A-A+ B-B+ C-C+ F

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 ContPeriodic p DFT (DFS) defined for DiscPeriodic p

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

Properties 5 Duality

Example

Properties 6 Convolution

Properties 7Delay

Properties 8Modulation

Discrete-time Fourier Transform (DTFT)

Example Rectangle

Example Upsampling

Fourier series 012 1

Discrete Fourier Transform (DFT) 07 1

Fourier series and DTFT

SummaryDefinitions CTFT InvCTFT DTFT InvDTFT

FS InvFS DFT InvDFT

Properties for CTFT Linearity Symmetry Duality Convolution Time-scale Multiplication Modulation Delay

Properties for DTFT Periodic Convolution Multiplication Delay Modulation Upsampling

Important signals Delta Complex exponentials, sinusoids Rectangular pulse, sinc Exponentials