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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.**Increase Toto infinity**(periodic) aperiodic (See derivation in the note)**To→Infinity**Δωo reduces to dω when To increases to infinity.**Derive the Fourier Transform of a rectangular pulse**The nonperiodicrectangular 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.