Aliasing

Introduction

Aliasing is the phenomenon where high-frequency signal components masquerade as lower frequencies after sampling — they take on a false identity (an "alias"). It occurs whenever the sampling rate is insufficient to capture the signal's highest frequency content, violating the Nyquist criterion. Once aliasing occurs, the original frequencies cannot be recovered; the damage is permanent and irreversible.

You have likely seen aliasing in everyday life without realizing it. When a car wheel appears to rotate backward in a video (the "wagon wheel effect"), that is aliasing — the camera's frame rate is too low to capture the true rotation speed. When a digital photograph of a striped shirt shows strange ripple patterns (moire), that is spatial aliasing. When a digital recording of a cymbal produces harsh metallic artifacts, that is frequency aliasing. All these phenomena share the same mathematical cause.

Mathematical Description

When a continuous signal $x(t)$ with spectrum $X(j\omega)$ is sampled at rate $f_s$, the spectrum of the sampled signal is:

$$X_s(j\omega) = \frac{1}{T_s}\sum_{k=-\infty}^{\infty}X(j(\omega - k\omega_s))$$

If the signal contains frequencies above $f_s/2$ (half the sampling rate), the spectral replicas overlap. In the overlap regions, frequency components from adjacent replicas add together, creating new spectral content that did not exist in the original baseband spectrum.

The Folding Frequency

The frequency $f_s/2$ is called the folding frequency (or Nyquist frequency). A signal component at frequency $f > f_s/2$ aliases to the frequency:

$$f_{alias} = |f - k \cdot f_s|$$

where $k$ is chosen to place $f_{alias}$ in the range $[0, f_s/2]$.

More specifically, for a component at frequency $f_0$ sampled at rate $f_s$: $$f_{alias} = \left|f_0 \mod f_s - \frac{f_s}{2}\right| - \frac{f_s}{2}$$

Or more intuitively: the frequency "folds" back from the Nyquist frequency like a reflection in a mirror.

Worked Example

Problem: A signal contains components at 100 Hz, 400 Hz, and 900 Hz. It is sampled at $f_s = 1000$ Hz. Determine aliased frequencies.

Solution: The Nyquist frequency is $f_s/2 = 500$ Hz.

• 100 Hz < 500 Hz → no aliasing, appears at 100 Hz ✓
• 400 Hz < 500 Hz → no aliasing, appears at 400 Hz ✓
• 900 Hz > 500 Hz → aliases to $|900 - 1000| = 100$ Hz

The 900 Hz component folds to 100 Hz, becoming indistinguishable from the legitimate 100 Hz component. The result is a corrupted 100 Hz component with amplitude and phase that combine the original 100 Hz and the aliased 900 Hz contributions.

Another Example: Audio Recording

Problem: A 22 kHz tone is recorded at $f_s = 44.1$ kHz (CD sampling rate). What is the perceived frequency?

Solution: Nyquist frequency = 22.05 kHz. Since 22 kHz < 22.05 kHz, this component is captured correctly without aliasing. This is exactly why the CD sampling rate was chosen to be 44.1 kHz — it accommodates the full human hearing range (up to ~20 kHz) with a small margin.

Now consider recording a 25 kHz tone at 44.1 kHz: $f_{alias} = |25 - 44.1| = 19.1$ kHz. This ultrasonic tone aliases into the audible range, which is why an anti-aliasing filter removing everything above 20 kHz is mandatory before the ADC in CD recording equipment.

Visual Aliasing Examples

Wagon Wheel Effect

A wheel rotating at $f_{rot}$ revolutions per second, filmed at $f_s$ frames per second:

• If $f_{rot} < f_s/2$: wheel appears to rotate at correct speed
• If $f_{rot} = f_s$: wheel appears stationary
• If $f_{rot}$ slightly exceeds $f_s$: wheel appears to rotate slowly backward

Moire Patterns

When a digital camera photographs a periodic pattern (like fabric weave or building facades), spatial aliasing creates large-scale patterns not present in the original scene. Digital camera manufacturers include optical low-pass (anti-aliasing) filters over the sensor to prevent this.

Stroboscopic Effect

A strobe light at 100 Hz illuminating a 99 Hz rotating disk makes the disk appear to rotate at 1 Hz — the difference frequency. This is temporal aliasing used deliberately as a measurement tool.

Prevention: Anti-Aliasing Filters

The standard prevention method is to filter the signal before sampling:

$$x_{filtered}(t) = x(t) * h_{AA}(t)$$

where $h_{AA}(t)$ is a low-pass filter with cutoff at or below $f_s/2$.

Analog Anti-Aliasing Filter Requirements

• Passband: 0 to $f_{max}$ with minimal distortion
• Stopband: $f_s/2$ and above with sufficient attenuation (typically 60-80 dB)
• Transition band: $f_{max}$ to $f_s/2$ — narrower transition requires higher filter order

Filter Order Trade-offs

A sharper filter (more attenuation in the transition band) requires higher order, which introduces:

• Phase distortion (group delay variation)
• Ringing artifacts (time-domain overshoot)
• Higher cost and complexity

This is why oversampling is attractive: by sampling at 4× or 8× the Nyquist rate, the transition band becomes very wide, and a simple 2nd or 3rd order filter suffices.

Aliasing in Discrete-Time Processing

Aliasing also occurs in purely digital contexts:

Decimation (downsampling): Reducing the sampling rate by factor $M$ effectively makes the new Nyquist frequency $f_s/(2M)$. Any energy above this frequency aliases. Solution: apply a digital low-pass filter before decimation.

Nonlinear operations: Squaring a signal doubles its bandwidth. If the resulting frequencies exceed $f_s/2$, aliasing occurs within the discrete-time representation.

DFT/FFT spectral analysis: Finite-length DFT imposes implicit periodicity, which can create time-domain aliasing if the signal is not properly windowed.

Intentional Aliasing: Bandpass Sampling

In some applications, aliasing is deliberately exploited. Bandpass sampling (undersampling) intentionally aliases a narrowband signal centered at a high carrier frequency down to baseband. This is used in software-defined radio receivers to avoid high-speed ADCs.

For a signal occupying $[f_L, f_H]$ with bandwidth $B = f_H - f_L$, sampling at rate $f_s = 2B$ (much less than $2f_H$) correctly captures the signal if the sampling rate is chosen so that spectral replicas do not overlap the signal band.

Key Takeaways

• Aliasing occurs when $f_s < 2f_{max}$ — high frequencies fold back into the baseband
• Aliased frequencies appear at $f_{alias} = |f_0 - k \cdot f_s|$ for appropriate integer $k$
• The damage is permanent — once aliased, original frequencies cannot be recovered
• Prevention: analog anti-aliasing filter before the ADC, cutting all energy above $f_s/2$
• Oversampling relaxes anti-aliasing filter requirements by widening the transition band
• Aliasing occurs in visual, audio, and purely digital domains (decimation, nonlinear processing)
• Bandpass sampling intentionally exploits aliasing for efficient narrowband digitization