Signal and Systems Prof. H. Sameti Chapter #2: 1) Representation of DT signals in terms of shifted unit samples System properties and examples 2) Convolution sum representation of DT LTI systems 3) Examples 4) The unit sample response and properties of DT LTI systems 5) Representation of CT Signals in terms of shifted unit impulses 6) Convolution integral representation of CT LTI systems 7) Properties and Examples 8) The unit impulse as an idealized pulse that is “short enough”: The operational definition of δ(t)

Exploiting Superposition and Time- Invariance Book Chapter#: Section# Computer Engineering Department, Signal and Systems 2

Representation of DT Signals Using Unit Samples Book Chapter#: Section# Computer Engineering Department, Signal and Systems 3

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 4 Coefficients Basic Signals

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 5

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 6

Convolution Sum Representation of Response of LTI Systems Book Chapter#: Section# Computer Engineering Department, Signal and Systems 7

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 8 prod of overlap for prod of overlap for

Calculating Successive Values: Shift, Multiply, Sum Book Chapter#: Section# Computer Engineering Department, Signal and Systems 9

Properties of Convolution and DT LTI Systems Book Chapter#: Section# Computer Engineering Department, Signal and Systems 10

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 11 - An Accumulator Unit Sample response Example 2:

The Commutative Property Book Chapter#: Section# Computer Engineering Department, Signal and Systems 12

The Distributive Property Book Chapter#: Section# Computer Engineering Department, Signal and Systems 13

The Associative Property Book Chapter#: Section# Computer Engineering Department, Signal and Systems 14

Properties of LTI Systems

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 16

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 17 Properties of LTI Systems

Representation of CT Signals Book Chapter#2: Section# Computer Engineering Department, Signal and Systems 18  Approximate any input x(t) as a sum of shifted, scaled pulses

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 19 has unit area

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 20 The Shifting Property of the Unit Impulse limit as

Response of CT LTI system Book Chapter#: Section# Computer Engineering Department, Signal and Systems 21 Impulse response : Taking limits Convolution Integral

Operation of CT Convolution Book Chapter#: Section# Computer Engineering Department, Signal and Systems 22 FlipSlideMultiply Integrate

PROPERTIES AND EXAMPLES Commutativity Shifting property Example: An integrator Step response: Book Chapter#: Section# Computer Engineering Department, Signal and Systems 23 So if input output

DISTRIBUTIVITY Book Chapter#: Section# Computer Engineering Department, Signal and Systems 24

ASSOCIATIVITY Book Chapter#: Section# Computer Engineering Department, Signal and Systems 25

Causality and Stability Computer Engineering Department, Signal and Systems Book Chapter#: Section# 26

The impulse as an idealized “short” pulse Book Chapter#: Section# Computer Engineering Department, Signal and Systems 27 Consider response from initial rest to pulses of different shapes and durations, but with unit area. As the duration decreases, the responses become similar for different pulse shapes.

The Operational Definition of the Unit Impulse δ(t)  δ(t) —idealization of a unit-area pulse that is so short that, for any physical systems of interest to us, the system responds only to the area of the pulse and is insensitive to its duration Book Chapter#: Section# Computer Engineering Department, Signal and Systems 28 Operationally: The unit impulse is the signal which when applied to any LTI system results in an output equal to the impulse response of the system. That is, δ(t) is defined by what it does under convolution. for all h(t)

The Unit Doublet —Differentiator Book Chapter#: Section# Computer Engineering Department, Signal and Systems 29 Impulse response = unit doublet The operational definition of the unit doublet:

Triplets and beyond! Book Chapter#: Section# Computer Engineering Department, Signal and Systems 30 n is number of differentiations n times Operational definitions

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 31 “-1 derivatives" = integral ⇒I.R.= unit step Impulse response: Operational definition: Cascade of n integrators :

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 32 Integrators (continued) the unit ramp More generally, for n>0

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 33 Then E.g. Define n and m can be positive or negative

Sometimes Useful Tricks Book Chapter#: Section# Computer Engineering Department, Signal and Systems 34 Differentiate first, then convolve, then integrate

Example Book Chapter#: Section# Computer Engineering Department, Signal and Systems 35

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 36 Example (continued)