Linear Algerbra assignment.py

# This code was written for an assignment done during a course at the University of Waterloo. 
# Due to copyright and trademark rules, the questions for which the code was written for is not provided. 

import math
import random
import numpy as np
class BoxMueller:
  call_count = 0;
  second_random_variable = 0.0;
    
  @classmethod
  def randn( cls ):
    cls.call_count += 1;

    if cls.call_count%2 == 0:
      return cls.second_random_variable;
    
    u1 = 1.0 - random.random();
    u2 = 1.0 - random.random();
    
    c = math.sqrt( -2.0*math.log(u1) );

    cls.second_random_variable = c*math.sin( 2*math.pi*u2 );
    return c*math.cos( 2*math.pi*u2 );

white_noise = [None] * 16

# Question 1
random.seed(132456  )     
for i in range( 0, 16 ):
  white_noise[i] = BoxMueller.randn();
pray = np.array(white_noise)
np.sqrt(np.sum(pray*pray))
np.mean(pray)


print( white_noise )
random.seed( 278567 )     
for i in range( 0, 16 ):
  white_noise[i] = BoxMueller.randn();
pray = np.array(white_noise)
np.sqrt(np.sum(pray*pray))
np.mean(pray)

print( white_noise )
random.seed( 354879)     
for i in range( 0, 16 ):
  white_noise[i] = BoxMueller.randn();
pray = np.array(white_noise)
np.sqrt(np.sum(pray*pray))
np.mean(pray)

print( white_noise )
random.seed(  4568742 )     
for i in range( 0, 16 ):
  white_noise[i] = BoxMueller.randn();
pray = np.array(white_noise)
np.sqrt(np.sum(pray*pray))
np.mean(pray)

print( white_noise )
random.seed( 5765892)     
for i in range( 0, 16 ):
  white_noise[i] = BoxMueller.randn();
pray = np.array(white_noise)
np.sqrt(np.sum(pray*pray))
np.mean(pray)
print( white_noise )

#Question 3
x = np.arange(16)
print(x)
m1 = np.cos(2.*np.pi * 1. * (x+0.5)/16.)
print(m1)
m2 = np.cos(2.*np.pi * 2. * (x+0.5)/16.)
print(m2)
m3 = np.cos(2.*np.pi * 3. * (x+0.5)/16.)
print(m3)
m4 = np.cos(2.*np.pi * 4. * (x+0.5)/16.)
print(m4)
m5 = np.cos(2.*np.pi * 5. * (x+0.5)/16.)
print(m5)
m6 = np.cos(2.*np.pi * 6. * (x+0.5)/16.)
print(m6)
m7 = np.cos(2.*np.pi * 7. * (x+0.5)/16.)
print(m7)
m8 = np.cos(2.*np.pi * 8. * (x+0.5)/16.)
print(m8)
m9 = np.sin(2.*np.pi * 1. * (x+0.5)/16.)
print(m9)
m10 = np.sin(2.*np.pi * 2. * (x+0.5)/16.)
print(m10)
m11 = np.sin(2.*np.pi * 3. * (x+0.5)/16.)
print(m11)
m12 = np.sin(2.*np.pi * 4. * (x+0.5)/16.)
print(m12)
m13 = np.sin(2.*np.pi * 5. * (x+0.5)/16.)
print(m13)
m14 = np.sin(2.*np.pi * 6. * (x+0.5)/16.)
print(m14)
m15 = np.sin(2.*np.pi * 7. * (x+0.5)/16.)
print(m15)
m16 = np.sin(2.*np.pi * 8. * (x+0.5)/16.)
print(m16)
m0 = np.ones(16)
print(m0)
# Question 4
m1_nor = np.sqrt(np.sum(m1*m1))
print("m1 nor:", m1_nor)
m2_nor = np.sqrt(np.sum(m2*m2))
print("m2 nor:", m2_nor)
m3_nor = np.sqrt(np.sum(m3*m3))
print("m3 nor:", m3_nor)
m4_nor = np.sqrt(np.sum(m4*m4))
print("m4 nor:", m4_nor)
m5_nor = np.sqrt(np.sum(m5*m5))
print("m5 nor:", m5_nor)
m6_nor = np.sqrt(np.sum(m6*m6))
print("m6 nor:", m6_nor)
m7_nor = np.sqrt(np.sum(m7*m7))
print("m7 nor:", m7_nor)
m9_nor = np.sqrt(np.sum(m9*m9))
print("m9 nor:", m9_nor)
m10_nor = np.sqrt(np.sum(m10*m10))
print("m10 nor:", m10_nor)
m11_nor = np.sqrt(np.sum(m11*m11))
print("m11 nor:", m11_nor)
m12_nor = np.sqrt(np.sum(m12*m12))
print("m12 nor:", m12_nor)
m13_nor = np.sqrt(np.sum(m13*m13))
print("m13 nor:", m13_nor)
m14_nor = np.sqrt(np.sum(m14*m14))
print("m14 nor:", m14_nor)
m15_nor = np.sqrt(np.sum(m15*m15))
print("m15 nor:", m15_nor)
m16_nor = np.sqrt(np.sum(m16*m16))
print("m16 nor:", m16_nor)
m0_nor = np.sqrt(np.sum(m0*m0))
print("m0 nor:", m0_nor)
m1n = m1/m1_nor
m2n = m2/m2_nor
m3n = m3/m3_nor
m4n = m4/m4_nor
m5n = m5/m5_nor
m6n = m6/m6_nor
m7n = m7/m7_nor
m9n = m9/m9_nor
m10n = m10/m10_nor
m11n = m11/m11_nor
m12n = m12/m12_nor
m13n = m13/m13_nor
m14n = m14/m14_nor
m15n = m15/m15_nor
m16n = m16/m16_nor
m0n = m0/m0_nor
# Question 5
print("m0 in m1",np.sum(m0n*m1n))
print("m0 in m2",np.sum(m0n*m2n))
print("m0 in m3",np.sum(m0n*m3n))
print("m0 in m4",np.sum(m0n*m4n))
print("m0 in m5",np.sum(m0n*m5n))
print("m0 in m6",np.sum(m0n*m6n))
print("m0 in m7",np.sum(m0n*m7n))
print("m0 in m9",np.sum(m0n*m9n))
print("m0 in m10",np.sum(m0n*m10n))
print("m0 in m11",np.sum(m0n*m11n))
print("m0 in m12",np.sum(m0n*m12n))
print("m0 in m13",np.sum(m0n*m13n))
print("m0 in m14",np.sum(m0n*m14n))
print("m0 in m15",np.sum(m0n*m15n))
print("m0 in m16",np.sum(m0n*m16n))
print("m1 in m2",np.sum(m1n*m2n))
print("m1 in m3",np.sum(m1n*m3n))
print("m1 in m4",np.sum(m1n*m4n))
print("m1 in m5",np.sum(m1n*m5n))
print("m1 in m6",np.sum(m1n*m6n))
print("m1 in m7",np.sum(m1n*m7n))
print("m1 in m9",np.sum(m1n*m9n))
print("m1 in m10",np.sum(m1n*m10n))
print("m1 in m11",np.sum(m1n*m11n))
print("m1 in m12",np.sum(m1n*m12n))
print("m1 in m13",np.sum(m1n*m13n))
print("m1 in m14",np.sum(m1n*m14n))
print("m1 in m15",np.sum(m1n*m15n))
print("m1 in m16",np.sum(m1n*m16n))
print("m2 in m3",np.sum(m2n*m3n))
print("m2 in m4",np.sum(m2n*m4n))
print("m2 in m5",np.sum(m2n*m5n))
print("m2 in m6",np.sum(m2n*m6n))
print("m2 in m7",np.sum(m2n*m7n))
print("m2 in m9",np.sum(m2n*m9n))
print("m2 in m10",np.sum(m2n*m10n))
print("m2 in m11",np.sum(m2n*m11n))
print("m2 in m12",np.sum(m2n*m12n))
print("m2 in m13",np.sum(m2n*m13n))
print("m2 in m14",np.sum(m2n*m14n))
print("m2 in m15",np.sum(m2n*m15n))
print("m2 in m16",np.sum(m2n*m16n))
print("m3 in m4",np.sum(m3n*m4n))
print("m3 in m5",np.sum(m3n*m5n))
print("m3 in m6",np.sum(m3n*m6n))
print("m3 in m7",np.sum(m3n*m7n))
print("m3 in m9",np.sum(m3n*m9n))
print("m3 in m10",np.sum(m3n*m10n))
print("m3 in m11",np.sum(m3n*m11n))
print("m3 in m12",np.sum(m3n*m12n))
print("m3 in m13",np.sum(m3n*m13n))
print("m3 in m14",np.sum(m3n*m14n))
print("m3 in m15",np.sum(m3n*m15n))
print("m3 in m16",np.sum(m3n*m16n))
print("m4 in m5",np.sum(m4n*m5n))
print("m4 in m6",np.sum(m4n*m6n))
print("m4 in m7",np.sum(m4n*m7n))
print("m4 in m9",np.sum(m4n*m9n))
print("m4 in m10",np.sum(m4n*m10n))
print("m4 in m11",np.sum(m4n*m11n))
print("m4 in m12",np.sum(m4n*m12n))
print("m4 in m13",np.sum(m4n*m13n))
print("m4 in m14",np.sum(m4n*m14n))
print("m4 in m15",np.sum(m4n*m15n))
print("m4 in m16",np.sum(m4n*m16n))
print("m5 in m6",np.sum(m5n*m6n))
print("m5 in m7",np.sum(m5n*m7n))
print("m5 in m9",np.sum(m5n*m9n))
print("m5 in m10",np.sum(m5n*m10n))
print("m5 in m11",np.sum(m5n*m11n))
print("m5 in m12",np.sum(m5n*m12n))
print("m5 in m13",np.sum(m5n*m13n))
print("m5 in m14",np.sum(m5n*m14n))
print("m5 in m15",np.sum(m5n*m15n))
print("m5 in m16",np.sum(m5n*m16n))
print("m6 in m7",np.sum(m6n*m7n))
print("m6 in m9",np.sum(m6n*m9n))
print("m6 in m10",np.sum(m6n*m10n))
print("m6 in m11",np.sum(m6n*m11n))
print("m6 in m12",np.sum(m6n*m12n))
print("m6 in m13",np.sum(m6n*m13n))
print("m6 in m14",np.sum(m6n*m14n))
print("m6 in m15",np.sum(m6n*m15n))
print("m6 in m16",np.sum(m6n*m16n))
print("m7 in m9",np.sum(m7n*m9n))
print("m7 in m10",np.sum(m7n*m10n))
print("m7 in m11",np.sum(m7n*m11n))
print("m7 in m12",np.sum(m7n*m12n))
print("m7 in m13",np.sum(m7n*m13n))
print("m7 in m14",np.sum(m7n*m14n))
print("m7 in m15",np.sum(m7n*m15n))
print("m7 in m16",np.sum(m7n*m16n))
print("m9 in m10",np.sum(m9n*m10n))
print("m9 in m11",np.sum(m9n*m11n))
print("m9 in m12",np.sum(m9n*m12n))
print("m9 in m13",np.sum(m9n*m13n))
print("m9 in m14",np.sum(m9n*m14n))
print("m9 in m15",np.sum(m9n*m15n))
print("m9 in m16",np.sum(m9n*m16n))
print("m10 in m11",np.sum(m10n*m11n))
print("m10 in m12",np.sum(m10n*m12n))
print("m10 in m13",np.sum(m10n*m13n))
print("m10 in m14",np.sum(m10n*m14n))
print("m10 in m15",np.sum(m10n*m15n))
print("m10 in m16",np.sum(m10n*m16n))
print("m11 in m12",np.sum(m11n*m12n))
print("m11 in m13",np.sum(m11n*m13n))
print("m11 in m14",np.sum(m11n*m14n))
print("m11 in m15",np.sum(m11n*m15n))
print("m11 in m16",np.sum(m11n*m16n))
print("m12 in m13",np.sum(m12n*m13n))
print("m12 in m14",np.sum(m12n*m14n))
print("m12 in m15",np.sum(m12n*m15n))
print("m12 in m16",np.sum(m12n*m16n))
print("m13 in m14",np.sum(m13n*m14n))
print("m13 in m15",np.sum(m13n*m15n))
print("m13 in m16",np.sum(m13n*m16n))
print("m14 in m15",np.sum(m14n*m15n))
print("m14 in m16",np.sum(m14n*m16n))
print("m15 in m16",np.sum(m15n*m16n))
# question 6 also add the whitnoise inner product with the zero vector
random.seed(132456)     
for i in range( 0, 16 ):
  white_noise[i] = BoxMueller.randn();
pray = np.array(white_noise)
np.sqrt(np.sum(pray*pray))
np.mean(pray)
print("w10 in m1",np.sum(white_noise*m1n))
print("w10 in m2",np.sum(white_noise*m2n))
print("w10 in m3",np.sum(white_noise*m3n))
print("w10 in m4",np.sum(white_noise*m4n))
print("w10 in m5",np.sum(white_noise*m5n))
print("w10 in m6",np.sum(white_noise*m6n))
print("w10 in m7",np.sum(white_noise*m7n))
print("w10 in m9",np.sum(white_noise*m9n))
print("w10 in m10",np.sum(white_noise*m10n))
print("w10 in m11",np.sum(white_noise*m11n))
print("w10 in m12",np.sum(white_noise*m12n))
print("w10 in m13",np.sum(white_noise*m13n))
print("w10 in m14",np.sum(white_noise*m14n))
print("w10 in m15",np.sum(white_noise*m15n))
print("w10 in m16",np.sum(white_noise*m16n))
print("w10 in m0",np.sum(white_noise*m0n))

random.seed( 278567 )     
for i in range( 0, 16 ):
  white_noise[i] = BoxMueller.randn();
pray = np.array(white_noise)
np.sqrt(np.sum(pray*pray))
np.mean(pray)
print("w20 in m1",np.sum(white_noise*m1n))
print("w20 in m2",np.sum(white_noise*m2n))
print("w20 in m3",np.sum(white_noise*m3n))
print("w20 in m4",np.sum(white_noise*m4n))
print("w20 in m5",np.sum(white_noise*m5n))
print("w20 in m6",np.sum(white_noise*m6n))
print("w20 in m7",np.sum(white_noise*m7n))
print("w20 in m9",np.sum(white_noise*m9n))
print("w20 in m10",np.sum(white_noise*m10n))
print("w20 in m11",np.sum(white_noise*m11n))
print("w20 in m12",np.sum(white_noise*m12n))
print("w20 in m13",np.sum(white_noise*m13n))
print("w20 in m14",np.sum(white_noise*m14n))
print("w20 in m15",np.sum(white_noise*m15n))
print("w20 in m16",np.sum(white_noise*m16n))
print("w20 in m0",np.sum(white_noise*m0n))

random.seed( 354879)     
for i in range( 0, 16 ):
  white_noise[i] = BoxMueller.randn();
pray = np.array(white_noise)
np.sqrt(np.sum(pray*pray))
np.mean(pray)
print("w30 in m1",np.sum(white_noise*m1n))
print("w30 in m2",np.sum(white_noise*m2n))
print("w30 in m3",np.sum(white_noise*m3n))
print("w30 in m4",np.sum(white_noise*m4n))
print("w30 in m5",np.sum(white_noise*m5n))
print("w30 in m6",np.sum(white_noise*m6n))
print("w30 in m7",np.sum(white_noise*m7n))
print("w30 in m9",np.sum(white_noise*m9n))
print("w30 in m10",np.sum(white_noise*m10n))
print("w30 in m11",np.sum(white_noise*m11n))
print("w30 in m12",np.sum(white_noise*m12n))
print("w30 in m13",np.sum(white_noise*m13n))
print("w30 in m14",np.sum(white_noise*m14n))
print("w30 in m15",np.sum(white_noise*m15n))
print("w30 in m16",np.sum(white_noise*m16n))
print("m30 in v0",np.sum(white_noise*m0n))

random.seed(  4568742 )     
for i in range( 0, 16 ):
  white_noise[i] = BoxMueller.randn();
pray = np.array(white_noise)
np.sqrt(np.sum(pray*pray))
np.mean(pray)
print("w40 in m1",np.sum(white_noise*m1n))
print("w40 in m2",np.sum(white_noise*m2n))
print("w40 in m3",np.sum(white_noise*m3n))
print("w40 in m4",np.sum(white_noise*m4n))
print("w40 in m5",np.sum(white_noise*m5n))
print("w40 in m6",np.sum(white_noise*m6n))
print("w40 in m7",np.sum(white_noise*m7n))
print("w40 in m9",np.sum(white_noise*m9n))
print("w40 in m10",np.sum(white_noise*m10n))
print("w40 in m11",np.sum(white_noise*m11n))
print("w40 in m12",np.sum(white_noise*m12n))
print("w40 in m13",np.sum(white_noise*m13n))
print("w40 in m14",np.sum(white_noise*m14n))
print("w40 in m15",np.sum(white_noise*m15n))
print("w40 in m16",np.sum(white_noise*m16n))
print("w40 in m0",np.sum(white_noise*m0n))

random.seed( 5765892)     
for i in range( 0, 16 ):
  white_noise[i] = BoxMueller.randn();
pray = np.array(white_noise)
np.sqrt(np.sum(pray*pray))
np.mean(pray)
print("w50 in m1",np.sum(white_noise*m1n))
print("w50 in m2",np.sum(white_noise*m2n))
print("w50 in m3",np.sum(white_noise*m3n))
print("w50 in m4",np.sum(white_noise*m4n))
print("w50 in m5",np.sum(white_noise*m5n))
print("w50 in m6",np.sum(white_noise*m6n))
print("w50 in m7",np.sum(white_noise*m7n))
print("w50 in m9",np.sum(white_noise*m9n))
print("w50 in m10",np.sum(white_noise*m10n))
print("w50 in m11",np.sum(white_noise*m11n))
print("w50 in m12",np.sum(white_noise*m12n))
print("w50 in m13",np.sum(white_noise*m13n))
print("w50 in m14",np.sum(white_noise*m14n))
print("w50 in m15",np.sum(white_noise*m15n))
print("w50 in m16",np.sum(white_noise*m16n))
print("w50 in m0",np.sum(white_noise*m0n))
# Question 7 
x = np.arange(16)
A = 20
Fee = 2
phi= 0.9197057831
sig = A*np.sin(2*np.pi*Fee*(x+0.5)/16. + phi)
coef1= np.sum(sig*m1n)
coef2=np.sum(sig*m2n)
coef3 =np.sum(sig*m3n)
coef4 =np.sum(sig*m4n)
coef5 = np.sum(sig*m5n)
coef6 = np.sum(sig*m6n)
coef7 = np.sum(sig*m7n)
coef9 = np.sum(sig*m9n)
coef10 = np.sum(sig*m10n)
coef11 = np.sum(sig*m11n)
coef12 = np.sum(sig*m12n)
coef13 = np.sum(sig*m13n)
coef14 = np.sum(sig*m14n)
coef15 = np.sum(sig*m15n)
coef16 = np.sum(sig*m16n)
coef0 = np.sum(sig*m0n)
print ("sig of 2-Norm", np.sqrt(np.sum(sig*sig)))
print(coef1)
print(coef2)
print(coef3)
print(coef4)
print(coef5)
print(coef6)
print(coef7)
print(coef9)
print(coef10)
print(coef11)
print(coef12)
print(coef13)
print(coef14)
print(coef15)
print(coef16)
print(coef0)
print ("sig. on the coef. of 2-Norm", np.sqrt((coef1*coef1)+(coef9*coef9)))

# Question 8
f1 = 1.398745
sig_1 = A*np.sin(2*np.pi*f1*(x+0.5)/16. + phi)
coef1x= np.sum(sig_1*m1n)
coef2x=np.sum(sig_1*m2n)
coef3x =np.sum(sig_1*m3n)
coef4x =np.sum(sig_1*m4n)
coef5x = np.sum(sig_1*m5n)
coef6x = np.sum(sig_1*m6n)
coef7x = np.sum(sig_1*m7n)
coef9x = np.sum(sig_1*m9n)
coef10x = np.sum(sig_1*m10n)
coef11x = np.sum(sig_1*m11n)
coef12x = np.sum(sig_1*m12n)
coef13x = np.sum(sig_1*m13n)
coef14x = np.sum(sig_1*m14n)
coef15x = np.sum(sig_1*m15n)
coef16x = np.sum(sig_1*m16n)
coef0x = np.sum(sig_1*m0n)
print(coef1x)
print(coef2x)
print(coef3x)
print(coef4x)
print(coef5x)
print(coef6x)
print(coef7x)
print(coef9x)
print(coef10x)
print(coef11x)
print(coef12x)
print(coef13x)
print(coef14x)
print(coef15x)
print(coef16x)
print(coef0x)
print ("2-Norm of Sig.", np.sqrt(np.sum(sig_1*sig_1)))
print ("sig. on the coef. of 2-Norm", np.sqrt((coef1x*coef1x)+(coef9x*coef9x)+(coef2x*coef2x)+(coef3x*coef3x)+(coef4x*coef4x)+(coef5x*coef5x)+(coef6x*coef6x)+(coef7x*coef7x)+(coef10x*coef10x)+(coef11x*coef11x)+(coef12x*coef12x)+(coef13x*coef13x)+(coef14x*coef14x)+(coef15x*coef15x)+(coef16x*coef16x)+(coef0x*coef0x)))
# The coefficients does provide usefull info on the 16 norxalized dimentional vector as it provides the 2 norm of the signal_1 vectors and provides some what of the frequicency as we can give a rough apporximation 
# Question 9 
White_noise1 = np.array(( -1.90, -6.05, -2.48, 2.82, -.324, 3.99, 5.22, -2.90, -.599, -1.64, -3.76, -2.97, 3.39, 4.23, 3.26, 3.35  ))
coef1m= np.sum(White_noise1*m1n)
coef2m=np.sum(White_noise1*m2n)
coef3m =np.sum(White_noise1*m3n)
coef4m =np.sum(White_noise1*m4n)
coef5m = np.sum(White_noise1*m5n)
coef6m = np.sum(White_noise1*m6n)
coef7m = np.sum(White_noise1*m7n)
coef9m = np.sum(White_noise1*m9n)
coef10m = np.sum(White_noise1*m10n)
coef11m = np.sum(White_noise1*m11n)
coef12m = np.sum(White_noise1*m12n)
coef13m = np.sum(White_noise1*m13n)
coef14m = np.sum(White_noise1*m14n)
coef15m = np.sum(White_noise1*m15n)
coef16m = np.sum(White_noise1*m16n)
coef0m = np.sum(White_noise1*m0n)
print(coef1m)
print(coef2m)
print(coef3m)
print(coef4m)
print(coef5m)
print(coef6m)
print(coef7m)
print(coef9m)
print(coef10m)
print(coef11m)
print(coef12m)
print(coef13m)
print(coef14m)
print(coef15m)
print(coef16m)
print(coef0m)
# The frequencies are 5 and 2 
#Question 10
t1 = np.array ((-0.1591651345,-0.9730117519,-0.5855458235,0.5248543817,0.9872519740,0.2307555723,-0.8106393076,-0.8511920292,0.1591651223,0.9730117520,0.5855458262,-0.5248543789,-0.9872519761,-0.2307555755,0.8106393115,0.8511920414))
coef1e= np.sum(t1*m1n)
coef2e=np.sum(t1*m2n)
coef3e =np.sum(t1*m3n)
coef4e =np.sum(t1*m4n)
coef5e = np.sum(t1*m5n)
coef6e = np.sum(t1*m6n)
coef7e = np.sum(t1*m7n)
coef9e = np.sum(t1*m9n)
coef10e = np.sum(t1*m10n)
coef11e = np.sum(t1*m11n)
coef12e = np.sum(t1*m12n)
coef13e = np.sum(t1*m13n)
coef14e = np.sum(t1*m14n)
coef15e = np.sum(t1*m15n)
coef16e = np.sum(t1*m16n)
coef0e = np.sum(t1*m0n)
print(coef1e)
print(coef2e)
print(coef3e)
print(coef4e)
print(coef5e)
print(coef6e)
print(coef7e)
print(coef9e)
print(coef10e)
print(coef11e)
print(coef12e)
print(coef13e)
print(coef14e)
print(coef15e)
print(coef16e)
print(coef0e)
# The frequensies is 3