Fast Fourier Transform

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Fast Fourier Transform

• Rejean Lau, M.Eng

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Fast Fourier Transform

• Efficient algorithm for calculating the Discrete
  Fourier Transform
• Presented by J.W Cooley and J.W. Tukey in paper
  An algorithm for the machine calculation of
  complex Fourier Series, Mathematics Compuation,
  Vol 19, 1965, pg. 297-301

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FFT

• Based on the complex DFT

In the Real DFT, the length of the output
frequency vectors is N/21, but in the Complex
DFT, it is N
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Periodic Nature of DFT

• Q. Why does the Complex DFT frequency array have
  length N as opposed the Real DFT with length
  N/21?
• The Complex DFT doubles number of frequency
  values by including negative frequencies which
  mirror the positive frequencies
• (0 frequency accounts for the 1)

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Negative Frequency Generation

• Can use a program to generate the negative
  frequencies to fill out the rest of the complex
  array (positions N/2 1 N)

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How the FFT works

• The FFT executes the following steps
• (1) Decomposing an N point time domain signal
  into N time domain signals each composed of a
  single point
• (2) Calculate the N frequency spectra
  corresponding to the N time domain signals
• (3) The N spectra are synthesized into a
  single frequency spectrum

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(1) Time domain decomposition
Log2N stages are required in the
decomposition Ie. A 16 point signal (24)
requires 4 stages A 512 point signal (27)
requires 7 stages A 4096 point signal (212)
requires 12 stages
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(1) Time domain decomposition

• The decomposition is simply a reordering of the
  samples in the signal

Notice that the reordering of the samples is done
in the order of bit-reversal of the binary
equivalent of the sample number The FFT time
domain decomposition can be carried out using a
bit reversal sorting algorithm
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(2) Calculate the N frequency Spectra

• The frequency of a 1 point signal is itself
• That is, the value of the signal is unchanged
• No computation is required in this step!
• Each of the 1 point signals that were time domain
  signals are now 1 point frequency domain signals

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(3) Synthesize the frequency spectrum signal

• Last step in the FFT is to combine the N
  frequency spectra in the exact reverse order that
  the time domain decomposition took place
• The bit reversal trick is not applicable in this
  case, and we must go back one stage at a time
• Continuing from our example
• First stage 16 frequency spectra (1 point each)
  are synthesized into 8 frequency spectra (2
  points each)
• Second stage 8 frequency spectra (2 points each)
  are synthesized into 4 frequency spectra (4
  points each)
• Third stage 4 frequency spectra (4 points each)
  are synthesized into 2 frequency spectra (8
  points each)
• Fourth stage 2 frequency spectra (8 points each)
  are synthesized into 1 frequency spectra (16
  points)

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3) Synthesize the frequency spectrum signal

• The frequency spectra are combined in the FFT by
  duplicating them, and
• then adding the duplicated spectra together
• The odd frequency spectrum must also be
  multiplied by a sinusoid
• (the shift in time domain corresponds to this
  multiplication)

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FFT Butterfly

• The butterfly is the basic computational element
  of the FFT, transforming two complex points into
  two other complex points

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Flow diagram of FFT

• Frequency Domain synthesis requires 3 loops
• Outer loop runs through the Log2N stages
• Middle loop moves through each of the individual
  spectra in the stage being worked on
• Innermost loop uses the butterfly to calculate
  the points in each frequency spectra
• The overhead in the boxes
• determine the beginning and
• ending indexes for the loops, as
• well as calculating the sinusoids
• needed in the butterflies

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FFT Program in BASIC
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FFT Programs

• In-place computation same arrays are used for
  the input, intermediate storage and output ?
  efficient use of memory
• In the BASIC program, data enter and leave the
  subroutine in the arrays REX and IMX , with
  the samples running from index 0 to N-1.
• Upon return from the subroutine, REX and IMX
  are overwritten with the frequency domain data.
• The length of the DFT must also be passed to the
  subroutines.
• BASIC program uses variable N

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Inverse FFT

• Complex DFT has strong duality
• This means there is symmetry between the time and
  frequency domains
• The Inverse DFT is nearly identical to the
  Forward DFT
• Easiest way to calculate an Inverse FFT is to
  calculate a Forward FFT, and then adjust the data

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Testing the FFT

• Two tests
• Start with some arbitrary time domain signal
  input, and run it through the FFT.
• Now run the resultant frequency spectrum
  through the Inverse FFT.
• - The result should be identical with the input
• Symmetry test.
• Frequency domain of the complex DFT should be
  symmetrical around samples 0 and N/2.
• -Results in the Inverse DFT producing a time
  domain that has
• an imaginary part compromising of
  all zeros

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FFT Performance

• Time required for bit reversal negligible
• Each of the Log2N stages has N butterfly
  computations

Where kFFT is a constant related to the processor
speed ie. kFFT 10 µs on a 100 MHz Pentium