102_1_Week-8-Recap - 35 WEEK 8 RECAP FOURIER TRANSFORMS 23...

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35 WEEK 8 RECAP FOURIER TRANSFORMS 23. Basic Properties f ( t ) := F - 1 { F ( i ω ) } ←- | -→ F ( i ω ) := F{ f ( t ) } f ( t ) = 1 2 π -∞ e i ω t F ( i ω ) d ω := F - 1 { F ( i ω ) } | F ( i ω ) = -∞ e - i ω t f ( t ) dt := F{ f ( t ) } 1. Linearity : af 1 ( t ) + bf 2 ( t ) | aF 1 ( i ω ) + bF 2 ( i ω ) 2. Complex Conjugate f ( t ) | F ( - i ω ) ( f ( t ) Real | F ( i ω ) = F ( - i ω ) ) 3. Duality Time Signal: F ( it ) | FT: 2 π f ( - ω ) 4. Scaling f ( α t ) | ( α > 0) : 1 α F i ω α , ( α < 0) : 1 - α F i ω α 5. Time-Shifting f ( t t 0 ) | e i ω t 0 F ( i ω ) 6. Frequency-Shifting e ± i ω 0 t f ( t ) | F ( i [ ω ω 0 ] ) 7. Time-Convolution -∞ h ( t - τ ) x ( τ ) d τ | H ( i ω ) · X ( i ω ) 8. Frequency-Convolution f ( t ) · ( t ) | 1 2 π -∞ F ( i [ ω - ζ ] ) L ( ζ ) d ζ 9. Di ff erentiation d n dt n f ( t ) | ( i ω ) n F ( i ω ) , n 0 F ( i ω ) = | F ( i ω ) | e i Θ ( ω ) , | F ( i ω ) | := Amplitude , Θ ( ω ) := Phase , ω R F ( i ω ) = -∞ e - i ω t f ( t ) dt = -∞ cos ω t f ( t ) dt - i -∞ sin ω t f ( t ) dt f ( t ) : Real Re F ( i ω ) = -∞ cos ω t f ( t ) dt , Im F ( i ω ) = - -∞ sin ω t f ( t ) dt
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36 NHAN LEVAN f ( t ) : Real and Even ? , f ( t ) : Real and Odd ?
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