Fourier Transform

1. Fourier Transform

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

3. Increase Toto infinity (periodic) aperiodic (See derivation in the note)

4. To→Infinity Δωo reduces to dω when To increases to infinity.

5. 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.

6. 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

7. Sufficient Condition for Fourier Transform Pair Dirichlet conditions

8. 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.

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

10. 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.