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Numerical MethodsFourier Transform PairPart: Frequency and Time Domainhttp://numericalmethods.eng.usf.edu
slide2
For more details on this topic
  • Go to http://numericalmethods.eng.usf.edu
  • Click on keyword
  • Click on Fourier Transform Pair
you are free
You are free
  • to Share – to copy, distribute, display and perform the work
  • to Remix – to make derivative works
under the following conditions
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.
chapter 11 03 fourier transform pair frequency and time domain

Chapter 11.03: Fourier Transform Pair: Frequency and Time Domain

Lecture # 5

Major: All Engineering Majors

Authors: Duc Nguyen

http://numericalmethods.eng.usf.edu

Numerical Methods for STEM undergraduates

6/5/2014

http://numericalmethods.eng.usf.edu

5

slide6

Example 1

6

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frequency and time domain
Frequency and Time Domain

The amplitude (vertical axis) of a given periodic function

can be plotted versus time (horizontal axis), but it can

also be plotted in the frequency domain as shown in

Figure 2.

Figure 2 Periodic function (see Example 1 in Chapter 11.02 Continuous Fourier Series) in frequency domain.

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frequency and time domain cont
Frequency and Time Domain cont.

Figures 2(a) and 2(b) can be described with the following equations from chapter 11.02,

(39, repeated)

where

(41, repeated)

8

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frequency and time domain cont1
Frequency and Time Domain cont.

For the periodic function shown in Example 1 of Chapter 11.02 (Figure 1), one has:

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frequency and time domain cont2
Frequency and Time Domain cont.

Define:

or

10

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frequency and time domain cont3
Frequency and Time Domain cont.

Also,

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frequency and time domain cont4
Frequency and Time Domain cont.

Thus:

Using the following Euler identities

12

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frequency and time domain cont5

1

Noting that

for any integer

Frequency and Time Domain cont.

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frequency and time domain cont6
Frequency and Time Domain cont.

Also,

Thus,

14

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frequency and time domain cont7
Frequency and Time Domain cont.

From Equation (36, Ch. 11.02), one has

(36, repeated)

Hence; upon comparing the previous 2 equations, one concludes:

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frequency and time domain cont8
Frequency and Time Domain cont.

For

the values for

and

(based on the previous 2 formulas) are exactly

identical as the ones presented earlier in Example

1 ofChapter 11.02.

16

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frequency and time domain cont9
Frequency and Time Domain cont.

Thus:

17

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frequency and time domain cont10
Frequency and Time Domain cont.

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frequency and time domain cont11
Frequency and Time Domain cont.

19

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frequency and time domain cont12
Frequency and Time Domain cont.

In general, one has

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the end
The End

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acknowledgement
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

slide23

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.

slide25
Numerical MethodsFourier Transform Pair Part: Complex Number in Polar Coordinateshttp://numericalmethods.eng.usf.edu
slide26
For more details on this topic
  • Go to http://numericalmethods.eng.usf.edu
  • Click on keyword
  • Click on Fourier Transform Pair
you are free1
You are free
  • to Share – to copy, distribute, display and perform the work
  • to Remix – to make derivative works
under the following conditions1
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.
chapter 11 03 complex number in polar coordinates contd

In Cartesian (Rectangular) Coordinates, a complex number can be expressed as:

In Polar Coordinates, a complex number can be expressed as:

Lecture # 6

Chapter 11.03: Complex number in polar coordinates (Contd.)

29

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complex number in polar coordinates cont
Complex number in polar coordinates cont.

Thus, one obtains the following relations between the Cartesian and polar coordinate systems:

This is represented graphically in Figure 3.

Figure 3. Graphical representation of the complex number system in polar coordinates.

30

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complex number in polar coordinates cont1
Complex number in polar coordinates cont.

Hence

and

31

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complex number in polar coordinates cont2
Complex number in polar coordinates cont.

Based on the above 3 formulas, the complex numbers

can be expressed as:

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complex number in polar coordinates cont3
Complex number in polar coordinates cont.

Notes:

  • The amplitude and angle are 0.59 and
  • 2.14 respectively (also see Figures 2a, and
  • 2b in chapter 11.03).

(b) The angle (in radian) obtained from

will be 2.138 radians (=122.48o).

However based on

Then = 1.004 radians (=57.52o).

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complex number in polar coordinates cont4
Complex number in polar coordinates cont.

Since the Real and Imaginary components of are

negative and positive, respectively, the proper selection

for should be 2.1377 radians.

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complex number in polar coordinates cont5
Complex number in polar coordinates cont.

35

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complex number in polar coordinates cont6
Complex number in polar coordinates cont.

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the end1
The End

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acknowledgement1
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

slide39

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.

slide41
Numerical MethodsFourier Transform Pair Part: Non-Periodic Functionshttp://numericalmethods.eng.usf.edu
slide42
For more details on this topic
  • Go to http://numericalmethods.eng.usf.edu
  • Click on keyword
  • Click on Fourier Transform Pair
you are free2
You are free
  • to Share – to copy, distribute, display and perform the work
  • to Remix – to make derivative works
under the following conditions2
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.
chapter 11 03 non periodic functions contd

Lecture # 7

Chapter 11. 03: Non-Periodic Functions (Contd.)

Recall

(39, repeated)

(41, repeated)

Define

(1)

45

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non periodic functions
Non-Periodic Functions

Then, Equation (41) can be written as

And Equation (39) becomes

From above equation

or

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non periodic functions cont
Non-Periodic Functions cont.

From Figure 4,

Figure 4. Frequency are discretized.

47

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non periodic functions cont1
Non-Periodic Functions cont.

Multiplying and dividing the right-hand-side of the

equation by , one obtains

; inverse Fourier

transform

Also, using the definition stated in Equation (1), one gets

; Fourier transform

48

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the end2
The End

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acknowledgement2
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

slide51

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.