Psych/NPB 163

Fourier Analysis

Why frequency analysis?

• Many signals in the natural environment are conveniently described in terms of a superposition of wobbly functions (e.g., many sounds are produced by vibrating membranes).
  
  
• Sines and cosines are eigenfunctions of linear, time-invariant systems. Thus, they can be used to conveniently characterize the response of a linear system. Also, convolution, which is a complicated signal transformation in the time or space domain, is performed by simple multiplication in the frequency domain.
  
  
  

• Joseph Fourier’s theorem, in its most general form, states that any function may be described in terms of a superposition of odd and even functions. Specifically, Fourier proposed that a signal may be decomposed into a summation of sinewaves of different frequencies, amplitudes and phases:
  
  .
  
  
• What is remarkable about this is that can be anything- the waveform produced by a bird chirping, the sound of your dishwasher, electromagnetic waves, etc.
  
  
• The amplitudes tell you how much of each frequency is present in the signal. For example, a pure tone (e.g., the waveform emitted by a tuning fork) would have equal to zero for all frequencies except for one. The sound produced when you say “shhh” would have amplitudes distributed across many frequencies.
  
  
• The Fourier series tells us that a sound may be decomposed in terms of sinewaves, but it doesn’t tell us how to do it - i.e., it doesn’t tell us what amplitudes A to assign to each f. For this we need the Fourier transform.
  
  

• The Fourier transform basically provides a way of representing a signal in a different space - i.e., in the frequency domain. You put into the Fourier transform a function of time or space, , and you get out a function of frequency, . The Fourier transform is formally defined as follows:
  
  .
  
  
• Thus, the Fourier transform is essentially the inner-product of the signal with the complex exponential , evaluated at different values of f.
  
  
• The complex exponential is just a real cosine-wave and an imaginary sine-wave:
  
  .
  
  Thus, the Fourier integral may be written alternatively as
  
  
  
  
• Here we can see that is a complex number. The real part tells us the result of multiplying our signal together with a cosine-wave, the imaginary part tells us the result of multiplying the signal together with a sine-wave.
  
  
• The amplitude A of each frequency component contained in is given by the modulus of S, which is defined as
  
  
  
  
• The phase can be extracted from the ratio of the real an imaginary components of as follows:
  
  
  
  
  

• The convolution of two functions,
  
  
  
  may be performed in the frequency domain via multiplication:
  
  .
  
  where , , and are Fourier transforms of , , and respectively.
  
  
• That this is so is due to the fact that 1) sines and cosines are eigenfunctions of linear time-invariant systems, and 2) any function may be represented as a sum of sines and cosines.