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2

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

below:

Figure 2: Simplified Sample-And-Hold Circuit

The RC time constant of this circuit will be proportional

to the aperture time. This implies that

Ψ has the form

indicated below:

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 >

do not contribute to the result.

Since

Ψ has been

this is a constant

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.

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

an input terminal. In this case, due to the symmetry of

the sampling process, we can expect

Ψ to be an even

function:

such as a gaussian, or a rectangular pulse then the

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:

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,

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.

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:

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.

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.

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