Spectrum AnalysisSpectrum Analysis

• Sound AnalysisSound Analysis• What are we going to do?What are we going to do?

• Record a soundRecord a sound

recorded sound

analog-to-digitalconverter

samples

time-varyingFourier Analysis

amplitudes and phases• Analyze the soundAnalyze the sound

Additive Synthesis

resynthesized sound

• Resynthesize the soundResynthesize the sound• Play a musical selection Play a musical selection

demonstrating the instrument demonstrating the instrument designdesign

• Prepare the soundPrepare the sound

Spectrum AnalysisSpectrum Analysissoundfile.wav PC.wav-format

soundfile

pvan.exe

soundfile.pvn

interactive programfor spectrum analysis

analysis file withamplitudes and frequencies

pvan.exe

graphs ofspectra

interactive programfor spectrum display

Synthetic TrumpetSynthetic Trumpet• Real musical instruments produce Real musical instruments produce

almost-harmonic soundsalmost-harmonic sounds• The waveform of this synthetic trumpet The waveform of this synthetic trumpet

repeats more exactly than that of a real repeats more exactly than that of a real instrument instrument

Spectrum of a SoundSpectrum of a Sound• For any periodic waveform, we can find the For any periodic waveform, we can find the

spectrum of the waveform.spectrum of the waveform.• The spectrum is the relative amplitudes of The spectrum is the relative amplitudes of

the harmonics that make up the waveform.the harmonics that make up the waveform.• The plural form of the word "spectrum" is The plural form of the word "spectrum" is

"spectra.""spectra."

Spectrum of a SoundSpectrum of a Sound• Example: amp1 = 1, amp2 = .5, and amp3 = .25, Example: amp1 = 1, amp2 = .5, and amp3 = .25,

the spectrum = {1, .5, .25}.the spectrum = {1, .5, .25}.• The following graphs show the usual ways to The following graphs show the usual ways to

represent the spectrum:represent the spectrum:

Frequency Harmonic Number

Finding the Spectrum of a SoundFinding the Spectrum of a Sound

1.1. isolate one period of the waveformisolate one period of the waveform

2.2. Discrete Fourier Transform of the Discrete Fourier Transform of the period.period.

• These steps together are called These steps together are called spectrum analysis.spectrum analysis.

Time-Varying Fourier AnalysisTime-Varying Fourier Analysis

• User specifies the User specifies the fundamental frequency for fundamental frequency for ONE toneONE tone• Automatically finding the Automatically finding the

fundamental frequency is fundamental frequency is called pitch tracking — a called pitch tracking — a current research problemcurrent research problem

• For example, for middle C:For example, for middle C:

ff11=261.6=261.6

sound

time-varyingFourier Analysis

Fourier Coefficients

Math

amplitudesand phases

Time-Varying Fourier AnalysisTime-Varying Fourier Analysis• Construct a window function that spans two Construct a window function that spans two

periods of the waveform.periods of the waveform.• The most commonly used windows are called The most commonly used windows are called

Rectangular (basically no window), Hamming, Rectangular (basically no window), Hamming, Hanning, Kaiser and Blackman.Hanning, Kaiser and Blackman.

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

Time-Varying Fourier AnalysisTime-Varying Fourier Analysis• The window function isolates the samples of The window function isolates the samples of

two periods so we can find the spectrum of two periods so we can find the spectrum of the sound.the sound.

Time-Varying Fourier AnalysisTime-Varying Fourier Analysis• The window function will smooth samples at the The window function will smooth samples at the

window endpoints to correct the inaccurate user-window endpoints to correct the inaccurate user-specified fundamental frequency.specified fundamental frequency.• For example, if the user estimates fFor example, if the user estimates f11=261.6, but it really is =261.6, but it really is

259 Hz.259 Hz.

Time-Varying Fourier AnalysisTime-Varying Fourier Analysis• Samples are only non-zero in windowed Samples are only non-zero in windowed

region, and windowed samples are zero at region, and windowed samples are zero at endpoints.endpoints.

Time-Varying Fourier AnalysisTime-Varying Fourier Analysis• Apply window and Fourier Transform to Apply window and Fourier Transform to

successive blocks of windowed samples.successive blocks of windowed samples.• Slide blocks one period each time.Slide blocks one period each time.

Spectrum AnalysisSpectrum Analysis• We analyze the tone (using the Fourier transform) We analyze the tone (using the Fourier transform)

to find out the strength of the harmonic partialsto find out the strength of the harmonic partials• Here is a snapshot of a Here is a snapshot of a [i:37][i:37] trumpet tone one trumpet tone one

second after the start of the tonesecond after the start of the tone

Trumpet's First HarmonicTrumpet's First Harmonic• The trumpet's first harmonic fades in and out as The trumpet's first harmonic fades in and out as

shown in this amplitude envelope:shown in this amplitude envelope:

Spectral Plot of Trumpet's First Spectral Plot of Trumpet's First 20 Harmonics20 Harmonics

Spectra of Other InstrumentsSpectra of Other Instruments• [i:38][i:38] English horn: English horn:

pitch is E3, 164.8 Hertz

Spectra of Other InstrumentsSpectra of Other Instruments• [i:39][i:39] tenor voice: tenor voice:

pitch is G3, 192 Hertz

Spectra of Other InstrumentsSpectra of Other Instruments• [i:40][i:40] guitar: guitar:

pitch is A2, 110 Hertz

Spectra of Other InstrumentsSpectra of Other Instruments• [i:41][i:41] pipa: pipa:

pitch is G2, 98 Hertz

Spectra of Other InstrumentsSpectra of Other Instruments• [i:42][i:42] cello: cello:

pitch is Ab3, 208 Hertz

Spectra of Other InstrumentsSpectra of Other Instruments• [i:43][i:43] E-mu's synthesized cello: E-mu's synthesized cello:

pitch is G2, 98 Hertz