Anti-Aliasing Filter

An anti-aliasing filter is a low-pass filter applied to a continuous signal before sampling to remove frequency components above the Nyquist limit. Without it, high-frequency content folds back into the baseband and permanently corrupts the sampled data, a problem seen in audio recording, radar, and sensor acquisition systems.

Core Concept

Aliasing happens because sampling is periodic in the frequency domain. Spectral copies appear at every integer multiple of the sampling frequency fs. If the original signal has energy above fs/2, those copies overlap the baseband copy. Overlapping spectra cannot be separated after sampling, so the original signal is unrecoverable.

The anti-aliasing filter (AAF) is a low-pass filter with cutoff at or below fs/2. It is placed before the analog-to-digital converter. The filter attenuates all frequencies above fs/2 to a level below the noise floor of the ADC. After filtering, the signal is safely below the Nyquist limit.

The transition bandwidth of the AAF determines how sharply the filter must roll off. A signal sampled at exactly 2W Hz requires a brick-wall filter. In practice, oversampling gives the filter room to roll off gradually. Sampling at 4W Hz allows the filter to roll off over a bandwidth of W Hz, which requires far fewer filter poles.

Key Formula

The aliased frequency f_alias of a component at frequency f_in when sampled at fs is:

f_alias = | f_in - round(f_in / fs) * fs |

For example, a 1.3 kHz tone sampled at 1 kHz aliases to |1300 - 1*1000| = 300 Hz. The AAF must attenuate the 1.3 kHz component before sampling. The required stopband attenuation equals the ADC dynamic range, typically 60 to 100 dB for 10 to 16 bit converters.

Problem: A signal has components at 200 Hz and 900 Hz. ADC sampling rate = 1000 Hz.
  Determine which component aliases and to what frequency.

Step 1: Nyquist limit = fs/2 = 500 Hz

Step 2: Check 200 Hz component
  200 Hz < 500 Hz => No aliasing. Component is safe.

Step 3: Check 900 Hz component
  900 Hz > 500 Hz => Aliasing occurs.
  f_alias = |900 - round(900/1000)*1000|
           = |900 - 1*1000|
           = | -100 | = 100 Hz

Step 4: The 900 Hz tone appears as a 100 Hz tone in the sampled signal.
  It overlaps any real signal at 100 Hz.

Step 5: AAF requirement
  AAF must attenuate 900 Hz to below ADC noise floor
  before the sampling instant.

Final Answer: 900 Hz aliases to 100 Hz. AAF cutoff must be at or below 500 Hz.

Exam Tip: In GATE problems, always check if each signal component satisfies f < fs/2. If a component at frequency f_0 violates this, find its alias using f_alias = |f_0 - k*fs| where k is the nearest integer to f_0/fs. Remember that the AAF is applied before sampling; a post-sampling filter cannot remove aliases that have already folded into the baseband.

Properties Summary

• Aliasing condition: occurs whenever a signal has energy above fs/2.
• Alias frequency formula: f_alias = |f_in mod fs| mapped to the range [0, fs/2].
• AAF specification: passband up to the signal bandwidth W, stopband above fs/2, attenuation equal to ADC dynamic range.
• Oversampling benefit: sampling at N*2W Hz gives a transition band of (N-1)*W Hz, relaxing the filter order needed.
• Filter order trade-off: higher oversampling ratio means a lower-order (simpler, cheaper) analog AAF.
• Sigma-delta ADCs: use extreme oversampling (64x to 256x) so that the analog AAF only needs to be a simple first-order RC filter.

Quick Revision

• AAF is placed before the ADC to remove frequency content above fs/2.
• Nyquist limit: fs/2. Signal bandwidth W must satisfy W < fs/2.
• Alias of a tone at f_in: f_alias = |f_in - nearest multiple of fs|.
• Required stopband attenuation = ADC resolution in bits * 6 dB (approximately).
• Oversampling relaxes the analog filter transition band requirement.
• Aliasing is irreversible once sampling has occurred.
• Exam trap: thinking that a digital LPF applied after sampling can remove aliases. It cannot, because the aliased frequency is already inside the passband.