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Signals and Systems Fall 2003 Lecture #3 11 September ) Representation of DT signals in terms of shifted unit samples 2) Convolution sum representation of DT LTI systems 3) Examples 4)The unit sample response and properties of DT LTI systems

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Exploiting Superposition and Time-Invariance Question:Are there sets of basic signals so that: a) We can represent rich classes of signals as linear combinations of these building block signals. b) The response of LTI Systems to these basic signals are both simple and insightful. Fact: For LTI Systems (CT or DT) there are two natural choices for these building blocks Focus for now: DT Shifted unit samples CT Shifted unit impulses

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Representation of DT Signals Using Unit Samples

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That is.. Coefficients Basic Signals SignalsThe Sifting Property of the Unit Sample

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Suppose the system is linear, and define hk[n] as the response to δ[n -k]: From superposition:

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Now suppose the system is LTI, and define the unit sample response h[n]: From TI: From LTI: Convolution Sum

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Convolution Sum Representation of Response of LTI Systems Interpretation Sum up responses over all k

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Visualizing the calculation of Choose value of n and consider it fixed View as functions of k with n fixed prod of overlap for prod of overlap for

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Calculating Successive Values: Shift, Multiply, Sum

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A DT LTI System is completely characterized by its unit sample response Properties of Convolution and DT LTI Systems There are many systems with this repsonse to There is only one LTI Systems with this repsonse to

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Unit Sample response - An Accumulator

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Step response of an LTI system input Unit Sample response of accumulator step input

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The Distributive Property Interpretation

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The Associative Property Implication (Very special to LTI Systems)

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Properties of LTI Systems Causality Stability

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