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**Signals and Systems Fall 2003 Lecture #3 11 September 2003**

1) Representation of DT signals in terms of shifted unit samples 2) Convolution sum representation of DT LTI systems 3) Examples 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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**SignalsThe Sifting Property of the Unit Sample**

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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**Properties of Convolution and DT LTI Systems**

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

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- An Accumulator Unit Sample response

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

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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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