Download - Continuous-time Signals
Continuous-time Signals
ELEC 309
Prof. Siripong Potisuk
Signal Transformations
Operations Performed on the Independent and Dependent Variables1) Reflection or Time Reversal or Folding2) Time Shifting3) Time Scaling4) Amplitude Scaling5) Amplitude ShiftingNote: The independent variable is assumed to be t representing time.
Time reversal or Folding or Reflection about t = 0
)()( txty
For CT signals, replace t by –t
x(-t) is the reflected version of x(t) obtained from x(t) by a reflection about t =0
Time Shifting (Advance or Delay)
(Advance) by left the toshifted )( 0
(Delay) by right the toshifted )( 0
shift. ofamount theis ),()(
00
00
00
ttxt
ttxt
tttxty
Time Scaling
versionstretched a 10
versioncompressed a 1
)( )(
factor aby )( of version scaled- timea is )(
by Replace
a
a
atxty
atxatx
att
Amplitude Scaling
)()( tAxty
Multiply x(t) by A, where A is the scaling factor
If A is negative, the original signal x(t) is alsoreflected about the horizontal axis.
Amplitude Shifting
Atxty )()(
Add a constant A to x(t), where A is the amount of shift (upward or downward)
Signal Characteristics
Deterministic vs. Random
Finite-length vs. Infinite-length
Right-sided/ Left-sided/ Two-sided
Causal vs. Anti-causal
Periodic vs. Aperiodic (Non-periodic)
Real vs. Complex
Conjugate-symmetric vs. Conjugate-antisymmetric
Even vs. Odd
Even & Odd CT Signals
).()( if ricantisymmet Conjugate
),()( if symmetric Conjugate*
*
txtx
txtx
A complex-valued signal x(t) is said to be
A real-valued signal x(t) is said to be
).()( if odd
),()( ifeven
txtx
txtx
Even-Odd Signal Decomposition
A CT signal can be decomposed into its even and odd parts.
2)()(
(t)
2)()(
(t)
txtxx
txtxx
o
e
CT Periodic Signals
A CT periodic signal x(t) is a function of timethat satisfies the condition
IktkTtxTtxtx , ),()()( 0
T is a positive constant
T0 is the smallest value of T called the fundamental period of x(t)f0 = 1/T0 called the fundamental frequency0 = 2f0 called the angular frequency
A Sum of CT Periodic Signals
number rational a
i.e., , , and )( of
period theis where, if periodic be will
)()()( signal The ly.respective , and
periods with signals periodic be )( and )(Let
2
1
21
2121
21
m
n
T
T
Inmtx
TTnTmT
txtxtxTT
txtx
CT Aperiodic (nonperiodic) Signals
Does not satisfy the condition
IktkTtxTtxtx , ),()()( 0
Periodic extension accomplished by using summation and time-shifting operation
Signal Metrics
Energy
Power
Magnitude
Area
Energy
An infinite-length signal with finite amplitude may or may not have finite energy.
dttxEx2
)(
A finite-length signal with finite amplitude have finite energy.
Power
A finite energy signal with zero average power is called an ENERGY signal.
dttxT
PT
TT
x
2|)(|
2
1lim
An infinite energy signal with finite average power is called a POWER signal.
Average Power
Average over one period for periodic signal, e.g.,
Root-mean-square power:
02 any for |)(|
1 0
0
tdttxT
PTt
t
x
xrms PP
Magnitude & Area
x
tx
M
txM
if Bounded
)(max :Magnitude
)( :Area dttxAx
