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Signals and Systems Fall 2003 Lecture #3 11 September 2003

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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Signals and Systems Fall 2003 Lecture #3 11 September 2003

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  1. 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

  2. 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

  3. Representation of DT Signals Using Unit Samples

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

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

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

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

  8. 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

  9. Calculating Successive Values: Shift, Multiply, Sum

  10. 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

  11. - An Accumulator Unit Sample response

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

  13. The Distributive Property Interpretation

  14. The Associative Property Implication (Very special to LTI Systems)

  15. Properties of LTI Systems Causality Stability

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