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1 Numerical Methods Fourier Transform Pair Part: Frequency and Time Domain
2 For more details on this topic Go to Clic on eyword Clic on Fourier Transform Pair
3 You are free to Share to copy, distribute, display and perform the wor to Remix to mae derivative wors
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5 Lecture # 5 Chapter.03: Fourier Transform Pair: Frequency and Time Domain Major: All Engineering Majors Authors: Duc Nguyen Numerical Methods for STEM undergraduates 9/4/00 5
6 Example f ( t) t for 0 < t for t < ( T ) 6 f ( t) a0 + acos( t) + bsin( t) f ( t) a + a Cos( t) + b Sin 0 ( t) + a Cos (t) + b Sin (t) f ( t) a + a Cos( t) + b Sin( t) + a Cos(t) + b Sin(t) a Cos 3t) + b Sin(3t) + a Cos(4t) + b Sin(4 ) ( t
7 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. 7 Figure Periodic function (see Example in Chapter.0 Continuous Fourier Series) in frequency domain.
8 Frequency and Time Domain cont. Figures (a) and (b) can be described with the following equations from chapter.0, ~ iw0t f ( t) C e where (39, repeated) ~ C T { } f ( t) e dt T 0 iw0t (4, repeated) 8
9 9 For the periodic function shown in Example of Chapter.0 (Figure ), one has: 0 T f w { } + 0 ~ dt e dt e t T C it it Frequency and Time Domain cont.
10 Frequency and Time Domain cont. Define: dt e i e i t dt e t A it it it or [ ] + + e e i e e i A i i i i 0
11 Frequency and Time Domain cont. Also, B e it dt ( it e ) i B i i [ i i ] [ i i e e e e ]
12 Frequency and Time Domain cont. Thus: { } B A C + ~ + + ~ i i e i i i e C Using the following Euler identities ) cos( ) sin( ) cos( ) sin( ) cos( i i e i + ) cos( ) sin( ) cos( i e i
13 3 Noting that ) cos( for any integer + ) ( ) ( ~ Cos i Cos C Frequency and Time Domain cont.
14 Frequency and Time Domain cont. Also, +,4,6,8,...) (,3,5,7,...) ( ) cos( even number for number odd for Thus, ( ) + i C ~ ( ) [ ] i C + ~ 4
15 Frequency and Time Domain cont. 5 From Equation (36, Ch..0), one has a ib ~ C (36, repeated) Hence; upon comparing the previous equations, one concludes: a b [ ( ) ]
16 Frequency and Time Domain cont. For,,3,4...8; the values for a and b (based on the previous formulas) are exactly identical as the ones presented earlier in Example of Chapter.0. 6
17 Thus: Frequency and Time Domain cont. C a ib i( ) ~ + i 0 i ~ a ib C i 7
18 C Frequency and Time Domain cont. a ib i i ~ a ib C ~ i 6 i 8 C a ib i ~ i
19 C Frequency and Time Domain cont. a ib 0 i 6 ~ i 9 C a ib i ~ i ~ a ib C i 4 i
20 Frequency and Time Domain cont. In general, one has + i for odd number,3,5,7,.. ~ C i for,4,6,8,.. even number 0
21 THE END
22 Acnowledgement This instructional power point brought to you by Numerical Methods for STEM undergraduate Committed to bringing numerical methods to the undergraduate
23 For instructional videos on other topics, go to /videos/ This material is based upon wor supported by the National Science Foundation under Grant # 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.
24 The End - Really
25 Numerical Methods Fourier Transform Pair Part: Complex Number in Polar Coordinates
26 For more details on this topic Go to Clic on eyword Clic on Fourier Transform Pair
27 You are free to Share to copy, distribute, display and perform the wor to Remix to mae derivative wors
28 Under the following conditions Attribution You must attribute the wor 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 wor). Noncommercial You may not use this wor for commercial purposes. Share Alie If you alter, transform, or build upon this wor, you may distribute the resulting wor only under the same or similar license to this one.
29 Lecture # 6 Chapter.03: Complex number in polar coordinates (Contd.) In Cartesian (Rectangular) Coordinates, a complex number can be expressed as: C ~ ~ C R + ( I )i 9 In Polar Coordinates, a complex number be expressed as: ~ C Ae iθ A C ~ can { cos( ) isin( )} { Acos( )} { Asin( )}i θ + θ θ + θ
30 Complex number in polar coordinates cont. Thus, one obtains the following relations between the Cartesian and polar coordinate systems: R Acos( θ ) I Asin( θ ) This is represented graphically in Figure Figure 3. Graphical representation of the complex number system in polar coordinates.
31 Complex number in polar coordinates cont. Hence R ( ) ( ) [ θ + A sin θ cos ( θ ) + sin ( θ )] + I A cos A cos( θ ) R A θ cos R A and sin( θ ) I A θ sin I A 3
32 Complex number in polar coordinates cont. Based on the above 3 formulas, the complex numbers can be expressed as: C ~ C ~ i( ) + i ( ) e 3
33 Complex number in polar coordinates cont. 33 Notes: ~ (a) The amplitude and angle C are 0.59 and.4 respectively (also see Figures a, and b in chapter.03). θ (in radian) obtained from R Cos(θ ) will be.38 radians (.48 o ). A I Sin(θ ) A θ (b) The angle However based on Then.004 radians (57.5 o ). I m θ R e
34 Since the Real and Imaginary components of are negative and positive, respectively, the proper selection for should be.377 radians. θ Complex number in polar coordinates cont. θ C 4 ~ i ( ) 0 + i (0.5) e (0.5) e i C 9 6 ~ i( ) i ( ) e
35 Complex number in polar coordinates cont. C 8 C + i ( ~ i ( ) i (0.5) e (0.5) e i ~ i( ) 5 ) e C ~ i ( ) i ( ) e ( ) e i 35
36 Complex number in polar coordinates cont. C 49 4 ~ i(.66495) i ( ) e 7 + ~ C8 0 + i (0.065) e 6 i 36
37 THE END
38 Acnowledgement This instructional power point brought to you by Numerical Methods for STEM undergraduate Committed to bringing numerical methods to the undergraduate
39 For instructional videos on other topics, go to /videos/ This material is based upon wor supported by the National Science Foundation under Grant # 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.
40 The End - Really
41 Numerical Methods Fourier Transform Pair Part: Non-Periodic Functions
42 For more details on this topic Go to Clic on eyword Clic on Fourier Transform Pair
43 You are free to Share to copy, distribute, display and perform the wor to Remix to mae derivative wors
44 Under the following conditions Attribution You must attribute the wor 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 wor). Noncommercial You may not use this wor for commercial purposes. Share Alie If you alter, transform, or build upon this wor, you may distribute the resulting wor only under the same or similar license to this one.
45 Recall Chapter. 03: Non-Periodic Functions (Contd.) ~ ( (39, repeated) iw0t f t) C e Lecture # 7 ~ C Define T { } f ( t) e dt T Fˆ ( iw 0 T iw0t 0 ) ( ) T f t e iw 0t dt (4, repeated) () 45
46 46 or Then, Equation (4) can be written as ~ C Fˆ ( iw 0 ) T And Equation (39) becomes From above equation T ˆ ( iw0t f ( t) F iw0) e iw0t f np ( t) lim f ( t) lim ( f ) F iw0 ) e T f 0 or f 0 f np ( t) lim f 0 Non-Periodic Functions ( f ) Fˆ ( i f ) e ˆ ( i ft
47 From Figure 4, Non-Periodic Functions cont. f f f np ( t) df Fˆ ( i f ) e i ft f np ( t) Fˆ ( if ) e i ft df Figure 4. Frequency are discretized. 47
48 f Non-Periodic Functions cont. Multiplying and dividing the right-hand-side of the equation by, one obtains np ) ˆ iw0t ( t) F( iw ) e d( w 0 0 ; inverse Fourier transform Also, using the definition stated in Equation (), one gets ˆ iw0t ( iw0 ) f np ( t) e d( t F ) ; Fourier transform 48
49 THE END
50 Acnowledgement This instructional power point brought to you by Numerical Methods for STEM undergraduate Committed to bringing numerical methods to the undergraduate
51 For instructional videos on other topics, go to /videos/ This material is based upon wor supported by the National Science Foundation under Grant # 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.
52 The End - Really
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