10/10,13,16,17 - Fourier Transform Properties

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Properties of the Fourier Transform

Linearity

Time Invariance (Delay)

Let and
Why isn't this

Frequency Shifting

Double Sideband Modulation

Differentiation in Time

Thus is a linear filter with transfer function

The Game (frequency domain)

  • You can play the game in the frequency or time domain, but it's not advisable to play it in both at same time
Input LTI System Output Reason
Given
Proportionality
Superposition
Time Invariance
Proportionality
Superposition
  • Having trouble seeing

The Game (Time Domain??)

Input LTI System Output Reason
Proportionality
from 10/3,6 - The Game
Proportionality
Superposition

Relation to the Fourier Series

Let and reverse the order of summation
Note that is the complex conjugate of
  • How can we assume that the answer exists in the real domain?

Aside: Polar coordinates

Remember from 10/02 - Fourier Series that

  • Rectangular coordinates:
  • Polar coordinates:

Building up to

Euler's Identity
Real odd function of t
&
= Real odd. Integrates out over symmetric limits.
Imaginary Odd function of
Real even function of t
&
= Real odd. Integrates out over symmetric limits.
Real Even function of

Definitions

Can't x(t) have parts that aren't even or odd? You can break any function down into a Taylor series. There are even and odd powers in the series.