recorded sound
Spectrum Analysis
• Sound Analysis
• What are we going to do?
• Record a sound
• Prepare the sound
• Analyze the sound
• Resynthesize the sound
• Play a musical selection
demonstrating the instrument
design
analog-to-digital
converter
samples
time-varying
Fourier Analysis
amplitudes and phases
Additive Synthesis
resynthesized sound
Spectrum Analysis
soundfile.wav
PC.wav-format
soundfile
pvan.exe
interactive program
for spectrum analysis
soundfile.pvn
analysis file with
amplitudes and frequencies
pvan.exe
interactive program
for spectrum display
graphs of
spectra
Synthetic Trumpet
• Real musical instruments produce
almost-harmonic sounds
• The waveform of this synthetic trumpet
repeats more exactly than that of a real
instrument
Spectrum of a Sound
• 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."
Spectrum of a Sound
• 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
Finding the Spectrum of a Sound
1. isolate one period of the waveform
2. Discrete Fourier Transform of the
period.
• These steps together are called
spectrum analysis.
Time-Varying Fourier Analysis
• User specifies the
fundamental frequency for
ONE tone
sound
time-varying
Fourier Analysis
• Automatically finding the
fundamental frequency is
called pitch tracking — a
current research problem
• For example, for middle C:
Fourier Coefficients
f1=261.6
amplitudes
and phases
Math
Time-Varying Fourier Analysis
• 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:
Time-Varying Fourier Analysis
• The window function isolates the samples of
two periods so we can find the spectrum of
the sound.
Time-Varying Fourier Analysis
• The window function will smooth samples at the
window endpoints to correct the inaccurate userspecified fundamental frequency.
• For example, if the user estimates f1=261.6, but it really is
259 Hz.
Time-Varying Fourier Analysis
• Samples are only non-zero in windowed
region, and windowed samples are zero at
endpoints.
Time-Varying Fourier Analysis
• Apply window and Fourier Transform to
successive blocks of windowed samples.
• Slide blocks one period each time.
Spectrum Analysis
• 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
Trumpet's First Harmonic
• The trumpet's first harmonic fades in and out as
shown in this amplitude envelope:
Spectral Plot of Trumpet's First
20 Harmonics
Spectra of Other Instruments
• [i:38] English horn:
pitch is E3, 164.8 Hertz
Spectra of Other Instruments
• [i:39] tenor voice:
pitch is G3, 192 Hertz
Spectra of Other Instruments
• [i:40] guitar:
pitch is A2, 110 Hertz
Spectra of Other Instruments
• [i:41] pipa:
pitch is G2, 98 Hertz
Spectra of Other Instruments
• [i:42] cello:
pitch is Ab3, 208 Hertz
Spectra of Other Instruments
• [i:43] E-mu's synthesized cello:
pitch is G2, 98 Hertz