We think you have liked this presentation. If you wish to download it, please recommend it to your friends in any social system. Share buttons are a little bit lower. Thank you!
Presentation is loading. Please wait.
Published byMavis Montgomery
Modified over 4 years ago
Leo Lam © 2010-2013 Signals and Systems EE235
Fourier Transform: Leo Lam © 2010-2013 2 Fourier Formulas: Inverse Fourier Transform: Fourier Transform: Time domain to Frequency domain
Fourier Transform (delta function): Leo Lam © 2010-2013 3 Fourier Transform of Standard Fourier Transform pair notation
Fourier Transform (rect function): Leo Lam © 2010-2013 4 Fourier Transform of Plot for T=1? t-T/2 0 T/2 1 Define
Fourier Transform (rect function): Leo Lam © 2010-2013 5 Fourier Transform of Observation: –Wider pulse (in t) taller narrower spectrum –Extreme case: Peak=pulse width (example: width=1) Zero-Crossings:
Fourier Transform - Inverse relationship Leo Lam © 2010-2013 6 Inverse relationship between time/frequency
Fourier Transform - Inverse Leo Lam © 2010-2013 7 Inverse Fourier Transform (Synthesis) Example:
Fourier Transform - Inverse Leo Lam © 2010-2013 8 Inverse Fourier Transform (Synthesis) Example: Single frequency spike in : exponential time signal with that frequency in t A single spike in frequencyComplex exponential in time
Fourier Transform Properties Leo Lam © 2010-2013 9 A Fourier Transform “Pair”: f(t) F() Re-usable! Scaling Additivity Convolution Time shift time domain Fourier transform
How to do Fourier Transform Leo Lam © 2010-2013 10 Three ways (or use a combination) to do it: –Solve integral –Use FT Properties (“Spiky signals”) –Use Fourier Transform table (for known signals)
FT Properties Example: Leo Lam © 2010-2013 11 Find FT for: We know the pair: So: -8 0 8 G()
More Transform Pairs: Leo Lam © 2010-2013 12 More pairs: time domain Fourier transform
Periodic signals: Transform from Series Leo Lam © 2010-2013 13 Integral does not converge for periodic f n s: We can get it from Fourier Series: How? Find x(t) if Using Inverse Fourier: So
Periodic signals: Transform from Series Leo Lam © 2010-2013 14 We see this pair: More generally, if X(w) has equally spaced impulses: Then: Fourier Series!!!
Periodic signals: Transform from Series Leo Lam © 2010-2013 15 If we know Series, we know Transform Then: Example: We know: We can write:
Leo Lam © 2010-2013 Summary Fourier Transform Pairs FT Properties
Duality of Fourier Transform Leo Lam © 2010-2013 17 Duality (very neat): Duality of the Fourier transform: If time domain signal f(t) has Fourier transform F(), then F(t) has Fourier transform 2 f(-) i.e. if: Then: Changed sign
Duality of Fourier Transform (Example) Leo Lam © 2010-2013 18 Using this pair: Find the FT of –Where T=5
Duality of Fourier Transform (Example) Leo Lam © 2010-2013 19 Using this pair: Find the FT of
Math Review with Matlab:
Lecture 7: Basis Functions & Fourier Series
Chapter 5 The Fourier Transform. Basic Idea We covered the Fourier Transform which to represent periodic signals We assumed periodic continuous signals.
Leo Lam © Signals and Systems EE235. Transformers Leo Lam ©
Properties of continuous Fourier Transforms
EE-2027 SaS 06-07, L11 1/12 Lecture 11: Fourier Transform Properties and Examples 3. Basis functions (3 lectures): Concept of basis function. Fourier series.
Autumn Analog and Digital Communications Autumn
Lecture 8: Fourier Series and Fourier Transform
EE-2027 SaS, L11 1/13 Lecture 11: Discrete Fourier Transform 4 Sampling Discrete-time systems (2 lectures): Sampling theorem, discrete Fourier transform.
Leo Lam © Signals and Systems EE235. Leo Lam © Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to.
EE313 Linear Systems and Signals Fall 2010 Initial conversion of content to PowerPoint by Dr. Wade C. Schwartzkopf Prof. Brian L. Evans Dept. of Electrical.
Leo Lam © Signals and Systems EE235 Lecture 27.
Chapter 4 The Fourier Transform EE 207 Dr. Adil Balghonaim.
Leo Lam © Signals and Systems EE235 Lecture 23.
Leo Lam © Signals and Systems EE235. So stable Leo Lam ©
Leo Lam © Signals and Systems EE235. Leo Lam © x squared equals 9 x squared plus 1 equals y Find value of y.
ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete Objectives: Linearity Time Shift and Time Reversal Multiplication Integration.
Fourier Transforms Section Kamen and Heck.
Outline Fourier transforms (FT) Forward and inverse Discrete (DFT) Fourier series Properties of FT: Symmetry and reciprocity Scaling in time.
© 2020 SlidePlayer.com Inc. All rights reserved.