Discrete Fourier Transform

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Discrete Fourier Transform

• Prepared by Rejean Lau
• M.Eng

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Fourier Analysis

• Named after mathematician Jean Baptiste Joseph
  Fourier (1768-1830)
• Fourier claimed that any continuous periodic
  signal could be represented as the sum of
  properly chosen sinusoidal waves

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Fourier Analysis
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Types of Signals
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Types of Signals
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Discrete Fourier Transform

• The only type of Fourier Transform that can be
  used in DSP is the DFT.
• Why?
• Digital computes can only work with information
  that is discrete and finite in length.

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Transforms

• Examples of other types of transforms Fourier,
  Laplace, Z, Hilbert, Discrete Cosine etc.
• What is a transform?
• A function which allows both the input and output
  to have multiple values.
• Eg. A signal composed of 100 samples. A transform
  changes the 100 samples into another 100 samples.

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Discrete Fourier Transform

• The discrete Fourier transform changes an N point
  input signal into two point output signals.
• The input signal contains the signal being
  decomposed, while the two output signals contain
  the amplitudes of the component sine and cosine
  waves
• The input signal is said to be in time domain,
  while the output signals are said to be in
  frequency domain.

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Discrete Fourier Transform

• The inverse DFT performs the reverse of the DFT
• Transform a frequency domain signal to time
  domain
• The input length N is usually selected to be a
  power of 2 . Ie. 128,256,512, 1024
• This is a requirement by the most efficient
  algorithm which calculates the DFT, called the
  FFT (fast fourier transform)

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DFT Notation

• Lower case letters represent time domain signals
• ie. x , y , z
• Time domain runs from x0 to xN-1
• Upper case letters represent the corresponding
  frequency domain signals
• ie. X , Y , Z
• Frequency signal X consists of two parts, each
  an array of N/2 1 samples.
• ReX Real part of X -amplitude of cos
  wave
• - runs from ReX0 to ReXN/2
• ImX Imaginary part of X -amplitude of sin
  wave
• - runs from ImX0 to ImXN/2

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Discrete Fourier Transform
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DFT Basis Functions

• The sine and cosine waves used in the DFT are
  called DFT basis functions.
• The index value of ReX and ImX represented
  by k, is the amplitude of the corresponding basis
  function.

k is also equal to the number of cycles that
occur over the N points of the signal
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DFT Basis Functions
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Synthesis Equation, Inverse DFT

• Given ReX and ImX (the frequency
  components), determine x (the original time
  signal).

Basis functions
Synthesis equation to determine xi
ReXk, ImX needs to be scaled before
inserting in synthesis equation
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Inverse DFT
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Spectral Density

• The scaling is because the frequency components
  are expressed as spectral density
• ReX / (2/N) ReX where 2/N is the spectral
  density
• Therefore ReX ReX (2/N)
• The difference in scaling of the first and last
  frequency components ReX0 and ReXN/2 they
  have half the spectral density 1/N
• ReX / (1/N) ReX where 1/N is
  the spectral density
• Therefore ReX ReX (1/N)

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Two ways the synthesis equation can be
programmed Method (1) Each of the scaled
sinusoids are generated one at a time and added
to an accumulation array, which ends up becoming
time domain signal Method (2) Each sample in the
time domain signal is calculated one at a time,
as the sum of all the corresponding samples in
the cosine and sine waves