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lec03-handout-6up - 6.003: Signals and Systems Lecture 3...

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6.003: Signals and Systems Lecture 3 September 17, 2009 1 6.003: Signals and Systems Feedback, Poles, and Fundamental Modes September 17, 2009 Last Time: Multiple Representations of 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 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. Delay + p 0 X Y 1 0 1 2 3 4 n x [ n ] = δ [ n ] 1 0 1 2 3 4 n y [ n ] Each cycle creates another sample in the output. The response will persist even though the input is transient. Geometric Growth: Poles These system 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. 1 0 1 2 3 4 n y [ n ] 1 0 1 2 3 4 n y [ n ] 1 0 1 2 3 4 n y [ n ] p 0 = 0 . 5 p 0 = 1 p 0 = 1 . 2 Check Yourself What value of p 0 represents the signal below? y [ n ] n 1. p 0 = 0 . 8 2. p 0 = 0 . 8 3. p 0 = 0 . 64 interspersed with p 0 = 0 . 64 4. none of the above
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6.003: Signals and Systems Lecture 3 September 17, 2009 2 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 < p 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
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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.

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lec03-handout-6up - 6.003: Signals and Systems Lecture 3...

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