{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Chapter 10 : Fourier Series"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 10.1, page no. 327"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"finding the fourier series of given function\n",
"enter the no of terms up to each of sin or cos terms in the expansion : 3\n",
"0.158857730350203*sin(x) + 0.127086184280162*sin(2*x) + 0.0953146382101218*sin(3*x) + 0.158857730350203*cos(x) + 0.0635430921400812*cos(2*x) + 0.0317715460700406*cos(3*x) + 0.158857730350203\n"
]
}
],
"source": [
"import sympy,math,numpy\n",
"\n",
"print \"finding the fourier series of given function\"\n",
"x = sympy.Symbol('x')\n",
"ao = 1/math.pi*sympy.integrate(sympy.exp(-1*x),(x,0,2*math.pi))\n",
"s = ao/2\n",
"n = int(raw_input(\"enter the no of terms up to each of sin or cos terms in the expansion : \"))\n",
"for i in range (1,n+1):\n",
" ai = 1/math.pi*sympy.integrate(sympy.exp(-x)*sympy.cos(i*x),(x,0,2*math.pi))\n",
" bi = 1/math.pi*sympy.integrate(sympy.exp(-x)*sympy.sin(i*x),(x,0,2*math.pi))\n",
" s = s+float(ai)*sympy.cos(i*x)+float(bi)*sympy.sin(i*x)\n",
"print s"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 10.2, page no. 328"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"To find the fourier transform of given function\n",
"Piecewise((2, s == 0), (1.0*I*exp(-1.0*I*s)/s - 1.0*I*exp(1.0*I*s)/s, True))\n",
"1.57079632679490\n"
]
}
],
"source": [
"import numpy,math,sympy\n",
"\n",
"print \"To find the fourier transform of given function\"\n",
"x = sympy.Symbol('x')\n",
"s = sympy.Symbol('s')\n",
"F = sympy.integrate(sympy.exp(1j*s*x),(x,-1,1))\n",
"print F\n",
"F1 = sympy.integrate(sympy.sin(x)/x,(x,0,numpy.inf))\n",
"print F1"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 10.3, page no. 328"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"finding the fourier series of given function\n",
"enter the no of terms up to each of sin or cos terms in the expansion : 3\n",
"3.0*sin(x) - 0.5*sin(2*x) + 1.0*sin(3*x) - 0.636619772367581*cos(x) - 0.0707355302630646*cos(3*x) - 0.785398163397448\n"
]
}
],
"source": [
"import numpy,sympy,math\n",
"\n",
"print \"finding the fourier series of given function\"\n",
"x = sympy.Symbol('x')\n",
"ao = 1/math.pi*(sympy.integrate(-1*math.pi*x**0,(x,-math.pi,0))+sympy.integrate(x,(x,0,math.pi)))\n",
"s = ao/2\n",
"n = int(raw_input(\"enter the no of terms up to each of sin or cos terms in the expansion : \"))\n",
"for i in range (1,n+1):\n",
" ai = 1/math.pi*(sympy.integrate(-1*math.pi*sympy.cos(i*x),(x,-1*math.pi,0))+sympy.integrate(x*sympy.cos(i*x),(x,0,math.pi)))\n",
" bi = 1/math.pi*(sympy.integrate(-1*math.pi*x**0*sympy.sin(i*x),(x,-1*math.pi,0))+sympy.integrate(x*sympy.sin(i*x),(x,0,math.pi)))\n",
" s = s+float(ai)*sympy.cos(i*x)+float(bi)*sympy.sin(i*x)\n",
"print s"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 10.4, page no. 329"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"finding the fourier series of given function\n",
"enter the no of terms up to each of sin or cos terms in the expansion :3\n",
"(exp(l) - exp(-l))/(2*l)\n"
]
}
],
"source": [
"import sympy,math\n",
"\n",
"print \"finding the fourier series of given function\"\n",
"x = sympy.Symbol('x')\n",
"l = sympy.Symbol('l')\n",
"ao = 1/l*sympy.integrate(sympy.exp(-1*x),(x,-l,l))\n",
"s = ao/2\n",
"n = int(raw_input(\"enter the no of terms up to each of sin or cos terms in the expansion :\"))\n",
"for i in range (1,n+1):\n",
" ai = 1/l*sympy.integrate(sympy.exp(-x)*sympy.cos(i*math.pi*x/l),(x,-l,l))\n",
" bi = 1/l*sympy.integrate(sympy.exp(-x)*sympy.sin(i*math.pi*x/l),(x,-l,l))\n",
" s = s+float(ai)*sympy.cos(i*math.pi*x/l)+float(bi)*sympy.sin(i*math.pi*x/l)\n",
"print s"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 10.5, page no. 330"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"finding the fourier series of given function\n",
"enter the no of terms up to each of terms in the expansion : 3\n",
"2.0*sin(x) - 1.0*sin(2*x) + 0.666666666666667*sin(3*x)\n"
]
}
],
"source": [
"import math,sympy\n",
"\n",
"print \"finding the fourier series of given function\"\n",
"x = sympy.Symbol('x')\n",
"l = sympy.Symbol('l')\n",
"s = 0\n",
"n = int(raw_input(\"enter the no of terms up to each of terms in the expansion : \"))\n",
"for i in range(1,n+1):\n",
" bi = 2/math.pi*sympy.integrate(x*sympy.sin(i*x),(x,0,math.pi))\n",
" s = s+float(bi)*sympy.sin(i*x)\n",
"print s"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 10.6, page no. 332"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"finding the fourier series of given function\n",
"enter the no of terms up to each of sin or cos terms in the expansion :3\n",
"l**2/3\n"
]
}
],
"source": [
"import math,sympy\n",
"\n",
"print \"finding the fourier series of given function\"\n",
"x = sympy.Symbol('x')\n",
"l = sympy.Symbol('l')\n",
"ao = 2/l*sympy.integrate(x**2,(x,0,l))\n",
"s = ao/2\n",
"n = int(raw_input(\"enter the no of terms up to each of sin or cos terms in the expansion :\"))\n",
"for i in range(1,n+1):\n",
" ai = 2/l*sympy.integrate(x**2*sympy.cos(i*math.pi*x/l),(x,0,l))\n",
" s = s+float(ai)*sympy.cos(i*math.pi*x/l)\n",
"print s"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": true
},
"source": [
"## Example 10.7, page no. 333"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"finding the fourier series of given function\n",
"enter the no of terms up to each of sin or cos terms in the expansion :2\n",
"7.06789929214115e-17*cos(x) + 0.424413181578387*cos(2*x) + 0.636619772367581\n"
]
}
],
"source": [
"import sympy,math\n",
"\n",
"print \"finding the fourier series of given function\"\n",
"x = sympy.Symbol('x')\n",
"ao = 2/math.pi*(sympy.integrate(sympy.cos(x),(x,0,math.pi/2))+sympy.integrate(-sympy.cos(x),(x,math.pi/2,math.pi)))\n",
"s = ao/2\n",
"n = int(raw_input(\"enter the no of terms up to each of sin or cos terms in the expansion :\"))\n",
"for i in range(1,n+1):\n",
" ai = 2/math.pi*(sympy.integrate(sympy.cos(x)*sympy.cos(i*x),(x,0,math.pi/2))+sympy.integrate(-sympy.cos(x)*sympy.cos(i*x),(x,math.pi/2,math.pi)))\n",
" s = s+float(ai)*sympy.cos(i*x)\n",
"print s"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": true
},
"source": [
"## Example 10.8, page no. 334"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"finding the fourier series of given function\n",
"enter the no of terms up to each of sin or cos terms in the expansion : 3\n",
"0.810569469138702*cos(x) + 7.79634366503875e-17*cos(2*x) + 0.0900632743487446*cos(3*x)\n"
]
}
],
"source": [
"import sympy,math\n",
"\n",
"print \"finding the fourier series of given function\"\n",
"x = sympy.Symbol('x')\n",
"ao = 2/math.pi*(sympy.integrate((1-2*x/math.pi),(x,0,math.pi)))\n",
"s = ao/2\n",
"n = int(raw_input(\"enter the no of terms up to each of sin or cos terms in the expansion : \"))\n",
"for i in range(1,n+1):\n",
" ai = 2/math.pi*(sympy.integrate((1-2*x/math.pi)*sympy.cos(i*x),(x,0,math.pi)))\n",
" s = s+float(ai)*sympy.cos(i*x)\n",
"print s"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 10.9, page no. 336"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"finding the fourier series of given function\n",
"enter the no of terms up to each of sin or cos terms in the expansion : 3\n",
"0.0\n"
]
}
],
"source": [
"import sympy,math\n",
"\n",
"print \"finding the fourier series of given function\"\n",
"x = sympy.Symbol('x')\n",
"l = sympy.Symbol('l')\n",
"s = 0\n",
"n = int(raw_input(\"enter the no of terms up to each of sin or cos terms in the expansion : \"))\n",
"for i in range(1,n+1):\n",
" bi = sympy.integrate(x*sympy.sin(i*math.pi*x/2),(x,0,2)) \n",
" s = s+float(bi)*sympy.sin(i*math.pi*x/2)\n",
"print float(s)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 10.10, page no. 337"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"finding the fourier series of given function\n",
"enter the no of terms up to each of sin or cos terms in the expansion : 5\n",
"1.0\n"
]
}
],
"source": [
"import sympy,math\n",
"\n",
"print \"finding the fourier series of given function\"\n",
"x = sympy.Symbol('x')\n",
"ao = 2/2*(sympy.integrate(x,(x,0,2)))\n",
"s = ao/2\n",
"n = int(raw_input(\"enter the no of terms up to each of sin or cos terms in the expansion : \"))\n",
"for i in range(1,n+1):\n",
" ai = 2/2*(sympy.integrate(x*sympy.cos(i*math.pi*x/2),(x,0,2)))\n",
" s = s +float(ai)*sympy.cos(i*math.pi*x/2)\n",
"print float(s)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 10.11, page no. 338"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"finding the fourier series of given function\n",
"enter the no of terms up to each of sin or cos terms in the expansion : 2\n",
"0.0\n"
]
}
],
"source": [
"import sympy,math\n",
"\n",
"print \"finding the fourier series of given function\"\n",
"x = sympy.Symbol('x')\n",
"ao = 0\n",
"s = ao\n",
"n = int(raw_input(\"enter the no of terms up to each of sin or cos terms in the expansion : \"))\n",
"for i in range(1,n+1):\n",
" bi = 2/1*(sympy.integrate((1/4-x)*sympy.sin(i*math.pi*x),(x,0,1/2))+sympy.integrate((x-3/4)*sympy.sin(i*math.pi*x),(x,1/2,1)))\n",
" s = s+float(bi)*sympy.sin(i*math.pi*x)\n",
"print float(s)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 10.12, page no. 339"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"finding the fourier series of given function\n",
"enter the no of terms up to each of sin or cos terms in the expansion : 3\n",
"-4.0*cos(x) + 0.999999999999999*cos(2*x) - 0.444444444444444*cos(3*x) + 3.28986813369645\n"
]
}
],
"source": [
"import sympy,math\n",
"\n",
"print \"finding the fourier series of given function\"\n",
"x = sympy.Symbol('x')\n",
"ao = 1/math.pi*sympy.integrate(x**2,(x,-math.pi,math.pi))\n",
"s = ao/2\n",
"n = int(raw_input(\"enter the no of terms up to each of sin or cos terms in the expansion : \"))\n",
"for i in range(1,n+1):\n",
" ai = 1/math.pi*sympy.integrate((x**2)*sympy.cos(i*x),(x,-math.pi,math.pi))\n",
" bi = 1/math.pi*sympy.integrate((x**2)*sympy.sin(i*x),(x,-math.pi,math.pi))\n",
" s = s+float(ai)*sympy.cos(i*x)+float(bi)*sympy.sin(i*x)\n",
"print s"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 10.13, page no. 341"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"The complex form of series is summation of f(n,x) where n varies from −%inf to %inf and f(n,x) is given by : \n",
"0\n"
]
}
],
"source": [
"import sympy,math\n",
"\n",
"print \"The complex form of series is summation of f(n,x) where n varies from −%inf to %inf and f(n,x) is given by : \"\n",
"n = sympy.Symbol('n')\n",
"x = sympy.Symbol('x')\n",
"a = sympy.exp(-x)\n",
"b = sympy.exp(-1j*math.pi*n*x)\n",
"cn = 1/2*sympy.Integral(sympy.exp(-x)*sympy.exp(-1j*math.pi*n*x),(x,-sympy.oo,sympy.oo))\n",
"fnx = float(cn)*sympy.exp(1j*n*math.pi*x) \n",
"print fnx"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 10.14, page no. 342"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Practical harmoninc analysis\n",
"No of sin or cos term in expansion : 5\n",
"-0.522944001594253*sin(x) - 0.410968418886797*sin(2*x) + 0.0927272727272729*sin(3*x) + 0.111796006670355*sin(4*x) - 0.126146907496654*sin(5*x) - 0.373397459621556*cos(x) + 0.159090909090909*cos(2*x) - 0.258181818181819*cos(3*x) - 0.257272727272728*cos(4*x) - 0.546602540378445*cos(5*x) + 1.30454545454545\n"
]
}
],
"source": [
"import sympy, math, numpy\n",
"\n",
"print \"Practical harmoninc analysis\"\n",
"x = sympy.Symbol('x')\n",
"xo = numpy.array([math.pi/6, math.pi/3, math.pi/2, 2*math.pi/3, 5*math.pi/6, math.pi, 7*math.pi/6, 4*math.pi/3, \\\n",
" 3*math.pi/2, 5*math.pi/3, 11*math.pi/6])\n",
"yo = numpy.array([1.10, 0.30, 0.16, 1.50, 1.30, 2.16, 1.25, 1.30, 1.52, 1.76, 2.00])\n",
"ao = 2*numpy.sum(yo)/len(xo)\n",
"s = ao/2\n",
"n = int(raw_input('No of sin or cos term in expansion : '))\n",
"for i in range(1,n+1):\n",
" an = 2*sum(yo*numpy.cos(i*xo))/len(yo)\n",
" bn = 2*sum(yo*numpy.sin(i*xo))/len(yo)\n",
" s = s+float(an)*sympy.cos(i*x)+float(bn)*sympy.sin(i*x)\n",
"print s"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": true
},
"source": [
"## Example 10.15, page no. 342"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Practical harmonic analysis\n",
"No of sin or cos term in expansion :5\n",
"-1.61653377487451e-16*sin(6.28318530717959*x/T) + 1.5*cos(6.28318530717959*x/T) + 0.75\n",
"1.5 -1.61653377487e-16\n",
"Direct current :\n",
"1.5\n"
]
}
],
"source": [
"import math,sympy,numpy\n",
"\n",
"print \"Practical harmonic analysis\"\n",
"x = sympy.Symbol('x')\n",
"T = sympy.Symbol('T')\n",
"xo = numpy.array([1/6,1/3,1/2,2/3,5/6,1])\n",
"yo = numpy.array([1.30,1.05,1.30,-0.88,-0.25,1.98])\n",
"ao = 2*sum(yo)/len(xo)\n",
"s = ao/2\n",
"n = int(raw_input(\"No of sin or cos term in expansion :\"))\n",
"i = 1\n",
"an = 2*sum(yo*numpy.cos(i*xo*2*math.pi))/len(yo)\n",
"bn = 2*sum(yo*numpy.sin(i*xo*2*math.pi))/len(yo)\n",
"s = s+float(an)*sympy.cos(i*x*2*math.pi/T)+float(bn)*sympy.sin(i*x*2*math.pi/T)\n",
"print s\n",
"print \"Direct current :\"\n",
"i = math.sqrt(an**2+bn**2)\n",
"print i"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 10.16, page no. 343"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Practical harmonic analysis\n",
"Input xo matrix (in factor of T):1\n",
"No of sin or cos term in expansion :5\n",
"-3.67394039744206e-16*sin(6.28318530717959*x/T) + 1.5*cos(6.28318530717959*x/T) + 4.5\n",
"Direct current :\n",
"1.5\n"
]
}
],
"source": [
"import math,sympy,numpy\n",
"\n",
"print \"Practical harmonic analysis\"\n",
"x = sympy.Symbol('x')\n",
"T = sympy.Symbol('T')\n",
"xo = int(raw_input(\"Input xo matrix (in factor of T):\"))\n",
"yo = numpy.array([1.30,1.05,1.30,-0.88,-0.25,1.98])\n",
"ao = 2*sum(yo)/xo \n",
"s = ao/2\n",
"n = int(raw_input(\"No of sin or cos term in expansion :\"))\n",
"i = 1\n",
"an = 2*sum(yo*numpy.cos(i*xo*2*math.pi))/len(yo)\n",
"bn = 2*sum(yo*numpy.sin(i*xo*2*math.pi))/len(yo)\n",
"s = s+float(an)*sympy.cos(i*x*2*math.pi/T)+float(bn)*sympy.sin(i*x*2*math.pi/T)\n",
"print s\n",
"print \"Direct current :\"\n",
"i = math.sqrt(an**2+bn**2)\n",
"print i"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 10.17, page no. 344"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Practical harmonic analysis\n",
"Input xo matrix (in factor of T):1\n",
"No of sin or cos term in expansion :4\n",
"-3.67394039744206e-16*sin(6.28318530717959*x/T) + 1.5*cos(6.28318530717959*x/T) + 4.5\n",
"Direct current :\n",
"1.5\n"
]
}
],
"source": [
"import math,sympy,numpy\n",
"\n",
"print \"Practical harmonic analysis\"\n",
"x = sympy.Symbol('x')\n",
"T = sympy.Symbol('T')\n",
"xo = int(raw_input(\"Input xo matrix (in factor of T):\"))\n",
"yo = numpy.array([1.30,1.05,1.30,-0.88,-0.25,1.98])\n",
"ao = 2*sum(yo)/xo \n",
"s = ao/2\n",
"n = int(raw_input(\"No of sin or cos term in expansion :\"))\n",
"i = 1\n",
"an = 2*sum(yo*numpy.cos(i*xo*2*math.pi))/len(yo)\n",
"bn = 2*sum(yo*numpy.sin(i*xo*2*math.pi))/len(yo)\n",
"s = s+float(an)*sympy.cos(i*x*2*math.pi/T)+float(bn)*sympy.sin(i*x*2*math.pi/T)\n",
"print s\n",
"print \"Direct current :\"\n",
"i = math.sqrt(an**2+bn**2)\n",
"print i"
]
}
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