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

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Properties of the Fourier Transform

Linearity

Time Invariance (Delay)

Let and

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 not both at the same time
    • Then how can you use the Fourier Transform, but can't build up to it?
Input LTI System Output Reason
Given
Proportionality
Superposition
Time Invariance
Proportionality
Superposition
  • Having trouble seeing
  • Since we were dealing in the frequency domain, is that the reason why multiplying one side did not result in a convolution on the other?


Now back in the time domain

Input LTI System Output Reason
Proportionality