Module 17 Inverse Laplace Transform and Waveform Synthesis ...
Transcript of Module 17 Inverse Laplace Transform and Waveform Synthesis ...
Module 17
Inverse Laplace Transform and Waveform Synthesis
Objective:(i) To describe how to obtain inverse Laplace transform making use of the
knowledge of properties of Laplace Transform and properties of ROC.
(ii) To Apply waveform synthesis to find Laplace forms of certain functions
Introduction :
Inverse Laplace transform maps a function in s-domain back to the time domain. One
application is to convert a system response to an input signal from s-domain back to the time
domain.
Since system analysis is usually easier in s-domain, the process is to convert the
system time domain representation to s-domain (both system and inputs),perform system
analysis in s-domain and then convert back to the time domain representation for the
response. The reason to do this process in this convoluted way is that due to its properties, the
Laplace transform converts the differential equations that describe system behaviour to a
polynomial. Also the convolution operation which describes the system action on the input
signals is converted to a multiplication operation. These two properties make it much easier
to do systems analysis in the s-domain.
Inverse Laplace transform is performed using Partial Fraction Expansion that split up
a complicated fraction into forms that are in the Laplace Transform table.
Description :
Concept of Inverse Laplace Transform
We are aware that the Laplace transform of a continuous signal x(t) is given by
π π = π₯(π‘)πβπ π‘ππ‘
β
ββ
Since s=Ο+jΟ
π Ο + jΟ = π₯(π‘)πβ(Ο+jΟ)π‘ππ‘
β
ββ
= π₯(π‘)πβΟπ‘ πβjΟπ‘ππ‘
β
ββ
= β± π₯(π‘)πβΟπ‘
π₯ π‘ πβΟπ‘ = β±β1 π Ο + jΟ =1
2π π Ο + jΟ πjΟπ‘πΟ
β
ββ
We can recover x(t) from its Laplace transform evaluated for a set of values of s=Ο+jΟin the
ROC, with Οfixed and Ο varying from -β to +β. Recovering s(t) from X(s) is done by
changing the variable of integration in the above equation from Οto s and using the fact that Ο
is constant, so that ds=jdΟ.
π₯ π‘ =1
2π π π πΟπ‘πjΟπ‘πΟ =
1
2π π π πsπ‘πΟ =
1
2ππ π π πsπ‘ππ
Ο+jβ
Οβπβ
β
ββ
β
ββ
The contour of integration in above equation is a straight line in the s-plane corresponding to
all points s satisfying Re{s}=Ο. This line is parallel to the jΟ-axis. Therefore, we can choose
any value of Ο such that π Ο + jΟ converges.
Partial Fraction Expansion
As we know that the rational form of X(s) can be expanded into partial fractions,
Inverse Laplace transform can be taken according to location of poles and ROC of X(s). The
roots of denominator polynomial, i.e., poles can be simple and real, complex or multiple.
We know that X(s) is expanded in partial fractions as
π π =π0
π β π 0+
π1
π β π 1+
π2
π β π 2+ β― +
πππ β π π
Here the roots s0,s1,s2,...sn can be real, complex or multiple. Then the values of
k0,k1,k2,...kn constants are calculated accordingly.
In order to find the appropriate time domain function, ROC should be indicated for
the s-domain function. Otherwise we may have multiple time-domain functions based on
different possible ROCs.
Illustration
Example for Real roots:
Problem 1:Find out the partial fraction expansion and hence Inverse Laplace transform of the
function π π =π 2+2π β2
π (π +2)(π β3) , ROC: Re{s} > 3
Solution:
The function π π =π 2+2π β2
π (π +2)(π β3) can be written as,
π π =π΄
π +
π΅
π + 2+
πΆ
π β 3
The constants calculated are A=1/3, B=-1/5, C=13/15
π π =1/3
π +
1/5
π + 2+
13/15
π β 3
π₯ π‘ =1
3ββ1
1
π +
1
5ββ1
1
π + 2 +
13
15ββ1
1
π β 3
From the given ROC: Re{s}>3, the resultant signal x(t) should be right sided.
Therefore, π₯ π‘ =1
3π’ π‘ +
1
5πβ2π‘π’(π‘) +
13
15π3π‘π’(π‘)
Example for Complex roots:
Problem 2: Obtain right sided time domain signal for the function π π =π 2+2π +1
(π +2)(π 2+4)
Solution:
We can write the given function as π π =π 2+2π +1
(π +2)(π 2+4)=
π΄
π +2+
π΅π +πΆ
π 2+22
The constants can be calculated as A=1/8, B=0.874, C=0.5
Therefore, π π =1
8
1
π +2+
0.874π +0.5
π 2+22 =1
8
1
π +2 + 0.874
π
π 2+22 + 0.25 2
π 2+22
Finally π₯(π‘) =1
8ββ1
1
π +2 + 0.874ββ1
π
π 2+22 + 0.25ββ1 2
π 2+22
i.e., π₯(π‘) =1
8πβ2π‘π’(π‘) + 0.874 cos 2π‘ π’(π‘) + 0.25π ππ 2π‘ π’(π‘)
Example for Multiple roots:
Problem 3: Find out the inverse Laplace transform of π π =π β2
π (π +1)3, ROC:Re{s}<-1
Solution:
We can write the given function as π π =π β2
π (π +1)3 =π΄
π +1 3 +π΅
π +1 2 +πΆ
(π +1)+
π·
π
The constants can be calculated as A=3, B=2, C=2, D=-2
Therefore, π π = 3 1
π +1 3 + 2 1
π +1 2 + 2 1
(π +1) β 2
1
π
From the given ROC: Re{s}<-1, the resultant signal x(t) should be left sided.
Finally,π₯ π‘ = 3ββ1 1
π +1 3 + 2ββ1 1
π +1 2 + 2ββ1 1
(π +1) β 2ββ1
1
π
From the result of Laplace transform
βπ‘π
π!πβππ‘π’ βπ‘
β
1
π + π π+1, π ππΆ: π π π < βπ
i.e.,π₯ π‘ = 3βπ‘2
2πβπ‘π’ βπ‘ β 2π‘πβπ‘π’ βπ‘ β 2πβπ‘π’ βπ‘ + 2π’ βπ‘
Laplace transform using Waveform Synthesis
In waveform synthesis, the unit step function u(t) and other functions serve as building blocks
in constructing other waveforms. Once the waveforms are synthesized in the form of other
functions, Laplace transform is found and simplified.
Illustration
For example, we may describe a pulse waveform in terms of unit step functions. A pulse of
unit amplitude from t=a to t=b can be formed by taking the difference between the two step
functions
i.e.,x(t) = u(t-a) - u(t-b)
Hence π π = πβπ π 1
π β πβπ π 1
π
Examples:
Solved Problems:
Problem 1: Find the Inverse Laplace transform of π(π ) =3π +7
π 2β2π β3 ROC: Re{s}>3
Solution:
We know that πβππ‘π’(π‘)β
1
π +π, π ππΆ: π π π > βπ
Writing X(s) in the form of partial fraction expansion
π π =π΄
π β 3+
π΅
π + 1
The constants can be calculated as A=4, B=-1
Therefore, π π =π΄
π β3+
π΅
π +1= 4
1
π β3 β
1
π +1
From the given ROC: Re{s}>3, the resultant signal x(t) should be right sided.
i.e., π₯ π‘ = 4ββ1 1
π β3 β ββ1
1
π +1 = 4π3π‘π’ π‘ β πβπ‘π’ π‘
Problem 2: Find the Inverse Laplace transform of π(π ) =β3
(π +2)(π β1) ROC: Re{s}<-2
Solution:
We know that βπβππ‘π’(βπ‘)β
1
π +π, π ππΆ: π π π < βπ
Writing X(s) in the form of partial fraction expansion
π π =π΄
π + 2+
π΅
π β 1
The constants can be calculated as A=1, B=-1
Therefore, π π =π΄
π +2+
π΅
π β1=
1
π +2 β
1
π β1
From the given ROC: Re{s}<-2, the resultant signal x(t) should be left sided.
i.e., π₯ π‘ = ββ1 1
π +2 β ββ1
1
π β1 = βπβ2π‘π’ βπ‘ + ππ‘π’ βπ‘
Problem 3: Find the Inverse Laplace transform of π(π ) =1
π 2+3π +2 ROC: -2<Re{s}<-1
Solution:
We know that
πβππ‘π’(π‘)β
1
π + π, π ππΆ: π π π > βπ
βπβππ‘π’(βπ‘)β
1
π + π, π ππΆ: π π π < βπ
Writing X(s) in the form of partial fraction expansion
π π =π΄
π + 1+
π΅
π + 2
The constants can be calculated as A=1, B=-1
Therefore, π π =π΄
π +1+
π΅
π +2=
1
π +1 β
1
π +2
From the given ROC: -2<Re{s}<-1, the two derived conditions are Re{s}<-1 which suits for
1
π +1 and Re{s}>-2 which suits for
1
π +2 . Therefore, the resultant signal x(t) should be two-
sided.
i.e., π₯ π‘ = ββ1 1
π +1 β ββ1
1
π +2 = βπβπ‘π’ βπ‘ β πβ2π‘π’ π‘
Problem 4:Determine right-sided x(t) if π(π ) =1
π 2(π 2βπ2) using convolution theorem
Solution:
The given function is π π =1
π 2 π 2βπ2 = π1 π .π2(π )
π1 π =1
π 2 and π2 π =
1
π 2βπ2 =
1
2π
1
π βπβ
1
π +π (Done using partial fraction expansion)
π₯1 π‘ = ββ1 1
π 2 = π‘π’(π‘)
π₯2 π‘ =1
2π ββ1
1
π β π β ββ1
1
π + π =
1
2π πππ‘ β πβππ‘ π’(π‘)
From the Convolution propertyπ₯1 π‘ β π₯2 π‘ β π1 π .π2(π )
i.e., π₯ π‘ = π₯1 π‘ β π₯2 π‘ = π₯1 π π₯2 π‘ β π ππβ
ββ=
1
2π π ππ(π‘βπ) β πβπ(π‘βπ) ππ
π‘
0
π₯ π‘ =1
2π β
2π‘
π+
1
π2 πππ‘ β πβππ‘ ; π‘ > 0
Problem 5:Find right sided x(t)ifπ π =1+πβ2π
3π 2+2π
Solution:
Givenπ π =1+πβ2π
3π 2+2π =
1
π (3π +2)+
πβ2π
π (3π +2)=
1
2
1
π β
1
π +2/3 +
1
2πβ2π
1
π β
1
π +2/3
Taking inverse Laplace transform and using time shifting property
π₯ π‘ =1
2 1 β πβ
2
3π‘ π’(π‘) +
1
2 1 β πβ
2
3 π‘β2
π’(π‘ β 2)
Problem 6:Given, π₯(π‘) = πβπ‘π’(π‘), find the Inverse Laplace transform of πβ3π π(2π )
Solution:
From Time scaling property
If π₯(π‘)β π(π )
then π₯ ππ‘ β
1
π π
π
π
with a=1/2, π₯ π‘
2
β
11
2 π
π 1
2 = 2π(2π )
Therefore, 1
2π₯
π‘
2
β π(2π )
Time shifting property states that then π₯ π‘ β π β πβπ ππ π . Applying this property
to above equation with Ο=3,
1
2π₯
π‘ β 3
2
β πβ3π π(2π )
Applying the transformation to π₯(π‘) = πβπ‘π’(π‘) given by above equation,
ββ1 πβ3π π(2π ) =1
2π₯
π‘ β 3
2 =
1
2πβ
π‘β3
2 π’
π‘ β 3
2
Problem 7: Find the steady state response of the following system to unit step excitation
π» π =π + 1
π 2 + 3π + 2
Solution:
We know that for a linear system outputy(t) =h(t)*x(t)
As the input isπ₯(π‘) = π’(π‘)β
1
π
Output π¦ π‘ = β π‘ β π’ π‘ β π π =
π»(π )
π
Therefore, π π =π»(π )
π =
π +1
π (π 2+3π +2)=
π +1
π (π +1)(π +2)=
1
π (π +2)
Using partial fraction expansion
π π =1
2 1
π β
1
π + 2
Therefore, step response π¦ π‘ = 1βπβ2π‘
2 π’(π‘)
Problem 8: Knowing that πβπ π
ππ‘π¦(π‘) = π π π β π¦(0β)
Solve the differential equation π
ππ‘π¦ π‘ + 5π¦ π‘ = π₯(π‘), with initial condition y(0
+)=-2 and
input x(t)=3e-2t
u(t)
Solution:
The given differential equation is, π
ππ‘π¦ π‘ + 5π¦ π‘ = π₯(π‘)
Taking Unilateral Laplace transform of above equation
sY(s)-y(0-)+5Y(s)=X(s)
Substituting initial condition y(0+)=y(0
-)=-2 and input π₯ π‘ = 3πβ2π‘π’ π‘
β π π = 3
1
π +2
and applying partial fraction expansion
π π =3
(π + 2)(π + 5)β
2
π + 5=
1
π + 2β
1
π + 5β
2
π + 5=
1
π + 2β
3
π + 5
Taking inverse Laplace transform of the above equation
π¦ π‘ = πβ2π‘π’ π‘ β 3πβ5π‘π’ π‘
Problem 9: Obtain the impulse response of the system shown and hence prove that the
system is BIBO stable
Solution:
Let us transform all the elements to their Laplace equivalents assuming zero initial
conditions. The Laplace equivalent circuit is shown below
Using voltage divider formula we can write,
π π =
1
π πΆ
π +1
π πΆ
π π =1
π π πΆ + 1π(π )
Therefore, transfer function
π» π =π(π )
π(π )=
1
π π πΆ + 1=
1
π πΆ
1
π +1
π πΆ
Taking inverse Laplace transform of above equation, impulse response is
β π‘ =1
π πΆπβ
π‘
π πΆπ’(π‘)
H(s) has one pole located at s=-1/RC
This pole lies in left half of the s-plane. Hence the system is both causal and stable.
This can also be proved checking the absolute integrability of impulse response
β(π‘) ππ‘ < ββ
ββ
β(π‘) ππ‘ =1
π πΆ πβ
π‘
π πΆ
β
0
ππ‘ =1
π πΆ
β
ββ
πβ
π‘
π πΆ
β1
π πΆ
0
β
= β πββ β π0 = 1 < β
Hence BIBO stable
Problem 10:Obtain the Laplace transform of the triangular pulse as shown below using
waveform synthesis
Solution:
As shown in the figure above, a ramp of slope βAβ starting at t=0 is taken as f1(t). The
function f2(t) is the ramp of slope -2A starting at t=1. When we add f1(t) and f2(t) we get the
signal shown in figure (c). Observe that a negative going ramp of slope (A-2A=-A) starts at
t=1. To cancel the part of this ramp after tβ₯2, a positive going ramp of slope +A in figure (d)
is added. Then we get the required triangular pulse of Figure (e).
With the help of step functions, the ramp functions in the figure above can be expressed as
follows:
f1(t)=A t u(t) { The function u(t)=1 for t>0; It indicates that ramp is present only for tβ₯0}
f2(t)=-2A (t-1) u(t-1) { The function u(t-1)=1 for t>1; It indicates that ramp is present only for
tβ₯1}
f3(t)=A(t-2) u(t-2)
Therefore, f(t)=f1(t)+f2(t)+f3(t)=A t u(t)-2A (t-1) u(t-1)+A(t-2) u(t-2)
Since u(t), u(t-1) and u(t-2) have values of β1β and they just represent the time shifts and
directions of ramp functions, they can be dropped in this expression. Laplace transform of
above equation becomes
β π π‘ = π΄1
π 2β 2π΄
πβπ
π 2+ π΄
πβ2π
π 2=
π΄
π 2 1 β 2πβπ + πβ2π
Assignment:
Problem 1:Find the inverse Laplace transform of π π = 4π 2 + 15π +8
π +2 2 π +1 .Assuming
signal is causal.
Problem 2:The transfer function of the system is given as π» π =2
π +3+
1
π β2. Determine the
impulse response if the system is (i) stable (ii) causal
Problem 3:Find the current i(t) in a series RLC circuit as shown in figure below, when a
voltage of 100Volts is switched on across the terminals aaβ at t=0.
Problem 4: The response h(t) of a linear time invariant system to an impulse Ξ΄(t), under
initially relaxed condition is h(t) = eβt
+ eβ2t
. Find the step response of this system.
Problem 5: Find the Inverse Laplace transform of πΊ π =π
(π β3)(π 2β4π +5), π < 2
Problem 6:Given f (t) and g(t) as show below:
Express g(t) in terms of f(t) and hence find Laplace transform
Problem 7: Find the Inverse Laplace transform of πΊ π =10π 2
(π +1)(π +3), Ο > 0
Problem 8:Let the Laplace transform of a function f (t) which exists for t > 0 be F1(s)and the
Laplace transform of its delayed version f (t β Ο) be F2(s). Let πΉ1β(π )be the complex
conjugate of F1(s) with the Laplace variable set s = Ο + jΟ.If πΊ π =πΉ2 π πΉ1
β(π )
πΉ1(π ) 2then Find the
inverse Laplace transform of G(s).
Problem 9:Find the Laplace transform of periodic sawtooth waveform using waveform
synthesis
Problem 10:Consider the LTI system for which we are given the following information:
π π =π + 2
π β 2
π₯ π‘ = 0 ; π‘ > 0
and π¦ π‘ = β2
3π2π‘π’ βπ‘ +
1
3πβπ‘π’ π‘
(a) Determine H(s) and its ROC
(b) Determine h(t)
Simulation:
The command one uses now is ilaplace. One also needs to define the symbols t and s.
Letβs calculate the inverse of the previous function F(s),
πΉ π =π β 5
π (π + 2)2
>>syms t s
>> F=(s-5)/(s*(s+2)^2);
>>ilaplace(F)
ans =
-5/4+(7/2*t+5/4)*exp(-2*t)
>> simplify(ans)
ans =
-5/4+7/2*t*exp(-2*t)+5/4*exp(-2*t)
>> pretty(ans)
- 5/4 + 7/2 t exp(-2 t) + 5/4 exp(-2 t)
Which corresponds to
π π‘ = β1.25 + 3.5π‘πβ2π‘ + 1.25πβ2π‘
Alternatively, one can write >>ilaplace((s-5)/(s*(s+2)^2))
Here is another example.
πΉ π =10(π + 2)
π (π 2 + 4π + 5)
>> F=10*(s+2)/(s*(s^2+4*s+5));
>>ilaplace(F)
ans =
-4*exp(-2*t)*cos(t)+2*exp(-2*t)*sin(t)+4
Another example is shown below
>>syms s ;
>>T = 1;
>>F1 = 1 - exp( -T * s )+ exp(-3*T * s/2);
>>F2 = s^2 ;
>>F = F1 / F2;
>>f = ilaplace( F )
f =
t - heaviside(t - 1)*(t - 1) + heaviside(t - 3/2)*(t - 3/2)
>>pretty( simplify ( f ) )
t - heaviside(t - 1) (t - 1) + heaviside(t - 3/2) (t - 3/2)
>>ezplot( f )
References:
[1] Alan V.Oppenheim, Alan S.Willsky and S.Hamind Nawab, βSignals & Systemsβ, Second
edition, Pearson Education, 8th
Indian Reprint, 2005.
[2] M.J.Roberts, βSignals and Systems, Analysis using Transform methods and MATLABβ,
Second edition,McGraw-Hill Education,2011
[3] John R Buck, Michael M Daniel and Andrew C.Singer, βComputer explorations in
Signals and Systems using MATLABβ,Prentice Hall Signal Processing Series
[4] P Ramakrishna rao, βSignals and Systemsβ, Tata McGraw-Hill, 2008
[5] Tarun Kumar Rawat, βSignals and Systemsβ, Oxford University Press,2011