Rectangular Function Impulse Function Continuous Time Systems 2.4 &2.6

How do you represent a unit rectangular function mathematically?

Fundamental(1) u(t)

Fundamental(2) u(t-0.5) u(-t-0.5) u(t+0.5) u(-t+0.5)

Block Function (window) rect(t/T) Can be expressed as u(T/2-t)-u(-T/2-t) – Draw u(t+T/2) first; then reverse it! Can be expressed as u(t+T/2)-u(t-T/2) Can be expressed as u(t+T/2)u(T/2-t) -T/2T/2 1 -T/2T/2 1 -T/2T/2 1 -T/2T/2

Application The rectangular pulse can be used to extract part of a signal

A Simple Cell Phone Charger Circuit (R1 is necessary) Another Application: Signal strength indicator

Mathematical Modeling Modify the unit rectangular pulse: 1.Shift to the right by To/4 2.The period is To/2 V1(t)V1(t-To)V1(t-2To)

Application of Impulse Function The unit impulse function is used to model sampling operation, i.e. the selection of a value of function at a particular time instant using analog to digital converter.

Generation of an Impulse Function Ramp function epsilon approaches 0

Shifted Impulse Function 0 0to  (t)  (t-to)

The Impulse Function We use a vertical arrow to represent 1/ε because g(t) Increases dramatically as ε approaches 0.

Another Definition of the Impulse Function

Mathematica Connection

Property f[t] f[t-2]

Property

Shifted Unit Step Function Slope is sharp at t=2

Property

g(at), a>1, e.g. 2 area: 1/ ε ε /2=1/2 1/ ε 2ε=2 t o /2 t o /2+ ε/2 1/ ε g(2t), 1 1/2 δ(t)

Property g(at), a<1, e.g. 1/2 area: 1/ ε 2ε=2 2t o 2t o + 2ε 1/ ε g(2t), 1 δ(t) 2 2t o

Property  (t)

Example

A system is an operation for which cause-and-effect relationship exists – Can be described by block diagrams – Denoted using transformation T[.] System behavior described by mathematical model Continuous-Time Systems T [.] X(t)y(t) (meat grinder)

Inverting Amplifier Vout=-(R1/R2)

Inverting Summer Example Vout=-RF(V1/R1+V2/R2) If RF/R1=1, RF/R2=1 Vout=-(V1+V2)

Multiplier

Parallel Connection

Cascade Connection

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