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

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6.003: Signals and Systems Continuous-Time Systems September 29, 2009

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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.
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

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Multiple Representations of Continuous-Time Systems Today we start to develop similarly useful representations for CT.
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 ...”

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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.
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 ?

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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 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
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 ? 1 time

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Check Yourself ˙ r 1 ( t ) = α ( r 0 ( t ) r 1 ( t )) The rates r 0 ( t ) and r 1 ( t ) have dimensions r 0 ( t ) r 1 ( t ) quantity of water time Quantity of water could be mass or volume, but must be consistent.
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