10: Fourier Analysis- 110:FOURIER ANALYSIS OFCOMPLEX SOUNDSAmplitude, loudness, and decibels Several weeks ago we found that we could synthesize complex sounds with a particular frequency, f,by adding together sine waves from the harmonic series with a fundamental equal to the frequency (f, 2f, 3f, 4f,…).We can reverse the process: a complex sound with particular frequency can be analyzed and quantified by the amplitude spectrum: the relative amplitudes of the harmonics. Our goal today is to understand how the pitch, loudness, and timbre of a sound are represented in a spectrum. However, rather than use analog Band Filter, like last week, we will use thePRAATprogram, which has powerful analysis software built in to calculate and graph the spectrum of a recorded sound.For the last couple of weeks we have used the amplitudes to represent the different waves. We can represent these in a table (the choice for the amplitude of the fundamental wave to be 2V is completely arbitrary): Another way to represent the relative amplitudes, which is used byPRAAT, is to graph the power for each harmonic. Since the sound power is proportional to the square of the amplitude for each harmonic, these numbers often get quite small for the higher harmonics. For this reason, the power is often expressed as a decibel. The power for each harmonic in decibels(dB) is:Relative Power(dB) = Pn= 10 log10[ (An/A1)2] = 20 log10[ (An/A1)] Since the logarithm of zero is negative infinity, the Relative Power (dB) for the even harmonics is "Error".
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