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exercises

Up to now we have dealt with simple periodic sound waves, as sine
tones are. However, sine tones are not the signals we
encounter
more frequently in our everyday life (on the contrary, we hardly ever
do). Actually, a tuning fork produces a
sinusoidal sound wave, but we normally
experience a wide range of different kinds of sounds, both periodic and
aperiodic, which do not have a sinusoidal sound wave.

Periodic signals show (by graphically reproducing them, for instance by means of an oscillogram) a repeating pattern in their waveform: the period (T). Besides sine tones, also the so called complex signals are periodic. Instead, aperiodic signals are signals such as noise and impulse (i.e. the noise of a door being slammed), whose wave form is not characterised by any repeating pattern.

Complex signals result from the addition of several signals. It was just mentioned introducing the concept of phase that sound signals can be combined and that the obtained signal results from adding amplitude, frequency and phase of the signal sound waves. A Fourier analysis (named after Jean Baptiste Joseph Fourier) decomposes any periodic complex (non sinusoidal) signal into its sine components (or Fourier components), each having a given frequency, amplitude and phase.

The figure below illustrates the oscillogram of a complex sound signal (at the bottom) and the three sines that make it up. As you can see, the period of the resulting complex wave corresponds to the smallest common multiple of the sines' periods (and the frequency corresponds to their maximum common denominator of their frequencies): as a matter of fact, once every period (in this case once every 0,01 s), the three sines have the same phase position.

Periodic signals show (by graphically reproducing them, for instance by means of an oscillogram) a repeating pattern in their waveform: the period (T). Besides sine tones, also the so called complex signals are periodic. Instead, aperiodic signals are signals such as noise and impulse (i.e. the noise of a door being slammed), whose wave form is not characterised by any repeating pattern.

Complex signals result from the addition of several signals. It was just mentioned introducing the concept of phase that sound signals can be combined and that the obtained signal results from adding amplitude, frequency and phase of the signal sound waves. A Fourier analysis (named after Jean Baptiste Joseph Fourier) decomposes any periodic complex (non sinusoidal) signal into its sine components (or Fourier components), each having a given frequency, amplitude and phase.

The figure below illustrates the oscillogram of a complex sound signal (at the bottom) and the three sines that make it up. As you can see, the period of the resulting complex wave corresponds to the smallest common multiple of the sines' periods (and the frequency corresponds to their maximum common denominator of their frequencies): as a matter of fact, once every period (in this case once every 0,01 s), the three sines have the same phase position.

Being the combination of single signals, every complex signal has different frequency components, namely those of the sines composing it (even though we will soon seen that this assertion is not completely correct). The lowest of those frequency components is called fundamental frequency (F0), namely what we just called the minumum common denominator of the single frequencies of the sines. For instance, the F0 of a complex signal composed by sine tones with, respectively, a 50, 100 and 150 Hz is 50 Hz. However, if two sines of 200 and 300 Hz combine, we get a 100 Hz F0 in the complex signal, which is not present in the original sines.

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