180 likes | 336 Vues
Pre-Class Music. Paul Koonce Whitewash (1992). Intro to Spectral Processing. Representing Audio Data. Time Domain: Changing Amplitude over Time Frequency Domain: Amplitude over Frequency. Amp. Time. Amp. Frequency. Domains. The domain is always the x-axis.
E N D
Pre-Class Music • Paul KoonceWhitewash (1992)
Representing Audio Data • Time Domain: Changing Amplitude over Time • Frequency Domain: Amplitude over Frequency Amp Time Amp Frequency
Domains • The domain is always the x-axis. • The value you are graphing (representing, etc.) is always the y-axis.
What’s Missing? • Time Domain • Frequency • Frequency Domain • Time
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
Reading Assignment • Roads, pp. 536 - 563, particular attention to Spectrum Analysis, starting on p. 545.
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.
Old Ways of Calculating • 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.
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) imposes a sequence of time windows.
The Fast Fourier Transform • The FFT relies on a mathematical trick: • A signal that is a power of 2 length 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 usually a STFT, meaning windows of power of 2 size (in samples) are applied to the waveform to be analyzed.
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) • Each collection of amplitude/phase pairs for a sampled time window is a frame, analogous to frames in a film.
Analysis Frequencies • Simplest form, think of the FFT as a bank of filters equally spaced from 0 Hz to the SR, at integer multiples ofSR / Nwhere N is the size of the analyzed time window. • Only half of the filters (amplitude/phase pairs) are usable, due to aliasing. (More later)
Windowing and Overlaps • Windows provide some temporal specificity to frequency analysis (when something happened) • Overlapping helps to capture the signal without gaps, like overlapping grains in granular synthesis helps smooth out the synthesized sound. • Other reasons later.
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.
Do Some Math • Time / Frequency Trade-off
Reading • Roads: pp. 536 - 577