Signal CharacteristicsCommon Signal in Engineering

Singularity Function

Section 2.2-2.3

Signal Characteristics

• Review• Even function• Odd function• Periodic Signal

Represent xe(t) in terms of x(t)

• Xe(t)– X(t)=Xe(t)+Xo(t)

– Xe(t)=X(t)-Xo(t)• Xo(t)=-Xo(-t)• X(-t)=Xe(-t)+Xo(-t)

– Xe(t)=X(t)-Xo(-t)=X(t)+X(-t)-Xe(-t)

• Therefore Xe(t)=[X(t)+X(-t)]/2• Similarly Xo(t)=[X(t)-X(-t)]/2

Even Function Example(1)

• Xe(t)=X(t)+Xo(t)

– X(t) is the sum of an even part and an odd part. (X(t)=Xe(t)+Xo(t))

– Let X(t) be a unit step function

Even Function Example(2)

X(t) X(-t)

(X(t)+X(-t))/2 gives you an even function!

Odd Function Example(1)

X(t) X(-t)

(X(t)-X(-t))/2 gives you an odd function!

Odd Function Example

• Mathemtica function: – Use Exp[-t/2] to represent exponential– Use UnitStep[t] to zero out t<0

• Generate an odd and an even function

Answer

Periodic Signal

• X(t) is period if X(t)=X(t+T), T>0– T is the period– To is the minimum value of T that satisfies the

definition• A signal that is not period is aperiodic.

To

Is This Signal Periodic?

A Systematic Procedure

The sum of continuous-time periodic signal is period if and only if the ratios of the periods of the individual signals are ratios of integers Example: x(t)=x1(t)+x2(t)+x3(t)

Is This Signal Periodic?

x(t)=x1(t)+x2(t)+x3(t)+x4(t)

π is irrational, aperiodic

Common Signals in Engineering

X(t)=Ceat occurs frequently in circuits!C and a can be complex!1. C and a are real2. C is complex and a is imaginary3. C and a are complex

Euler’s Formula

Mathematica Example

Complex Exponential in Polar Form

Case 1: C and a are real

(Bacterial growth)

τ=𝐿 /𝑅

Case 2: C=complex, a is imaginary

Application Example

Case 3: C=Complex and a=complex

Singular Functions

• Unit Step Function• Rectangular Function• Impulse Response

Unit Step Function

Properties of Unit Step Function

u(2t-1)

u(t-1/2)

u(at-1)=u(t-1/a)

u(t)=1-u(-t)

Multiple Plots Using Mathematica