Aliasing

Aliasing occurs when a continuous-time signal is sampled at a rate too low to represent its high-frequency content, causing those frequencies to fold back and appear as false lower-frequency components. Every analog-to-digital converter in audio interfaces, oscilloscopes, and SDR receivers must sample above the Nyquist rate to avoid this distortion.

Core Concept

Sampling a continuous-time signal x(t) at rate fs creates a train of impulses in the frequency domain. The spectrum of the sampled signal consists of the original spectrum X(f) repeated at every integer multiple of fs. If the signal has frequency content up to fmax, these copies are spaced fs apart. They do not overlap only when fs is at least 2*fmax.

When fs is less than 2*fmax, adjacent spectral copies overlap. The overlapping tail of one copy adds onto the baseband spectrum. After reconstruction, this addition appears as a new sinusoid at a frequency different from any component in the original signal. That false sinusoid is the alias. A 1 kHz tone sampled at 1.5 kHz produces an alias at 0.5 kHz.

The solution is an anti-aliasing filter placed before the sampler. This lowpass filter attenuates all frequency content above fs/2 before sampling occurs. In audio ADCs such as the PCM1804 used in studio equipment, a sharp analog lowpass filter limits the signal to below half the sample rate, then oversampling and digital filtering refine the passband further.

Key Formula

Nyquist-Shannon sampling theorem: a signal with bandwidth fmax can be perfectly reconstructed from samples taken at rate fs >= 2*fmax. The minimum allowed rate 2*fmax is the Nyquist rate.

Alias frequency formula: if a sinusoid at frequency f is sampled at fs, the alias appears at |f - round(f/fs)*fs|. For a simpler single-period case: alias = |f - fs| when f is between fs/2 and fs.

f: frequency of the original sinusoid in Hz. fs: sampling frequency in Hz. fmax: highest frequency component of the signal. Nyquist frequency: fs/2.

Problem: A 9 kHz sinusoid is sampled at fs = 8 kHz. Find the alias frequency.

Given: f = 9 kHz, fs = 8 kHz
Nyquist frequency = fs/2 = 4 kHz
Since f > fs/2, aliasing occurs.

Step 1 — Find alias:
  alias = |f - n*fs| for integer n that brings alias into [0, fs/2]
  n=1: |9 kHz - 1*8 kHz| = 1 kHz
  1 kHz is within [0, 4 kHz] -> this is the alias.

Step 2 — Interpret:
  The 9 kHz tone appears as a 1 kHz tone in the sampled data.
  After playback, the listener hears 1 kHz, not 9 kHz.

Final Answer: Alias frequency = 1 kHz.
To avoid this, sampling rate must be at least 18 kHz (2 * 9 kHz).

Exam Tip: GATE problems on aliasing often give a sampling frequency and ask which input frequency produces a specific alias. Use the folding formula: alias = |f mod fs| folded to [0, fs/2]. Also know that a signal sampled at exactly the Nyquist rate fs = 2*fmax is theoretically sufficient but practically risky; real filters cannot have infinitely sharp cutoffs, so practical systems sample at 10% to 20% above Nyquist to allow filter roll-off guard bands.

Properties Summary

• Nyquist rate: fs = 2*fmax; sampling at this rate is the minimum for perfect reconstruction.
• Nyquist frequency: fs/2; no frequency above this can be represented without aliasing.
• Alias formula: apparent frequency = |f - n*fs| for the integer n bringing the result to [0, fs/2].
• Anti-aliasing filter: lowpass filter with cutoff <= fs/2 applied before sampling.
• Oversampling: sampling well above Nyquist relaxes the anti-aliasing filter requirements.
• Reconstruction filter (ideal): lowpass filter with cutoff fs/2 applied after the DAC to remove spectral copies.
• Frequency resolution after sampling: delta_f = fs/N for an N-point DFT of the sampled sequence.

Quick Revision

• Aliasing: high-frequency components fold back to appear as lower frequencies.
• Nyquist rate = 2 * highest frequency in signal.
• Sampling below Nyquist rate causes aliasing; sampling at or above prevents it.
• Alias frequency = |f - fs| for a tone f between fs/2 and fs.
• Anti-aliasing LPF must be applied before the ADC, not after.
• Human audio band is 20 Hz to 20 kHz; CD sampling at 44.1 kHz satisfies Nyquist.
• Oversampling in sigma-delta ADCs (used in TI ADS1256) pushes aliases far from baseband, easing filter design.
• Exam trap: students compute Nyquist rate as fmax instead of 2*fmax; the Nyquist rate is twice the maximum frequency, not equal to it.