Intro to Fourier Analysis

• Definition

• Analysis of periodic waves

• Analysis of aperiodic waves

• Digitization

• Time-frequency uncertainty

The Fourier series

• Any continuous waveform can be partitioned into a sum of sinusoidal waves

• P(t) = Po + Pn cos (2fnt + n)

• Po is the ambient pressure

• Pn is the pressure of the nth cosine wave

• fn is the frequency of the nth cosine wave

• n is the phase of the nth cosine wave

Sound spectrum

Frequencyspectrum

Time domain

Frequency domain

Types of periodic waveforms

• Amplitude varies in a repeating manner - amplitude modulation

• Frequency varies in a repeating manner - frequency modulation

• The shape of the waveform varies in a repeating manner - nonsinusoidal periodic wave

Amplitude modulation (AM)

Carrier frequencyplus side bands

Frequency modulation (FM)

Modulationdetermines thenumber of sidebands

Periodic nonsinusoidal signals

Harmonic series

Harmonic series• Harmonic frequencies

are integer multiples of

the fundamental frequency,

i.e. w, 2w, 3w, 4w …

• Dirichlet’s rule states that the energy in higher harmonics falls off exponentially with the frequency of the harmonic

• Note, however, that some animals alter the amplitude of harmonics by selective filtering during sound production

Compound signals

• Nonsinusoidal modulation of a sine wave

• Sinusoidal modulation of a nonsinusoidal carrier wave

• Nonsinusoidal modulation of a nonsinusoidal carrier wave

Pulsed sine wave (frog or insect)

Fourier analysis of aperiodic signals

• Most natural signals have a short, not infinite, duration

• The more aperiodic a signal is, the more frequency components are needed to describe the signal with a Fourier series

• In the limit, an infinitely short signal has constant amplitude at all frequencies, a delta pulse

Finite sounds and Fourier ‘lobes’

Fourier “windows”

Bartlett

Hamming

Digitizing sound

Aliasing

Nyquist frequency = 1/2 the sampling frequency

Fourier window size and bandwidth

Time-frequency uncertainty

Sound spectrum “waterfall”

Song sparrow

Sonagram

Song sparrow

Narrow bandwidth analysis is good for frequency measurements, but not accurate for time measurements