1.12 Amplitude quantization

The Sampling Theorem says that if we sample a bandlimitedsignal $s(t)$ fast enough, it can be recovered without error from its samples $s(n{T}_s)$ , $n\in \{\dots , -1, 0, 1, \dots \}$ . Sampling is only the first phase of acquiring data into acomputer: 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 .

A phenomenon reminiscent of the errors incurred in representing numbers on a computer prevents signal amplitudesfrom being converted with no error into a binary number representation. In analog-to-digital conversion, the signal isassumed to lie within a predefined range. Assuming we can scale the signal without affecting the information itexpresses, we'll define this range to be $\left[-1 , 1\right]$ . Furthermore, the A/D converter assigns amplitude values inthis range to a set of integers. A $B$ -bit converter produces one of the integers $\{0, 1, \dots , 2^B-1\}$ for each sampled input. [link] shows how a three-bit A/D converter assigns input values tothe integers.We define a quantization interval to be the range of values assigned to the same integer. Thus, for our examplethree-bit A/D converter, the quantization interval $\Delta$ is $0.25$ ; in general, it is $\frac{2}{2^B}$ .

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 anamplitude equal to the value lying halfway in the quantization interval. The integer 6 would be assigned to the amplitude0.625 in this scheme. The error introduced by converting a signal fromanalog to digital form by sampling and amplitude quantization then back again would be half the quantizationinterval for each amplitude value. Thus, the so-called A/D error equals half the width of a quantization interval: $\frac{1}{2^B}$ . As we have fixed the input-amplitude range, the more bitsavailable in the A/D converter, the smaller the quantization error.

To analyze the amplitude quantization error more deeply, we need to compute the signal-to-noise ratio, which equals the ratio of the signal power and the quantizationerror power. Assuming the signal is a sinusoid, the signal power is the square of the rms amplitude: $\mathrm{power}(s)=\left(\frac{1}{\sqrt{2}}\right)^2=\frac{1}{2}$ . The illustration details a single quantization interval.

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