Signals and Systems Fall 2003 Lecture #3 11 September 2003
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Transcript of Signals and Systems Fall 2003 Lecture #3 11 September 2003
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Signals and SystemsFall 2003 Lecture #3
11 September 2003
1) Representation of DT signals in terms of shifted unit samples 2) Convolution sum representation of DT LTI systems3) Examples4) 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
stepinput
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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