The Discrete Fourier Transform

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The Discrete Fourier Transform
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The Fourier Transform

• The Fourier transform is a mathematical
  operation with many applications in physics and
  engineering that expresses a mathematical
  function of time as a function of frequency,
  known as its frequency spectrum.
• from http//en.wikipedia.org/wiki/Fourier_transfor
  m

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The Fourier Transform

• For instance, the transform of a musical chord
  made up of pure notes (without overtones)
  expressed as amplitude as a function of time, is
  a mathematical representation of the amplitudes
  and phases of the individual notes that make it
  up.
• from http//en.wikipedia.org/wiki/Fourier_transfor
  m

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Amplitude phase

• f(x) ? sin( ?x ? ) ?
• ? is the amplitude
• ? is the frequency
• ? is the phase
• ? is the DC offset

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More generally

• f(x) ?1 sin( ? 1 x ? 1 ) ?2 sin( ? 2 x ?
  2 ) ?

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The Fourier Transform

• The function of time is often called the time
  domain representation, and the frequency spectrum
  the frequency domain representation.
• from http//en.wikipedia.org/wiki/Fourier_transfor
  m

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Applications

• differential equations
• geology
• image and signal processing
• optics
• quantum mechanics
• spectroscopy

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Review of complex numbers
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Complex numbers

• Complex numbers . . .
• extend the 1D number line to the 2D plane
• are numbers that can be put into the rectangular
  form, abi where i2 -1, and a and b are real
  numbers.

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Complex numbers(rectangular form)
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Complex numbers

• Complex numbers . . .
• a is the real part b is the imaginary part
• If a is 0, then abi is purely imaginary if b is
  0, then abi is a real number.
• originally called fictitious by Girolamo
  Cardano in the 16th century

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Complex arithmetic

• add/subtract
• add/subtract the real and imaginary parts
  separately

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Complex arithmetic

• complex conjugate
• often denoted as
• negate only the imaginary part

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Complex arithmetic

• inverse
• where
• z is a complex number
• z bar is the length or magnitude of z
• a is the real part
• b is the imaginary part

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Complex arithmetic

• multiplication (FOIL)

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Complex arithmetic

• division

complex conjugate of denominator
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Complex numbers(polar form)
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exponential vs. trigonometric
(phasor form)
Leonhard Euler 1707-1783
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DFT(Discrete Fourier Transform)
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DFT

• Say we have a sequence of N (possibly complex)
  numbers, x0 xN-1.
• The DFT produces a sequence of N (typically
  complex) numbers, X0 XN-1, via the following

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DFT IDFT

• The DFT (Discrete Fourier Transform) produces a
  sequence of N (typically complex) numbers, X0
  XN-1, via the following
• The IDFT (Inverse DFT) is defined as follows

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Calculating the DFT

• So how can we actually calculate ?

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Calculating the DFT

• So how can we calculate ?
• Lets use this relationship
• Then
• So what does this mean?

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Interpretation of DFT

• Back to the polar form
• r/N is the amplitude and ? is the phase of a
  sinusoid with frequency k/N into which xn is
  decomposed

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Calculating the DFT using excel
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Check w/ matlab/octave

• see http//www.mathworks.com/help/matlab/ref/fft
  .html
• N 256 of samples
• n (0N-1) subscripts
• b1 0.5 freq 1
• b2 2.5 freq 2
• xn 0.5 sin( b1n ) 0.2 sin( b2n )
• plot( xn )
• Xn fft( xn )
• plot( abs(Xn(1N/2)) )

• X0real xn . cos( -2pin0/N )
• X0imag xn . sin ( -2pin0/N )
• X1real xn . cos( -2pin1/N )
• X1imag xn . sin ( -2pin1/N )
• X2real xn . cos( -2pin2/N )
• X2imag xn . sin ( -2pin2/N )
• X3real xn . cos( -2pin3/N )
• X3imag xn . sin ( -2pin3/N )
• .
• .
• .

Note . is element-wise (rather than matrix)
multiplication in matlab.
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Add random noise.

• see http//www.mathworks.com/help/matlab/ref/fft
  .html
• N 256 of samples
• n (0N-1) subscripts
• b1 0.5 freq 1
• b2 2.5 freq 2
• r randn( 1, N ) noise
• xn 0.5 sin( b1n ) 0.2 sin( b2n ) 0.5
  r
• plot( xn )
• Xn fft( xn )
• plot( abs(Xn(1N/2)) )

• X0real xn . cos( -2pin0/N )
• X0imag xn . sin ( -2pin0/N )
• X1real xn . cos( -2pin1/N )
• X1imag xn . sin ( -2pin1/N )
• X2real xn . cos( -2pin2/N )
• X2imag xn . sin ( -2pin2/N )
• X3real xn . cos( -2pin3/N )
• X3imag xn . sin ( -2pin3/N )
• .
• .
• .

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Signal without and with noise.
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Signal with noise. FFT of noisy signal (two major
components are still apparent).
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Example of differences in phase. xn 0.5 sin(
b1n ) 0.2 sin( b2n ) xn 0.5
sin( b1n 0.5 ) 0.2 sin( b2n )
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Computational complexityDFT vs. FFT

• The DFT is O(N2) complex multiplications.
• In 1965, Cooley (IBM) and Tukey (Princeton)
  described the FFT, a fast way (O(N log2 N)) to
  compute the FT using digital computers.
• It was later discovered that Gauss described this
  algorithm in 1805, and others had discovered it
  as well before Cooley and Tukey.
• With N 106, for example, it is the difference
  between, roughly, 30 seconds of CPU time and 2
  weeks of CPU time on a microsecond cycle time
  computer. from Numerical Recipes in C

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Extending the DFT to 2D(and higher)

• Let f(x,y) be a 2D set of sampled points. Then
  the DFT of f is the following
• (Note that engineers often use i for amps
  (current) so they use j for ?-1 instead.)

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Extending the DFT to 2D(and higher)

• In fact, the 2D DFT is separable so it can be
  decomposed into a sequence of 1D DFTs.
• And this can be generalized to higher and higher
  dimensions as well.

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The classical Gibbs phenomenon

• Visit http//en.wikipedia.org/wiki/Square_wave.
• Hear it at http//www.youtube.com/watch?vuIuJTWS2
  uvY.