EECE 301 Note Set 15 FT Properties

EECE 301 Note Set 15 FT Properties - EECE 301 Signals &...

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
1/39 EECE 301 Signals & Systems Prof. Mark Fowler Note Set #15 • C-T Signals: Fourier Transform Properties • Reading Assignment: Section 3.6 of Kamen and Heck
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2/39 Ch. 1 Intro C-T Signal Model Functions on Real Line D-T Signal Model Functions on Integers System Properties LTI Causal Etc Ch. 2 Diff Eqs C-T System Model Differential Equations D-T Signal Model Difference Equations Zero-State Response Zero-Input Response Characteristic Eq. Ch. 2 Convolution C-T System Model Convolution Integral D-T System Model Convolution Sum Ch. 3: CT Fourier Signal Models Fourier Series Periodic Signals Fourier Transform (CTFT) Non-Periodic Signals New System Model New Signal Models Ch. 5: CT Fourier System Models Frequency Response Based on Fourier Transform New System Model Ch. 4: DT Fourier Signal Models DTFT (for “Hand” Analysis) DFT & FFT (for Computer Analysis) New Signal Model Powerful Analysis Tool Ch. 6 & 8: Laplace Models for CT Signals & Systems Transfer Function New System Model Ch. 7: Z Trans. Models for DT Signals & Systems Transfer Function New System Model Ch. 5: DT Fourier System Models Freq. Response for DT Based on DTFT New System Model Course Flow Diagram The arrows here show conceptual flow between ideas. Note the parallel structure between the pink blocks (C-T Freq. Analysis) and the blue blocks (D-T Freq. Analysis).
Background image of page 2
3/39 Fourier Transform Properties Note: There are a few of these we won’t cover…. see Table on Website or the inside front cover of the book for them. As we have seen, finding the FT can be tedious (it can even be difficult ) But…there are certain properties that can often make things easier. Also, these properties can sometimes be the key to understanding how the FT can be used in a given application. So… even though these results may at first seem like “just boring math” they are important tools that let signal processing engineers understand how to build things like cell phones, radars, mp3 processing, etc. I prefer that you use the tables on the website… they are better than the book’s
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
4/39 1. Linearity (Supremely Important) If & then ) ( ) ( ω X t x ) ( ) ( Y t y [] [ ] ) ( ) ( ) ( ) ( bY aX t by t ax + + To see why : {} dt e t by t ax t by t ax t j + = + ) ( ) ( ) ( ) ( F dt e t y b dt e t x a t j t j + = ) ( ) ( By standard Property of Integral of sum of functions Use Defn of FT ) ( X = ) ( Y = By Defn of FT { } { } ) ( ) ( ) ( ) ( t y b t x a t by t ax F F F + = + Another way to write this property: Gets used virtually all the time!!
Background image of page 4
5/39 Example Application of “Linearity of FT”: Suppose we need to find the FT of the following signal… ) ( t x 1 2 2 2 t 1 Finding this using straight-forward application of the definition of FT is not difficult but it is tedious: {} dt e dt e dt e t x t j t j t j + + = 2 1 1 1 1 2 2 ) ( ω F 1 So… we look for short-cuts: • One way is to recognize that each of these integrals is basically the same • Another way is to break x ( t ) down into a sum of signals on our table!!!
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
6/39 ) ( t x 1 2 2 2 t + = π ω sinc 2 2 sinc 4 ) ( X
Background image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 02/29/2012 for the course EECE 301 taught by Professor Fowler during the Fall '08 term at Binghamton University.

Page1 / 39

EECE 301 Note Set 15 FT Properties - EECE 301 Signals &...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online