import numpy as np

f_1 = 5 
f_2 = 5E3
omega_1 = 2*np.pi*f_1
omega_2 = 2*np.pi*f_2

A_c = 10
a = 1

def m(t):
    return 0.2*np.cos(omega_1*t) + 0.8*np.cos(omega_2*t)

def dm_dt(t):
    return -0.2*omega_1*np.sin(omega_1*t) - 0.8*omega_2*np.sin(omega_2*t)

def ddm_dtt(t):
    return -0.2*omega_1**2*np.cos(omega_1*t) - 0.8*omega_2**2*np.cos(omega_2*t)

def g(t):
    return A_c + A_c*m(t)

# first derivative of g(t)
def dg_dt(t):
    return A_c*dm_dt(t)

# second derivative of g(t)
def ddg_dtt(t):
    return A_c**2*ddm_dtt(t)

# use Newton's method to find the maximum of the function
def newton_method(t):
    for i in range(10):
        t = t-dg_dt(t)/ddg_dtt(t)
    print("Newton method:", t, np.min(m(t)))
    return np.min(g(t))

# Sample g(t) at the maxima and minima of the carrier signal
def sample_method():
    f_s = 4*max(f_1, f_2)
    T_s = 1/f_s
    T_0 = 1/min(f_1, f_2)
    
    t = np.arange(0,T_0,T_s)

    t = t[np.argmin(m(t))]
    print("Sampling method:", t, m(t))
    return t

if __name__ == '__main__':
    newton_method(0.1)