0The sampling theorem tells us how to convert a continuous signal into a set of samples without losing information. It helps us understand how often we need to measure a signal to keep it accurate.
Fs >= 2 * Fmax where: Fs = Sampling frequency (how often we take samples) Fmax = Highest frequency in the signal
The sampling frequency must be at least twice the highest frequency present in the signal.
If sampling is done below this rate, the signal can be distorted (called aliasing).
Fmax = 1000 # highest frequency in Hz Fs = 2 * Fmax # minimum sampling frequency print(Fs) # Output: 2000
Fs = 1500 Fmax = 1000 if Fs >= 2 * Fmax: print("Sampling frequency is sufficient.") else: print("Sampling frequency is too low, aliasing may occur.")
This program shows a 5 Hz sine wave sampled at three different rates: below, at, and above the Nyquist rate (which is 10 Hz here). The plot helps visualize how undersampling causes distortion, while sampling at or above the Nyquist rate captures the signal correctly.
import numpy as np import matplotlib.pyplot as plt # Define a continuous signal: a sine wave with frequency 5 Hz f = 5 # frequency in Hz T = 1 # duration in seconds t_continuous = np.linspace(0, T, 1000) # continuous time points signal_continuous = np.sin(2 * np.pi * f * t_continuous) # Sampling frequency less than Nyquist rate (undersampling) Fs_low = 8 # sampling frequency in Hz (less than 2*f=10) n_samples_low = int(Fs_low * T) t_samples_low = np.linspace(0, T, n_samples_low, endpoint=False) samples_low = np.sin(2 * np.pi * f * t_samples_low) # Sampling frequency equal to Nyquist rate Fs_nyquist = 2 * f # 10 Hz n_samples_nyquist = int(Fs_nyquist * T) t_samples_nyquist = np.linspace(0, T, n_samples_nyquist, endpoint=False) samples_nyquist = np.sin(2 * np.pi * f * t_samples_nyquist) # Sampling frequency higher than Nyquist rate Fs_high = 20 # 20 Hz n_samples_high = int(Fs_high * T) t_samples_high = np.linspace(0, T, n_samples_high, endpoint=False) samples_high = np.sin(2 * np.pi * f * t_samples_high) # Plotting plt.figure(figsize=(10, 6)) plt.plot(t_continuous, signal_continuous, label='Continuous Signal', color='black') plt.stem(t_samples_low, samples_low, linefmt='r-', markerfmt='ro', basefmt='r-', label='Undersampling (8 Hz)') plt.stem(t_samples_nyquist, samples_nyquist, linefmt='g-', markerfmt='go', basefmt='g-', label='Nyquist Rate (10 Hz)') plt.stem(t_samples_high, samples_high, linefmt='b-', markerfmt='bo', basefmt='b-', label='Oversampling (20 Hz)') plt.title('Sampling Theorem (Nyquist-Shannon) Example') plt.xlabel('Time (seconds)') plt.ylabel('Amplitude') plt.legend() plt.grid(True) plt.show()
Sampling below the Nyquist rate causes aliasing, which means the signal looks like a different frequency.
In practice, signals are often filtered to remove frequencies above half the sampling rate before sampling.
Nyquist rate is the theoretical minimum; sampling faster can improve quality and simplify processing.
The sampling theorem tells us how often to measure a signal to keep all information.
Sampling frequency must be at least twice the highest frequency in the signal.
Sampling below this rate causes errors called aliasing.
fs?highest_freq = 500 sampling_rate = ?
fs = 800
nyquist = fs * 0.5
print("Nyquist frequency is", nyquist)Pass the test and this pattern moves unknown strong while earning +1 Grip toward your certificate.