MIT6_003S10_lec03_handout

MIT6_003S10_lec03_handout - 6.003: Signals and Systems...

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6.003: Signals and Systems Lecture 3 February 9, 2010 6.003: Signals and Systems Last Time: Multiple Representations of DT Systems Verbal descriptions: preserve the rationale. Feedback, Poles, and Fundamental Modes “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 February 9, 2010 Last Time: Feedback, Cyclic Signal Paths, and Modes Systems with signals that depend on previous values of the same signal are said to have feedback . Example: The accumulator system has feedback. Delay + X Y By contrast, the difference machine does not have feedback. 1 Delay + X Y Last Time: Feedback, Cyclic Signal Paths, and Modes The effect of feedback can be visualized by tracing each cycle through the cyclic signal paths. X Delay + p 0 Y x [ n δ [ n ] y [ n ] n n 10123 4 4 Each cycle creates another sample in the output. The response will persist even though the input is transient. Geometric Growth: Poles These unit-sample responses can be characterized by a single number the pole which is the base of the geometric sequence. Delay + p 0 X Y y [ n p n 0 , if n> =0 ; 0 , otherwise. 4 n y [ n ] 4 n y [ n ] 4 n y [ n ] p 0 . 5 p 0 =1 p 0 . 2 Check Yourself How many of the following unit-sample responses can be represented by a single pole? n n n n n 1
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6.003: Signals and Systems Lecture 3 February 9, 2010 Geometric Growth The value of p 0 determines the rate of growth. y [ n ] y [ n ] y [ n ] y [ n ] 1 0 1 z p 0 < 1 : magnitude diverges, alternating sign 1 <p 0 < 0 : magnitude converges, alternating sign 0 0 < 1 : magnitude converges monotonically p 0 > 1 : magnitude diverges monotonically Second-Order Systems The unit-sample responses of more complicated cyclic systems are more complicated. R R 1 . 6 0 . 63 + X Y 1 0 123 4 56 7 8 n y [ n ] Not geometric. This response grows then decays.
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This note was uploaded on 11/07/2011 for the course ELECTRICA 6.003 taught by Professor Staff during the Summer '10 term at MIT.

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MIT6_003S10_lec03_handout - 6.003: Signals and Systems...

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