operations on signals

1. Chapter 2
Updated 10/09/14

2. Outline
• Transformation of Continuous-Time
Signal
– Time Reversal
– Time Scaling
– Time Shifting
– Amplitude Transformation
• Signal Characteristics

3. Time reversal:
X(t) Y=X(-t)
Time
Reversal

4. Mathematica Example
Shift+<Enter> to execute

5. Time scaling
X(t) Y=X(at)
Time
Scaling

6. Time scaling
• Given y(t),
– find w(t) = y(3t)
– v(t) = y(t/3).

7. Circuit Example
• LC Tank Oscillator

8. 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
X(t) Y=X(t-to)
Time
Shifting

9. Connection to Circuits

10. Note: Unit Step Function
Unit Step function
(a discontinuous continuous-time signal):

11. Mathematica Example (1)

12. Mathematica Example (2)

13. Draw
• x(t) = u(t+1)- u(t-2)
u(t+1)- u(t-2)
t=0

14. Mathematica Example (2)

15. 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),

16. 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

17. Method 1
Find x(-t) first, then shift t by t-2.

18. Method 2
find x(2+t), then apply time reversal

19. X(2-t)+x(t)
X(2-t)
X(t)
X(2-t)+x(t)

20. Combination of Scaling and
Shifting
Method 1:
Shift then scale

21. Combination of Scaling and
Shifting
Method 2:
Scale then shift

22. Amplitude Operations
In general:
y(t)=Ax(t)+B
B>0  Shift up
B<0  Shift down
|A|>1 Gain
|A|<1 Attenuation
A>0NO reversal
A<0 reversal
Reversal
Scaling
Scaling

23. Y(t)=AX(t)+B Example

24. Input and Output
Vin, m=1 mV Vout, m=46 mV

25. Define a Piecewise Function
in Mathematica

26. Example 2-1
X(t)
Advance: X(t+1)
Advance,scaling
&reversal
X(-t/2+1)
Advance & Scaling
X(t/2+1)

27. Signal Characteristics
• Even Function
Xe(-t) = Xe(t)

28. Signal Characteristics
• Odd Function
Xo(t) =- Xo(-t)

29. 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

30. 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

31. 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!
Change t -t
x t x t x t
= ´ ®
( ) ( ) ( )
1 2
x t x t x t
- = - ´ - =
( ) ( ) ( )
( ) ( ) ( )
1 2
1 2
x t ´ x t =
x t
(even) (odd)
x t x t x t
= ´ ®
( ) ( ) ( )
1 2
x t x t x t
- = - ´ - =
( ) ( ) ( )
x t x t x t
´- = - =
( ) ( ) ( )
1 2
x ( - t )
¬
Odd
1 2