Post on 05-Jan-2016
transcript
Chapter 2
Updated 04/20/23
Outline
• Transformation of Continuous-Time Signal– Time Reversal– Time Scaling– Time Shifting– Amplitude Transformation
• Signal Characteristics
Time reversal:
Time Reversal
X(t) Y=X(-t)
Mathematica Example
Shift+<Enter> to execute
Time scaling
TimeScaling
X(t) 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
• 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
TimeShifting
X(t) 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)
u(t+1)- u(t-2)
t=0
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(t)
X(2-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
B>0 Shift upB<0 Shift down
|A|>1 Gain |A|<1 Attenuation
A>0NO reversal A<0 reversal
Reversal
Scaling
Scaling
Y(t)=AX(t)+B Example
Input and Output
Vout, m=46 mVVin, m=1 mV
Define a Piecewise Function in Mathematica
Example 2-1
X(t)
Advance: X(t+1)
Advance & ScalingX(t/2+1)
Advance,scaling &reversalX(-t/2+1)
Signal Characteristics
• Even Function
Xe(-t) = Xe(t)
Signal Characteristics
• Odd Function
Xo(t) =- Xo(-t)
Signal Characteristics
Xe + Ye = ZeXo + Yo = ZoXe + Yo = Ze + Zo
Xe * Ye = ZeXo * Yo = ZeXe * Yo = Zo
Any signal can be represented in terms of a odd function and an even function.
x(t)=xo(t)+xe(t)
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.
• Prove that product of two odd signals is even.
• What is the product of an even signal and an odd signal? Prove it!
)()()(
)()()(
)()()(
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txtxtx
txtxtx
txtxtx
Oddtx
txtxtx
txtxtx
txtxtx
)(
)()()(
)()()(
)()()(
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Change t -t
(even) (odd)