Frequency Response & DTFT

Reference: https://www.youtube.com/watch?v=-UqXHjpH3lA 

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2D Frequency Response

 

Impulse Response vs. Frequency Response vs. Transfer Function

• Impulse Response: 
• Frequency Response:
• Transfer Function: 

Example

• 2D FFT could be turn to the product of two 1D FFT.

 

Fourier Transform

• Normalized case: -1/2 to 1/2

FFT Shift

1. Visualization of FFT

• Red horizontal line: Anything on the red horizontal line has only content that is only variation in the x-axis, and 0 variation in the y-axis.
• Blue vertical line: Only variation in the y-axis and 0 variation in the x-axis.
• Green dots in the corner: the highest frequency variation.

 

DTFT (Discrete time Fourier Transform)

• Continuous in the frequency domain, but discrete signal.
• Shift Does not changing the output of the signal
• Convolution of the time domain/spatial domain is the same as the product in the frequency domain.

• Separability maintains in the spatial domain and the frequency domain.

 

Parseval's Relation

• Inner product between two 2D signals in the frequency domain is equla to the inner product between their fourier transform in the value.
• Total Energy preserves both domains in this case.