The addition operation: spectrum synthesis

The frequency components of a complex signal may be approximated as a series of simple sinusoids whose values are added together for each point in time. The restriction `approximation' is needed in practice; the mathematical basis of this operation is precise, but unfortunately requires consideration of signals of infinite length, and an infinite series of such signals. In practice, limits have to be defined, and therefore an `approximation' is required.

Addition of signal values for each point in time is the operation by which complex signals are constructed from simple sinusoids, as shown in the Figures, which show:

1. Harmonically related sinusoids (the fundamental frequency and the first two harmonics).
2. An approximation to a sawtooth wave (the sum of the harmonics, here the first six); the more harmonics are involved, the sharper the peak and the flatter the flanks of the wave. The sawtooth waveform is particurly important in speech signal processing, as it approximates to the shape of the signal produced by the source of the speech signal in phonation.
3. A square wave, composed of the sum of odd harmonics (here the first three odd multiples of the fundamental frequency). The more odd harmonics are involved, the closer the corners are to right angles, and the flatter the flanks and the top. The signals produced by rapid switching and signal editing may approximate to square waves: the sharper the cut-off flank, the higher the frequency components which are introduced by the switching or editing operation as `spurious signals'. For this reason, editing is usually performed at points of zero amplitude in the signal, zero crossings.

In a signal whose components are all harmonically related, the frequencies of the harmonics are integer multiples of the fundamental frequency, , and are counted by these integers: the third harmonic has the frequency , for example. This entails that the fundamental frequency, , equals the difference between the frequency values of neighbouring harmonics. This relation is also used in transformations which determine the fundamental frequency from a complex signal if it can be assumed to have harmonic structure.

  
Figure 5: Sine wave (frequency = F, amplitude = A).

  
Figure 6: Sine wave (frequency = 2F, amplitude = A/2).

  
Figure 7: Sine wave (frequency = 3F, amplitude = A/3).

  
Figure 8: Sawtooth: sum of harmonics.

  
Figure 9: Square wave: sum of odd harmonics.

If you are curious about what happens when signals are subtracted, the answer is quite simple: the subtraction operation is equivalent to simply changing the phase of the subtrahend by and adding.