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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.

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Presentation Transcript
fourier series vs 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
Increase Toto infinity

(periodic)

aperiodic

(See derivation in the note)

to infinity
To→Infinity

Δωo reduces to dω when To increases to infinity.

derive the fourier transform of a rectangular pulse
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
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
clarification
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
Examples
  • Function that does not satisfy Dirichlet condition, but still have Fourier Transform pair.
    • Unit Step function
    • Periodic function
power condition
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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