CHAPTER 5 Signals, Systems, and Transforms
5.9 The Laplace Transform
DRAFT - Version 2 - The Laplace Transform
This concludes our discussion of the Fourier transform. We will now turn our attention to the more general Laplace trans- form, which shares all the properties of the Fourier transform, and can be applied to wider range of functions.
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5.9.1 Poles, Zeroes, and the Region of convergence
We now examine, in greater depth, the somewhat mysterious notion of a region of convergence by referring to a concrete example. Observe that the unit step signal u(t) is not absolutely integrable (the area under this signal is infinite). Therefore it does not satisfy the Dirichlet conditions and a Fourier transform is not guaranteed. Nevertheless, it can be shown that its Fou- rier transform is . In the next example, we will compute its Laplace transform.
EXAMPLE 20: LAPLACETRANSFORMOFAUNITSTEP
Compute the Laplace transform of the unit step signal.
Solution: By definition, . Now, =0 iff . Assuming this is the case,
we can evaluate the integral as . Thus, . The region of convergence of this trans- form is the set of all values of s where the condition Re(s) > 0 holds. Recall that s is complex, so this is the right half plane of the complex plane.
The next example reinforces this notion.
EXAMPLE 21: LAPLACETRANSFORMOFAREALEXPONENTIAL
Compute the Laplace transform of the signal where a is a real constant.
Solution: By definition, . As before, is 0 iff
, that is, . If this condition holds, then the integral evaluates to . Therefore,
. In this case, the region of convergence is the complex half plane defined by Re(s) > a.
What if s = a? In this case, the denominator becomes zero and the transform’s value is infinite. This is called a pole of the system (the pole in the previous example was at 0).
It is easy to show that the transform pair is valid even for a complex a, where a = as long as .
The values of s for which the transform vanishes are called the zeroes of the transform. This is illustrated by the next exam- ple.
EXAMPLE 22: LAPLACETRANSFORMOFASINUSOID
Compute the Laplace transform of the sinusoid .
Solution: We use Euler’s formula to rewrite the signal as . By definition, πδ ω( ) 1
jω--- +
X s( ) u t( )e–stdt
∞ –
∞
∫
e–stdt0
∞
∫
e---––sst0
= = = ∞ e–s∞ Re s( )>0
1
---s Re s( )>0 u t( ) 1 ---s
↔ Re s( )>0
x t( ) = u t( )eat
X s( ) u t( )eate–stdt
∞ –
∞
∫
e–(s–a)tdt0
∞
∫
e---––((ss––aa)t)0
= = = ∞ e–(s–a)∞
Re s( –a)>0 Re s( )>a 1
s–a
( )
---
u t( )e–at 1 s–a
( )
---
↔ Re s( )>a
σ+jω Re s( ) σ>
u t( )cosω1t
u t( ) ejω1t+e–jω1t ---2
⎝ ⎠
⎛ ⎞
DRAFT - Version 2 - The Laplace Transform
where, in the last step, we used the result from the previous example, and we are assuming that . This reduces to , with the region of convergence is . Note that the transform becomes infinite for (these are the poles of the transform) and is zero for s = 0, which is the zero of the transform.
It is important to keep track of the region of convergence of the transform. Two time-domain functions that are completely different may have an identical transform and may differ only in their region of convergence. Therefore, the Laplace trans- form is unique only if the region is also specified.
If the region of convergence of the Laplace transform of a signal includes the imaginary axis, then the Fourier transform of the signal is defined and can be obtained by setting . Otherwise, the Laplace transform of the signal exists, but not its Fourier transform.
5.9.2 Properties of the Laplace transform
The Fourier transform is a special case of the Laplace transform. Therefore, all the properties of the Fourier transform, namely linearity, time-shifting, time-scaling, duality, differentiation, and the convolution property, also hold for the Laplace transform, though in a slightly different form. We summarize the corresponding properties in Table 3.
Property Pre-condition Post condition Notes
Linearity a and b are arbitrary
constants and can be complex
Time scal- ing
An compression in the time scale expands the fre- quency scale X s( ) u t( ) ejω1t+e–jω1t
---2
⎝ ⎠
⎛ ⎞e–stdt
∞ –
∞
∫
=
ejω1te–stdt
0
∞
∫
e–jω1te–stdt0
∞
∫
+
⎝ ⎠
⎜ ⎟
⎜ ⎟
⎛ ⎞
---2
=
1 2--- 1
s–jω1 --- 1
s+jω1 ---
⎝ + ⎠
⎛ ⎞
=
Re s( )>0 s
s2+ω12
--- Re s( )>0 s = ±jω1
s = jω
x1( )t ↔X1( )s α1<Re s( ) β< 1 x2( )t ↔X2( )s α2<Re s( ) β< 2
ax1( )t +bx2( )t ↔aX1( )s +bX2( )s max(α1,α2)<Re s( )<min(β1,β2)
x t( )↔X s( ) α<Re s( ) β< x at( ) 1 ---Xa s
a---
⎝ ⎠⎛ ⎞
↔
aα<Re s( )<aβ
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Frequency scaling
An compression in the frequency scale expands the time scale
Time shift- ing
Delaying by a time t0 multiplies the transform by Frequency
shifting
Note the change in the region of con- vergence due to fre- quency shifting Differenti-
ation
Differentiation in the time domain corresponds to mul- tiplication by a fac- tor of s in the transform domain
Integration Integration in the
time domain corre- sponds to division by a factor of s in the transform domain. Note that the region of con- vergence also changes Convolu-
tion in time domain
y(t) is a convolu- tion of two func- tions x1(t) and x2(t).
The product of their transforms X1(s) and Xs(s) determine Y(s), the transform of y(t), and its region of conver- gence
Property Pre-condition Post condition Notes
x t( )↔X s( ) α<Re s( ) β< 1 ---xa t
a---
⎝ ⎠⎛ ⎞ ↔X as( ) α
---a Re s( ) β a---
< <
x t( )↔X s( ) α<Re s( ) β<
x t( –t0)↔e–st0X s( )
α<Re s( ) β< e–st0 x t( )↔X s( ) α<Re s( ) β< eatx t( )↔X s( –a)
α–Re a( )<Re s( ) β< –Re a( ) x t( )↔X s( ) α<Re s( ) β< dnx t( )
dtn
---↔snX s( ) α<Re s( ) β<
x t( )↔X s( ) α<Re s( ) β<
x
∞ –
t
∫
( )drr ↔X s---( )s max(α,0)<Re s( ) β<x1( )t ↔X1( )s α1<Re s( ) β< 1
x2( )t ↔X2( )s α2<Re s( ) β< 2 y t( ) = x1( )t ⊗x2( )t
y t( )↔Y s( ) Y s( ) = X1( )X ss ( )2 max(α1,α2)<Re s( )<min(β1,β2)
DRAFT - Version 2 - The Laplace Transform
TABLE 3. Properties of the Laplace transform
These properties, along with the table of common transforms (Table 4) allow us to derive the transform of most common sig- nals without having to derive them from first principles. Note that these transforms are defined for functions that exist only for t > 0, so that the Laplace integral has limits from 0 to . This is also called the unilateral Laplace transform. In a practi- cal system, a causal signal is 0 for t < 0, so the unilateral transform suffices for all practical systems.
TABLE 4. Some standard Laplace transforms
EXAMPLE 23: SOLVINGASYSTEMUSINGTHE LAPLACETRANSFORM Multiplica-
tion in time domain
y(t) is a product of two functions x1(t) and x2(t). The con- volution of their transforms X1(s) and Xs(s) determine Y(s), the transform of y(t), and its region of conver- gence
Final value theorem
The limiting value of x(t) in the time domain as is given by finding the limit of sX(s) as
in the trans- form domain
No. Signal x(t) X(s) Region of convergence
1 Delta or unit impulse 1 All s
2 Unit step u(t) 1/s Re(s) > 0
3 Delayed delta All s
4 Ramp tu(t) Re(s) > 0
5 Exponential decay
6 Nth power decay
7 Sine Re(s) > 0
8 Cosine Re(s) > 0
9 Exponentially modu- lated sine
10 Exponentially modu- late cosine
Property Pre-condition Post condition Notes
x1( )t ↔X1( )s α1<Re s( ) β< 1
x2( )t ↔X2( )s α2<Re s( ) β< 2 y t( ) = x1( )t x2( )t
y t( )↔Y s( ) Y s( ) 1
2πj ---X
1( )s ⊗X2( )s
=
α1+α2<Re s( ) β< 1+β2
X s( ) lim x t( )
t→∞ lim sX s( ) s→0
=
t→∞
s→0
∞
δ( )t
δ(t–t0) e–t0s 1 s2 ---
eαtu t( ) 1
s–α
--- Re s( ) α>
tn
n!---eαtu t( ) 1
s–α ( )n+1
--- Re s( ) α>
ωt ( )u t( )
sin ω
s2+ω2 --- ωt
( )
cos u t( ) s
s2+ω2 --- eαtsin( )u tωt ( ) ω s–α ( )2+ω2
--- Re s( ) α>
eαtcos( )ωt u t( ) s–α s–α ( )2+ω2
--- Re s( ) α>
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153
Consider a system such that the Laplace transform H(s) of its transfer function h(t) is . What is its response to the input
?
Solution: Since , from Row 5 of Table 4, . Therefore, . Using the
method of partial fractions (i.e., writing the expression as and solving for a and b - also see Section 8.12 on page 255), we find that . To find the inverse transform, recall that the Laplace transform is linear, so we only need to find the inverse transform of each term in isolation. From Row 5 of Table 4 again, we get
.
This analysis assumes that the system is at rest at time zero. If this is not the case, then the actual response would be the sum of the natural response, which is the way the system behaves assuming that there is no external input, and the forced response, which is the way the system behaves assuming that its initial state is at rest. The details of this analysis are beyond the scope of this text.
This concludes our study of the Laplace transform. We will now focus on two transforms that deal with discrete signals rather then continuous signals, as we have done so far.