EE3TP4_16_GeneralizedFT_v1_Lecture 24

EE3TP4_16_GeneralizedFT_v1_Lecture 24 - Fourier Transform...

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X ω = −∞ x t e jωt dt Fourier Transform A sufficient condition for the existence of the Fourier Transform of x(t) includes : −∞ x t dt ∞ Generalized FT This section allows us to apply FT to an even broader class of signals that includes some periodic signals and others. The trick is to allow the delta function to be a part of a valid FT
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F { δ t } = −∞ δ t e jωt dt = e 0 = 1 Sifting property δ(t) ↔ 1 δ(t) t F { δ(t) } = 1 ω . . . . . . 1 FT of an Impulse Function! This has to be true . .. recall that the FT of the zero state output of a LTI system is equal to the product of the FT of the input and the FT of the impulse response of the system.
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δ(t) ↔ 1 1 δ(ω) “A DC signal” has FT concentrated at 0 Hz DC = 0 Hz ω X ω = 2 πδ ω x ( t ) = 1 t . . . . . . So we now know:
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EE3TP4_16_GeneralizedFT_v1_Lecture 24 - Fourier Transform...

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