Chapter 8: Sampling

Chapter 9: Partial Differential Equations

Chapter 10: Wavelets

Chapter 11: Musical Tones

Chapter 12: Probability

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

Figure 8.1: An audio signal for the spoken word Fourier.

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.

Figure 8.2: An 80-sample segment of the e sound from the spoken word Fourier.

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...