2. Spectrum 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-digital converter samples time-varying Fourier Analysis amplitudes and phases Analyze the soundAnalyze the sound Additive Synthesis resynthesized sound Resynthesize the soundResynthesize the sound Play a musical selection demonstrating the instrument designPlay a musical selection demonstrating the instrument design Prepare the soundPrepare the sound

3. Spectrum Analysis soundfile.wav PC.wav-format soundfile pvan.exe soundfile.pvn interactive program for spectrum analysis analysis file with amplitudes and frequencies pvan.exe graphs of spectra interactive program for spectrum display

4. Synthetic Trumpet Real musical instruments produce almost-harmonic soundsReal musical instruments produce almost-harmonic sounds The waveform of this synthetic trumpet repeats more exactly than that of a real instrumentThe waveform of this synthetic trumpet repeats more exactly than that of a real instrument

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

6. Spectrum of a Sound Example: amp1 = 1, amp2 =.5, and amp3 =.25, the spectrum = {1,.5,.25}.Example: amp1 = 1, amp2 =.5, and amp3 =.25, the spectrum = {1,.5,.25}. The following graphs show the usual ways to represent the spectrum:The following graphs show the usual ways to represent the spectrum: FrequencyHarmonic Number

7. 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.These steps together are called spectrum analysis.

8. Time-Varying Fourier Analysis User specifies the fundamental frequency for ONE toneUser specifies the fundamental frequency for ONE tone Automatically finding the fundamental frequency is called pitch tracking — a current research problemAutomatically finding the fundamental frequency is called pitch tracking — a current research problem For example, for middle C:For example, for middle C: f 1 =261.6 sound time-varying Fourier Analysis Fourier Coefficients Math amplitudes and phases

9. Time-Varying Fourier Analysis Construct a window function that spans two periods of the waveform.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.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:Except for the Rectangular window, most look like half a period of a sine wave:

10. Time-Varying Fourier Analysis The window function isolates the samples of two periods so we can find the spectrum of the sound.The window function isolates the samples of two periods so we can find the spectrum of the sound.

11. Time-Varying Fourier Analysis The window function will smooth samples at the window endpoints to correct the inaccurate user- specified fundamental frequency.The window function will smooth samples at the window endpoints to correct the inaccurate user- specified fundamental frequency. For example, if the user estimates f 1 =261.6, but it really is 259 Hz.For example, if the user estimates f 1 =261.6, but it really is 259 Hz.

12. Time-Varying Fourier Analysis Samples are only non-zero in windowed region, and windowed samples are zero at endpoints.Samples are only non-zero in windowed region, and windowed samples are zero at endpoints.

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

14. Spectrum Analysis We analyze the tone (using the Fourier transform) to find out the strength of the harmonic partialsWe 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 toneHere is a snapshot of a [i:37] trumpet tone one second after the start of the tone

15. Trumpet's First Harmonic The trumpet's first harmonic fades in and out as shown in this amplitude envelope:The trumpet's first harmonic fades in and out as shown in this amplitude envelope:

16. Spectral Plot of Trumpet's First 20 Harmonics

17. Spectra of Other Instruments [i:38] English horn:[i:38] English horn: pitch is E3, 164.8 Hertz

18. Spectra of Other Instruments [i:39] tenor voice:[i:39] tenor voice: pitch is G3, 192 Hertz

19. Spectra of Other Instruments [i:40] guitar:[i:40] guitar: pitch is A2, 110 Hertz

20. Spectra of Other Instruments [i:41] pipa:[i:41] pipa: pitch is G2, 98 Hertz

21. Spectra of Other Instruments [i:42] cello:[i:42] cello: pitch is Ab3, 208 Hertz

22. Spectra of Other Instruments [i:43] E-mu's synthesized cello:[i:43] E-mu's synthesized cello: pitch is G2, 98 Hertz