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Fourier Transform

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Fourier Series Vs. Fourier Transform We use Fourier Series to represent periodic signals We will use Fourier Transform to represent non-period signal.

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Increase T o to infinity (periodic) aperiodic (See derivation in the note)

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To→Infinity Δω o reduces to dω when T o increases to infinity.

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Derive the Fourier Transform of a rectangular pulse The nonperiodic rectangular pulse has the same form as the envelope Of the Fourier series representation of periodic rectangular pulse train.

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Sufficient Versus Necessary Condition Something (x) is sufficient for something else (y) if the occurrence of x guarantees y. – For example, getting an A in a class guarantees passing the class. So getting an A is a sufficient condition for passing. If x is sufficient for y, then whenever you have x, you have y; you can't have x without y. For example, you can't get an A and not pass. Note that getting an A is not a necessary condition for passing, since you can pass without getting an A

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Sufficient Condition for Fourier Transform Pair Dirichlet conditions

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Clarification Something (x) is sufficient for something else (y) if the occurrence of x guarantees y. If a function satisfies Dirichlet conditions, then it must have F(ω) Getting an A is not a necessary condition for passing, since you can pass without getting an A If a function does not satisfy Dirichlet condition, it can still have Fourier Transform pair.

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Examples Function that does not satisfy Dirichlet condition, but still have Fourier Transform pair. – Unit Step function – Periodic function

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Power Condition Signals that have infinite energy, but contain a finite amount of power, but meet other Dirichlet condition have a valid Fourier Transform. Unit Step, periodic function and signum function have Fourier Transform pair under this less stringent requirement.

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