College of Santa Fe Auditory Theory

Lecture 005 Sound IV


  1. Damped and driven oscillation
  2. Coupled Oscillations
  3. Time and frequency domains
  4. Fourrier Synthesis
  5. Online Fourrier Synthesizer
  6. Square wave example
  7. The effect of phase
  8. The spectrum of non periodic sound waves.
  9. Analyzing spectra
  10. Low Pass Filter (LPF)
  11. High Pass Filter (HPF)
  12. Band Pass Filter (BPF)
  13. Band Eliminate Filter (BEF)
  14. Fourrier Analysis
  15. Filter time response
  16. Time responses of acoustic systems
  17. Time and frequency representations of sounds
  18. Brain Bullets


Damped and Driven Oscillation - Resonance (Video Demo)

Coupled Oscillation (Video Demo)

1.6 Time and frequency domains 51

Time domain plots amplitude against time.

Frequency domain plots ampolitude against frequency.

Time and Frequency Domains (Video Demo)

Because a sine wave represents a single frequency we can display it like this and know most of what we need to know about it:

Time is on the X axis (across) and amplitude is on the Y axis. This is called a time domain display because it plots amplitude against time. In the real world we hear very few sine waves unless they are test tones or emergency warnings. Real world sounds are complex and contain many harmonics. Curiously our ears have the ability to synthesize multiple harmonics into a single sound as the following audio demonstration shows.

Cancelled Harmonics (Audio Demo)

1.6.1 The spectrum of periodic sound waves

Here are four natural waveforms:

Solo Violin

Solo Trumpet

Solo Flute

Solo Oboe.

All the works of Beethoven can be respresented by a single squiggly line (two if we need stereo) so these displays do accurately respresent the sound however it is difficult to look at them and deduce the sounds that went into creating their complex structures.

The Fourrier Theorem

Fourrier theorem states that all complex waveforms can be built up by using an appropriate set of sine waves of different frequencies and phases.

For pure tones the amplitude of each harmonic is 1/ Harmonic number2

Fourrier Synthesis (Video Demo)

Online Fourrier Synthesizer

Square wave example.

Click the green button to see and hear the result of merging the first 14 partials of a square wave. Note that the first harmonic is a perfect sine wave which is gradually moved more and more towards a square wave with the addition of each harmonic.

The time domain display for a square wave looks like this:

The frequency domain display for a square wave looks like this:

Here frequency is on the X axis and amplitude is on the Y axis. Each vertical line represents a sine wave harmonic which makes up the square wave. Its height represents its amplitude.

A square wave it is formed by summing together sine waves which are odd multiples of its fundamental frequency. So if the fundamental Frequency is N the harmonic series of a square wave goes N, 3N, 5N, 7N, 9N, 11N, 13N, 15N. A Sawtooth wave contains all partials for that in a sawtooth the harmonic series goes N, 2N, 3N, 4N 5N

Calculating harmonics

Fundamental (f0): Hz 1
Second Harmonic(f2): Hz (f0 x 2) 1/4
Third Harmonic (f3): Hz (f0 x 3) 1/9
Fourth Harmonic (f4): Hz (f0 x 4) 1/16
Fifth Harmonic (f5): Hz (f0 x 5) 1/25
Sixth Harmonic (f6): Hz (f0 x 6) 1/36
Seventh Harmonic (f7): Hz (f0 x 7) 1/49
Eighth Harmonic (f8): Hz (f0 x 8) 1/64
Ninth Harmonic (f9): Hz (f0 x 9) 1/81
Tenth Harmonic (f10): Hz (f0 x 10) 1/100
Eleventh Harmonic (f11): Hz (f0 x 11) 1/121
Twelfth Harmonic (f12): Hz (f0 x 12) 1/144
Thirteenth Harmonic (f13): Hz (f0 x 13) 1/169
Fourteenth Harmonic (f14): Hz (f0 x 14) 1/196
Fifteenth Harmonic (f15): Hz (f0 x 15) 1/225
Sixteenth Harmonic (f16): Hz (f0 x 16) 1/256

1.6.2 The effect of phase.

In addition to the pitch and amplitude of a harmonic the phase of the harmonic can have a radical effect on the shape and often the sound of a wave

Curiously the sound can sometimes be the same since the ear, although extremely sensitive to relative phases is insensitive to absolute phase.


1.6.3 The spectrum of non periodic sound waves.


If sine waves which do not have integer relationships between their frequencies are combined the result may rapidly phase cancel any pattern and create an aperiodic soundwave.



1.7 Analyzing spectra 56

In order to understand how we analyze spectra it is important to understand a little about sound filters. These are devices which allow some frequencies to pass and block others.

Here is the audio spectrum

Low Pass Filter (LPF)

Here is the same spectrum with a Low Pass Filter (LPF) applied

High Pass Filter (HPF)

Here is the same spectrum with a High Pass Filter (HPF) applied

The "knee" of the filter is called the "Cutoff" and can be moved up or down (right or left in these diagrams" to allow more or less of the sound to pass through. As the names imply a low pass filter cuts out highs and allows lows to pass while a High pass filter cuts out lows and allows highs to pass.

Filitered Noise (Audio Demo)


Band Pass Filter (BPF)

If a low pass and a high pass are arranged in series (one after the other) the result is a band pass filiter which allows audio through between their cutoffs. As you can see the Lowpass shaves off the highs first and the highpass shaves off the lows second. What is left is a band of sound between them.


Band Eliminate Filter (BEF)

If a LPF and a HPF are arranged in parallel the result is a band eliminate filter. Sound passes above and below the space between their cutoffs but none in the middle.

Filter Bank

If audio is run into a bank of Band Pass Filters it is then possible to measure the energy which flows through each one. This gives an indication of the spectral content of the sound.

Fourrier Analysis (Video Demo)

1.7.2 Filter time response

One problem with filtering a signal to derive its content is that the response time of the filter depends upon the frequency content. The rise time of a low frequency sine wave is much longer than the rise time of a high frequency sine wave.

A similar argument can be used for bandpass filters. The speed of variation of the envelope depends upon the number of, and total frequency occupied by, the sine wave components in the output of the filter. As it is the amplitude variation in the output of the filter that carries the information it too has an inherent time response which is a function of its band width. Wide bandwidth filters are fast while narrow bandwidth filters are slow.

All filters have a response time which is a function of

This is an inherent problem which cannot be solved by technology. The response time of a filter is inversely proportional to the bandwidth so a narrow band filter has a slow rise and fall time whereas a wide band filter has a fast rise and fall time.

In practice this means that if the frequency resolution of a spectrum analyzer is good then its response time will be slow creating a smearing of the sound image. On the other hand if the response time is good then the frequency resolution will be poor.

1.7.3 Time responses of acoustic systems

The argument can be reversed to show that when the ouput of an acoustic instrument such as a musical instrument changes slowly then its spectrum occupies a narrow bandwidth whereas if it changes quickly then it must occupy a larger bandwidth.


1.7.4 Time and frequency representations of sounds.

The waterfall display is one useful way of displaying both time and frequency characteristics. In this display amplitude is represented by the darkenss of the grey scale image, frequency is on the Y axis and time is on the X axis. However it does suffer from the time smearing effects discussed above

Note the smearing of the attacks in the narrow band and the blurring of harmonics in the wide band.

Reading Assignment

Before next class please read Sections

(pages 65 to 79) of Acoustics and Psychoacoustics. We will have a brief quiz on these sections at the beginning of the next class.

You Need to Know