Eeng 360 1

1
Chapter 2 Discrete Fourier Transform (DFT)

• Topics
• Discrete Fourier Transform.
• Using the DFT to Compute the Continuous Fourier
  Transform.
• Comparing DFT and CFT
• Using the DFT to Compute the Fourier Series

Huseyin Bilgekul Eeng360 Communication Systems
I Department of Electrical and Electronic
Engineering Eastern Mediterranean University
2
Discrete Fourier Transform (DFT)

• Definition The Discrete Fourier Transform (DFT)
  is defined by

The Inverse Discrete Fourier Transform (IDFT) is
defined by
The Fast Fourier Transform (FFT) is a fast
algorithm for evaluating the DFT.
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Using the DFT to Compute the Continuous Fourier
Transform

• Suppose the CFT of a waveform w(t) is to be
  evaluated using DFT.
• The time waveform is first windowed (truncated)
  over the interval (0, T) so that only a finite
  number of samples, N, are needed. The windowed
  waveform ww(t) is
• The Fourier transform of the windowed waveform is
• Now we approximate the CFT by using a finite
  series to represent the integral,
• t k?t, f n/T, dt ?t, and ?t T/N

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Computing CFT Using DFT

• We obtain the relation between the CFT and DFT
  that is,

• The sample values used in the DFT computation
  are x(k) w(k?t),

• If the spectrum is desired for negative
  frequencies
• the computer returns X(n) for the positive n
  values of 0,1, , N-1
• It must be modified to give spectral values
  over the entire
• fundamental range of -fs/2 lt f ltfs/2.
• For positive frequencies we use For
  Negative Frequencies

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Comparison of DFT and the Continuous Fourier
Transform (CFT)

• Relationship between the DFT and the CFT involves
  three concepts
• Windowing,
• Sampling,
• Periodic sample generation

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Comparison of DFT and the Continuous Fourier
Transform (CFT)

• Relationship between the DFT and the CFT involves
  three concepts
• Windowing,
• Sampling,
• Periodic sample generation

7
Fast Fourier Transform

• The Fast Fourier Transform (FFT) is a fast
  algorithm for evaluating DFT.

Block diagrams depicting the decomposition of an
inverse DTFS as a combination of lower order
inverse DTFSs. (a) Eight-point inverse DTFS
represented in terms of two four-point inverse
DTFSs. (b) four-point inverse DTFS
represented in terms of two-point inverse DTFSs.
(c) Two-point inverse DTFS.
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Using the DFT to Compute the Fourier Series

• The Discrete Fourier Transform (DFT) may also be
  used to compute the complex Fourier series.
• Fourier series coefficients are related to DFT
  by,

• Block diagram depicting the sequence of
  operations involved in approximating the FT with
  the DTFS.

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Ex. 2.17 Use DFT to compute the spectrum of a
Sinusoid
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Ex. 2.17 Use DFT to compute the spectrum of a
Sinusoid
Spectrum of a sinusoid obtained by using the
MATLAB DFT.
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Using the DFT to Compute the Fourier Series
The DTFT and length-N DTFS of a 32-point cosine.
The dashed line denotes the CFT. While the stems
represent NXk. (a) N 32 (b) N 60 (c) N
120.
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Using the DFT to Compute the Fourier Series
The DTFS approximation to the FT of x(t)
cos(2?(0.4)t) cos(2?(0.45)t). The stems denote
Yk, while the solid lines denote CFT. (a) M
40. (b) M 2000. (c) Behavior in the vicinity of
the sinusoidal frequencies for M 2000. (d)
Behavior in the vicinity of the sinusoidal
frequencies for M 2010