lec06 - 6.003: Signals and Systems Continuous-Time Systems...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
6.003: Signals and Systems Continuous-Time Systems September 29, 2009
Background image of page 1

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

View Full DocumentRight Arrow Icon
Mid-term Examination #1 Wednesday, October 7, 7:30-9:30pm, Walker Memorial. No recitations on the day of the exam. Coverage: DT Signals and Systems Lectures 1–5 Homeworks 1–4 Homework 4 will include practice problems for mid-term 1. However, it will not collected or graded. Solutions will be posted. Closed book: 1 page of notes ( 8 1 2 × 11 inches; front and back). Designed as 1-hour exam; two hours to complete. Review sessions during open office hours. Conflict? Contact [email protected] before Friday, October 2, 5pm.
Background image of page 2
Multiple Representations of Discrete-Time Systems We have developed several useful representations for DT systems. Verbal descriptions: preserve the rationale. “To reduce the number of bits needed to store a sequence of large numbers that are nearly equal, record the first number, and then record successive differences.” Difference equations: mathematically compact. y [ n ] = x [ n ] x [ n 1] Block diagrams: illustrate signal flow paths. 1 Delay + x [ n ] y [ n ] Operator representations: analyze systems as polynomials. Y = (1 −R ) X
Background image of page 3

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

View Full DocumentRight Arrow Icon
Multiple Representations of Continuous-Time Systems Today we start to develop similarly useful representations for CT.
Background image of page 4
Representing Continuous-Time Systems As with DT systems, we begin with verbal descriptions. Example: Leaky Tank r 0 ( t ) r 1 ( t ) h 1 ( t ) Verbal descriptions of continuous-time systems typically specify relations among signals (e.g., inputs and outputs) and their rates of change. “Water flows into a tank at rate r 0 ( t ) and flows out at a rate proportional to the depth of water in the tank. Determine . ..”
Background image of page 5

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

View Full DocumentRight Arrow Icon
Representing Continuous-Time Systems Translate the verbal description into differential equations. r 0 ( t ) r 1 ( t ) h 1 ( t ) Assume r 1 ( t ) hydraulic pressure at the bottom of the tank: r 1 ( t ) ρgh 1 ( t ) Height h 1 ( t ) will grow in proportion to r 0 ( t ) r 1 ( t ) : ˙ h 1 ( t ) r 0 ( t ) r 1 ( t ) Combining first equation with second: ˙ r 1 ( t ) ˙ h 1 ( t ) r 0 ( t ) r 1 ( t ) ˙ r 1 ( t ) = α ( r 0 ( t ) r 1 ( t ) ) Derivatives play key role in CT, analogous to time shifts in DT.
Background image of page 6
Check Yourself ˙ r 1 ( t ) = α ( r 0 ( t ) r 1 ( t ) ) r 0 ( t ) r 1 ( t ) h 1 ( t ) What are the dimensions of α ?
Background image of page 7

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

View Full DocumentRight Arrow Icon
Check Yourself ˙ r 1 ( t ) = α ( r 0 ( t ) r 1 ( t )) Flow rates r 0 ( t ) and r 1 ( t ) have dimensions r 0 ( t ) r 1 ( t ) ± quantity of water time ² Quantity of water could be mass or volume, but must be consistent. The derivative of r 1 ( t ) has dimensions ˙ r 1 ( t ) quantity of water time time Therefore, α has dimensions α ± 1 time ²
Background image of page 8
Check Yourself ˙ r 1 ( t ) = α ( r 0 ( t ) r 1 ( t ) ) r 0 ( t ) r 1 ( t ) h 1 ( t ) What are the dimensions of α ?
Background image of page 9

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

View Full DocumentRight Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 08/24/2011 for the course EECS 6.003 taught by Professor Dennism.freeman during the Spring '11 term at MIT.

Page1 / 47

lec06 - 6.003: Signals and Systems Continuous-Time Systems...

This preview shows document pages 1 - 10. Sign up to view the full document.

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