1. Pre-Class Music Paul Lansky Six Fantasies on a Poem by Thomas Campion

2. Intro to Spectral Processing

3. Representing Audio Data Time Domain: Changing Amplitude over Time Frequency Domain: Amplitude over Frequency Amp Time Amp Frequency

4. What’s Missing? Time Domain  Frequency Frequency Domain  Time

5. Converting Domains Fourier Transform  Converts a time-domain representation into a frequency domain representation Inverse Fourier Transform  Converts a frequency-domain representation into a time domain representation

6. Reading Assignment Roads, pp. 536 - 563, particular attention to Spectrum Analysis, starting on p. 545.

7. Fourier Analysis

8. Background Theory developed in 1822 by Jean Baptiste Joseph, Baron de Fourier Any arbitrary periodic signal can be represented as a sum of many simultaneous sine waves.  a periodic signal repeats at regular intervals of time Any arbitrary periodic waveform can be deconstructed into combinations of simple sine waves of different amplitudes, frequencies, and phases. The idea was so controversial, and attacked so severely, it wasn’t published for 165 years.

9. Old Ways of Calculating By hand (measurement of curves and area by hand) Mechanical Springs (1870) Analog Filter Banks (1930) Computer Analysis (1940) (Discrete Fourier Transform, DFT) Fast Fourier Transform (FFT) developed in 1960s greatly reduced number of calculations needed, and made the process practical to use.

10. Working towards the FFT The Fourier Transform (FT) is applied to a continuous (analog) waveform The Discrete Fourier Transform (DFT) is applied to a digital signal (series of samples) The Short Time Fourier Transform (STFT) divides a digital signal into a sequence of overlapping time windows.

11. The Fast Fourier Transform The FFT relies on a mathematical trick:  Any signal that is a power-of-2 in length (samples) can be analyzed as a whole, in halves, in fourths, in eighths, etc., until you reach one sample in length.  As complicated as this sounds, it takes far fewer calculations than a DFT. The FFT is a common form of a STFT, meaning windows of power-of-2 size (in samples) are applied to the digital waveform to be analyzed.  If needed, zero amplitude values can be added to the end of the final window for signals that are not of power- of-2 size.

12. What the FFT does Measures energy at specific equally spaced frequencies. For each frequency, you get  Amplitude (or magnitude)  Phase  (not as straightforward as what you might hope for - x, y coordinates, which usually require conversion to a polar format for processing.) Each amplitude and phase pair (or x, y coordinates) constitute a bin. The collection of all the bins for an entire window of time is a frame.  You can think of FFT frames analogously with motion picture frames. Both represent static data that when played back (converted) at the proper speed can recreate, or simulate, the original motion-based phenomenon.

13. Analysis Frequencies In its simplest conceptual form, think of the FFT as a bank of filters equally spaced from 0 Hz to the SR, at integer multiples of SR / N where N is the size of the analyzed time window. Only half of the filters (amplitude/phase pairs) are usable, due to aliasing. (More later)

14. Windowing and Overlaps Windows provide some temporal specificity to frequency analysis. With appropriate amplitude envelopes, windows eliminate discontinuities between the beginning and end of the sample window, which could produce spurious analysis results. Overlapping helps to capture the signal without gaps, like overlapping grains in granular synthesis helps smooth out the synthesized sound. Other reasons later.

15. Problems with FFT (FT) Periodicity implies infinity, that a sound exists forever, without beginning or end, without any changes. Time/Frequency Uncertainty (Trade-off) (Quantum Physics)  high resolution in time domain sacrifices resolution in the frequency domain  high resolution in frequency domain sacrifices resolution in time domain.

16. Time / Frequency Trade-off Do some math. Assuming a 44,100 Hz Sampling Rate:  A 512-sample window gives frequency bands spaced approximately 86 Hz (44,100/512), and a temporal resolution of approximately 11.5 ms (512/44100).  Using a 2048-sample window increases the frequency resolution by a factor of 4 to approx. 21.5 Hz, but decreases the temporal resolution by the same factor to approx. 46.5 ms.