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 byMarshall Warren
Modified over 4 years ago
Leo Lam © 2010-2011 Signals and Systems EE235
Leo Lam © 2010-2011 Speed of light
Leo Lam © 2010-2011 Today’s menu Fourier Series (Exponential form) Fourier Transform!
Fourier Series: Circuit Application Leo Lam © 2010-2011 4 Rectified sinusoids Now we know: Circuit is an LTI system: Find y(t) Remember: +-+- sin(t) full wave rectifier y(t) f(t) Where did this come from? S Find H(s)!
Fourier Series: Circuit Application Leo Lam © 2010-2011 5 Finding H(s) for the LTI system: e st is an eigenfunction, so Therefore: So: Shows how much an exponential gets amplified at different frequency s
Fourier Series: Circuit Application Leo Lam © 2010-2011 6 Rectified sinusoids Now we know: LTI system: Transfer function: To frequency: +-+- sin(t) full wave rectifier y(t) f(t)
Fourier Series: Circuit Application Leo Lam © 2010-2011 7 Rectified sinusoids Now we know: LTI system: Transfer function: System response: +-+- sin(t) full wave rectifier y(t) f(t)
Fourier Series: Dirichlet Conditon Leo Lam © 2010-2011 8 Condition for periodic signal f(t) to exist has exponential series: Weak Dirichlet: Strong Dirichlet (converging series): f(t) must have finite maxima, minima, and discontinuities in a period All physical periodic signals converge
End of Fourier Series Leo Lam © 2010-2011 9 We have accomplished: –Introduced signal orthogonality –Fourier Series derivation –Approx. periodic signals: –Fourier Series Properties Next: Fourier Transform
Fourier Transform: Introduction Leo Lam © 2010-2011 10 Fourier Series: Periodic Signal Fourier Transform: extends to all signals Recall time-scaling:
Fourier Transform: Leo Lam © 2010-2011 11 Recall time-scaling: 0 Fourier Spectra for T, Fourier Spectra for 2T,
Fourier Transform: Leo Lam © 2010-2011 12 Non-periodic signal: infinite period T 0 Fourier Spectra for T, Fourier Spectra for 2T,
Fourier Transform: Leo Lam © 2010-2011 13 Fourier Formulas: For any arbitrary practical signal And its “coefficients” (Fourier Transform): F( ) is complex Rigorous math derivation in Ch. 4 (not required reading, but recommended) Time domain to Frequency domain
Fourier Transform: Leo Lam © 2010-2011 14 Fourier Formulas compared: Fourier transform coefficients: Fourier transform (arbitrary signals) Fourier series (Periodic signals): Fourier series coefficients: and
Fourier Transform (example): Leo Lam © 2010-2011 15 Find the Fourier Transform of What does it look like? If a <0, blows up magnitude varies with phase varies with
Fourier Transform (example): Leo Lam © 2010-2011 16 Fourier Transform of Real-time signals magnitude: even phase: odd magnitudephase
Fourier Transform (Symmetry): Leo Lam © 2010-2011 17 Real-time signals magnitude: even – why? magnitude Even magnitude Odd phase Useful for checking answers
Fourier Transform/Series (Symmetry): Leo Lam © 2010-2011 18 Works for Fourier Series, too! Fourier transform (arbitrary practical signal) Fourier series (periodic functions) Fourier coefficients Fourier transform coefficients magnitude: even & phase: odd
Fourier Transform (example): Leo Lam © 2010-2011 19 Fourier Transform of F( ) is purely real F() for a=1
Leo Lam © 2010-2011 Summary Fourier Transform intro Inverse etc.
Fourier Transform Periodicity of Fourier series
Signals and Systems Fall 2003 Lecture #5 18 September Complex Exponentials as Eigenfunctions of LTI Systems 2. Fourier Series representation of.
Leo Lam © Signals and Systems EE235 Leo Lam.
Lecture 7: Basis Functions & Fourier Series
Leo Lam © Signals and Systems EE235 Lecture 16.
Leo Lam © Signals and Systems EE235. Transformers Leo Lam ©
EECS 20 Chapter 8 Part 21 Frequency Response Last time we Revisited formal definitions of linearity and time-invariance Found an eigenfunction for linear.
Lecture 8: Fourier Series and Fourier Transform
Lecture 14: Laplace Transform Properties
Lecture 12: Laplace Transform
University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2006 Don Fussell Fourier Transforms.
Leo Lam © Signals and Systems EE235. Leo Lam © Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to.
Leo Lam © Signals and Systems EE235 Lecture 27.
Chapter 15 Fourier Series and Fourier Transform
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.
Leo Lam © Signals and Systems EE235. Leo Lam © Merry Christmas! Q: What is Quayle-o-phobia? A: The fear of the exponential (e).
© 2020 SlidePlayer.com Inc. All rights reserved.