David Howard and Jamie Angus - April 09, 2008
1.6 Time and frequency domains
So far we have mainly considered a sound wave to be a sinusoidal wave at a particular frequency. This is useful as it allows us to consider aspects of sound propagation in terms of the wavelength.
However, most musical sounds have a waveform which is more complex than a simple sine wave and a selection is shown in Figure 1.45. How can we analyse real sound waveforms, and make sense of them in acoustical terms? The answer is based on the concept of superposition and a technique called Fourier analysis.
Figure 1.45 Waveforms from musical instruments.
1.6.1 The spectrum of periodic sound waves
Fourier analysis states that any waveform can be built up by using an appropriate set of sine waves of different frequencies, amplitudes and phases. To see how this might work consider the situation shown in Figure 1.46.
Figure 1.46 The effect of adding several harmonically related sine waves together.
This shows four sine waves whose frequencies are 1F Hz, 3F Hz, 5F Hz, and 7F Hz, whose phase is zero (that is, they all start from the same value, as shown by the dotted line in Figure 1.46) and whose amplitude is inversely proportional to the frequency. This means that the 3F Hz component is 1/3 the amplitude of the component at 1F Hz and so on. When these sine waves are added together, as shown in Figure 1.46, the result approximates a square wave, and, if more high frequency components were added, it would become progressively closer to an ideal square wave.
The higher frequency components are needed in order to provide the fast rise, and sharp corners, of the square wave. In general, as the rise time gets faster, and/or the corners get sharper, then more high frequency sine waves are required to represent the waveform accurately. In other words we can look at a square wave as a waveform that is formed by summing together sine waves which are odd multiples of its fundamental frequency and whose amplitudes are inversely proportional to frequency.
The spectrum of periodic sound waves (cont.)
Figure 1.47 The frequency domain representation, or spectrum, of a square wave.
This representation is called the frequency domain representation, or spectrum, of a waveform and the waveform's amplitude versus time plot is called its time domain representation.The individual sine wave components of the waveform are often called the partials of the waveform. If they are integer related, as in the square wave, then they can be called harmonics. The lowest frequency is called the fundamental, or first harmonic, and the higher frequency harmonics are labelled according to their frequency multiple relative to the fundamental. Thus the second harmonic is twice the frequency of the fundamental and so on.
Partials on the other hand need not be harmonically related to the fundamental, and are numbered in their order of appearance with frequency. However, as we shall see later, this results in a waveform that is aperiodic.
So for the square wave the second partial is the third harmonic and the third partial is the fifth harmonic. Other waveforms have different frequency domain representations, because they are made up of sine waves of different amplitudes and frequencies. Some examples of other waveforms in both the time and frequency domains are shown in Chapter 3.
1.6.2 The effect of phase
The phase, which expresses the starting value of the individual sine wave components, also affects the waveshape. Figure 1.48 shows what happens to a square wave if alternate partials are subtracted rather than added, and this is equivalent to changing the phase of these components by 180°. That is, alternate frequency components start from halfway around the circle compared with the other components, as shown by the dotted line in Figure 1.48. However, although the time domain waveform is radically different the frequency domain is very similar, as the amplitudes are identical, only the phase of some of the harmonics have changed.
Figure 1.48 The effect of adding harmonically related sine waves together with different phase shifts.
Interestingly, in many cases, the resulting wave is perceived as sounding the same, even though the waveform is different. This is because the ear, as we will see later, appears to be less sensitive to the phase of the individual frequency compared to the relative amplitudes.
However, if the phase changes are extreme enough we can hear a difference (see Schroeder 1975). Because of this, often only the amplitude of the frequency components are plotted in the spectrum and, in order to handle the range of possible amplitudes and because of the way we perceive sound, the amplitudes are usually plotted as decibels. For example, Figure 4.24 in Chapter 4 shows the waveform and spectrum plotted in this fashion for middle C played on a clarinet and tenor saxophone.
The spectrum of non-periodic sound waves
1.6.3 The spectrum of non-periodic sound waves
So far, only the spectrum of waveforms which are periodic, that is, have pitch, has been considered. However, some instruments, especially percussion, do not have pitch and hence are non-periodic, or aperiodic. How can we analyse these instruments in terms of a basic waveform, such as a sine wave, which is inherently periodic? The answer is shown in Figure 1.49.
Figure 1.49 The effect of adding several non-harmonically related sine waves together.
Here the square wave example discussed earlier has had four more sine waves added to it. However, these sine waves are between the harmonics of the square wave and so are unrelated to the period, but they do start off in phase with the harmonics. The effect of these other components is to start cancelling out the repeat periods of the square waves, because they are not related in frequency to them.
By adding more components which sit in between the harmonics, this cancellation of the repeats becomes more effective so that when in the limit, the whole space between the harmonics is filled with sine wave components of the appropriate amplitude and phase. These extra components will add constructively only at the beginning of the waveform and will interfere with successive cycles due to their different frequencies. Therefore, in this case, only one square wave will exist.
Thus the main difference between the spectrum of periodic and aperiodic waveforms is that periodic waveforms have discrete partials, which can be represented as lines in the spectrum with a spacing which is inversely proportional to the period of the waveform. Aperiodic waveforms by contrast will have a spectrum which is continuous and therefore does not have any discrete components. However, the envelope of the component amplitudes as a function of frequency will be the same for both periodic and aperiodic waves of the same shape, as shown in Figure 1.50. Figure 3.6 in Chapter 3 shows the aperiodic waveform and spectrum of a brushed snare.
Figure 1.50 The frequency domain representation, or spectrum, of an aperiodic square wave.
Coming up in Part 8: Analyzing spectra.
Printed with permission from Focal Press, a division of Elsevier. Copyright 2006. "Acoustics and Psychoacoustics" by David Howard and Jamie Angus. For more information about this title, please visit www.focalpress.com.
Acoustics and Psychoacoustics: Introduction to Sound, Part 1: Pressure waves and sound transmission | Part 2: Sound intensity, power and pressure level | Part 3: Adding sounds together | Part 4: The inverse square law | Part 5: Sound Interactions | Part 6: Sound Interactions (cont.)
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