Chapter 9 Computation of the Discrete Fourier Transform

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Chapter 9 Computation of the Discrete Fourier
Transform
Biomedical Signal processing

• Zhongguo Liu
• Biomedical Engineering
• School of Control Science and Engineering,
  Shandong University

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Chapter 9 Computation of the Discrete Fourier
Transform
9.0 Introduction 9.1 Efficient Computation of
Discrete Fourier Transform 9.2 The Goertzel
Algorithm 9.3 decimation-in-time FFT
Algorithms 9.4 decimation-in-frequency FFT
Algorithms 9.5 practical considerations(software
realization)
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9.0 Introduction

• Implement a convolution of two sequences by the
  following procedure

• Why not convolve the two sequences directly?

• There are efficient algorithms called Fast
  Fourier Transform (FFT) that can be orders of
  magnitude more efficient than others.

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9.1 Efficient Computation of Discrete Fourier
Transform

• The DFT pair was given as

• Baseline for computational complexity
• Each DFT coefficient requires
• N complex multiplications
• N-1 complex additions
• All N DFT coefficients require
• N2 complex multiplications
• N(N-1) complex additions

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9.1 Efficient Computation of Discrete Fourier
Transform

• Complexity in terms of real operations
• 4N2 real multiplications
• 2N(N-1) real additions (approximate 2N2)

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9.1 Efficient Computation of Discrete Fourier
Transform

• Most fast methods are based on Periodicity
  properties
• Periodicity in n and k Conjugate symmetry

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9.2 The Goertzel Algorithm

• Makes use of the periodicity
• Multiply DFT equation with this factor

• using xn0 for nlt0 and ngtN-1

• Xk can be viewed as the output of a filter to
  the input xn
• Impulse response of filter
• Xk is the output of the filter at time nN

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9.2 The Goertzel Algorithm

• Goertzel Filter

• Computational complexity
• 4N real multiplications 4N real additions
• Slightly less efficient than the direct method

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Second Order Goertzel Filter

• Goertzel Filter

• Multiply both numerator and denominator

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Second Order Goertzel Filter

• Complexity for one DFT coefficient ( x(n) is
  complex sequence).
• Poles 2N real multiplications and 4N real
  additions
• Zeros Need to be implement only once
• 4 real multiplications and 4 real additions
• Complexity for all DFT coefficients
• Each pole is used for two DFT coefficients
• Approximately N2 real multiplications and 2N2
  real additions

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Second Order Goertzel Filter

• If do not need to evaluate all N DFT coefficients
• Goertzel Algorithm is more efficient than FFT if
• less than M DFT coefficients are needed,M lt log2N

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9.3 decimation-in-time FFT Algorithms

• Makes use of both periodicity and symmetry
• Consider special case of N an integer power of 2
• Separate xn into two sequence of length N/2
• Even indexed samples in the first sequence
• Odd indexed samples in the other sequence

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9.3 decimation-in-time FFT Algorithms

• Substitute variables n2r for n even and n2r1
  for odd

• Gk and Hk are the N/2-point DFTs of each
  subsequence

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9.3 decimation-in-time FFT Algorithms

• Gk and Hk are the N/2-point DFTs of each
  subsequence

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8-point DFT using decimation-in-time
Figure 9.3
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computational complexity

• Two N/2-point DFTs
• 2(N/2)2 complex multiplications
• 2(N/2)2 complex additions
• Combining the DFT outputs
• N complex multiplications
• N complex additions
• Total complexity
• N2/2N complex multiplications
• N2/2N complex additions
• More efficient than direct DFT

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9.3 decimation-in-time FFT Algorithms

• Repeat same process , Divide N/2-point DFTs into
• Two N/4-point DFTs
• Combine outputs

N8
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9.3 decimation-in-time FFT Algorithms

• After two steps of decimation in time

• Repeat until were left with two-point DFTs

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9.3 decimation-in-time FFT Algorithms

• flow graph for 8-point decimation in time

• Complexity
• Nlog2N complex multiplications and additions

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Butterfly Computation

• Flow graph constitutes of butterflies

• We can implement each butterfly with one
  multiplication

• Final complexity for decimation-in-time FFT
• (N/2)log2N complex multiplications and additions

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9.3 decimation-in-time FFT Algorithms

• Final flow graph for 8-point decimation in time

• Complexity
• (Nlog2N)/2 complex multiplications and Nlog2N
  additions

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9.3.1 In-Place Computation????

• Decimation-in-time flow graphs require two sets
  of registers
• Input and output for each stage

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9.3.1 In-Place Computation????

• Note the arrangement of the input indices
• Bit reversed indexing(????)

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cause of bit-reversed order
binary coding for position 000 001 010 011 100
101 110 111
Figure 9.13
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9.3.2 Alternative forms

• Note the arrangement of the input indices
• Bit reversed indexing(????)

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9.3.2 Alternative forms
strongpointin-place computations shortcomingnon-
sequential access of data
Figure 9.14
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Figure 9.15
shortcomingnot in-place computation
non-sequential access of data
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Figure 9.16
shortcomingnot in-place computation
strongpoint sequential access of data
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9.3 decimation-in-time FFT Algorithms

• Substitute variables n2r for n even and n2r1
  for odd

Review

• Gk and Hk are the N/2-point DFTs of each
  subsequence

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9.3.1 In-Place Computation????

• Bit reversed indexing(????)

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9.3.2 Alternative forms
strongpointin-place computations shortcomingnon-
sequential access of data
Figure 9.14
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9.4 Decimation-In-Frequency FFT Algorithm

• The DFT equation

• Split the DFT equation into even and odd
  frequency indexes

• Substitute variables

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9.4 Decimation-In-Frequency FFT Algorithm

• The DFT equation

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• decimation-in-frequency decomposition of an
  N-point DFT to N/2-point DFT

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• decimation-in-frequency decomposition of an
  8-point DFT to four 2-point DFT

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• 2-point DFT

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• Final flow graph for 8-point DFT decimation in
  frequency

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9.4.1 In-Place Computation????
DIF FFT
DIT FFT
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9.4.1 In-Place Computation????
DIF FFT
DIT FFT
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9.4.2 Alternative forms

• decimation-in-frequecy Butterfly Computation

• decimation-in-time Butterfly Computation

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The DIF FFT is the transpose of the DIT FFT
DIF FFT
DIT FFT
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9.4.2 Alternative forms
DIF FFT
DIT FFT
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9.4.2 Alternative forms
DIF FFT
DIT FFT
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Figure 9.24 erratum
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9.4.2 Alternative forms
DIF FFT
DIT FFT
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Chapter 9 HW

• 9.1, 9.2, 9.3,

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