lecture24

# lecture24 - – Make impulse response of a discrete-time...

This preview shows pages 1–4. Sign up to view the full content.

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 and Computer Engineering The University of Texas at Austin Z and Laplace Transforms

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

View Full Document
18 - 2 Z and Laplace Transforms Transform difference/differential equations into algebraic equations that are easier to solve Are complex-valued functions of a complex frequency variable Laplace: s = σ + j 2 π f Z : z = r e j ϖ Transform kernels are complex exponentials Laplace: e s t = e σ t + j 2 π f t = e σ t e j 2 π f t Z : z k = ( r e j ϖ ) k = r k e j ϖ k dampening factor oscillation term
18 - 3 Z and Laplace Transforms No unique mapping from Z to Laplace domain or from Laplace to Z domain Mapping one complex domain to another is not unique One possible mapping is impulse invariance

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

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

Unformatted text preview: – Make impulse response of a discrete-time linear time-invariant (LTI) system be a sampled version of the continuous-time LTI system. H ( z ) y [ k ] f [ k ] Z H ( s ) ( 29 t f ~ ( 29 t y ~ Laplace T s e z z H s H | ) ( ) ( = = 18 - 4 Impulse Invariance Mapping • Impulse invariance mapping is z = e s T Laplace Domain Z Domain Left-hand plane Inside unit circle Imaginary axis Unit circle Right-hand plane Outside unit circle 1 Im{ z } Re{ z } s = -1 ± j ⇒ z = 0.198 ± j 0.31 ( T = 1 s) s = 1 ± j ⇒ z = 1.469 ± j 2.287 ( T = 1 s) π 1 2 1 1 max max ⇒ = ⇒ = s f f ω 1 1-1-1 Im{ s } Re{ s } f j s 2 = lowpass, highpass bandpass, bandstop allpass, or notch filter?...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern