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) = P o + P n cos (2 f n t + n ) P o is the ambient pressure P n is the pressure of the nth cosine wave f n is the frequency of the nth cosine wave n is the phase of the nth cosine wave
Sound spectrum Frequency spectrum 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 frequency plus side bands
Frequency modulation (FM) Modulation determines the number of side bands
Periodic nonsinusoidal signals 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