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{
"metadata": {
"name": "",
"signature": "sha256:5d045633a134a6b88641045e0705da6aba6d8735cb400399955dbb6bc55628cc"
},
"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
{
"cells": [
{
"cell_type": "heading",
"level": 1,
"metadata": {},
"source": [
"Chapter 7 - Sampling Theory and Pulse modulation"
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 1 - pg 324"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#calculate the nyquist rate\n",
"\n",
"import math\n",
"#given\n",
"#analog signal x(t) = 3*cos(50*math.pi*t) + 10*sin(300*math.pi*t) - cos(100*math.pi*t)\n",
"#comparing signal with x(t) = 3*cos(w_1*t) + 10*sin(w_2*t) - cos(w_3*t)\n",
"#therefore\n",
"w_1 = 50*math.pi;#first frequency in rad/sec\n",
"w_2 = 300*math.pi;#second frequency in rad/sec\n",
"w_3 = 100*math.pi;#third frequency in rad/sec\n",
"\n",
"#calculations\n",
"f_1 = w_1/(2*math.pi);#first frequency in Hz\n",
"f_2 = w_2/(2*math.pi);#second frequency in Hz\n",
"f_3 = w_3/(2*math.pi);#third frequency in Hz\n",
"f_m = f_2#maximum frequency \n",
"f_s = 2*f_m#nyquist rate for a signal\n",
"\n",
"#results\n",
"print \"Nyquist rate (Hz) = \",f_s\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Nyquist rate (Hz) = 300.0\n"
]
}
],
"prompt_number": 2
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 2 - pg 325"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#calculate the Nyquist rate and interval\n",
"import math\n",
"#given\n",
"#x(t) = (1/()2*math.pi))*cos(4000*math.pi*t)*cos(1000*math.pi*t)\n",
"#expanding\n",
"print(\"x(t) = (1/(2*math.pi)*cos(4000*math.pi*t)*cos(1000*math.pi*t)\");\n",
"print(\"x(t) = (1/(4*math.pi)*2*cos(4000*math.pi*t)*cos(1000*math.pi*t)\");\n",
"print(\"x(t) = (1/(4*math.pi))*[cos(4000*math.pi*t + 1000*pi*t)*cos(4000*math.pi*t - 1000*math.pi*t)]\")\n",
"print(\"x(t) = (1/(4*math.pi))*[cos(5000*math.pi*t + cos(3000*math.pi*t))]\")\n",
"#by comparing above equation with x(t) = (1/(4*math.pi))*[cos(w_1*t) + cos(w_2*t)] \n",
"w_1 = 5000.*math.pi\n",
"w_2 = 3000.*math.pi\n",
"\n",
"#calculations\n",
"f_1 = w_1/(2*math.pi);\n",
"f_2 = w_2 /(2*math.pi);\n",
"f_m = f_1\n",
"f_s = 2*f_m#Nyquist rate\n",
"T_s = 1/f_s#Nyquist interval\n",
"\n",
"#results\n",
"print \"Nyquist rate (Hz) = \",f_s\n",
"print \"Nyquist interval (msec) = \",T_s*1000\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"x(t) = (1/(2*math.pi)*cos(4000*math.pi*t)*cos(1000*math.pi*t)\n",
"x(t) = (1/(4*math.pi)*2*cos(4000*math.pi*t)*cos(1000*math.pi*t)\n",
"x(t) = (1/(4*math.pi))*[cos(4000*math.pi*t + 1000*pi*t)*cos(4000*math.pi*t - 1000*math.pi*t)]\n",
"x(t) = (1/(4*math.pi))*[cos(5000*math.pi*t + cos(3000*math.pi*t))]\n",
"Nyquist rate (Hz) = 5000.0\n",
"Nyquist interval (msec) = 0.2\n"
]
}
],
"prompt_number": 9
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 3 - pg 326"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#calculate the discrete time signal for all conditions\n",
"\n",
"#given\n",
"#x(t) = 8*cos(200*%pi*t)\n",
"f= 100.#highest frequency component of continuous time signal in hertz\n",
"f_s2 = 400.#sampling frequency in hertz for second condition\n",
"f_s3 = 400.#sampling frequency in hertz for third condition\n",
"f_s4 = 150.#sampling frequency in hertz for fourth condition since 0 < f_s4 < f_s2/2 \n",
"\n",
"#calcultions\n",
"NR = 2*f#Nyquist rate\n",
"F_1 = f/NR;\n",
"F_2 = f/f_s2;\n",
"F_3 = f/f_s3;\n",
"F_4 = f/f_s4;\n",
"f_4 = f_s4*F_4;\n",
"\n",
"#results\n",
"print \"The discrete time signal x(n) for the first condition is x(n) = 8*cos(2*pi*\",F_1,\"*n)\"\n",
"print \"the discrete time signal x(n) for the second condition is x(n) = 8*cos(2*pi*\",F_2,\"*n)\"\n",
"print \"the discrete time signal x(n) for the third condition is x(n) = 8*cos(2*pi*\",F_3,\"*n)\"\n",
"print \"The discrete time signal x(n) for the fourth condition is x(n) = 8*cos(2*pi*\",f_4,\"*t)\"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The discrete time signal x(n) for the first condition is x(n) = 8*cos(2*pi* 0.5 *n)\n",
"the discrete time signal x(n) for the second condition is x(n) = 8*cos(2*pi* 0.25 *n)\n",
"the discrete time signal x(n) for the third condition is x(n) = 8*cos(2*pi* 0.25 *n)\n",
"The discrete time signal x(n) for the fourth condition is x(n) = 8*cos(2*pi* 100.0 *t)\n"
]
}
],
"prompt_number": 6
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 4 - pg 327"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#calculate the nyquist rate\n",
"import math\n",
"#given\n",
"#x(t) = 6*cos(50*math.pi*t) + 20*sin(300*math.pi*t) - 10*cos(100*math.pi*t)\n",
"#by comparing with standard eqn x(t) = A_1*cos(w_1*t) + A_2*sin(w_2*t) + A_3*cos(w_3*t) we get \n",
"w_1 = 50*math.pi#frequency in rad/sec\n",
"w_2 =300*math.pi#frequency in rad/sec\n",
"w_3 = 100*math.pi#frequency in rad/sec\n",
"\n",
"#calculations\n",
"f_1 = w_1/(2*math.pi)#frequency in hertz\n",
"f_2 = w_2/(2*math.pi)#frequency in hertz\n",
"f_3 = w_3/(2*math.pi)#frequency in hertz\n",
"if f_1 > f_2 and f_1> f_3:\n",
" f_max = f_1\n",
"elif f_2 > f_1 and f_2> f_3:\n",
" f_max = f_2\n",
"elif f_3 > f_1 and f_3> f_2:\n",
" f_max = f_3\n",
"\n",
"f_s = 2*f_max;#nyquist rate\n",
"\n",
"#results\n",
"print \"Nyquist rate for a continuous signal (Hz) = \",f_s\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Nyquist rate for a continuous signal (Hz) = 300.0\n"
]
}
],
"prompt_number": 5
}
],
"metadata": {}
}
]
}
|
