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# Analog-to-Digital Conversion

Module by: Don Johnson. E-mail the author

Summary: Analog-to-digital conversion.

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The Sampling Theorem says that if we sample a bandlimited signal st s t fast enough, it can be recovered without error from its samples sn T s s n T s , n=101 n 1 0 1 Sampling is only the first phase of acquiring data into a computer: Computational processing further requires that the samples be quantized: analog values are converted into digital form. In short, we will have performed analog-to-digital (A/D) conversion.

## Exercise 1

Recalling the plot of average daily highs in Problem FIX ME why is this plot so jagged? Interpret this effect in terms of analog-to-digital conversion.

### Solution

The plotted temperatures were quantized to the nearest degree. Thus, the high temperature's amplitude was quantized as a form of A/D conversion.

A phenomenon reminiscent of the errors incurred in representing numbers on a computer prevents signal amplitudes from being converted with no error into a binary number representation. In analog-to-digital conversion, the signal is assumed to lie within a predefined range. Assuming we can scale the signal without affecting the information it expresses, we'll define this range to be 1 1 1 1 . Furthermore, the A/D converter assigns amplitude values in this range to a set of integers. A B B -bit converter produces one of the integers 012B1 0 1 2 B 1 for each sampled input. Figure 1 shows how a two-bit A/D converter assigns input values to the integers.

We define a quantization interval to be the range of values assigned to the same integer. Thus, for our example two-bit A/D converter, the quantization interval is 0.5 0.5 ; in general, it is 22B 2 2 B . Because values lying anywhere within a quantization interval are assigned the same value for computer processing, the original amplitude value cannot be recovered without error. Typically, the D/A converter, the device that converts integers to amplitudes, assigns an amplitude equal to the value lying halfway in the quantization interval. The integer 0 would be assigned to the amplitude 0.75 0.75 in this scheme. The error introduced by converting a signal from analog to digital form by sampling and amplitude quantization then back again would be half the quantization interval for each amplitude value. Thus, the so-called A/D error equals half the width of a quantization interval: 12B 1 2 B . As we have fixed the input-amplitude range, the more bits available in the A/D converter, the smaller the quantization error.

## Exercise 2

How many bits would be required in the A/D converter to ensure that the maximum amplitude quantization error was less than 60 db smaller than the signal's peak value?

### Solution

Solving 2B=.001 2 B .001 results in B=10 B 10  bits.

Once we have acquired signals with an A/D converter, we can process them using digital hardware or software. It can be shown that if the computer processing is linear, the result of sampling, computer processing, and unsampling is equivalent to some analog linear system. Why go to all the bother if the same function can be accomplished using analog techniques? Knowing when digital processing excels and when it does not is an important issue.

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