4.6  From the time domain to the frequency domain

In the previous sections, you saw that complicated waveforms can be created by combining simple sine waves in a systematic way. Remarkably, any waveform can be represented as a combination of sine waves of appropriately chosen amplitude (loudness), frequency (pitch) and phase. This result, demonstrated by the French mathematician and physicist Joseph Fourier (1768–1830), lies at the heart of the design of electronic systems to reproduce high-fidelity sound.

The most familiar representation of waves is in what is called the time domain, i.e. the changing value of the signal through time. As you have seen, complicated waves can be formed from combinations of sine waves of given frequency and amplitude. This allows another representation in what is called the frequency domain. To see this, enter the following waves into Interactive 4:

• Wave 1: A = 0.7, f = 110 Hz
• Wave 2: A = 0.2, f = 330 Hz
• Wave 3: A = 0.1, f = 550 Hz
• Wave 4: A = 0.05, f = 770 Hz.

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Interactive 4  Waves represented in the time domain and the frequency domain

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This is an interactive that may be used to add two or more sine waves together, display a visual representation of the result in the time and frequency domains, and also hear its sound.

To start with, three pairs of two graphs are shown in a vertical column. The left-hand graph in each row is the representation in the time domain. It displays voltage between minus 1 and 1 volts against time between 0 and 20 milliseconds. The right-hand graph in each row is the representation in the frequency domain. It displays amplitude against frequency in hertz.

Between each pair of graphs is a loudspeaker button. Above each of the top two pairs of graphs is its corresponding sine wave equation, with type-in text boxes to change the amplitude, frequency and period. Finally, an ‘add another sine wave’ button is positioned below the loudspeaker button belonging to the second pair of graphs.

When values are input to the sine wave equations above the top two pairs of graphs, a visual representation of the sine wave in the time and frequency domains is shown on the graphs, and you can also listen to it by clicking on the loudspeaker button. In addition, the result of adding the two sine waves is shown on the bottom graphs, and this can also be listened to by clicking on the corresponding loudspeaker button. Clicking on the ‘add another sine wave’ button inserts another pair of graphs with sine wave parameters above the bottom pair, allowing you to sum three sine waves instead of two, and so on.

This interactive changes according to the parameters you enter and therefore you may need a sighted helper to describe the waveforms to you.

Interactive 4  Waves represented in the time domain and the frequency domain

Figure 30 shows the composite wave formed from the four sine waves listed above. On the right of this is a graph showing amplitude against frequency. There is a vertical line of height A = 0.7 corresponding to frequency f = 110 Hz, a vertical line of height A = 0.2 corresponding to f = 330 Hz, a vertical line of height A = 0.1 corresponding to f = 550 Hz and a vertical line of length A = 0.05 corresponding to f = 770 Hz. This set of lines is called the frequency spectrum of the signal on the left.

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Figure 30  Transforming a signal in the time domain into the frequency domain

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This is a screenshot from Interactive 4. On the left, the time-domain graph shows a wave like those seen previously – in this case, approximating a square wave. On the right, the frequency-domain graph shows lines of decreasing height (amplitude) at frequencies 110, 330, 550 and 770 hertz. The sound of the wave when the loudspeaker button is pressed is relatively harmonious.

Figure 30  Transforming a signal in the time domain into the frequency domain

One of the many problems with reproducing sound is that the desired sound may be contaminated by noise. For example, if you record a video and commentary on your phone in a crowded place, you will record what you want (you speaking) but also what you may not want (the hubbub surrounding you). Engineers talk about the part of the recording that is wanted as the signal and the part that is not wanted as noise.

For example, suppose you are recording yourself playing the piano and an emergency vehicle goes past with its high-pitched siren blaring. By using electronics, it may be possible to filter out that unwanted noise, or at least make it a lot less intrusive.

To represent noise, add a fifth sine wave to your signal with A = 0.3 and f = 935 Hz. This distorts the signal, as shown in Figure 31. Click on the loudspeaker button and you will hear that the sound is now quite discordant. It is not very clear what the noise is doing in the time-domain representation on the left, but the rogue frequency of 935 Hz can be seen clearly as a vertical bar of length 0.3 in the frequency-domain representation on the right. Now it is easy to imagine that one could take a pair of scissors and snip out that unwanted part of the composite signal in the frequency domain, before reassembling everything as the ‘cleaned’ signal in the time domain.

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Figure 31  Noise with frequency f = 935 Hz shows clearly in the frequency domain

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This is a screenshot from Interactive 4 showing the same graphs as Figure 30, but this time a ‘noise’ wave has been added. This is shown in the frequency domain as a vertical line at 935 hertz that is between the first two lines (110 and 330 hertz) in height. On the left, the wave has been disrupted from its almost square appearance and now has lots of large high-frequency peaks and troughs. This time when the loudspeaker button is pressed, the high-frequency ‘noise’ is heard as background to the harmonious square wave.

Figure 31  Noise with frequency f = 935 Hz shows clearly in the frequency domain

Apart from removing unwanted noise, Fourier’s theory can also be used to manipulate signals in the frequency domain to give interesting new aesthetic results when the signals in the time domain are played through a loudspeaker. This has stimulated research and technology that has been widely used in the music industry for more than half a century, since the invention of the electric guitar.