2. Outline
⢠Transformation of Continuous-Time
Signal
â Time Reversal
â Time Scaling
â Time Shifting
â Amplitude Transformation
⢠Signal Characteristics
8. Time Shifting
⢠The original signal x(t) is
shifted by an amount to
.
Time Shift: y(t)=x(t-to)
⢠X(t)âX(t-to) // to>0 â
Signal Delayed â Shift
to the right
⢠X(t) â X(t+to) // to<0 â
Signal Advanced â
Shift to the left
X(t) Y=X(t-to)
Time
Shifting
15. Time Shifting
Example
⢠Given x(t) = u(t+2) -u(t-2),
â find
â˘x(t-t0)=
â˘x(t+t0)=
Answer:
â˘x(t-t0)= u(t-to+2) -u(t-to-2),
â˘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:
â Observation: Rewrite x(2-t) as x(-(t-2))
â Find x(-t) first, then shift t by t-2.
⢠Method 2:
â Observation: X(2-t) implies time reversal.
â So find x(2+t), then apply time reversal
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
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 tď -t
x t x t x t
= ´ Ž
( ) ( ) ( )
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
= ´ Ž
( ) ( ) ( )
1 2
x t x t x t
- = - ´ - =
( ) ( ) ( )
x t x t x t
´- = - =
( ) ( ) ( )
1 2
x ( - t )
ÂŹ
Odd
1 2
Hinweis der Redaktion
Example: Playing a tape recorder backward
Beatle Revolution 9 is said be played backward!
Example: fast forwarding / slow playing
Example Delay â at the air port / radio stations / LRC circuits
Time advancing is not physically realizble!