Transcript Signal Characteristics Common Signal in Engineering Singularity Function Section 2.2-2.3 Signal Characteristics • • • • Review Even function Odd function Periodic Signal.

Signal Characteristics
Common 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 real
2. C is complex and a is imaginary
3. 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(at-1)=u(t-1/a)
u(t-1/2)
u(t)=1-u(-t)
Multiple Plots Using
Mathematica