Post on 08-Nov-2014
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1
Singularity functions
1-2-1 The unit-step function
The continuous-time unit-step function
The continuous-time unit-step function is denoted as ( )u t and is defined mathematically by:
( )0, for 0
1, for 0
tu t
t
<= ≥
which have the zero amplitude for all 0t < and the amplitude of 1 for all 0t ≥ , and its plot is shown in Figure 1-10
t
( )u t
1
0
2 Fundamental of signal processing
Figure 1-10: The continuous-time unit step function
The discrete-time unit-step function
The discrete-time unit-step function is denoted as [ ]u n , and is defined mathematically by:
[ ]0 for 1, 2, 3,
1 for 0,1, 2,3, 4,
nu n
n
= − − −= =
and its plot is shown in Figure 1-11.
n
( )u n
1•••••
1 2 31−2−3−4− 0
Figure 1-11: The discrete-time unit step function
The amplitude scaling
If A , is an arbitrary nonzero real number, than ( )Au t is step function with amplitude of A for all 0t ≥ and zero for all 0t < as
( )0, for 0
, for 0
tAu t
A t
<= ≥
and its plot is shown in Figure 1-12.
SIGNALS 3
t
( )Au t
A
0
Figure 1-12: The continuous-time generic step function with amplitude of A .
The causality property of unit step function
The signal ( )f t defined over time domain of t−∞ ≤ ≤ +∞ , starts at t = −∞ . If there is a desire that the signal be in causal form (starts at 0t = ), it can be described as ( ) ( )f t u t . The product ( ) ( )f t u t of any signal ( )f t is equal to ( )f t for all 0t ≥ and 0 for all
0t < is given by:
( ) ( )( )
0, for 0
, for 0
tf t u t
f t t
<= ≥
Note that the signal ( )f t exist over t−∞ < < ∞ and by multiplying the function ( )f t by unit-step function ( )u t , any nonzero value of ( )f t in the time interval of
0t−∞ < < will be forced to zero, and the signal will be turned on at 0t = . The plot of ( ) ( )f t u t is shown in Figure 1.13.
4 Fundamental of signal processing
( )f t
t
t
( ) ( )f t u t
SIGNALS 5
Figure 1-13:
The anti-causality property of unit step function
( )f t
t
t
( ) ( )f t u t−
0
0
6 Fundamental of signal processing
The non-causal signal
The time-shifting operation
The time shift to the right with 0t of unit step function sets a signal to “turn on” at time 0t rather than 0t = .
( ) 00
0
1, for0, for
t tu t t
t t≥
− = <
t
( )0u t t−
1
0t
Figure 1-14: The continuous-time unit step function time shifted to the right by 0t
The time shift to the left with 0t of unit step function sets a signal to “turn on” earlier than at time 0t = at the time 0t− .
( ) 00
0
1, for0, for
t tu t t
t t≥ −
+ = < −
SIGNALS 7
t
( )0u t t+
1
0t− 0
Figure 1-15: The continuous-time unit step function time shifted to the left by 0t
The time reversal
( ) ( ) 00 0
0
1, for( )
0, fort t
u t t u t tt t≤
− = − − = >
t
( )0u t t+
1
0t
Figure 1-16: The continuous-time unit step function time shifted to the right by 0t and time reflected
EXAMPLE 1:
Sketch the generic step function ( )0Au t t− .
SOLUTION:
8 Fundamental of signal processing
The unit step function ( )u t is scaled by scalar factor of A and time shifted by 0t to the right
( ) 00
0
0 t tAu t t
A t t
<− = ≥
t
( )u t
A
0t
Figure 1-17: The continuous-time generic step function.
The symmetrical unit rectangular pulse
The symmetrical unit rectangular pulse can be constructed by two unit step function ( )u t shifted 1 2 to the left and 1 2 to the right.
( ) 1 12 2
t u t u t = + − −
∏
Also alternative presentation is
( )112
0 otherwise
tt
≤=
∏
SIGNALS 9
t
( )t∏
1
12
12
−
Figure 1-17: The continuous-time unit pulse function
Note that the symmetrical unit rectangular pulse is an even function.
EXAMPLE 2:
Write an analytical expression to describe the waveform shown in Figure 1.8.
t
( )f t
2
53
Figure 1-18: The continuous-time pulse function time shifted to the right
SOLUTION:
10 Fundamental of signal processing
t
( )2 3u t −
2
53
t3
( )2 5u t− −
t5+ =
( )f t
35
2
2−
2
t
( )f t
2−
Figure 1-19: The continuous-time pulse function time shifted to the right
( ) ( ) ( )2 3 5f t u t u t= − − −
The signum or sign function
( )1 0
sgn 0 0
1 0
t
t t
t
>= =− <
( ) ( ) ( )sgn t u t u t= − −
SIGNALS 11
t
( )u t
1
1−
Figure 1-20: The continuous-time pulse function time shifted to the right
The unit ramp function
The continuous-time unit ramp function denoted as ( )r t is the integral of the unit step function
( ) ( )t
r t u dτ τ−∞
= ∫
and can be defined as:
( ) ( )0 0
0
tr t tu t
t t
<= = ≥
Alternatively, note that the step function is the derivative of unit ramp function
( ) ( )r tu t
t∂
=∂
12 Fundamental of signal processing
t
( )r t
1 2
12
Figure 1-21: The continuous-time ramp function
The time –shifted to the right unit ramp function
t
( )0r t t−
0t
Figure 1-22: The continuous-time ramp function time shifted to the right by 0t
The time-shifted ramp function having slope of m is denoted by ( )0r mt t− , and mathematically defined by:
SIGNALS 13
( )0
0
000
tmt t for tmr mt ttfor tm
− ≥− = <
and it is plotted in Figure 1-23.
t
( )0r mt t−
0t−
0tm
Figure 1-23: Ramp function with slope of m and time-shifted by 0t .
0
0
0
0
0
y mt tt y t
ty tm
= −= = −
= =
14 Fundamental of signal processing
t
( )S t
2
3
Figure 1-24: The continuous-time saw tooth function
The signal between the interval of 0 2t≤ ≤ is a line between two points
( )1 1 10, 0P t s= = and ( )2 2 22, 3P t s= = given by equation:
( ) ( ) ( )2 11
2 1
3 0 30 for 0 22 0 2
s sS t t t t t tt t− −
= − = − = ≤ ≤− −
( ) 0 otherwiseS t =
( ) ( ) ( )3 22
S t t u t u t= − −
t
( )tΛ
1−
1
1
SIGNALS 15
Figure 1-25: The continuous-time unit triangle function
( )1 for 1 1
0 for 1t t
tt
− − < <Λ = >
EXAMPLE:
t
( )f t
1−
2
1
1P
2P
3P
Figure 1-26:
( )1 1 1, 0P x y= − =
( )2 2 20, 2P x y= =
( )2 11 1
2 1
y yy y x xx x−
− = −−
16 Fundamental of signal processing
( ) ( )2 00 ( 1)0 1
y x−− = − −
− −
( )2 1y x= +
( ) ( )2 1f t t= +
( ) ( ) ( ) ( ) ( )2 1 1 4 2 1 1t u t tu t t u t+ + − + − −
t
( )f t
5
6
97
Figure 1-27:
( ) ( ) ( ) ( ) ( ) ( )6 6 65 5 2 7 7 9 92 2 2t u t t u t t u t − − − − − + − −
SIGNALS 17
The unit Impulse function (Dirac distribution or delta function)
The analytical expression for the unit impulse function is denoted as ( )tδ . Where
( )1 0
0 0
tt
tδ
== ≠
( )( ) 0
0 0
t dt tt
t
δδ
∞
−∞
== ≠
∫
( )unbounded 0
0 0
tt
tδ
== ≠
The impulse function is abstraction of the pulse with an infinitely large amplitude and infinitesimally small pulse width. The unit impulse can be visualized as a pulse with amplitude of 1 ε and width of ε , or as a triangle
( ) 1t dtδ∞
−∞=∫
The unit impulse function ( )tδ is not bounded at 0t =
t
( )tδ
1ε
2ε
2ε
−t
0ε →
1ε
εε− t0ε →
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Figure 1.30: The unit impulse function
( ) ( )0, for 01, for 0
t tu t d
tδ τ τ
−∞
<= = ≥∫
( ) utt
δ∂
=∂
Sampling or sifting properties of a function by an unit impulse function
( ) ( ) ( ) ( ) ( )0 0x t t dt x t dt xδ δ∞ ∞
−∞ −∞= =∫ ∫
( ) ( ) ( )0 0x t t t dt x tδ∞
−∞− =∫
The unit sample sequence (unit impulse sequence) The analytical expression for the unit impulse function is denoted as ( )nδ . Where
( )1, 0
0, 0
nn
nδ
== ≠
SIGNALS 19
n
( )nδ
Figure 1.31: The unit impulse function
properties of the unit impulse function
The sampling (sifting) property
( ) ( ) ( )0 0f t t t dt f tδ+∞
−∞− =∫
( ) ( ) ( ) ( ) ( ) ( ) ( )0 0
0 00 0 0 0 0
t t
t tf t t t dt f t t t dt f t t t dt f tδ δ δ+ +
− −
+∞
−∞− = − = − =∫ ∫ ∫
The time scaling property
( ) ( )1mt tm
δ δ=
( ) ( ) ( )1dmt dt dm mτ
δ δ τ δ τ τ+∞ +∞ +∞
−∞ −∞ −∞= =∫ ∫ ∫
20 Fundamental of signal processing
The real exponential function
The exponential signals are mathematically denoted
The sinusoidal signal
( ) ( )sinS t A t tω ϕ= + −∞ < < ∞
Where A is the amplitude or peak value,ω the angular frequency in radian per second ( )secrad , and ϕ , the phase in radian. The frequency f in Hertz ( )cycle second is
2f ω π= and 1f T= . The sinusoid is periodic with period of 2π ω .
( ) ( )sin 2S t A ft tπ ϕ= + −∞ < < ∞
( ) 2sin tS t A tTπ
ϕ = + −∞ < < ∞
t
( )sin A tω ϕ+
A
T
Sometimes the amplitude is considered as peak-to-peak value that is twice of peak value. The amplitude of a sine wave signal is given as of its root-mean-square (rms) value which is peak value divided by 2 .
SIGNALS 21
2peak
rms
VV =
For example peak value of home power supply in this country is 120 2 volts and rms value of 120 volts.
Frequency f is given in per second ( )1s or Hertz ( )Hz and period T is given in second
( )s .
The phase ϕ is with respect to an arbitrary time reference.
Sinusoidal signal can be
Signals can be represented as the sum of sinusoids. For example square wave signal
A
A−
t
( )f t
( ) 0 0 04 1 1sin sin 3 sin 5 ...
3 5Af t t t tω ω ωπ
= + + +
Where A is the amplitude of the square wave and 0ω is called the fundamental frequency 0
2Tπ
ω = , where T is period of the square wave.
22 Fundamental of signal processing
A
A−
t
( )f t