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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 Spectrum Analysis • 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 Spectrum Analysis soundfile.wav PC.wav-format soundfile**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**• 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 Time-Varying Fourier Analysis 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**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 user-specified 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:**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