Verifying the Perfect Reconstruction Formula in the Shannon/Nyquist Sampling Theorem

This article explores the perfect reconstruction formula derived from the Shannon/Nyquist sampling theorem:

xa(t) = 2 * T * fc * Σ(xa(n * T) * sinc(2 * fc * (t - n * T)))

where:

• xa(t) represents a given analog signal.* T denotes the sampling period.* fc is the cut-off frequency of the ideal low-pass filter used for reconstruction.

We'll investigate whether both sides of the equation remain consistent despite the right-hand side (RHS) containing T and fc while the left-hand side (LHS) does not.

(a) Computer Experiments for Verification

We can use Python to check the formula's validity through experimentation:pythonimport numpy as npimport matplotlib.pyplot as plt

def analog_signal(t): # Define xa(t), for example: return np.sin(2 * np.pi * t)

def sinc_function(t): # Define sinc(x) = sin(pix)/(pix), handling x = 0: if t == 0: return 1.0 else: return np.sin(np.pi * t) / (np.pi * t)

def perfect_reconstruction(T, fc, n_values): t_values = np.arange(n_values) * T xa_values = analog_signal(t_values) xa_samples = xa_values * np.sinc(2 * fc * (t_values - np.floor(t_values / T) * T)) xa_reconstructed = np.sum(xa_samples) return xa_reconstructed

Example usageT = 0.01 # Sampling periodfc = 10.0 # Cut-off frequencyn_values = 1000 # Number of samples

xa_reconstructed = perfect_reconstruction(T, fc, n_values)print('Reconstructed xa(t):', xa_reconstructed)

This code defines the analog signal, sinc function, and reconstruction logic. By modifying xa(t), T, fc, and n_values, you can observe how the reconstructed xa(t) behaves.

(b) Mathematical Proof

Let's analyze the equation mathematically. Substituting the RHS into the LHS:

xa(t) = 2 * T * fc * Σ(xa(n * T) * sinc(2 * fc * (t - n * T)))

Using sinc function properties and summation:

xa(t) = 2 * T * fc * Σ(xa(n * T) * sinc(2 * fc * t) * sinc(-2 * fc * n * T))``(sinc is even, sinc(x) = 1 for x = 0)``xa(t) = 2 * T * fc * sinc(2 * fc * t) * Σ(xa(n * T))``(Σ(xa(n * T)) is constant)

The RHS becomes a constant multiplied by the sinc function of 2 * fc * t. This demonstrates that the reconstructed signal is a scaled and shifted version of the sinc function, fulfilling the perfect reconstruction requirement.

Note: This explanation offers a simplified understanding. A rigorous proof might involve further steps and considerations.