eduwavepool.unizwa.edu.om€¦ · signals and systems with matlab applications, second edition 2-1...

Signals and Systems with MATLAB Applications, Second Edition 2-1

Orchard Publications

Chapter 2

The Laplace Transformation

his chapter begins with an introduction to the Laplace transformation, definitions, and proper-

ties of the Laplace transformation. The initial value and final value theorems are also discussed

and proved. It concludes with the derivation of the Laplace transform of common functions

of time, and the Laplace transforms of common waveforms.

2.1 Definition of the Laplace Transformation

The two-sided or bilateral Laplace Transform pair is defined as

(2.1)

(2.2)

where denotes the Laplace transform of the time function , denotes the

Inverse Laplace transform, and is a complex variable whose real part is , and imaginary part ,

that is, .

In most problems, we are concerned with values of time greater than some reference time, say

, and since the initial conditions are generally known, the two-sided Laplace transform

pair of (2.1) and (2.2) simplifies to the unilateral or one-sided Laplace transform defined as

(2.3)

(2.4)

The Laplace Transform of (2.3) has meaning only if the integral converges (reaches a limit), that is, if

(2.5)

To determine the conditions that will ensure us that the integral of (2.3) converges, we rewrite (2.5)

T

L f t !" # F s != f t !$–

$

% est–

dt=

L1–

F s !" # f t !=1

2&j-------- F s !

' j(–

' j(+

% est

ds=

L f t !" # f t ! L1–

F s !" #

s ' (

s ' j(+=

t

t t0 0= =

L f t !" # F= s ! f t !t0

$

% est–

dt f t !0

$

% est–

dt= =

L1–

F s !" # f= t !1

2&j-------- F s !

' j(–

' j(+

% est

ds=

f t !0

$

% est–

dt $)

Chapter 2 The Laplace Transformation

2-2 Signals and Systems with MATLAB Applications, Second Edition


as

(2.6)

The term in the integral of (2.6) has magnitude of unity, i.e., , and thus the condition

for convergence becomes

(2.7)

Fortunately, in most engineering applications the functions are of exponential order*. Then, we

can express (2.7) as,

(2.8)

and we see that the integral on the right side of the inequality sign in (2.8), converges if .

Therefore, we conclude that if is of exponential order, exists if

(2.9)

where denotes the real part of the complex variable .

Evaluation of the integral of (2.4) involves contour integration in the complex plane, and thus, it will

not be attempted in this chapter. We will see, in the next chapter, that many Laplace transforms can

be inverted with the use of a few standard pairs, and therefore, there is no need to use (2.4) to obtain

the Inverse Laplace transform.

In our subsequent discussion, we will denote transformation from the time domain to the complex

frequency domain, and vice versa, as

(2.10)

2.2 Properties of the Laplace Transform

1. Linearity Property

The linearity property states that if

have Laplace transforms

* A function is said to be of exponential order if .

f t !e "t–

0

#

$ ej%t–

dt #&

ej%t–

ej%t–

1=

f t !e "t–

0

#

$ dt #&

f t !

f t ! f t ! ke"0t

for all t 0'&

f t !e "t–

0

#

$ dt ke"0t

e"t–

0

#

$ dt&

" "0(

f t ! L f t !) *

Re s) * " "0(=

Re s) * s

f t ! F s !+

f1 t ! f2 t ! , fn t !- - -



Properties of the Laplace Transform

respectively, and

are arbitrary constants, then,

(2.11)

Proof:

Note 1:

It is desirable to multiply by to eliminate any unwanted non-zero values of for .

2. Time Shifting Property

The time shifting property states that a right shift in the time domain by units, corresponds to mul-

tiplication by in the complex frequency domain. Thus,

(2.12)

Proof:

(2.13)

Now, we let ; then, and . With these substitutions, the second integral

on the right side of (2.13) becomes

3. Frequency Shifting Property

The frequency shifting property states that if we multiply some time domain function by an

exponential function where a is an arbitrary positive constant, this multiplication will produce a

shift of the s variable in the complex frequency domain by units. Thus,

F1 s ! F2 s ! " Fn s !# # #

c1 c2 " cn# # #

c1 f1 t ! c2 f2 t ! " cn

fn

t !+ + + c1 F1 s ! c2 F2 s ! " cn

Fn

s !+ + +$

L c1 f1 t ! c2 f2 t ! " cn fn t !+ + +% & c1 f1 t ! c2 f2 t ! " cn fn t !+ + +' (t0

)

* dt=

c1 f1 t !t0

)

* est–

dt c2 f2 t !t0

)

* est–

dt " + cn fn t !t0

)

* est–

dt+ +=

c1 F1 s ! c2 F2 s ! " cn Fn s !+ + +=

f t ! u0 t ! f t ! t 0+

a

eas–

f t a– !u0 t a– ! eas–

F s !$

L f t a– !u0 t a– !% & 00

a

* est–

dt f t a– !a

)

* est–

dt+=

t a– ,= t , a+= dt d,=

f , !0

)

* es , a+ !–

d, eas–

f , !0

)

* es,–

d, eas–

F s != =

f t !

eat–

a




(2.14)

Proof:

Note 2:

A change of scale is represented by multiplication of the time variable by a positive scaling factor

. Thus, the function after scaling the time axis, becomes .

4. Scaling Property

Let be an arbitrary positive constant; then, the scaling property states that

(2.15)

Proof:

and letting , we get

Note 3:

Generally, the initial value of is taken at to include any discontinuity that may be present

at . If it is known that no such discontinuity exists at , we simply interpret as .

5. Differentiation in Time Domain

The differentiation in time domain property states that differentiation in the time domain corresponds

to multiplication by in the complex frequency domain, minus the initial value of at .

Thus,

(2.16)

Proof:

eat–

f t ! F s a+ !"

L eat–

f t !# $ eat–

f t !0

%

& est–

dt f t !0

%

& es a+ ! t–

dt F s a+ != = =

t

a f t ! f at !

a

f at !1

a---F

s

a---

' () *"

L f at !# $ f at !0

%

& est–

dt=

t + a,=

L f at !# $ f + !0

%

& es + a, !–

d+a---

' () * 1

a--- f + !

0

%

& es a, ! +–

d + !1

a---F

s

a---

' () *

= = =

f t ! t 0-

=

t 0= t 0-

= f 0- ! f 0 !

s f t ! t 0-

=

f ' t !d

dt----- f t != sF s ! f 0

- !–"

L f ' t !# $ f ' t !0

%

& est–

dt=




Using integration by parts where

(2.17)

we let and . Then, , , and thus

The time differentiation property can be extended to show that

(2.18)

(2.19)

and in general

(2.20)

To prove (2.18), we let

and as we found above,

Then,

Relations (2.19) and (2.20) can be proved by similar procedures.

We must remember that the terms , and so on, represent the initial conditions.

Therefore, when all initial conditions are zero, and we differentiate a time function times,

this corresponds to multiplied by to the power.

v ud uv u vd –=

du f ' t! "= v est–

= u f t! "= dv sest–

–=

L f ' t! "# $ f t! "est–

0%

&s f t! "

0%

&

est–

dt+ f t! "est–

0%

a

a &'lim sF s! "+= =

esa–

f a! " f 0%! "–( )

a &'lim sF s! "+ 0 f 0

%! "– sF s! "+==

d 2

dt2

-------- f t! " s 2F s! " sf 0%! "– f ' 0

%! "–*

d 3

dt3

-------- f t! " s3F s! " s2f 0%! "– sf ' 0

%! "– f '' 0%! "–*

dn

dtn

-------- f t! " snF s! " s

n 1–f 0

%! "– sn 2–

f ' 0%! "– + f–

n 1–0

%! "–*

g t! " f ' t! "d

dt----- f t! "= =

L g ' t! "# $ sL g t! "# $ g 0%! "–=

L f '' t! "# $ sL f ' t! "# $ f ' 0%! "– s sL f t! "( ) f 0

%! "–( ) f ' 0%! "–= =

s 2F s! " sf 0%! "– f ' 0

%! "–=

f 0%! " f ' 0

%! " f '' 0%! ",,

f t! " n

F s! " s nth




6. Differentiation in Complex Frequency Domain

This property states that differentiation in complex frequency domain and multiplication by minus one,

corresponds to multiplication of by in the time domain. In other words,

(2.21)

Proof:

Differentiating with respect to s, and applying Leibnitz’s rule* for differentiation under the integral, we

get

In general,

(2.22)

The proof for follows by taking the second and higher-order derivatives of with respect

to .

7. Integration in Time Domain

This property states that integration in time domain corresponds to divided by plus the initial

value of at , also divided by . That is,

(2.23)

* This rule states that if a function of a parameter is defined by the equation where f is some

known function of integration x and the parameter , a and b are constants independent of x and , and the par-

tial derivative exists and it is continuous, then .

f t ! t

tf t ! d

ds-----– F s !"

L f t !# $ F s ! f t !0

%

& est–

dt= =

' F ' ! f x '( ! xda

b

&=

' '

f) ')* dF

d'-------

x '( !)' !)

----------------- xda

b

&=

d

ds-----F s !

d

ds----- f t !

0

%

& est–

dts)

)

0

%

& est–

f t !dt t–

0

%

& est–

f t !dt= = =

tf t !+ ,0

%

& est–

dt– L tf t !+ ,–==

tnf t ! 1– !n d

n

dsn

--------F s !"

n 2- F s !

s

F s ! s

f t ! t 0.

= s

f / !%–

t

& d/ F s !s

-----------f 0

. !s

------------+"




Proof:

We express the integral of (2.23) as two integrals, that is,

(2.24)

The first integral on the right side of (2.24), represents a constant value since neither the upper, nor

the lower limits of integration are functions of time, and this constant is an initial condition denoted

as . We will find the Laplace transform of this constant, the transform of the second integral

on the right side of (2.24), and will prove (2.23) by the linearity property. Thus,

(2.25)

This is the value of the first integral in (2.24). Next, we will show that

We let

then,

and

Now,

(2.26)

and the proof of (2.23) follows from (2.25) and (2.26).

f ! "#–

t

$ d f ! "#–

0

$ d f ! "0

t

$ d +=

f 0%! "

L f 0%! "& ' f 0

%! "0

#

$ est–

dt f 0%! " e

st–

0

#

$ dt f 0%! "e

st–

s–--------

0

#

= = =

f 0%! " 0

f 0%! "

s------------–( )

* +–, f 0

%! "s

------------==

f ! "0

t

$ d F s! "

s------------

g t! " f ! "0

t

$ d =

g' t! " f ! "=

g 0! " f ! "0

0

$ d 0= =

L g' t! "& ' G s! " sL g t! "& ' g 0%! "– G s! " 0–= = =

sL g t! "& ' G s! "=

L g t! "& ' G s! "s

-----------=

L f ! "0

t

$ d . /0 12 3 F s! "

s-----------=




8. Integration in Complex Frequency Domain

This property states that integration in complex frequency domain with respect to corresponds to

division of a time function by the variable , provided that the limit exists. Thus,

(2.27)

Proof:

Integrating both sides from to , we get

Next, we interchange the order of integration, i.e.,

and performing the inner integration on the right side integral with respect to , we get

9. Time Periodicity

The time periodicity property states that a periodic function of time with period corresponds to

the integral divided by in the complex frequency domain. Thus, if we let

be a periodic function with period , that is, , for we get the trans-

form pair

(2.28)

s

f t ! tf t !

t--------

t 0"lim

f t !t

-------- F s ! sds

#

$%

F s ! f t !0

#

$ est–

dt=

s #

F s ! sds

#

$ f t !0

#

$ est–

dt sds

#

$=

F s ! sds

#

$ est–

s

#

$ sd f t ! td0

#

$=

s

F s ! sds

#

$ 1

t---– e

st–

s

#f t ! td

0

#

$f t !

t--------e

st–td

0

#

$ Lf t !

t--------

& '( )* +

= = =

T

f t !0

T

$ est–

dt 1 esT–

– ! f t !

T f t ! f t nT+ != n 1 2 3 ,- - -=

f t nT+ !

f t !0

T

$ est–

dt

1 esT–

–

-----------------------------%




Proof:

The Laplace transform of a periodic function can be expressed as

In the first integral of the right side, we let , in the second , in the third ,

and so on. The areas under each period of are equal, and thus the upper and lower limits of

integration are the same for each integral. Then,

(2.29)

Since the function is periodic, i.e., , we can write

(2.29) as

(2.30)

By application of the binomial theorem, that is,

(2.31)

we find that expression (2.30) reduces to

10. Initial Value Theorem

The initial value theorem states that the initial value of the time function can be found

from its Laplace transform multiplied by and letting .That is,

(2.32)

Proof:

From the time domain differentiation property,

or

L f t !" # f t !0

$

% est–

dt f t !0

T

% est–

dt f t !T

2T

% est–

dt f t !2T

3T

% est–

dt &+ + += =

t '= t ' T+= t ' 2T+=

f t !

L f t !" # f ' !0

T

% es'–

d' f ' T+ !0

T

% es ' T+ !–

d' f ' 2T+ !0

T

% es ' 2T+ !–

d' &+ + +=

f ' ! f ' T+ ! f ' 2T+ ! & f ' nT+ != = = =

L f ' !" # 1 esT–

e2sT– &+ + + ! f ' !

0

T

% es'–

d'=

1 a a2

a3 &+ + + +

1

1 a–------------=

L f ' !" #

f ' !0

T

% es'–

d'

' esT–

–

----------------------------------=

f 0( ! f t !

s s $)

f t !t 0)lim sF s !

s $)lim f 0

( != =

d

dt----- f t ! sF s ! f 0

( !–*




Taking the limit of both sides by letting , we get

Interchanging the limiting process, we get

and since

the above expression reduces to

or

11. Final Value Theorem

The final value theorem states that the final value of the time function can be found from

its Laplace transform multiplied by s, then, letting . That is,

(2.33)

Proof:

From the time domain differentiation property,

or

Taking the limit of both sides by letting , we get

Ld

dt----- f t !

" #$ %& '

sF s ! f 0( !–

d

dt----- f t !

0

)

* est–

dt= =

s )+

sF s ! f 0( !–, -

s )+lim

d

dt----- f t !

.

T

* est–

dtT )+. 0+

lims )+lim=

sF s ! f 0( !–, -

s )+lim

d

dt----- f t !

.

T

* est–

s )+lim dt

T )+. 0+

lim=

est–

s )+lim 0=

sF s ! f 0( !–, -

s )+lim 0=

sF s !s )+lim f 0

( !=

f ) ! f t !

s 0+

f t !t )+lim sF s !

s 0+lim f ) != =

d

dt----- f t ! sF s ! f 0

( !–/

Ld

dt----- f t !

" #$ %& '

sF s ! f 0( !–

d

dt----- f t !

0

)

* est–

dt= =

s 0+




and by interchanging the limiting process, we get

Also, since

the above expression reduces to

and therefore,

12. Convolution in the Time Domain

Convolution* in the time domain corresponds to multiplication in the complex frequency domain,

that is,

(2.34)

Proof:

(2.35)

We let ; then, , and . By substitution into (2.35),

* Convolution is the process of overlapping two signals. The convolution of two time functions and is

denoted as , and by definition, where is a dummy variable. We will

discuss it in detail in Chapter 6.

sF s ! f 0" !–# $

s 0%lim

d

dt----- f t !

&

T

' est–

dtT (%& 0%

lims 0%lim=

sF s ! f 0" !–# $

s 0%lim

d

dt----- f t !

&

T

' est–

s 0%lim dt

T (%& 0%

lim=

est–

s 0%lim 1=

sF s ! f 0" !–# $

s 0%lim

d

dt----- f t !

&

T

' dtT (%& 0%

lim f t !&

T

'T (%& 0%

lim= =

f T ! f & !–# $T (%& 0%

lim f ( ! f 0" !–==

sF s !s 0%lim f ( !=

f1 t ! f2 t !

f1 t !*f2 t ! f1 t !*f2 t ! f1 ) !f2 t )– !(–

(

' d)= )

f1 t !*f2 t ! F1 s !F2 s !*

L f1 t !*f2 t !+ , L f1 ) !f2 t )– !(–

(

' d) f1 ) !f2 t )– !0

(

' d)0

(

' est–

dt= =

f1 ) ! f2 t )– !0

(

' est–

dt0

(

' d)=

t )– -= t - )+= dt d-=




13. Convolution in the Complex Frequency Domain

Convolution in the complex frequency domain divided by , corresponds to multiplication in the

time domain. That is,

(2.36)

Proof:

(2.37)

and recalling that the Inverse Laplace transform from (2.2) is

by substitution into (2.37), we get

We observe that the bracketed integral is ; therefore,

For easy reference, we have summarized the Laplace transform pairs and theorems in Table 2.1.

2.3 The Laplace Transform of Common Functions of Time

In this section, we will present several examples for finding the Laplace transform of common func-

tions of time.

Example 2.1

Find

L f1 t !*f2 t !" # f1 $ ! f2 % !0

&

' es % $+ !–

d%0

&

' d$ f1 $ !e s$–d$

0

&

' f2 % !0

&

' es%–

d%= =

F1 s !F2 s !=

1 2(j)

f1 t !f2 t !1

2(j-------- F1 s !*F2 s !*

L f1 t !f2 t !" # f1 t !f2 t !0

&

' est–

dt=

f1 t !1

2(j-------- F1

+ j,–

+ j,+

' - !e-td-=

L f1 t !f2 t !" #1

2(j-------- F1

+ j,–

+ j,+

' - !e-td- f2 t !

0

&

' est–

dt=

1

2(j-------- F1

+ j,–

+ j,+

' - ! f2 t !0

&

' es -– !t–

dt d-=

F2 s -– !

L f1 t !f2 t !" #1

2(j-------- F1

+ j,–

+ j,+

' - !F2 s -– !d-1

2(j--------F1 s !*F2 s != =

L u0 t !" #



The Laplace Transform of Common Functions of Time

TABLE 2.1 Summary of Laplace Transform Properties and Theorems

Property/Theorem Time Domain Complex Frequency Domain

1 Linearity

2 Time Shifting

3 Frequency Shifting

4 Time Scaling

5 Time Differentiation

See also (2.18) through (2.20)

6 Frequency Differentiation

See also (2.22)

7 Time Integration

8 Frequency Integration

9 Time Periodicity

10 Initial Value Theorem

11 Final Value Theorem

12 Time Convolution

13 Frequency Convolution

c1 f1 t ! c2 f2 t !+

+ " cn fn t !+

c1 F1 s ! c2 F2 s !+

+ " cnFn s !+

f t a– !u0 t a– !e

as–F s !

eas–

f t !F s a+ !

f at ! 1

a---F

s

a---

# $% &

d

dt----- f t ! sF s ! f 0

' !–

tf t ! d

ds-----– F s !

f ( !)–

t

* d( F s !s

-----------f 0

' !s

------------+

f t !t

-------- F s ! sds

)*

f t nT+ !f t !

0

T

* est–

dt

1 esT–

–

--------------------------------

f t !t 0+lim

sF s !s )+

lim f 0' !=

f t !t )+

lim sF s !s 0+lim f ) !=

f1 t !*f2 t ! F1 s !F2 s !

f1 t !f2 t ! 1

2,j-------- F1 s !*F2 s !




Solution:

We start with the definition of the Laplace transform, that is,

For this example,

Thus, we have obtained the transform pair

(2.38)

for .*

Example 2.2

Find

Solution:

We apply the definition

and for this example,

We will perform integration by parts recalling that

(2.39)

We let

then,

* This condition was established in (2.9).

L f t !" # F s ! f t !0

$

% est–

dt= =

L u0 t !" # 10

$

% est–

dte

st–

s---------

0

$

01

s---–& '

( )–

1

s---= = = =

u0 t !1

s---*

Re s" # + 0,=

L u1 t !" #

L f t !" # F s ! f t !0

$

% est–

dt= =

L u1 t !" # L t" # t0

$

% est–

dt= =

u vd% uv v ud%–=

u t and dv est–

= =

du 1 and ve

st––

s-----------= =




By substitution into (2.39),

(2.40)

Since the upper limit of integration in (2.40) produces an indeterminate form, we apply L’ Hôpital’s

rule*, that is,

Evaluating the second term of (2.40), we get


(2.41)

for .

Example 2.3

Find

Solution:

To find the Laplace transform of this function, we must first review the gamma or generalized facto-

rial function which is defined as

(2.42)

* Often, the ratio of two functions, such as , for some value of x, say a, results in an indeterminate form. To

work around this problem, we consider the limit , and we wish to find this limit, if it exists. To find this

limit, we use L’Hôpital’s rule which states that if , and if the limit as x

approaches a exists, then,

L t !t– e

st–

s-------------

0

e

st––

s-----------

0

!– dtt– e

st–

s-------------

est–

s2

--------–

0

= =

f x" #g x" #----------

f x" #g x" #----------

x a$lim

f a" # g a" # 0= =d

dx------f x" # d

dx------g x" #%

f x" #g x" #----------

x a$lim

d

dx------f x" # d

dx------g x" #%& '

( )x a$lim=

test–

t $lim

t

est

------t $lim

td

dt" #

td

de

st" #----------------

t $lim

1

sest

--------t $lim 0= = = =

L t* + 1

s2

----=

t1

s2----,

- 0.

L tnu0 t" #* +

/ n" #

/ n" # xn 1–

ex–

xd0

!=




The integral of (2.42) is an improper integral* but converges (approaches a limit) for all .

We will now derive the basic properties of the gamma function, and its relation to the well known

factorial function

The integral of (2.42) can be evaluated by performing integration by parts. Thus, in (2.39) we let

Then,

and (2.42) is written as

(2.43)

With the condition that , the first term on the right side of (2.43) vanishes at the lower limit

. It also vanishes at the upper limit as . This can be proved with L’ Hôpital’s rule by dif-

ferentiating both numerator and denominator m times, where . Then,

Therefore, (2.43) reduces to

and with (2.42), we have

* Improper integrals are two types and these are:

a. where the limits of integration a or b or both are infinite

b. where f(x) becomes infinite at a value x between the lower and upper limits of integration inclusive.

n 0

f x! " xda

b

#

f x! " xda

b

#

n! n n 1–! " n 2–! " $ $ 3 2 1$ $=

u ex–

and dv xn 1–

==

du ex–

– dx and vx

n

n-----==

% n! " xne

x–

n------------

x 0=

&1

n--- x

ne

x–xd

0

&

#+=

n 0

x 0= x &'

m n(

xne

x–

n------------

x &'lim

xn

nex

--------x &'lim

xm

m

d

dx

n

xm

m

d

dne

x

-------------------x &'lim

xm 1–

m 1–

d

dnx

n 1–

xm 1–

m 1–

d

dne

x------------------------------------

x &'lim )= = = =

n n 1–! " n 2–! ") n m– 1+! "xn m–

nex

------------------------------------------------------------------------------------x &'lim

n 1–! " n 2–! ") n m– 1+! "

xm n–

ex

--------------------------------------------------------------------x &'lim 0= ==

% n! "1

n--- x

ne

x–xd

0

&

#=




(2.44)

By comparing the integrals in (2.44), we observe that

(2.45)

or

(2.46)

It is convenient to use (2.45) for , and (2.46) for . From (2.45), we see that becomes

infinite as .

For , (2.42) yields

(2.47)

and thus we have the important relation,

(2.48)

From the recurring relation of (2.46), we obtain

(2.49)

and in general

(2.50)

for

The formula of (2.50) is a noteworthy relation; it establishes the relationship between the

function and the factorial

We now return to the problem of finding the Laplace transform pair for , that is,

(2.51)

To make this integral resemble the integral of the gamma function, we let , or , and

n! " xn 1–

ex–

xd0

#

$1

n--- x

ne

x–xd

0

#

$= =

n! " n 1+! "n

---------------------=

n n! " n 1+! "=

n 0% n 0& n! "

n 0'

n 1=

1! " ex–

xd0

#

$ ex–

0

#– 1= = =

1! " 1=

2! " 1 ( 1! " 1= =

3! " 2 ( 2! " 2 1( 2!= = =

4! " 3 ( 3! " 3 2( 3!= = =

n 1+! " n!=

n 1 2 3 )* * *=

n! "

n!

tnu0t

L tnu0t+ , t

n

0

#

$ est–

dt=

st y= t y s-=




thus . Now, we rewrite (2.51) as

Therefore, we have obtained the transform pair

(2.52)

for positive integers of and .

Example 2.4

Find

Solution:

and using the sifting property of the delta function, we get

Thus, we have the transform pair

(2.53)

for all .

Example 2.5

Find

Solution:

and again, using the sifting property of the delta function, we get

dt dy s =

L tnu0t! " y

s--

# $% &

n

0

'

( ey–d

y

s--

# $% & 1

sn 1+

----------- yn

0

'

( ey–dy

) n 1+* +

sn 1+

--------------------n!

sn 1+

-----------= = = =

tnu0 t* +

n!

sn 1+

-----------,

n - 0.

L / t* +! "

L / t* +! " / t* +0

'

( est–

dt=

L / t* +! " / t* +0

'

( est–

dt es 0* +–

1= = =

/ t* + 1,

-

L / t a–* +! "

L / t a–* +! " / t a–* +0

'

( est–

dt=





(2.54)

for .

Example 2.6

Find

Solution:


(2.55)

for .

Example 2.7

Find

Solution:

For this example, we will use the transform pair of (2.52), i.e.,

(2.56)

and the frequency shifting property of (2.14), that is,

(2.57)

L t a–! "# $ t a–! "0

%

& est–

dt eas–

= =

t a–! " eas–'

( 0)

L eat–

u0 t! "# $

L eat–

u0 t! "# $ eat–

0

%

& est–

dt es a+! "t–

0

%

& dt==

1

s a+-----------–* +

, - es a+! "t–

0

%1

s a+-----------==

eat–

u0 t! "1

s a+-----------'

( a–)

L tne

at–

u0 t! ". /0 12 3

tnu0 t! " n!

sn 1+

-----------'

eat–

f t! " F s a+! "'




Then, replacing with in (2.56), we get the transform pair

(2.58)

where is a positive integer, and Thus, for , we get the transform pair

(2.59)

for .

For , we get the transform

(2.60)

and in general,

(2.61)

for

Example 2.8

Find

Solution:

and from tables of integrals*

* This can also be derived from , and the use of (2.55) where . By the lin-

earity property, the sum of these terms corresponds to the sum of their Laplace transforms. Therefore,

s s a+

tne

at–

u0 t! "n!

s a+! "n 1+-------------------------#

n $ a–% n 1=

teat–

u0 t! " 1

s a+! "2------------------#

$ a–%

n 2=

t2e

at–

u0 t! " 2!

s a+! "3------------------#

tne

at–

u0 t! " n!

s a+! "n 1+-------------------------#

$ a–%

L &t u0sin t! "' (

L &t u0sin t! "' ( &tsin! "0

)

* est–

dt &tsin! "0

a

* est–

dta )+lim= =

&tsin1

j2----- e

j&te

j&t––! "= e

at–u0 t! " 1

s a+-----------#

L &tu0sin t! ", - 1

j2----- 1

s j&–--------------

1

s j&+--------------–. /

0 1 &

s2 &2+

-----------------= =




Then,


(2.62)

for !

Example 2.9

Find

Solution:

and from tables of integrals*

Then,

* We can use the relation and the linearity property, as in the derivation of the transform

of on the footnote of the previous page. We can also use the transform pair ; this

is the time differentiation property of (2.16). Applying this transform pair for this derivation, we get

eax

bxsin" dxe

axa bxsin b bxcos–# $

a2

b2

+

------------------------------------------------------=

L %t u0sin t# $& ' est–

s– %tsin % %tcos–# $

s2 %2+

-----------------------------------------------------------

0

a

a ()lim=

eas–

s– %asin % %acos–# $

s2 %2+

--------------------------------------------------------------%

s2 %2+

-----------------+a ()lim

%

s2 %2+

-----------------==

%t u0tsin%

s2 %2+

-----------------*

+ 0,

L %cos t u0 t# $& '

L %cos t u0 t# $& ' %tcos# $0

(

" est–

dt %tcos# $0

a

" est–

dta ()lim= =

%tcos1

2--- e

j%te

j%t–+# $=

%sin td

dt----- f t# $ sF s# $ f 0

-# $–*

L %cos tu0 t# $. / L1

%----

d

dt----- %sin tu0 t# $

1

%----L

d

dt----- %sin tu0 t# $

1

%----s

%

s2 %2+

----------------- s

s2 %2+

-----------------= = = =

eax

bxcos" dxe

axbxacos b bxsin+# $

a2

b2

+

------------------------------------------------------=




Thus, we have the fransform pair

(2.63)

for !

Example 2.10

Find

Solution:

Since

using the frequency shifting property of (2.14), that is,

(2.64)

we replace with , and we get

(2.65)

for and .

Example 2.11

Find

Solution:

Since

L "cos t u0 t# $% &e

st–s– "tcos " "tsin+# $

s2

"2

+

-----------------------------------------------------------

0

a

a '(lim=

eas–

s– "acos " "asin+# $

s2

"2

+

---------------------------------------------------------------s

s2

"2

+

-----------------+a '(lim

s

s2

"2

+

-----------------==

"cos t u0ts

s2

"2

+

-----------------)

* 0+

L eat–

"t u0sin t# $% &

"tu0tsin"

s2

"2

+

-----------------)

eat–

f t# $ F s a+# $)

s s a+

eat–

"t u0sin t# $"

s a+# $2

"2

+

-------------------------------)

* 0+ a 0+

L eat–

"cos t u0 t# $% &

"cos t u0 t# $s

s2

"2

+

-----------------)



The Laplace Transform of Common Waveforms

using the frequency shifting property of (2.14), we replace with , and we get

(2.66)

for and .

For easy reference, we have summarized the above derivations in Table 2.2.

2.4 The Laplace Transform of Common Waveforms

In this section, we will present some examples for deriving the Laplace transform of several wave-

forms using the transform pairs of Tables 1 and 2.

TABLE 2.2 Laplace Transform Pairs for Common Functions

1

2

3

4

5

6

7

8

9

10

11

s s a+

eat–

cos t u0 t! "s a+

s a+! "2

2

+

-------------------------------#

$ 0% a 0%

f t! " F s! "

u0 t! " 1 s&

t u0 t! " 1 s2

&

tnu0 t! " n!

sn 1+

-----------

' t! " 1

' t a–! " eas–

eat–

u0 t! "1

s a+-----------

tne

at–

u0 t! "n!

s a+! "n 1+

-------------------------

t u0 t! "sin

s2

2

+

-----------------

cos t u0 t! " s

s2

2

+

-----------------

eat–

t u0 t! "sin

s a+! "2

2

+

-------------------------------

eat–

cos t u0 t! " s a+

s a+! "2

2

+

-------------------------------




Example 2.12

Find the Laplace transform of the waveform of Figure 2.1. The subscript stands for pulse.

Figure 2.1. Waveform for Example 2.12

Solution:

We first express the given waveform as a sum of unit step functions. Then,

(2.67)

Next, from Table 2.1,

and from Table 2.2,

For this example,

and

Then, by the linearity property, the Laplace transform of the pulse of Figure 2.1 is

Example 2.13

Find the Laplace transform for the waveform of Figure 2.2. The subscript stands for line.


fP t ! P

A

a

t

0

fP t !

fP t ! A u0 t ! u0 t a– !–" #=


F s !$

u0 t ! 1 s%$

Au0 t ! A s%$

Au0 t a– ! eas– A

s---$

A u0 t ! u0 t a– !–" #A

s--- e

as––

A

s---

A

s--- 1 e

as–– !=$

fL t ! L

1

t

0

1

2

fL t !




Solution:

We must first derive the equation of the linear segment. This is shown in Figure 2.3. Then, we

express the given waveform in terms of the unit step function.

Figure 2.3. Waveform for Example 2.13 with the equation of the linear segment

For this example,

From Table 2.1,

and from Table 2.2,

Therefore, the Laplace transform of the linear segment of Figure 2.2 is

(2.68)

Example 2.14

Find the Laplace transform for the triangular waveform of Figure 2.4.

Solution:

We must first derive the equations of the linear segments. These are shown in Figure 2.5. Then, we

express the given waveform in terms of the unit step function.


1

t

0

1

2

fL t ! t 1–

fL t ! t 1– !u0 t 1– !=


F s !"

tu0 t !1

s2

----"

t 1– !u0 t 1– ! es– 1

s2----"

fT t !

1t

0

1

2

fT t !




Figure 2.5. Waveform for Example 2.13 with the equations of the linear segments

For this example,

and collecting like terms,

From Table 2.1,

and from Table 2.2,

Then,

or

Therefore, the Laplace transform of the triangular waveform of Figure 2.4 is

(2.69)

Example 2.15

Find the Laplace transform for the rectangular periodic waveform of Figure 2.6.

1t

0

1

2

fT t !

t– 2+t

fT t ! t u0 t ! u0 t 1– !–" # t– 2+ ! u0 t 1– ! u0 t 2– !–" #+=

tu0 t ! tu0 t 1– !– tu0 t 1– !– 2u0 t 1– ! tu0 t 2– ! 2u0 t 2– !–+ +=

fT t ! tu0 t ! 2 t 1– !u0 t 1– ! t 2– !u0 t 2– !+–=


F s !$

tu0 t !1

s2

----$

tu0 t ! 2 t 1– !u0 t 1– ! t 2– !u0 t 2– !+–1

s2

---- 2e–

s– 1

s2

---- e2s– 1

s2

----+$

tu0 t ! 2 t 1– !u0 t 1– ! t 2– !u0 t 2– !+–1

s2

---- 1 2es–

– e2s–

+ !$

fT t !1

s2

---- 1 es–

– !2

$

fR t !





Solution:

This is a periodic waveform with period , and thus we can apply the time periodicity prop-

erty

where the denominator represents the periodicity of . For this example,

or

or

(2.70)

a

t

0

A

2a 3a

A

fR t! "

T 2a=

L f #! "$ %f #! "

0

T

& es#–

d#

1 esT–

–

-------------------------------=

f t! "

L fR t! "$ % 1

1 e2as–

–

--------------------- fR t! "0

2a

& est–

dt1

1 e2as–

–

--------------------- A0

a

& est–

dt A–! "a

2a

& est–

dt+= =

A

1 e2as–

–

--------------------- est–

–

s-----------

0

a

est–

s--------

a

2a

+=

L fR t! "$ % A

s 1 e2as–

–! "---------------------------- e

as–– 1 e

2as–e

as––+ +! "=

A

s 1 e2as–

–! "---------------------------- 1 2e

as–– e

2as–+! " A 1 e

as––! "

2

s 1 eas–

+! " 1 eas–

–! "--------------------------------------------------==

A

s---

1 eas–

–! "

1 eas–

+! "-----------------------

A

s---

eas 2'

eas 2'–

eas 2'–

eas 2'–

–

eas 2'

eas 2'–

eas 2'–

eas 2'–

+

---------------------------------------------------------------( )* +, -

=

( )* +, -

=

A

s---

eas 2'–

eas 2'–

--------------e

as 2'e

as 2'––

eas 2'

eas 2'–

+

---------------------------------( )* +, - A

s---

as 2'! "sinh

as 2'! "cosh-----------------------------==

fR t! "A

s---

as

2-----( ), -tanh.




Example 2.16

Find the Laplace transform for the half-rectified sine wave of Figure 2.7.


Solution:

This is a periodic waveform with period . We will apply the time periodicity property

where the denominator represents the periodicity of . For this example,

or

(2.71)

fHW t !

sint

1

" 2" 3" 4"

fHW t !

T 2"=

L f # !$ %

f # !0

T

& es#–

d#

1 esT–

–

-------------------------------=

f t !

L fHW t !$ %1

1 e2"s–

–

--------------------- f t !0

2"

& est–

dt1

1 e2"s–

–

--------------------- tsin0

"

& est–

dt= =

1

1 e2"s–

–

---------------------e

st–s t tcos–sin !

s2

1+

------------------------------------------

0

"

1

s2

1+ !------------------

1 e"s–

+ !

1 e2"s–

– !--------------------------==

L fHW t !$ %1

s2

1+ !------------------

1 e"s–

+ !

1 e"s–

+ ! 1 e"s–

– !-----------------------------------------------=

fHW t !1

s2

1+ ! 1 e"s–

– !------------------------------------------'



Summary

2.5 Summary

The two-sided or bilateral Laplace Transform pair is defined as

where denotes the Laplace transform of the time function , denotes

the Inverse Laplace transform, and is a complex variable whose real part is , and imaginary

part , that is, .

The unilateral or one-sided Laplace transform defined as

We denote transformation from the time domain to the complex frequency domain, and vice

versa, as

The linearity property states that

The time shifting property states that

The frequency shifting property states that

The scaling property states that

The differentiation in time domain property states that

L f t! "# $ F s! "= f t! "%–

%

& est–

dt=

L1–

F s! "# $ f t! "=1

2'j-------- F s! "

( j)–

( j)+

& est

ds=

L f t! "# $ f t! " L1–

F s! "# $

s (

) s ( j)+=

L f t! "# $ F= s! " f t! "t0

%

& est–

dt f t! "0

%

& est–

dt= =

f t! " F s! "*

c1 f1 t! " c2 f2 t! " + cn

fn

t! "+ + + c1 F1 s! " c2 F2 s! " + cn

Fn

s! "+ + +*

f t a–! "u0 t a–! " eas–

F s! "*

eat–

f t! " F s a+! "*

f at! " 1

a---F

s

a---

, -. /*

f ' t! "d

dt----- f t! "= sF s! " f 0

0! "–*

d 2

dt2

-------- f t! " s 2F s! " sf 00! "– f ' 0

0! "–*




and in general

where the terms , and so on, represent the initial conditions.

The differentiation in complex frequency domain property states that

and in general,

The integration in time domain property states that

The integration in complex frequency domain property states that

provided that the limit exists.

The time periodicity property states that

The initial value theorem states that

The final value theorem states that

d 3

dt3

-------- f t! " s3F s! " s2f 0#! "– sf ' 0

#! "– f '' 0#! "–$

dn

dtn

-------- f t! " snF s! " s

n 1–f 0

#! "– sn 2–

f ' 0#! "– % f–

n 1–0#! "–$

f 0#! " f ' 0

#! " f '' 0#! "&&

tf t! " d

ds-----– F s! "$

tnf t! " 1–! "n d

n

dsn

--------F s! "$

f '! "(–

t

) d' F s! "s

-----------f 0

#! "s

------------+$

f t! "t

-------- F s! " sds

(

)$

f t! "t

--------t 0*lim

f t nT+! "

f t! "0

T

) est–

dt

1 esT–

–

-----------------------------$

f t! "t 0*lim sF s! "

s (*lim f 0

#! "= =

f t! "t (*lim sF s! "

s 0*lim f (! "= =



Summary

Convolution in the time domain corresponds to multiplication in the complex frequency domain,

that is,

Convolution in the complex frequency domain divided by , corresponds to multiplication

in the time domain. That is,

The Laplace transforms of some common functions of time are shown below.

f1 t! "*f2 t! " F1 s! "F2 s! "#

1 2$j%

f1 t! "f2 t! "1

2$j-------- F1 s! "*F2 s! "#

u0 t! " 1 s%#

t 1 s2%#

tnu0 t! "

n!

sn 1+

-----------#

& t! " 1#

& t a–! " eas–

#

eat–

u0 t! "1

s a+-----------#

teat–

u0 t! "1

s a+! "2

------------------#

t2e

at–

u0 t! "2!

s a+! "3

------------------#

tne

at–

u0 t! "n!

s a+! "n 1+

-------------------------#

't u0tsin'

s2

'2

+

-----------------#

'cos t u0ts

s2

'2

+

-----------------#

eat–

't u0sin t! "'

s a+! "2

'2

+

-------------------------------#

eat–

'cos t u0 t! "s a+

s a+! "2

'2

+

-------------------------------#




The Laplace transforms of some common waveforms are shown below.

A

a

t

0

fP t! "

A u0 t! " u0 t a–! "–# $ A

s--- e

as––

A

s---

A

s--- 1 e

as––! "=%

1

t

0

1

2

fL t! "

t 1–! "u0 t 1–! " es– 1

s2----%

1t

0

1

2

fT t! "

fT t! "1

s2

---- 1 es–

–! "2

%

a

t

0

A

2a 3a

&A

fR t! "

fR t! " A

s---

as

2-----

' () *tanh%



Summary

sint

1

2 3 4

fHW t! "

fHW t! "1

s2

1+! " 1 e s–

–! "------------------------------------------#




2.6 Exercises

1. Find the Laplace transform of the following time domain functions:

a.

b.

c.

d.

e.


a.

b.

c.

d.

e.


a.

b.

c.

d.

e. Be careful with this! Comment and skip derivation.


a.

b.

c.

12

6u0 t !

24u0 t 12– !

5tu0 t !

4t5u0 t !

j8

j5 90"–#

5e5t–

u0 t !

8t7e

5t–u0 t !

15$ t 4– !

t3

3t2

4t 3+ + + !u0 t !

3 2t 3– !$ t 3– !

3 5tsin !u0 t !

5 3tcos !u0 t !

2 4ttan !u0 t !

3t 5tsin !u0 t !

2t2

3tcos !u0 t !

2e5t–

5tsin



Exercises

d.

e.


a.

b.

c.

d.

e.


a.

b.

c.

d.

e.


a.

b.

c.

d.

8e3t–

4tcos

tcos !" t # 4$– !

5tu0 t 3– !

2t2

5t 4+– !u0 t 3– !

t 3– !e 2t–u0 t 2– !

2t 4– !e 2 t 2– !u0 t 3– !

4te3t–

2tcos !u0 t !

td

d3tsin !

td

d3e

4t– !

td

dt

22tcos !

td

de

2t–2tsin !

td

dt

2e

2t–

!

tsin

t---------

%sin

%---------- %d

0

t

&

atsin

t------------

%cos

%----------- %d

t

'

&




e.

8. Find the Laplace transform for the sawtooth waveform of Figure 2.8.

Figure 2.8. Waveform for Exercise 8.

9. Find the Laplace transform for the full rectification waveform of Figure 2.9.

Figure 2.9. Waveform for Exercise 9

e –

------- d

t

!

"

fST t# $

A

a 2at

fST t# $

3a

fFR t# $

Full Rectified Waveform

sint1

a 2a 3a 4a

fFR t# $

eduwavepool.unizwa.edu.om€¦ · signals and systems with matlab applications, second edition 2-1...

Documents