Listen to Periodic Functions


The Problem

Countless technologies have been developed that use mathematical algorithms to analyze sound inputs. Such devices include hearing aids, microphones, speech recognition software, and more. The effectiveness of these and other technologies is dependent on their ability to analyze sound mathematically. While many of these devices have a great deal of technological sophistication, even something as simple as the beep of a touchtone telephone has interesting mathematics involved. This project addresses the following questions regarding the sounds made by a touchtone telephone:

  • How is a sound wave represented mathematically?
  • How are the sounds from touchtone phones created and analyzed?
  • Given a telephone number, how does a telephone generate the sounds corresponding to the numbers digits?
  • Given a sequence of sounds from a dialed telephone number, how can the digits dialed be determined?
  • In sound recordings, of telephone sounds or otherwise, how can the desired audio signal be distinguished from random noise?

Background Information

All sounds are caused by vibrating objects that create pressure disturbances in the surrounding air. As an object such as a tuning fork vibrates, the object exerts forces on the air around it, causing areas of higher-pressure air to propagate outward from the object. These high-pressure areas are called compressions, and the lower-pressure areas between the compressions are known as rarefactions. Together the compressions and rarefactions form a pressure wave in the air that we perceive as sound. When a pressure wave reaches the eardrum, the wave causes the eardrum to vibrate at the same frequency as the wave. The inner ear and the brain convert this vibration into a perceived sound.

Since sound is a periodic pressure wave, the sine function is often used to model sound mathematically. However, not nearly all sounds have graphs that can perfectly match a sine curve. A perfectly sinusoidal sound wave is called a pure tone. Almost all sounds heard in the natural world, on the other hand, are much more complex. Still, even complicated sounds are periodic waves, and the sine function can be a building block for representing all types of sound.

Sine functions are used to generate the sounds a telephone makes when a particular digit from the keypad is dialed. The tone produced by any number on a telephone keypad is actually the average of two sine waves. The telephone converts these sounds to electrical signals, which are then decoded to connect you to the telephone whose number you dialed.

The chart below shows the two sound frequencies that are combined to produce the tones for each number on a telephone keypad.

Frequency (Hz) 1209 1336 1477
697 1 2 3
770 4 5 6
852 7 8 9
941 * 0 #


Materials Included

  • Five sets of questions for students (“Sound.doc”)
    • Question 1: Introduction to Sound Waves
    • Question 2: Introduction to Touchtone Telephones
    • Question 3: Exploring Touchtone Telephones
    • Question 4: Generating Touchtone Telephone Numbers
    • Question 5: Reducing Background Noise
  • Sample solutions for teachers (“Sound Answers.doc”)
  • An interactive spreadsheet for student and teacher use (“Touchtone.xls”)
  • A guide explaining the spreadsheet’s features (“UsersGuide.doc”)