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Spectrum Analysis and PVan

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Spectrum Analysis

and PVan

analog-to-digital

converter

samples

time-varying

Fourier Analysis

- Analyze the sound

amplitudes and phases

- Resynthesize the sound

Additive Synthesis

resynthesized sound

recorded sound

- Sound Analysis
- What are we going to do?
- Record a sound

- Prepare the sound

- Play a musical selection demonstrating the instrument design

pvan.exe

interactive program

for spectrum analysis

analysis file with

amplitudes and frequencies

soundfile.pvn

interactive program

for spectrum display

pvan.exe

graphs of

spectra

soundfile.wav

PC.wav-format

soundfile

- Real musical instruments produce almost-harmonic sounds
- The waveform of this synthetic trumpet repeats more exactly than that of a real instrument

- For any periodic waveform, we can find the spectrum of the waveform.
- The spectrum is the relative amplitudes of the harmonics that make up the waveform.
- The plural form of the word "spectrum" is "spectra."

- Example: amp1 = 1, amp2 = .5, and amp3 = .25, the spectrum = {1, .5, .25}.
- The following graphs show the usual ways to represent the spectrum:

Frequency

Harmonic Number

- isolate one period of the waveform
- Discrete Fourier Transform of the period.
- These steps together are called spectrum analysis.

time-varying

Fourier Analysis

Fourier Coefficients

Math

amplitudes

and phases

sound

- User specifies the fundamental frequency for ONE tone
- Automatically finding the fundamental frequency is called pitch tracking — a current research problem
- For example, for middle C:

f1=261.6

- Construct a window function that spans two periods of the waveform.
- The most commonly used windows are called Rectangular (basically no window), Hamming, Hanning, Kaiser and Blackman.

- Except for the Rectangular window, most look like half a period of a sine wave:

- The window function isolates the samples of two periods so we can find the spectrum of the sound.

- The window function will smooth samples at the window endpoints to correct the inaccurate user-specified fundamental frequency.
- For example, if the user estimates f1=261.6, but it really is 259 Hz.

- Samples are only non-zero in windowed region, and windowed samples are zero at endpoints.

- Apply window and Fourier Transform to successive blocks of windowed samples.
- Slide blocks one period each time.

- We analyze the tone (using the Fourier transform) to find out the strength of the harmonic partials
- Here is a snapshot of a [i:37] trumpet tone one second after the start of the tone

- The trumpet's first harmonic fades in and out as shown in this amplitude envelope:

- [i:38] English horn:

pitch is E3, 164.8 Hertz

- [i:39] tenor voice:

pitch is G3, 192 Hertz

- [i:40] guitar:

pitch is A2, 110 Hertz

- [i:41] pipa:

pitch is G2, 98 Hertz

- [i:42] cello:

pitch is Ab3, 208 Hertz

- [i:43] E-mu's synthesized cello:

pitch is G2, 98 Hertz