Title: Numerical Methods Discrete Fourier Transform Part: Discrete Fourier Transform http://numericalmethods.eng.usf.edu
1Numerical MethodsDiscrete Fourier Transform
Part Discrete Fourier Transform
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5Chapter 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
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5
6Discrete 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
7Discrete Fourier Transform
then Eq. (39) becomes
(1)
8Discrete 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
9Discrete Fourier Transform cont.
(4)
(5)
10Discrete 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
11Discrete 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
12Discrete Fourier TransformCase 2
(10)
Define
(11)
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12
13Discrete Fourier TransformCase 2
Then, Eq. (10) can be expressed as
(12)
14Discrete 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
15Discrete Fourier TransformCase 2
(See Eq. (10))
(15)
(16)
16Discrete 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
17The End
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18Acknowledgement
- 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.
20The End - Really
21Numerical MethodsDiscrete Fourier Transform
Part Discrete Fourier Transform
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22- For more details on this topic
- Go to http//numericalmethods.eng.usf.edu
- Click on Keyword
- Click on Discrete Fourier Transform
23You are free
- to Share to copy, distribute, display and
perform the work - to Remix to make derivative works
24Under the following conditions
- Attribution You must attribute the work in the
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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.
25Lecture 9
Chapter 11.04 Discrete Fourier Transform (DFT)
Substituting Eq.(18) into Eq.(7), and then
referring to Eq.(6), one gets
(18A)
Thus
26Discrete Fourier TransformCase 2
Eq. (18A) can, therefore, be simplified to
(18B)
Thus
(19)
(1, repeated)
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26
27Discrete Fourier Transform cont.
Equations (19) and (1) can be rewritten as
(20)
(21)
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27
28Discrete Fourier Transform cont.
To avoid computation with complex numbers,
Equation (20) can be expressed as
(20A)
where
29Discrete 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
30The End
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31Acknowledgement
- 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.
33The End - Really
34Numerical MethodsDiscrete Fourier Transform
Part Aliasing Phenomenon Nyquist Samples,
Nyquist ratehttp//numericalmethods.eng.usf.edu
35- For more details on this topic
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- Click on Keyword
- Click on Discrete Fourier Transform
36You are free
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37Under the following conditions
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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.
38Chapter 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
39Aliasing Phenomenon, Nyquist samples, Nyquist
rate cont.
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39
40Aliasing Phenomenon, Nyquist samples, Nyquist
rate cont.
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40
41Windowing phenomenon
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41
42Windowing phenomenon cont.
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42
43Nyquist samples, Nyquist rate
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43
44Nyquist 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).
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44
45Nyquist samples, Nyquist rate
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45
46Nyquist samples, Nyquist rate
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46
47Nyquist samples, Nyquist rate
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47
48The End
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49Acknowledgement
- 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.
51The End - Really