rectangular function impulse function continuous time systems 2.4 &2.6

Rectangular FunctionImpulse 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/2 T/2

1

-T/2 T/2

1

-T/2 T/2

1

-T/2 T/2

Application

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

c03f10A 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/42. 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

0 to

d(t)

d(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

Property

Shifted Unit Step Function

Slope is sharp at t=2

Property

Property

g(at), a>1, e.g. 2

area: 1/ ε ε /2=1/21/ ε 2ε=2

to/2 to/2+ ε/2

1/ ε

g(2t),

11/2

δ(t)

Property

g(at), a<1, e.g. 1/2

area: 1/ ε 2ε=2

2to 2to+ 2ε

1/ ε

g(2t),

1

δ(t) 2

2to

Property

d(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=1Vout=-(V1+V2)

Multiplier

Parallel Connection

Cascade Connection

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