Sampling and the Nyquist Theorem

Sampling and the Nyquist Theorem

Digital signals are created by sampling a continuous signal at discrete time steps. The sampling rate $f_s$ (in Hz) determines what frequencies can be faithfully captured.

The Nyquist–Shannon Sampling Theorem

To perfectly reconstruct a signal with highest frequency $f_{\max}$, the sampling rate must satisfy:

$f_s \geq 2 f_{\max}$

The Nyquist rate is the minimum sufficient rate:

$f_{\text{Nyquist}} = 2 f_{\max}$

Example: CD audio captures frequencies up to 22,050 Hz, so it uses $f_s = 44,100$ Hz — exactly the Nyquist rate.

Aliasing

When a signal is sampled below the Nyquist rate, aliasing occurs: high-frequency components are "folded" into lower frequencies and become indistinguishable from them.

A frequency $f$ sampled at rate $f_s$ aliases to:

$f_{\text{alias}} = \left|f - \text{round}\!\left(\frac{f}{f_s}\right) \cdot f_s\right|$

Example: A 1300 Hz tone sampled at 1000 Hz aliases to $|1300 - 1 \cdot 1000| = 300$ Hz.

Your Task

Implement:

• nyquist_rate(f_max) — returns $2 f_{\max}$
• aliased_frequency(f, fs) — returns the aliased frequency when $f$ is sampled at $f_s$

def nyquist_rate(f_max):
    return 2 * f_max

def aliased_frequency(f, fs):
    return abs(f - round(f / fs) * fs)

print(nyquist_rate(22050))           # 44100
print(aliased_frequency(1300, 1000)) # 300