Transcript Chapter 2 Updated 11/7/2015 Outline • Transformation of Continuous-Time Signal – – – – Time Reversal Time Scaling Time Shifting Amplitude Transformation • Signal Characteristics.

Chapter 2
Updated 11/7/2015
Outline
• Transformation of Continuous-Time
Signal
–
–
–
–
Time Reversal
Time Scaling
Time Shifting
Amplitude Transformation
• Signal Characteristics
Time reversal:
X(t)
Time
Reversal
Y=X(-t)
Mathematica Example
Shift+<Enter> to execute
Time scaling
X(t)
Time
Scaling
Y=X(at)
Time scaling
• Given y(t),
– find w(t) = y(3t)
– v(t) = y(t/3).
Circuit Example
• LC Tank Oscillator
Time Shifting
X(t)
• The original signal x(t) is
shifted by an amount to
.
Time Shift: y(t)=x(t-to)
• X(t)→X(t-to) // to>0 →
Signal Delayed → Shift
to the right
• X(t) → X(t+to) // to<0 →
Signal Advanced →
Shift to the left
Time
Shifting
Y=X(t-to)
Connection to Circuits
Note: Unit Step Function
Unit Step function
(a discontinuous continuous-time signal):
Mathematica Example (1)
Mathematica Example (2)
Draw
• x(t) = u(t+1)- u(t-2)
t=0
u(t+1)- u(t-2)
Mathematica Example (2)
Time Shifting
Example
• Given x(t) = u(t+2) -u(t-2),
– find
• x(t-t0)=
• x(t+t0)=
Answer:
• x(t-t0)= u(t-to+2) -u(t-to-2),
• x(t+t0)= u(t+to+2) -u(t+to-2),
Problem
• Determine x(t) + x(2-t) , where x(t) =
u(t+1)- u(t-2
• Method 1:
– Observation: Rewrite x(2-t) as x(-(t-2))
– Find x(-t) first, then shift t by t-2.
• Method 2:
– Observation: X(2-t) implies time reversal.
– So find x(2+t), then apply time reversal
Method 1
Find x(-t) first, then shift t by t-2.
Method 2
find x(2+t), then apply time reversal
X(2-t)+x(t)
X(2-t)
X(2-t)+x(t)
X(t)
Combination of Scaling and
Shifting
Method 1:
Shift then scale
Combination of Scaling and
Shifting
Method 2:
Scale then shift
Amplitude Operations
In general:
y(t)=Ax(t)+B
Reversal
B>0  Shift up
B<0  Shift down
|A|>1 Gain
|A|<1 Attenuation
Scaling
A>0NO reversal
A<0 reversal
Scaling
Y(t)=AX(t)+B Example
Input and Output
Vin, m=1 mV
Vout, m=46 mV
Define a Piecewise Function
in Mathematica
Example 2-1
X(t)
Advance: X(t+1)
Advance,scaling
&reversal
X(-t/2+1)
Advance & Scaling
X(t/2+1)
Signal Characteristics
• Even Function
Xe(-t) = Xe(t)
Signal Characteristics
• Odd Function
Xo(t) =- Xo(-t)
Signal Characteristics
Any signal can be represented in terms of
a odd function and an even function.
x(t)=xo(t)+xe(t)
Xe + Ye = Ze
Xo + Yo = Zo
Xe + Yo = Ze + Zo
Xe * Ye = Ze
Xo * Yo = Ze
Xe * Yo = Zo
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
Proof Examples
• Prove that product of two
even signals is even.
Change t -t
x(t )  x1 (t )  x2 (t ) 
x(t )  x1 (t )  x2 (t ) 
x1 (t )  x2 (t )  x(t )
• Prove that product of two
odd signals is even.
(even)
• What is the product of an
even signal and an odd
signal? Prove it!
(odd)
x(t )  x1 (t )  x2 (t ) 
x(t )  x1 (t )  x2 (t ) 
x1 (t )   x2 (t )   x(t ) 
x(t )  Odd