Pretty much all of Fall 2024

7th-Semester-Fall-2024/ECOMMS/homework/homework-2/problem-1.md

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+# ECOMMS Homework 2 - Aidan Sharpe
+
+## Problem 1
+```python
+import numpy as np
+
+f_c = 1250
+f_m = 125
+A_c = 10
+a = 1
+
+def g(t):
+    return A_c * (1 + a*(0.2*np.cos(2*np.pi*f_m*t) + 0.5*np.sin(2*np.pi*f_c*t)))
+
+# First derivative of g(t)
+def dg_dt(t):
+    return -0.2*2*f_m*np.pi*A_c*a*np.sin(2*np.pi*f_m*t) \
+		   + 2*np.pi*0.5*f_c*A_c*np.cos(2*np.pi*f_c*t)
+
+# Second derivative of g(t)
+def ddg_dtt(t):
+    return -0.2*(2*np.pi*f_m)**2*A_c*a*np.cos(2*np.pi*f_m*t) \
+		   - 0.5*(2*np.pi*f_c)**2*A_c*a*np.sin(2*np.pi*f_c*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.max(g(t)))
+    return np.max(g(t))
+
+def dc_dt(t):
+    return A_c*a*np.pi*f_c*np.cos(2*np.pi*f_c*t)
+
+# Sample g(t) at the maxima and minima of the carrier signal
+def sample_method():
+    samples = f_c / f_m
+    n = np.arange(samples)
+    t = (2*n + 1) / (4*f_c)
+    t = t[np.argmax(g(t))]
+    print("Sampling method:", t, g(t))
+    return t
+
+if __name__ == '__main__':
+    t_max = sample_method()
+    A_max = newton_method(t_max)
+    a_coeff = (A_max - A_c) / A_c
+    print("Value of a where positive modulation is 90%:", 0.9/a_coeff)
+```
+
+### Sampling Method
+$t_max = 0.0002$, $g_max = 16.9754$
+### Newton Method
+$t_max = 0.000199206$, $g_max = 16.9755$
+
+Positive modulation is 90% when $a=1.2902$

7th-Semester-Fall-2024/ECOMMS/homework/homework-2/problem-1.py

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+import numpy as np
+
+f_c = 1250
+f_m = 125
+A_c = 10
+a = 1
+
+def g(t):
+    return A_c * (1 + a*(0.2*np.cos(2*np.pi*f_m*t) + 0.5*np.sin(2*np.pi*f_c*t)))
+
+# first derivative of g(t)
+def dg_dt(t):
+    return -0.2*2*f_m*np.pi*A_c*a*np.sin(2*np.pi*f_m*t) + 2*np.pi*0.5*f_c*A_c*np.cos(2*np.pi*f_c*t)
+
+# second derivative of g(t)
+def ddg_dtt(t):
+    return -0.2*(2*np.pi*f_m)**2*A_c*a*np.cos(2*np.pi*f_m*t) - 0.5*(2*np.pi*f_c)**2*A_c*a*np.sin(2*np.pi*f_c*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.max(g(t)))
+    return np.max(g(t))
+
+def dc_dt(t):
+    return A_c*a*np.pi*f_c*np.cos(2*np.pi*f_c*t)
+
+# Sample g(t) at the maxima and minima of the carrier signal
+def sample_method():
+    samples = f_c / f_m
+    n = np.arange(samples)
+    t = (2*n + 1) / (4*f_c)
+    t = t[np.argmax(g(t))]
+    print("Sampling method:", t, g(t))
+    return t
+
+if __name__ == '__main__':
+    t_max = sample_method()
+    A_max = newton_method(t_max)
+    a_coeff = (A_max - A_c) / A_c
+    print("Value of a where positive modulation is 90%:", 0.9/a_coeff)

7th-Semester-Fall-2024/ECOMMS/homework/homework-2/problem-2.py

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+import numpy as np
+
+f_m = 0.2
+f_c = 2*f_m
+A_c = 1
+a = 1
+
+def g(t):
+    return A_c*np.cos(2*np.pi*f_m*t) + A_c*np.cos(2*np.pi*f_c*t)
+
+# first derivative of g(t)
+def dg_dt(t):
+    return -(2*np.pi*f_m)*A_c*np.sin(2*np.pi*f_m*t) + -(2*np.pi*f_c)*A_c*np.sin(2*np.pi*f_c*t)
+
+# second derivative of g(t)
+def ddg_dtt(t):
+    return -(2*np.pi*f_m)**2*A_c*np.cos(2*np.pi*f_m*t) + -(2*np.pi*f_c)**2*A_c*np.cos(2*np.pi*f_c*t)
+
+# use Newton's method to find the maximum of the function
+def newton_method(t):
+    for i in range(3):
+        t = t - dg_dt(t)/ddg_dtt(t)
+        print(f"Iteration {i+1}: {t}\t{g(t)}")
+
+def dc_dt(t):
+    return A_c*a*np.pi*f_c*np.cos(2*np.pi*f_c*t)
+
+if __name__ == '__main__':
+    T_c = 1/f_c
+    t_min = T_c/2
+    newton_method(t_min)
+    
+    newton_method(0)

7th-Semester-Fall-2024/ECOMMS/homework/homework-3/problem-1.py

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+import numpy as np
+import scipy as sp
+import matplotlib.pyplot as plt
+
+f_c = 1
+f_m = f_c/4
+
+omega_m = 2*np.pi*f_m
+omega_c = 2*np.pi*f_c
+
+T_c = f_c
+
+t = np.arange(0,10,T_c/10)
+m = np.cos(omega_m * t)
+
+plt.plot(t, m)
+plt.show()
+
+D_p = np.pi
+D_f = np.pi
+
+S_p = np.cos(omega_c + D_p*m)
+plt.plot(t,S_p)
+plt.show()

7th-Semester-Fall-2024/ECOMMS/homework/homework-3/problem-2.py

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+import numpy as np
+import scipy as sp
+import matplotlib.pyplot as plt
+
+# Modulation index
+beta = 2.0
+
+# Number of impulses is 2*(beta+1) + 1.
+# For beta=2, the number of impulses is 3 either side of the center frequency
+# plus 1 for the center frequency, for a total of 7 impulses.
+
+# Message frequency
+f_m = 15E+3
+
+# Transmission bandwidth
+B_T = 2*(beta+1)*f_m
+
+n = np.arange(0,10,1)
+bessel_values = sp.special.jv(n,beta)
+bessel_power = np.cumsum(bessel_values)
+plt.stem(n, bessel_values)
+plt.plot(n, bessel_power)
+plt.hlines(0.98*bessel_power[-1], xmin=n[0], xmax=n[-1])
+plt.show()
+
+#