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Title: Numerical Methods Discrete Fourier Transform Part: Discrete Fourier Transform http://numericalmethods.eng.usf.edu


1
Numerical MethodsDiscrete Fourier Transform
Part Discrete Fourier Transform
http//numericalmethods.eng.usf.edu
2
  • For more details on this topic
  • Go to http//numericalmethods.eng.usf.edu
  • Click on Keyword
  • Click on Discrete Fourier Transform

3
You are free
  • to Share to copy, distribute, display and
    perform the work
  • to Remix to make derivative works

4
Under the following conditions
  • Attribution You must attribute the work in the
    manner specified by the author or licensor (but
    not in any way that suggests that they endorse
    you or your use of the work).
  • Noncommercial You may not use this work for
    commercial purposes.
  • Share Alike If you alter, transform, or build
    upon this work, you may distribute the resulting
    work only under the same or similar license to
    this one.

5
Chapter 11.04 Discrete Fourier Transform (DFT)
Lecture 8
Major All Engineering Majors Authors Duc
Nguyen http//numericalmethods.eng.usf.edu Numeri
cal Methods for STEM undergraduates
9/3/2013
http//numericalmethods.eng.usf.edu
5
6
Discrete Fourier Transform
Recalled the exponential form of Fourier series
(see Eqs. 39, 41 in Ch. 11.02), one gets
(39, repeated)
(41, repeated)

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6
7
Discrete Fourier Transform
then Eq. (39) becomes
(1)
8
Discrete Fourier Transform cont.
To simplify the notation, define
(2)
Then, Eq. (1) can be written as
(3)
Multiplying both sides of Eq. (3) by
, and performing
obtains (note l integer
number)

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8
9
Discrete Fourier Transform cont.
(4)
(5)
10
Discrete Fourier Transform cont.
Switching the order of summations on the
right-hand-side of Eq.(5), one obtains
(6)
Define
(7)

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10
11
Discrete Fourier TransformCase 1
Case(1) is a multiple integer of N, such
as or where
Thus, Eq. (7) becomes
(8)
Hence
(9)

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11
12
Discrete Fourier TransformCase 2
(10)
Define
(11)

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12
13
Discrete Fourier TransformCase 2
Then, Eq. (10) can be expressed as
(12)
14
Discrete Fourier TransformCase 2
From mathematical handbooks, the right side of
Eq. (12) represents the geometric series, and
can be expressed as
(13)
(14)

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14
15
Discrete Fourier TransformCase 2
(See Eq. (10))
(15)
(16)
16
Discrete Fourier TransformCase 2
Substituting Eq. (16) into Eq. (15), one gets
(17)
Thus, combining the results of case 1 and case 2,
we get
(18)

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16
17
The End
  • http//numericalmethods.eng.usf.edu

18
Acknowledgement
  • This instructional power point brought to you by
  • Numerical Methods for STEM undergraduate
  • http//numericalmethods.eng.usf.edu
  • Committed to bringing numerical methods to the
    undergraduate

19
  • For instructional videos on other topics, go to
  • http//numericalmethods.eng.usf.edu/videos/
  • This material is based upon work supported by the
    National Science Foundation under Grant
    0717624. Any opinions, findings, and conclusions
    or recommendations expressed in this material are
    those of the author(s) and do not necessarily
    reflect the views of the National Science
    Foundation.

20
The End - Really
21
Numerical MethodsDiscrete Fourier Transform
Part Discrete Fourier Transform
http//numericalmethods.eng.usf.edu
22
  • For more details on this topic
  • Go to http//numericalmethods.eng.usf.edu
  • Click on Keyword
  • Click on Discrete Fourier Transform

23
You are free
  • to Share to copy, distribute, display and
    perform the work
  • to Remix to make derivative works

24
Under the following conditions
  • Attribution You must attribute the work in the
    manner specified by the author or licensor (but
    not in any way that suggests that they endorse
    you or your use of the work).
  • Noncommercial You may not use this work for
    commercial purposes.
  • Share Alike If you alter, transform, or build
    upon this work, you may distribute the resulting
    work only under the same or similar license to
    this one.

25
Lecture 9
Chapter 11.04 Discrete Fourier Transform (DFT)
Substituting Eq.(18) into Eq.(7), and then
referring to Eq.(6), one gets
(18A)
Thus
26
Discrete Fourier TransformCase 2
Eq. (18A) can, therefore, be simplified to
(18B)
Thus
(19)
(1, repeated)

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26
27
Discrete Fourier Transform cont.
Equations (19) and (1) can be rewritten as
(20)
(21)

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27
28
Discrete Fourier Transform cont.
To avoid computation with complex numbers,
Equation (20) can be expressed as
(20A)
where
29
Discrete Fourier Transform cont.
(20B)
The above complex number equation is equivalent
to the following 2 real number equations
(20C)
(20D)

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29
30
The End
  • http//numericalmethods.eng.usf.edu

31
Acknowledgement
  • This instructional power point brought to you by
  • Numerical Methods for STEM undergraduate
  • http//numericalmethods.eng.usf.edu
  • Committed to bringing numerical methods to the
    undergraduate

32
  • For instructional videos on other topics, go to
  • http//numericalmethods.eng.usf.edu/videos/
  • This material is based upon work supported by the
    National Science Foundation under Grant
    0717624. Any opinions, findings, and conclusions
    or recommendations expressed in this material are
    those of the author(s) and do not necessarily
    reflect the views of the National Science
    Foundation.

33
The End - Really
34
Numerical MethodsDiscrete Fourier Transform
Part Aliasing Phenomenon Nyquist Samples,
Nyquist ratehttp//numericalmethods.eng.usf.edu
35
  • For more details on this topic
  • Go to http//numericalmethods.eng.usf.edu
  • Click on Keyword
  • Click on Discrete Fourier Transform

36
You are free
  • to Share to copy, distribute, display and
    perform the work
  • to Remix to make derivative works

37
Under the following conditions
  • Attribution You must attribute the work in the
    manner specified by the author or licensor (but
    not in any way that suggests that they endorse
    you or your use of the work).
  • Noncommercial You may not use this work for
    commercial purposes.
  • Share Alike If you alter, transform, or build
    upon this work, you may distribute the resulting
    work only under the same or similar license to
    this one.

38
Chapter 11.04 Aliasing Phenomenon, Nyquist
samples, Nyquist rate (Contd.)
Lecture 10
Figure 1 Function to be sampled and Aliased
sample problem.

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38
39
Aliasing Phenomenon, Nyquist samples, Nyquist
rate cont.

http//numericalmethods.eng.usf.edu
39
40
Aliasing Phenomenon, Nyquist samples, Nyquist
rate cont.

http//numericalmethods.eng.usf.edu
40
41
Windowing phenomenon

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41
42
Windowing phenomenon cont.

http//numericalmethods.eng.usf.edu
42
43
Nyquist samples, Nyquist rate

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43
44
Nyquist samples, Nyquist rate
Hence
which implies
Physically, the above equation states that one
must have at least 2 samples per cycle of the
highest frequency component present (Nyquist
samples, Nyquist rate).

http//numericalmethods.eng.usf.edu
44
45
Nyquist samples, Nyquist rate

http//numericalmethods.eng.usf.edu
45
46
Nyquist samples, Nyquist rate

http//numericalmethods.eng.usf.edu
46
47
Nyquist samples, Nyquist rate

http//numericalmethods.eng.usf.edu
47
48
The End
  • http//numericalmethods.eng.usf.edu

49
Acknowledgement
  • This instructional power point brought to you by
  • Numerical Methods for STEM undergraduate
  • http//numericalmethods.eng.usf.edu
  • Committed to bringing numerical methods to the
    undergraduate

50
  • For instructional videos on other topics, go to
  • http//numericalmethods.eng.usf.edu/videos/
  • This material is based upon work supported by the
    National Science Foundation under Grant
    0717624. Any opinions, findings, and conclusions
    or recommendations expressed in this material are
    those of the author(s) and do not necessarily
    reflect the views of the National Science
    Foundation.

51
The End - Really
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