EECE 301 Note Set 14 Fourier Transform

EECE 301 Note Set 14 Fourier Transform - 1/27 EECE 301...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full Document Right Arrow Icon

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

View Full Document Right Arrow Icon

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1/27 EECE 301 Signals & Systems Prof. Mark Fowler Note Set #14 • C-T Signals: Fourier Transform (for Non-Periodic Signals) • Reading Assignment: Section 3.4 & 3.5 of Kamen and Heck 2/27 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). 3/27 4.3 Fourier Transform Recall : Fourier Series represents a periodic signal as a sum of sinusoids Note : Because the FS uses “harmonically related” frequencies k ω , it can only create periodic signals ∑ ∞ −∞ = = k t j k k e c t x ω ) ( or complex sinusoids t jk e ω With arbitrary discrete frequencies… NOT harmonically related ∑ ∞ −∞ = = k t j k k e c t x ω ) ( The problem with is that it cannot include all possible frequencies! Q: Can we modify the FS idea to handle non-periodic signals? A: Yes!! What about ? That will give some non-periodic signals but not some that are important!! 4/27 How about: ∫ ∞ ∞ − = ω ω π ω d e X t x t j ) ( 2 1 ) ( Called the “Fourier Integral ” also, more commonly, called the “ Inverse Fourier Transform ” Plays the role of c k Plays the role of t jk e ω Integral replaces sum because it can “add up over the continuum of frequencies”! Okay… given x ( t ) how do we get X ( ω )? ∫ ∞ ∞ − − = dt e t x X t j ω ω ) ( ) ( Note: X ( ω ) is complex-valued function of ω ∈ (- ∞ , ∞ ) | X ( ω )| ) ( ω X ∠ Yes … this will work for any practical non-periodic signal!! Called the “ Fourier Transform ” of x ( t ) Need to use two plots to show it 5/27 Comparison of FT and FS Fourier Series : Used for periodic signals Fourier Transform : Used for non-periodic signals (although we will see later that it can also be used for periodic signals) ∑ ∞ −∞ = = n t jk k e c t x ) ( ω ∫ + − = T t t t jk k dt e t x T c ) ( 1 ω ∫ ∞ ∞ − = ω ω π ω d e X t x t j ) ( 2 1 ) ( ∫ ∞ ∞ − − = dt...
View Full Document

{[ snackBarMessage ]}

Page1 / 27

EECE 301 Note Set 14 Fourier Transform - 1/27 EECE 301...

This preview shows document pages 1 - 6. Sign up to view the full document.

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