import numpy as np import matplotlib.pyplot as plt PI = np.pi INF = 10 print("# Quiz 1 Preparation - Aidan Sharpe") print("""## Question 1 Find the energy and average power of $x(n) = e^{j 2\pi n / 3}$ """) def x_1(n): return np.exp(2j*PI*n/3) x_n = x_1(np.arange(-INF, INF)) E_x = np.sum( np.abs(x_n)**2 ) print("Energy of x[n]:", E_x if E_x < INF else "Infinity") N = 3 x_n = x_1(np.arange(0, N)) P_x = (1/N) * np.sum( np.abs(x_n)**2 ) print("Power of x[n]:", P_x if P_x < INF else "Infinity") print("""## Question 2 Is $x(n - \pi)$ a valid discrete signal? No""") print("""## Question 3 Consider the system $y[n] = S[x[n]] = e^{n+3}\cos(n-5)x[n-4]$. Is the system linear, time-invariant, causal, and BIBO stable? """) a_3 = 5 def S_3(x,n,a=1): return np.exp(n+3) * np.cos(n-5) * x(n-4) * a def x_3_1(n): return n*(n-1)*np.heaviside(n,0) def x_3_2(n): return 5*n*np.heaviside(n,0) def x_3_1_2(n): return x_3_1(n) + x_3_2(n) S_ax_3 = S_3(x_3_1, np.arange(-INF, INF), a=a_3) y_3 = S_3(x_3_1, np.arange(-INF, INF)) print("Homogeneous:", abs(sum(S_ax_3 - a_3*y_3)) < 0.01, '\n') S_x1_x2_3 = S_3(x_3_1_2, np.arange(-INF, INF)) y1_3 = S_3(x_3_1, np.arange(-INF,INF)) y2_3 = S_3(x_3_2, np.arange(-INF, INF)) print("Superposition:", abs(sum(S_x1_x2_3 - (y1_3 + y2_3))) < 0.01, '\n') print("""## Quesiton 4 Consider the LTI sstem with impulse response $h[n] = u[n]$. If the input x[n] = n (0.3)^n u[n]$ and the output is $y[n]$, calculate: ### a) $y(-1)$""")