numerical methods fourier transform pair part: frequency and time domain
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Numerical Methods
Fourier Transform Pair Part: Frequency and Time Domain
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Chapter 11.03: Fourier Transform Pair: Frequency and
Time Domain
Major: All Engineering Majors
Authors: Duc Nguyen
http://numericalmethods.eng.usf.edu
Numerical Methods for STEM undergraduateshttp://numericalmethods.eng.usf.edu 504/10/23
Lecture # 5
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)()()( 1101 tSinbtCosaatf
)2()2()()()( 221102 tSinbtCosatSinbtCosaatf
)2()2()()()( 221104 tSinbtCosatSinbtCosaatf
Example 1
)(2for
0for)(
Tt
tttf
)4()4()3()3( 4433 tSinbtCosatSinbtCosa
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Frequency and Time DomainThe 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.
Figures 2(a) and 2(b) can be described with the following equations from chapter 11.02,
k
tikw
keCtf 0~)( (39,
repeated)
where
Ttikw
k dtetfT
C0
0)(1~ (41,
repeated)
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For the periodic function shown in Example 1 of Chapter 11.02 (Figure 1), one has:
12
2220
Tfw
2
0
1~dtedtet
TC iktikt
k
Frequency and Time Domain cont.
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Frequency and Time Domain cont.
Define:
00
0
11dte
ike
iktdtetA iktiktikt
or
22
2
11
11
ke
ke
k
i
ek
eik
A
ikik
ikik
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Also,
2
2
ik
edteB iktikt
ikikikik eek
iee
ikB
22
Frequency and Time Domain cont.
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Frequency and Time Domain cont.
Thus: BACk
21~
2
22
11
2
1~ ikik
k ek
i
kk
i
kk
ieC
Using the following Euler identities
)cos(
)sin()cos(
)sin()cos(
k
kik
kike ik
)2cos()2sin()2cos(2 kkike ik
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Noting that 1)2cos( k for any integerk
)2(
11)(
2
1~22
kCosk
i
kkkCosCk
1
Frequency and Time Domain cont.
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Frequency and Time Domain cont.
Also,
,...)8,6,4,2(1
,...)7,5,3,1(1)cos(
numberevenkfor
numberoddkfork
Thus,
k
i
kkC
k
k
22
11
2
1~
ikk
C k
k
2
111
2
1~2
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From Equation (36, Ch. 11.02), one has
2~ kkk
ibaC
(36, repeated)
Hence; upon comparing the previous 2 equations, one concludes:
1)1(12
k
k ka
k
bk1
Frequency and Time Domain cont.
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Frequency and Time Domain cont.
For
;8...4,3,2,1k the values for ka kband
(based on the previous 2 formulas) are exactly
identical as the ones presented earlier in Example1 of Chapter 11.02.
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Frequency and Time Domain cont.
Thus:
iiiba
C2
11
2
)1(2
2~ 111
ii
ibaC
4
10
221
0
2~ 222
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ii
ibaC
8
10
241
0
2~ 444
ii
ibaC
10
1
25
1
251
252
2~ 555
ii
ibaC
6
1
9
1
231
92
2~ 333
Frequency and Time Domain cont.
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Frequency and Time Domain cont.
ii
ibaC
12
10
261
0
2~ 666
ii
ibaC
14
1
49
1
271
492
2~ 777
ii
ibaC
16
10
281
0
2~ 888
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In general, one has
numberevenkfori
k
numberoddkforikk
Ck
,..8,6,4,22
1
,..7,5,3,12
11
~ 2
Frequency and Time Domain cont.
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THE ENDhttp://numericalmethods.eng.usf.edu
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The End - Really
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Numerical Methods
Fourier Transform Pair Part: Complex Number in Polar Coordinates
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Chapter 11.03: Complex number in polar coordinates
(Contd.)
In Cartesian (Rectangular) Coordinates, a complex number can be expressed as:
kC~
iIRC kkk ~
In Polar Coordinates, a complex number can be expressed as:
kC~
iAAiAAeC i
k )sin()cos()sin()cos(~
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Lecture # 6
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Complex number in polar coordinates cont.
Thus, one obtains the following relations between the Cartesian and polar coordinate systems: cosARk sinAI k This is represented graphically in Figure 3.
Figure 3. Graphical representation of the complex number system in polar coordinates.
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Complex number in polar coordinates cont.
Hence
)(sin)(cossincos 222222222 AAAIR kk
A
R
A
R kk 1cos)cos(
A
I
A
I kk 1sin)sin(
and
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Based on the above 3 formulas, the complex numberscan be expressed as:
kC~
)13770783.2(
1 )59272353.0(2
11~ ieiC
Complex number in polar coordinates cont.
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eR
mI
Notes:(a)The amplitude and angle are 0.59 and 2.14 respectively (also see Figures 2a, and 2b in chapter 11.03).
1
~C
(b) The angle (in radian) obtained from
A
RCos k)( will be 2.138 radians (=122.48o).
However based on A
ISin k)(
Then = 1.004 radians (=57.52o).
Complex number in polar coordinates cont.
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)57079633.1(22 )25.0()25.0(
4
10
~ ii
eeiC
)77990097.1(
3 )17037798.0(6
1
9
1~ ieiC
Since the Real and Imaginary components of are negative and positive, respectively, the proper selectionfor should be 2.1377 radians.
Complex number in polar coordinates cont.
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)57079633.1(24 )125.0()125.0(
8
10
~ ii
eeiC
)69743886.1(
5 )100807311.0(10
1
25
1~ ieiC
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Complex number in polar coordinates cont.
)57079633.1(26 )08333333.0()08333333.0(
12
10
~ ii
eeiC
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)66149251.1(
7 )07172336.0(14
1
49
1~ ieiC
28 )0625.0(
16
10
~ i
eiC
Complex number in polar coordinates cont.
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Numerical Methods
Fourier Transform Pair Part: Non-Periodic Functions
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Chapter 11. 03: Non-Periodic Functions (Contd.)
Recall
k
tikw
keCtf 0~)( (39, repeated)
Ttikw
k dtetfT
C0
0)(1~ (41, repeated)
Define
2
2
00 )()(ˆ
T
T
tikw dtetfikwF
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Lecture # 7
(1)
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Then, Equation (41) can be written as
)(ˆ1~
0ikwFT
Ck
And Equation (39) becomes
k
tikweikwFT
tf 00 )(ˆ
1)(
From above equation
k
tikw
ffor
Tnp eikwFftftf 000
0
)(ˆ)(lim)(lim)(
k
ftik
fnp efikFftf 2
0)2(ˆ)(lim)(
or
Non-Periodic Functions
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Non-Periodic Functions cont.
Figure 4. Frequency are discretized.
From Figure 4,
ffk
fti
np efiFdftf 2)2(ˆ)(
dfefiFtf fti
np 2)2(ˆ)(
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Non-Periodic Functions cont.
Multiplying and dividing the right-hand-side of the equation by , one obtains2
; inverse Fourier transform
)()(ˆ2
1)( 0
00 wdeiwFtf tiw
np
Also, using the definition stated in Equation (1), one gets
)()()(ˆ 00 tdetfiwF tiw
np; Fourier transform
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