8A - 8A. ANALYSIS OF COMPLEX SOUNDS Amplitude, loudness,...

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P108 Lab 8a: page 1 8A. ANALYSIS OF COMPLEX SOUNDS Amplitude, loudness, and decibels Last week 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 an analog Band Filter, we will use the P RAAT program, which has powerful analysis software built in to calculate and graph the spectrum of a recorded sound. For the last two 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): Table 1: The spectrum of a square wave Error Error Error Error 440 880 1320 1760 2200 2640 3080 3520 3960 123456789 2.000 0.000 0.667 0.000 0.400 0.000 0.286 0.000 0.222 1.000 0.000 0.333 0.000 0.200 0.000 0.143 0.000 0.111 1.0000 0.0000 0.1111 0.0000 0.0400 0.0000 0.0204 0.0000 0.0123 0.0 -9.5 -14.0 -16.9 -19.1 Harmonic frequency (Hz) Harmonic number=fn/f1 Amplitude (Volt) Relative amplitude=An/A1 Relative Power=(An/A1)^2 Relative Power (dB) Another way to represent the relative amplitudes, which is used by P RAAT , 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) = P n = 10 log 10 [ ( A n /A 1 ) 2 ] = 20 log 10 [ ( A n /A 1 ) ] Since the logarithm of zero is negative infinity, the Relative Power (dB) for the even harmonics is "Error".
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P108 Lab 8a: page 2 We can display the harmonic power spectrum as a bar graph: Relative power (dB) 1 2 3 4 5 6 7 8 9 -40 -30 -20 -10 0 Harmonic number ( f n / f 1 ) Note that the bars for the even harmonics are missing since these harmonic are "missing" from a square wave. Analysis of a Triangle Wave A. Predictions based on known relative amplitudes Try to make a table similar to Table 1 for the triangle wave you synthesized several weeks ago. Assume the amplitude of the fundamental wave to be 2V. Remember that the even harmonics had zero amplitude and the amplitudes for the odd harmonics were given by: A n = A 1 /n 2 . Table 2: The spectrum of a 440 Hz triangle wave: 440 880 1320 1760 2200 2640 3080 3520 3960 123456789 Harmonic frequency (Hz) Harmonic number=fn/f1 Amplitude (Volt) Relative amplitude=An/A1 Relative Power=(An/A1)^2 Relative Power (dB)
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P108 Lab 8a: page 3 Now graph your results for the relative power in dB for each harmonic. Note that the fundamental is 0 dB by definition: Relative power (dB) 1 2 3 4 5 6 7 8 9 -40 -30 -20 -10 0 Harmonic number ( f n / f 1 ) B. "Warm up": Analysis of a measured sine wave Before using P RAAT
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This note was uploaded on 01/20/2012 for the course PHYSICS 108 taught by Professor Kesmodel during the Fall '08 term at Indiana.

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8A - 8A. ANALYSIS OF COMPLEX SOUNDS Amplitude, loudness,...

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