Pre-Class MusicPre-Class Music

• Paul Koonce

Whitewash (1992)

• Paul Koonce

Whitewash (1992)

Intro to Spectral Processing

Intro to Spectral Processing

Representing Audio DataRepresenting Audio Data• Time Domain: Changing Amplitude over Time

• Frequency Domain: Amplitude over Frequency

• Time Domain: Changing Amplitude over Time

• Frequency Domain: Amplitude over Frequency

Amp

Time

Amp

Frequency

DomainsDomains

• The domain is always the x-axis.

• The value you are graphing (representing, etc.) is always the y-axis.

• The domain is always the x-axis.

• The value you are graphing (representing, etc.) is always the y-axis.

What’s Missing?What’s Missing?

• Time Domain– Frequency

• Frequency Domain– Time

• Time Domain– Frequency

• Frequency Domain– Time

Converting DomainsConverting 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

• 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 AssignmentReading Assignment

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

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

Fourier AnalysisFourier Analysis

BackgroundBackground

• 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.

• 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 CalculatingOld 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.

• 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 FFTWorking 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 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 TransformThe 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.

• 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 doesWhat 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.

• 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 FrequenciesAnalysis Frequencies

• Simplest 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)

• Simplest 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)

Windowing and OverlapsWindowing 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.

• 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)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.

• 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 MathDo Some Math

• Time / Frequency Trade-off• Time / Frequency Trade-off

ReadingReading

• Roads: pp. 536 - 577• Roads: pp. 536 - 577