//Frequency components of a signal
//Build a noised signal samples at
1000Hz
clc
clf
sample_rate=1000
t=-3: 1/sample_rate:3
N=size(t,'*');//number of samples
s=exp(-t^2)
y=fft(s)//s is real so fft response is
conjugate symmetric and we retain
only the first N
f=sample_rate*(0:(N/2))/N;
n=size(f,'*')
subplot(2,1,1)
plot2d(t,s)
subplot(2,1,2)
a=gca();
plot2d(f,abs(y(1:n)))
a.data_bounds=[0,0;20,200]
--> exec('C:\Users\asekh\Documents\n th roots of unity.sce', -1)
Enter n=2
"rt is ="
0. + i
0. - i
"rt1 is="
2. + 3.i
--> exec('C:\Users\asekh\Documents\n th roots of unity.sce', -1)
Enter n=3
"rt is ="
-1. + 0.i
0.5 + 0.8660254i
0.5 - 0.8660254i
"rt1 is="
2. + 3.i
--> exec('C:\Users\asekh\Documents\n th roots of unity.sce', -1)
Enter n=4
"rt is ="
-0.7071068 + 0.7071068i
-0.7071068 - 0.7071068i
0.7071068 + 0.7071068i
0.7071068 - 0.7071068i
"rt1 is="
2. + 3.i
//Plotting Legendre Polynomial
clc;clf
x=-1:0.01:1
disp("Program to plot Legendre
Polynomial")
leg=legendre(0:7,0,x)
plot2d(x',leg',leg="P0@P1@P2@P
3@P4@P5@P6@P7@")
xlabel("x")
ylabel("Pn(x)")
//Orthogonality of Legendre Polynomial
clf
disp("Program for Orhtogonality of
Legendre POlynomial")
n1=input("Please enter the value of
n1=")
n2=input("Please enter the value of
n2=")
function r=orth(x)
r=legendre(n1,0,x)*legendre(n2,0,x)
endfunction
z=integrate('orth','x',-1,1,0.001)
if z<0.001 then
z=0
else
disp("2/(2*n+1)is")
disp("(2/(2*n2)+1)")
end
disp("The answer is "+string(z))
//Plot first six bessel functions of
first kind
clf
x=linspace(0,10,100)';
n=0:5;
y=besselj(n,x);
plot2d(x,y,leg="J0@J1@J2@J3@J4
@J5");
xtitle("First six Bessel functions of
first kind","x","Jn(x)")
clc
clf;
function dy=f(x, y)
dy(1)=y(2)
dy(2)=-y(1)-2*y(2)
endfunction
x=0:0.01:10
y0=[0;5]
y=ode(y0,0,x,f)
plot2d(x,y(1,:),2)
plot2d(x,y(2,:),3)
xlabel("x axis")
ylabel("y axis")
xtitle("solution of differential
equation")
h1=legend("y(1)","y(2) or, dy/dt")
clc
clf;
function dy=f(x, y)
dy(1)=y(2)
dy(2)=-exp(-x)*y(1)+x^2
endfunction
t=0:0.01:10
y0=[0;0]
y=ode(y0,0,t,f)
plot2d(t,y(1,:),2)
plot2d(t,y(2,:),3)
xlabel("x axis")
ylabel("y axis")
xtitle("solution of differential
equation")
h1=legend("y(t)","dy/dt")
clc
clf;
function dy=f(t, y)
dy(1)=y(2)
dy(2)=-y(1)-exp(-t)*y(1)
endfunction
t=0:0.01:10
y0=[0;5]
y=ode(y0,0,t,f)
plot2d(t,y(1,:),2)
plot2d(t,y(2,:),3)
xlabel("t axis")
ylabel("y axis")
xtitle("solution of differential
equation")
h1=legend("y(1)","y(2) or, dy/dt")
//Recurrence relation of Bessel Function
//2nJn(x)'/x=J(n-1)(x)+J(n+1)(x)
clf
x=-5:0.1:5
n=input("Enter n=")
function R=rhs(x)
R=(2*n/x)*besselj(n,x)
endfunction
function L=lhs(x)
L=besselj(n-1,x)+besselj(n+1,x)
endfunction
value_lhs=feval(x,lhs)
value_rhs=feval(x,rhs)
plot(x,value_rhs,'*b')
plot(x,value_lhs,'g')
l=legend('RHS','LHS')
title("Recurrence Relation")
xlabel("x")
ylabel("J","+string(n)")
clf
clc
funcprot(0)
function dy=f(x, y)
dy=exp(-x);
endfunction
x0=0;
y0=0;
x=0:0.1:20;
y=ode(x0,y0,x,f);
plot(x,y);
xlabel('x');
ylabel('y');
xtitle('x v/s f(x,y)');
legend("dy/dx =exp(-x)")
clf
funcprot(0)
s=input("Enter sigma=")
function y=dirac(x)
y=(1/sqrt(2*(%pi)*(s^2)))*(%e)^(((x-2)^2)/(2*(s^2)))*(x+3)
endfunction
I=integrate('dirac','x',0,4)
disp(I)
clc
clf
funcprot(0)
deff('a=f(x)','a=1')
function a=period(x)
L=1
if (x>=-L)&(x<=0) then
a=-f(x)
elseif(x>=0)&(x<=L) then
a=f(x)
elseif(x>=L) then
x_new=x-2*L
a=period(x_new)
elseif(x<=-L) then
x_new=x+2*L
a=period(x_new)
end
endfunction
xlabel("Time")
ylabel("Function")
xgrid
L=1
n=50
a0=(1/L)*intg(-L,L,period,1e-2)
mprintf('a_0=%f\n',a0)
for i=1:n
function b=period1(x, f)
b=period(x)*cos(i*%pi*x/L)
endfunction
function c=period2(x, f)
c=period(x)*sin(i*%pi*x/L)
endfunction
A(i)=(1/L)*intg(-L,L,period1,1e-2)
mprintf('coefficient of cos():a_%i=%f\t',i,A(i))
B(i)=(1/L)*intg(-L,L,period2,1e-2)
mprintf('coefficient of sin():b_%i=%f\t',i,B(i))
end
function series=solution(x)
series = a0/2
for i=1:n
series=series+A(i)*cos(i*%pi*x/L)+B(i)*sin(i*%pi*x/L)
end
endfunction
x=-1:0.001:1
plot(x,solution(x),'r')
plot(x,period,'b')
plot(solution(x),period,'g')
xlabel('X-Axis')
ylabel('Y-Axis')
//Plot first six modified bessel
functions of first kind
clf
x=linspace(0,10,100)';
n=0:5;
y=besseli(n,x);
plot2d(x,y,leg="I0@I1@I2@I3@I4
@I5");
xtitle("First six modified Bessel
functions of first kind","x","In(x)")
clc
clf
a=0;
b=2*%pi;
sigma=0.1;
m=200;
xd=linspace(a,b,m)';
yd=sin(xd)+grand(xd,"nor",0,sigm
a);
n=6;
x=linspace(a,b,n)';
[y,d]=lsq_splin(xd,yd,x);
ye=sin(xd);
ys=interp(xd,x,y,d);
plot2d(xd,[ye,yd,ys],style=[2,2,3],leg="Exact function
experimental measure (Gaussian
fitted spline)")
xtitle("Least square fitting")