Aliasing: The Spectral Impostor

From Sampling Frequency to Discrete Sequence

Given $x(t)$ and sampling frequency $F_s$, the discrete sequence is:

\[x[n] = x_a(nT) = x_a\left(\frac{n}{F_s}\right)\]

Example:

\(x(t) = \cos(2\pi \cdot 100 t), \quad F_s = 500 \text{ Hz}\) \(x[n] = \cos\left(2\pi \cdot 100 \cdot \frac{n}{500}\right) = \cos(0.4\pi n)\)

The digital frequency $\omega = 2\pi F / F_s$ (radians/sample) replaces analog frequency $F$.

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Aliasing: The Spectral Impostor

What Is Aliasing?

When $F_s < 2F_{\text{max}}$, high-frequency components masquerade as lower frequencies in the sampled signal. The spectrum “folds over” at multiples of $F_s/2$.

Frequency-Domain View

In the frequency domain, sampling replicates the original spectrum at intervals of $F_s$:

\[X_s(F) = F_s \sum_{k=-\infty}^{\infty} X(F - k F_s)\]

If these replicas overlap, aliasing occurs. The spectral content above $F_s/2$ folds back into the baseband $[0, F_s/2]$.

Aliasing Formula

A frequency component at $F$ appears at an alias frequency $F_{\text{alias}}$ in the sampled signal:

\[F_{\text{alias}} = |F - k F_s| \quad \text{for integer } k \text{ such that } F_{\text{alias}} \in [0, F_s/2]\]

Example: 60 Hz Power-Line Hum

• Signal contains 60 Hz interference
• Sampled at $F_s = 100$ Hz (below Nyquist rate of 120 Hz)

• Alias: $

  60 - 100

  = 40$ Hz

• The 60 Hz hum appears as a 40 Hz artifact in the digital signal!

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