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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. Spectrum Analysis. Sound Analysis What are we going to do?

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Spectrum Analysis and PVan

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Spectrum analysis and pvan

Spectrum Analysis

and PVan

Spectrum analysis





Fourier Analysis

  • Analyze the sound

amplitudes and phases

  • Resynthesize the sound

Additive Synthesis

resynthesized sound

recorded sound

Spectrum Analysis

  • Sound Analysis

  • What are we going to do?

    • Record a sound

  • Prepare the sound

  • Play a musical selection demonstrating the instrument design

Spectrum analysis1


interactive program

for spectrum analysis

analysis file with

amplitudes and frequencies


interactive program

for spectrum display


graphs of


Spectrum Analysis




Synthetic trumpet

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

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 sound1

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:


Harmonic Number

Finding the spectrum of a sound

Finding the Spectrum of a Sound

  • isolate one period of the waveform

  • Discrete Fourier Transform of the period.

  • These steps together are called spectrum analysis.

Time varying fourier analysis


Fourier Analysis

Fourier Coefficients



and phases

Time-Varying Fourier Analysis


  • 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:


Time varying fourier analysis1

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 analysis2

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 analysis3

Time-Varying Fourier Analysis

  • 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.

Time varying fourier analysis4

Time-Varying Fourier Analysis

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

Time varying fourier analysis5

Time-Varying Fourier Analysis

  • Apply window and Fourier Transform to successive blocks of windowed samples.

    • Slide blocks one period each time.

Spectrum analysis2

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

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

Spectral Plot of Trumpet's First 20 Harmonics

Spectra of other instruments

Spectra of Other Instruments

  • [i:38] English horn:

pitch is E3, 164.8 Hertz

Spectra of other instruments1

Spectra of Other Instruments

  • [i:39] tenor voice:

pitch is G3, 192 Hertz

Spectra of other instruments2

Spectra of Other Instruments

  • [i:40] guitar:

pitch is A2, 110 Hertz

Spectra of other instruments3

Spectra of Other Instruments

  • [i:41] pipa:

pitch is G2, 98 Hertz

Spectra of other instruments4

Spectra of Other Instruments

  • [i:42] cello:

pitch is Ab3, 208 Hertz

Spectra of other instruments5

Spectra of Other Instruments

  • [i:43] E-mu's synthesized cello:

pitch is G2, 98 Hertz

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