7. The Discrete and Fast Fourier Transform

1
7. The Discrete and Fast Fourier Transform




2
Contents

• 1. Basic theory of discrete Fourier transform
  (DFT)
• 2. Relationships among various types of
  Fourier transforms
• 3. Computation of DFT
• 4. Brief introduction to fast Fourier
  transform (FFT)

3
Discrete Fourier transform (DFT) and fast
Fourier transform (FFT)

• Discrete Fourier transform (DFT)
• For the frequency domain representation of
  discrete-time signals
• Calculation of Fourier transform by (digital)
  computer frequency domain should be made
  discrete
• Fast Fourier transform (FFT)
• An algorithm for the fast computation of DFT

4
Discrete Fourier series and (discrete-time)
Fourier transform (Ch.3)
Discrete Fourier series
(discrete-time) Fourier transform

• - For periodic signals
• Line spectrum
• Spectral coefficients ak

• - For aperiodic signals
• Spectrum continuous function of ?
• (X(W))

5
Discrete Fourier transform (DFT)
(7.1)
Fourier transform (discrete-time FT)
DFT
That is, in DFT, the frequency W (02p) is
represented by discrete N samples ? The spectrum
Xk is a discrete sequence
6
Discrete Fourier transform and Fourier series
7
Inverse discrete Fourier transform (IDFT)
(7.2)
Inverse Fourier transform
8
Inverse discrete Fourier transform (IDFT) and
Synthesis equation of Fourier series
9
DFT, discrete-time FT, and discrete Fourier series
(a) An aperiodic signal and its spectrum.
Spectrum continuous function of W, period 2p,
For real signal, it is symmetric with respect to p
(b) A period signal and its spectrum. Spectrum
line spectrum with period N. It can be obtained
as samples of X(W), or by Fourier series
10
DFT and IDFT

• DFT and IDFT same computation, except 1/N and
  the sign of the exponent
• Thus, same algorithm can be used to both DFT and
  IDFT

11
FT (X(W) or H(W)) and DFT (Xk or Hk)

• Digital computers or DSP hardwares cannot process
  continuous function
• DFT is used to represent X(?) or H(?) as a
  discrete function, by sampling
• Large sampling interval
• Small number of samples
• Appropriate representation of function is not
  possible
• Small sampling interval
• Excessive samples computational load

12
Sampling of frequency in DFT

• Sampling interval ? 2? / N
• ? Number of frequency samples 2? / ( 2? / N )
  N (a period 2?)
• ? Xk can be regarded as a sampled version of
  the FT X(?)
• ? Degree of freedom 2N (for a complex signal), N
  (for a real signal)
• For a real signal half of the spectral samples
  (Xk) can represent the whole spectrum, due to
  the symmetry of the spectrum

13
Properties of the DFT

• Linearity
• Time-shift
• Convolution
• Modulation

14
Properties of the DFT
15
Properties of the DFT

• Cosine even function, Sine odd function
• If xn is a real signal
• The real part of Xk even function of k
• The imaginary part of Xk odd function of k
• Due to this symmetry ? the overall spectrum can
  be represented by half of Xks
• If xn is a complex signal, there is no symmetry
• - All Xks should be computed

16
Properties of the DFT

• If xn is a real, and even function (xnx-n)
  
• ? Imaginary part of Xk becomes 0, there is only
  cosine term (purely real)
• If xn is real, and odd function (xn-x-n)
• ? Real part of Xk becomes 0, there is only sine
  term (purely imaginary)

17
Computation of DFT

• Computation time determined by algorithm,
  implementation of program, and hardware
• Implementation of digital signal processing
• General-purpose computer, high-level language
  (such as C)
• Special hardware (DSP Chip), high-level/low-level
  languages (such as assembly)
• Design of a special digital system (FPGA, Custom
  IC/VLSI)
• Multiplication
• Major source of slow computation
• Specialized DSP hardware should be designed and
  used
• It is not computationally efficient to calculate
  equations (7.1), (7.2) directly

18
Computation of DFT

• When the signal xn is real
• DFT
• Inverse DFT
• The only difference between DFT and IDFT 1/N and
  sign of the exponent ? single algorithm can be
  used for both

19
Computation of DFT (real signal)

• Magnitude and phase
• Real part
• Imaginary part

20
Computation of DFT (imaginary signal)

• When the signal xn is complex
• Real part
• Imaginary part

21
Computation time

• Computation time is slowed by the multiplication
• Eqs. (7.17) and (7.18) (when xn is complex)
• ? Each requires 4N2 multiplications of real
  numbers
• Eqs. (7.12) and (7.13) (when xn is real)
• ? Each requires 2N2 multiplications of real
  numbers
• Thus, the computation time is proportional to N2
• (Ex.) For 2048 or 4096 samples several minutes
  (1998? ??)

22
Faster computation of DFT

• An idea for faster computation to use the
  periodicity of the DFT and IDFT (due to the sin
  and cos functions)
• As the values of k and n change, same values of
  sin and cos occur (redundancy of calculation)
• ? We can calculate the values of sin and cos in
  advance
• A method for faster computation of DFT
• Calculation of sin and cos in advance
• Storage of those values in memory
• Table lookup
• Basic idea of FFT

23

• Ex. 64 samples
• cos(2?kn/64) The
  values of k, n changes from 0 to 63
• sin(2?kn/64) The cos
  and sin have only 64 different values
• 
  
• - Typical time savings due to the use of the
  table lookup on the order of 50

24
DFT computed by table lookup
25
Fast Fourier transform (FFT)
-

• Developed in the late 1960s
• Number of multiplication
• Direct computation of DFT N2
• FFT N log2 N
• Ratio of computation time N2 / N log2 N N/
  log2 N
• Ex When N 8, the ratio is 8/32.6666
• When N2048, the ratio is 2048/11186.18