S)
If H(s) is large in the area of interest then the noise is
attenuated in this same area. The result is that the
signal to noise ratio is increased.
If an analysis is done of this (which is beyond the scope
of this paper) it turns out that this is a much more effective method of reducing the noise than the simple oversampling and low pass filtering that is outlined above.
With a first order filter for H(s) then the SNR improvement
that can be realized with Σ∆ techniques is 9dB per octave
of oversampling as compared to the 3dB that we
obtained above. As the order of the filter used increases
the gains can be increased as well.
Conclusion
As digitizing systems increase in speed, aperture effects
play a larger and larger role in the total error budget of
the system. Techniques for analysis and prediction of
the errors have been presented. Techniques for the
reduction of aperture related errors have been presented.
3
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