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lec03 - 6.003 Signals and Systems Feedback Poles and...

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6.003: Signals and Systems Feedback, Poles, and Fundamental Modes September 17, 2009
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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
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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
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Check Yourself What value of p 0 represents the signal below?
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