Sampling and interpolation

Introduction

As I speak the word Fourier, a microphone converts the pressure wave from my voice into an electrical voltage f(t), 0 ≤ t ≤.5 sec. The sound card in my computer discretizes this signal by producing the samples

Figure 8.1 shows the (overlapping) line segments joining

You can identify the hissy initial consonant “f”, the three long vowels “o”, “e”, “a” (as in boat, beet, bait), and the semivowel “r” (as in burr).

An expanded 80-sample segment of the 10-msec interval .33 sec t 34 sec (a part of the “e” sound) is shown in Fig. 8.2. It would appear that we have enough sample points to construct a good approximation to the original audio signal.

In practice, we work with a quantized approximation of the sample f(nT). For example, the Fourier recording uses 1 byte ≔ 8 bits per sample, with one bit specifying the sign of f(nT) and with seven bits specifying the modulus |f(nT)| to within 1 part in 127 (of some fixed maximum modulus). It takes 4000 bytes of storage for the Fourier recording, more than we need for a page of this text with

A digitized sound file that uses 8000 8-bit samples/sec is of telephone quality. For high fidelity we must increase the sampling rate and reduce the quantization error. A compact disk recording uses 44100 16-bit samples/sec or

Digitized sound files are very big!

Shannon's hypothesis

Let f be a function on ℝ, let T > 0, and let

It is easy to produce a function y on R that interpolates f at the sample points, i.e., …