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Week 11. Chapter 10 Fourier Transforms
1. The Four Fourier transforms
• CTFT
• DTFT
• FS
• DFT/S
2. Relationships
3. Important properties
1. The Four Fourier transforms
• The signal spaces
• The transform definitions
2. Relationships
3. Important properties
The 4 Signal Spaces
CTFT is defined for ContSignals
DTFT is defined for DiscSignals
FS is defined for ContPeriodicp
DFT (DFS) defined for DiscPeriodicp
CTFT
CTFT Intuition
Properties 1 Sinusoids
Example FS & FT of periodic signals
Properties 2 Time change
Example
Properties 3 Symmetry
Properties 4 More symmetry
Example
Example
Properties 5 Duality
Example
Example
Properties 6 Convolution
Properties 7
Delay
Properties 8 Modulation
Discrete-time Fourier Transform (DTFT)
Example Rectangle
Example Upsampling
Fourier series
1
0
1
2
Discrete Fourier Transform (DFT)
1
0
7
Fourier series and DTFT
Summary Definitions
CTFT
InvCTFT
DTFT
InvDTFT
FS
InvFS
DFT
InvDFT
Properties for CTFT
Linearity
Symmetry
Duality
Convolution
Multiplication
Delay
Modulation
Time-scale
Properties for DTFT
Periodic
Convolution
Multiplication
Delay
Modulation
Upsampling
Important signals
Delta
Complex exponentials, sinusoids
Rectangular pulse, sinc
Exponentials