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. SignalsThe Sifting Property of the Unit Sample
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