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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