Waveform Synthesis

AIM: To perform waveform synthesis using Laplace Transforms of a given signal  

OBJECTIVE: To perform waveform synthesis using Laplace Transform of a given signal.

EQUIPMENT:

   PC with windows (95/98/XP/NT/2007).

   MATLAB Software

THEORY:

laplace transform :

The bilateral Laplace transform is defined as follows:  

%this program find the laplace transform of the given function f(t)

syms t

ft=input('define the function')

fs=laplace(ft);

disp('the laplace transform of the given function f(t) is')

disp(fs)

1. f(t)= (-4/3)e^-t+(1/3)e^2t.

define the function  

ft = -4/3*exp(-t)+1/3*exp(2*t)  

the Laplace transform of the given function f(t) is

-4/3/(s+1)+1/3/(s-2) i.e. –(4/3).(1/(s+1))+(1/3).(1/(s-2)).

2. f(t)=t^2.sin(t).

define the function  

ft = t^2*sin(t)  

the laplace transform of the given function f(t) is

2/(s^2+1)^3*(-1+3*s^2).

3. f(t)=t².e^-2t.

define the function  

ft = t^2*exp(-2*t)  

the laplace transform of the given function f(t) is

2/(s+2)^3

4. f(t)=3e^-2t- 2e^-t

define the function  

ft = 3*exp(-2*t)-2*exp(-t)  

the laplace transform of the given function f(t) is

3/(s+2)-2/(s+1)

5. define the function  

ft = dirac(t)  

the Laplace transform of the given function f(t) is 1

%this program finds the inverse Laplace transform of given f(s)

syms s t;

fs=input('enter the laplace transform ')

ft=ilaplace(fs);

disp('the  inverse Laplace transform of the given f(s) is')

disp(ft)

1.enter the Laplace transform  

fs =1/(s+4)

the  inverse Laplace transform of the given f(s) is

exp(-4*t)

2.enter the Laplace transform  

fs =36/(s^2+3*s+36)  

the  inverse Laplace transform of the given f(s) is

8/5*15^(1/2)*exp(-3/2*t)*sin(3/2*15^(1/2)*t)

3.enter the Laplace transform  

fs = (2*s^2+5*s+12)/(s^2+2*s+10)/(s+2)

the  inverse Laplace transform of the given f(s) is

exp(-t)*cos(3*t)+exp(-2*t)

4. enter the Laplace transform  

fs = 1/s

the  inverse Laplace transform of the given f(s) is 1

5. enter the Laplace transform  

fs = 1/s^3  

the  inverse Laplace transform of the given f(s) is 1/2*t^2

%This program performs the wave form synthesis of a  stair case waveform  

f=input('enter the sampling frequency')
T=1/f;
L=input('enter the lower bound for the time axis')
U=input('enter the upper bound for the time axis')
t=L:T:U;
x=zeros(1,length(t));
y=x;
for i=find(t==0):find(t==1)
   x(i)=1;
end
x(find(t==1))=0;
for i=find(t==1):find(t==2)
   x(i)=2;
end
x(find(t==2))=0;
for i=find(t==2):find(t==3)
   x(i)=3;
end
z=find(diff(x)==1);
y(z(1)+1:length(y))=1;
subplot(2,2,1)
plot(t,x,'b')
xlabel('time')
ylabel('amplitude')
title('u(t)+u(t-1)+u(t-2) with f=1000;L=-1;U=3')
%axis([ x-axis(min) x-axis(max) y-axis(min) y-axis(max)])
axis([ -1 4 -0 4])
grid
subplot(2,2,2)
plot(t,y,'r')
xlabel('time')
ylabel('amplitude')
title('The First Constituent Step Function')
grid
axis([ -1 4 0 2])
y=y-y;
y(z(2)+1:length(y))=1;
subplot(2,2,3)
plot(t,y,'r')
xlabel('time')
ylabel('amplitude')
title('The Second Constituent Step Function')
grid
axis([ -1 4 0 2])
y=y-y;
y(z(3)+1:length(y))=1;
subplot(2,2,4)
plot(t,y,'r')
xlabel('time')
ylabel('amplitude')
title('The Third Constituent Step Function')
grid
axis([ -1 4 0 2])

Output:

(ii)  x(t)=2u(t)-3u(t-2)+2u(t-4)

f=input('enter the sampling frequency')
T=1/f;
L=input('enter the lower bound for the time axis <0')
U=input('enter the upper bound for the time axis>4')
t=L:T:U;
x=zeros(1,length(t));
y=x;
for i=find(t==0):find(t==2)
   x(i)=2;
end
x(find(t==2))=0;
for i=find(t==2):find(t==4)
   x(i)=-3;
end
x(find(t==4))=0;
for i=find(t==4):find(t==U)
   x(i)=2;
end
z=find(diff(x)==2);
y(z(1)+1:length(y))=2;
subplot(4,1,1)
plot(t,x,'b')
xlabel('time')
ylabel('amplitude')
title('2u(t)-3u(t-2)+2u(t-4) with f=1000,L=-1,U=6')
grid
subplot(4,1,2)
plot(t,y,'r')
xlabel('time')
ylabel('amplitude')
title('The First Constituent Step Function')
grid
y=y-y;
z=find(diff(x)==-5);
y(z(1)+1:length(y))=-5;
subplot(4,1,3)
plot(t,y,'r')
xlabel('time')
ylabel('amplitude')
title('The Second Constituent Step Function')
grid
y=y-y;
z=find(diff(x)==5);
y(z(1)+1:length(y))=5;
subplot(4,1,4)
plot(t,y,'r')
xlabel('time')
ylabel('amplitude')
title('The Third Constituent Step Function',2)
grid

OUTPUT:

enter the sampling frequency1000

f =  1000

enter the lower bound for the time axis <0-1

L = -1

enter the upper bound for the time axis>46

U = 6

CONCLUSION:  In this experiment Laplace Transforms of various signals was computed  and  wave form synthesis was implemented.

OUTCOME: The Student must be able to understand the time domain to frequency domain in s-plan using MATLAB