# Fourier Transform. Fourier Series Vs. Fourier Transform We use Fourier Series to represent periodic signals We will use Fourier Transform to represent.

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Fourier Transform

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

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

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

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.

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

Sufficient Condition for Fourier Transform Pair Dirichlet conditions

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.

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

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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