Syncing to work on interview with entrepreneur essay

6th-Semester-Spring-2024/DSP/Labs/Lab-03/lab-03-1.py

@@ -1,11 +1,10 @@
 import numpy as np
 import matplotlib.pyplot as plt
-
+import scipy.signal
 n = np.arange(0,200)
-
 a = 0.9
 
-# x[n] = a^n u[n]
+# $x[n] = a^n u[n]$
 x = a**n * np.heaviside(n, 1)
 
 asums = np.zeros(len(n))
@@ -25,3 +24,42 @@ plt.plot(n, asums)
 plt.xlabel("$n$")
 plt.ylabel("$\sum_n x[n]$", rotation="horizontal")
 plt.show()
+
+# Compute DTFT of $x[n]$
+X = scipy.signal.freqz(1, (1, -a))
+omega, h = X
+
+# Plot DTFT of $x[n]$
+plt.plot(omega, np.abs(h))
+plt.ylabel("Amplitude")
+plt.xlabel("Frequency [rad/sample]")
+plt.show()
+
+# Set up plot for DTFT of $x[n]$ for comparrison with truncated DTFTs
+plt.plot(omega, np.abs(h), label="Actual DTFT")
+plt.ylabel("Amplitude")
+plt.xlabel("Frequency [rad/sample]")
+
+# Calculate the truncated DTFTs of $x[n]$ as a function of $K$, $\sum_{n=-K}^K x[n] e^{-j\omega n}$
+for K in (3, 10, 20):
+    # Finite geometric series formula
+    X_K = (1 - (a**(K+1) * np.exp(-1j * omega * (K+1)))) / (1 - a*np.exp(-1j*omega))
+    plt.plot(omega, np.abs(X_K), label=f"Truncated DTFT ($K={K}$)")
+
+plt.legend()
+plt.show()
+
+# Frequency of maximum difference between actual and truncated DTFT
+K_range = np.arange(1,200+1)
+max_diffs = np.zeros(K_range.shape)
+for K in K_range:
+    X_K = (1 - a**(K+1) * np.exp(-1j * omega * (K+1))) / (1 - a*np.exp(-1j*omega))
+    abs_diff = np.abs(X_K - h)
+    max_diffs[K-1] = np.max(abs_diff)
+
+plt.plot(K_range, max_diffs)
+plt.xlabel("K")
+plt.ylabel("Maximum Error")
+plt.show()
+
+