signal and system lecture a
TRANSCRIPT
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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 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 forthese 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
The Sifting Property of the Unit Sample
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DT Systemx[n] y[n]
Suppose the system is linear, and define hk
[n] as the
response to [n - k]:
From superposition:
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DT Systemx[n] y[n]
Now suppose the system is LTI, and define the unit
sample response h[n]:
From LTI:
From TI:
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Convolution Sum Representation of
Response of LTI Systems
Interpretation
n n
n n
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Visualizing the calculation of
y[0] = prod ofoverlap for
n = 0
y[1] = prod ofoverlap for
n = 1
Choose value ofn and consider it fixed
View as functions ofk with n fixed
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Calculating Successive Values: Shift, Multiply, Sum
-1
1 1 = 1
(-1) 2 + 0 (-1) + 1 (-1) = -3
(-1) (-1) + 0 (-1) = 1
(-1) (-1) = 1
4
0 1 + 1 2 = 2
(-1) 1 + 0 2 + 1 (-1) = -2
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Properties of Convolution and DT LTI Systems
1) A DT LTI System is completely characterized by its unit sample
response
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Unit Sample response
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The Commutative Property
Ex: Step response s[n] 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
1) Causality
2) Stability