Post on 19-Dec-2015
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
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
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
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