Download presentation

Presentation is loading. Please wait.

Published byMerryl Richards Modified over 4 years ago

1
Leo Lam © 2010-2012 Signals and Systems EE235 Lecture 23

2
Leo Lam © 2010-2012 Today’s menu Fourier Series Example Fourier Transform

3
Leo Lam © 2010-2012 Motivation

4
Fourier Series: Quick exercise Leo Lam © 2010-2012 4 Given: Find its exponential Fourier Series: (Find the coefficients d n and 0 )

5
Fourier Series: Fun examples Leo Lam © 2010-2012 5 Rectified sinusoids Find its exponential Fourier Series: t 0 f(t) =|sin(t)| Expand as exp., combine, integrate

6
Fourier Series: Circuit Application Leo Lam © 2010-2012 6 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)!

7
Fourier Series: Circuit Application Leo Lam © 2010-2012 7 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

8
Fourier Series: Circuit Application Leo Lam © 2010-2012 8 Rectified sinusoids Now we know: LTI system: Transfer function: To frequency: +-+- sin(t) full wave rectifier y(t) f(t)

9
Fourier Series: Circuit Application Leo Lam © 2010-2012 9 Rectified sinusoids Now we know: LTI system: Transfer function: System response: +-+- sin(t) full wave rectifier y(t) f(t)

10
Leo Lam © 2010-2012 Summary Fourier Series circuit example

11
Fourier Series: Dirichlet Conditon Leo Lam © 2010-2012 11 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

12
End of Fourier Series Leo Lam © 2010-2012 12 We have accomplished: –Introduced signal orthogonality –Fourier Series derivation –Approx. periodic signals: –Fourier Series Properties Next: Fourier Transform

13
Fourier Transform: Introduction Leo Lam © 2010-2012 13 Fourier Series: Periodic Signal Fourier Transform: extends to all signals Recall time-scaling:

14
Fourier Transform: Leo Lam © 2010-2012 14 Recall time-scaling: 0 Fourier Spectra for T, Fourier Spectra for 2T,

15
Fourier Transform: Leo Lam © 2010-2012 15 Non-periodic signal: infinite period T 0 Fourier Spectra for T, Fourier Spectra for 2T,

16
Fourier Transform: Leo Lam © 2010-2012 16 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

17
Fourier Transform: Leo Lam © 2010-2012 17 Fourier Formulas compared: Fourier transform coefficients: Fourier transform (arbitrary signals) Fourier series (Periodic signals): Fourier series coefficients: and

18
Fourier Transform (example): Leo Lam © 2010-2012 18 Find the Fourier Transform of What does it look like? If a <0, blows up magnitude varies with phase varies with

Similar presentations

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google