Numerical Methods Fourier Transform Pair Part: Frequency and Time Domain
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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
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) 4 0 + a Cos 3t) + b Sin(3t) + a Cos(4t) + b Sin(4 ) ( t 3 3 4 4
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.
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 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.
Frequency and Time Domain cont. Define: + 0 0 0 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
Frequency and Time Domain cont. Also, B e it dt ( it e ) i B i i [ i i ] [ i i e e e e ]
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
3 Noting that ) cos( for any integer + ) ( ) ( ~ Cos i Cos C Frequency and Time Domain cont.
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
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 [ ( ) ]
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
Thus: Frequency and Time Domain cont. C a ib i( ) ~ + i 0 i ~ a ib C 0 + 4 i 7
C Frequency and Time Domain cont. a ib i 9 3 0 i ~ a ib 4 4 4 C 4 0 + 9 ~ 3 3 + 3 8 i 6 i 8 C a ib i 5 5 5 ~ 5 5 + 5 0 i
C Frequency and Time Domain cont. a ib 0 i 6 ~ 6 6 0 + 6 i 9 C a ib i 49 7 49 ~ 7 7 + 7 0 i ~ a ib 8 8 8 C 0 + 8 6 i 4 i
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
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Numerical Methods Fourier Transform Pair Part: Complex Number in Polar Coordinates
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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
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 3. 30 Figure 3. Graphical representation of the complex number system in polar coordinates.
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
Complex number in polar coordinates cont. Based on the above 3 formulas, the complex numbers can be expressed as: C ~ C ~ i(.3770783) + i (0.597353) e 3
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
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 (.57079633) 0 + i (0.5) e (0.5) e i C 9 6 ~ i(.77990097) i (0.7037798) e 3 + 34
Complex number in polar coordinates cont. C 8 C + i (0.008073 5 0 ~ i (.57079633) 4 0 + i (0.5) e (0.5) e i ~ i(.69743886) 5 ) e C ~ i (.57079633) 6 0 + i (0.08333333) e (0.08333333) e i 35
Complex number in polar coordinates cont. C 49 4 ~ i(.66495) i (0.077336) e 7 + ~ C8 0 + i (0.065) e 6 i 36
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Numerical Methods Fourier Transform Pair Part: Non-Periodic Functions
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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 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
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
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
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