Decimation

Decimation is the process of reducing the sampling rate of a discrete-time signal by an integer factor M. It is a fundamental operation in multirate DSP, used whenever a signal sampled at a high rate needs to be processed or stored at a lower rate while retaining the information of interest.

Decimation is not simply discarding every M-th sample without precaution. Directly downsampling a wideband signal causes aliasing. The correct procedure always pairs downsampling with an anti-aliasing lowpass filter applied before the downsampler. This combination is what constitutes a complete decimation operation.

Core Concept Explanation

When a signal x[n] sampled at rate F_s is downsampled by M, the new sampling rate becomes F_s/M. By the Nyquist theorem, this lower rate can represent only frequencies up to F_s/(2M). If the original signal contains energy above this limit, downsampling folds those frequencies back into the baseband, corrupting the signal. This is aliasing.

The solution is the anti-aliasing lowpass filter applied at the original rate F_s before the downsampler. This filter must have a normalized cutoff frequency of pi/M (in radians per sample at the original rate). It removes all spectral energy above pi/M, ensuring the downsampled signal is free from aliasing.

After filtering, the decimated output y[m] is simply the filtered signal v[n] sampled at every M-th point: y[m] = v[mM]. The DTFT of the decimated signal can be expressed in terms of the original spectrum using the decimation spectral formula, which reveals M uniformly shifted copies of the filtered spectrum.

Mathematical Expression

The spectrum of the decimated signal y[m] with decimation factor M is:

Y(e^jω) = (1/M) * sum_{k=0}^{M-1} V(e^{j(ω - 2πk)/M})

where V(e^jω) is the DTFT of the filtered signal v[n]. This formula shows that the output spectrum is the sum of M frequency-shifted and compressed copies of V. If V is bandlimited to pi/M, these M copies do not overlap, and there is no aliasing. The new Nyquist frequency at the decimated rate is pi, corresponding to F_s/(2M) in Hz.

The anti-aliasing filter H(z) is an ideal lowpass with cutoff at omega_c = pi/M. In practice, a FIR linear phase filter or a Chebyshev IIR filter is designed with this cutoff frequency. The filter operates at the high rate F_s and is the computationally dominant part of the decimation system.

Practical Understanding

Decimation is used in audio processing when a signal recorded at 192 kHz is converted to 48 kHz for playback. The anti-aliasing filter removes audio above 24 kHz before 4x downsampling. In software-defined radio, wide-band signals sampled at hundreds of MHz are decimated to baseband rates for demodulation.

Computational efficiency is a key motivation. Processing at a lower rate requires fewer multiplications per second. Polyphase decomposition of the anti-aliasing filter is an efficient implementation technique that moves the filter computation to the lower rate, reducing the total arithmetic cost by factor M. This is widely used in hardware DSP implementations.

Given:
Original sampling rate F_s = 48000 Hz
Decimation factor M = 4
Desired anti-aliasing filter cutoff

Why this formula applies:
Filter cutoff must be pi/M in normalized frequency to prevent aliasing after downsampling.

Formula:
omega_c = pi / M  (normalized)
f_cutoff = F_s / (2M)  (in Hz)

Substitution:
omega_c = pi / 4 = 0.785 rad/sample
f_cutoff = 48000 / (2 * 4) = 48000 / 8

Calculation:
f_cutoff = 6000 Hz

Final Answer with units:
Anti-aliasing LPF cutoff = 6000 Hz (or pi/4 normalized)
New output rate = 48000 / 4 = 12000 Hz
Signal content up to 6000 Hz is preserved without aliasing.

Exam Tip: For decimation by M, the anti-aliasing filter cutoff is pi/M in normalized frequency, or equivalently F_s/(2M) in Hz. The output rate is F_s/M. GATE often tests whether you correctly apply both steps: filter first, then downsample. Downsampling alone without filtering always causes aliasing.

Mechanism Summary

• Decimation by M reduces sampling rate from F_s to F_s/M.
• Anti-aliasing LPF must be applied before downsampling. Cutoff = pi/M (normalized) or F_s/(2M) Hz.
• Downsampling: y[m] = v[mM] where v[n] is the filtered signal.
• Decimated spectrum = M shifted copies of the filtered spectrum. No aliasing if LPF is applied correctly.
• Polyphase decomposition reduces the LPF computation to the lower output rate, saving factor-M multiply-adds.
• Applications: audio rate conversion, SDR baseband processing, sensor data reduction.

Quick Revision

• Decimation = LPF (cutoff pi/M) followed by downsample by M.
• Output rate = F_s/M. Filter cutoff in Hz = F_s/(2M).
• Without LPF: aliasing. With LPF: alias-free decimation.
• Spectrum of decimated signal: (1/M) * sum of M frequency-shifted copies of V(e^jω).
• Polyphase implementation reduces computation by factor M.
• Trap: the anti-aliasing filter must operate at the high rate F_s, not at the decimated rate.
• Trap: decimation factor M must satisfy F_s/M >= 2 * signal_bandwidth to avoid losing signal content.