Aliasing (Under Sampling)

Table of Contents
1. Sampling and the Nyquist criterion
2. Aliasing (undersampling): definition and qualitative explanation
3. Mathematical description
4. Example: the stroboscopic effect (visual aliasing)
5. Practical consequences and methods to avoid aliasing
6. Advantages and controlled use of aliasing
7. When aliasing prevents reconstruction
8. Summary
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Sampling a continuous-time signal converts it into a sequence of values taken at regular time intervals. When the sampling rate is sufficiently high relative to the signal bandwidth, the original signal can be recovered. If the sampling rate is too low, however, spectral replicas overlap and the reconstructed signal may differ from the original. This phenomenon is called aliasing (or undersampling). The notes below explain the concept, give a mathematical description, present a simple physical example, and list practical consequences and uses.

Sampling and the Nyquist criterion

A continuous-time signal x(t) whose highest frequency component is B (Hz) is called band-limited with bandwidth B. The classical sampling theorem states that if the sampling frequency f_s satisfies

f_s ≥ 2B

then x(t) can be exactly reconstructed from its samples x[n] = x(nT), where T = 1/f_s, using an ideal low-pass interpolation filter. The value 2B is often called the Nyquist rate.

Aliasing (undersampling): definition and qualitative explanation

Aliasing occurs when the sampling frequency is less than twice the signal bandwidth (f_s < 2B) or when the sampling is otherwise ill-chosen so that spectral replicas overlap. In the frequency domain, sampling a signal produces periodic repetitions (replicas) of its spectrum shifted by integer multiples of the sampling angular frequency. If these shifted copies overlap in frequency, energy from different original frequencies adds together in the sampled spectrum and cannot be separated by linear reconstruction. As a result, high-frequency components are mapped (aliased) to lower frequencies and the reconstructed signal will differ from the original.

Frequency-domain picture

Sampling a signal with a periodic impulse train makes the spectrum of the sampled signal periodic. If X(ω) is the Fourier transform of x(t) and the sampling angular frequency is ω_s = 2πf_s, the sampled-signal spectrum X_s(ω) consists of shifted copies of X(ω) at multiples of ω_s. Overlap of these copies causes aliasing.

Mathematical description

Start with a continuous-time signal x(t) and an impulse sampling train p(t).

p(t) = Σ_n=-∞^∞ δ(t - nT)

x_s(t) = x(t) · p(t)

Let X(ω) be the Fourier transform of x(t) and X_s(ω) the Fourier transform of x_s(t).

X_s(ω) = (1/T) Σ_k=-∞^∞ X(ω - kω_s)

Here ω_s = 2π/T. The sampled spectrum is a sum of shifted versions of X(ω) scaled by 1/T.

If the shifted terms X(ω - kω_s) do not overlap, an ideal low-pass filter can extract the baseband copy and reconstruct x(t). If they overlap, the contributions from different k add for the same ω and cannot be separated, producing aliasing.

Example: the stroboscopic effect (visual aliasing)

A familiar physical example of aliasing is the stroboscopic effect. Consider a disc with a single radial line marked on it and a strobe light that flashes at a precise periodic rate. The strobe samples the continuous motion of the disc at discrete instants; the perceived motion is determined by the effective sampling rate relative to the rotational frequency.

• If the strobe frequency is much higher than the rotation frequency, the rotation appears correctly.
• If the strobe frequency equals the rotation frequency, the radial line appears stationary.
• If the strobe frequency is lower and particularly if it is below twice the rotational frequency, the apparent rotation speed seen under the strobe can be lower than the true speed; the disc may also appear to rotate in the reverse direction. These apparent slow or reversed motions are visual aliases of the true motion.

This example shows how sampling (strobe flashes) maps actual motion frequencies to perceived (aliased) frequencies when sampling is insufficient or chosen to produce those aliases deliberately.

Practical consequences and methods to avoid aliasing

• Anti-aliasing filter: In many practical systems an analogue low-pass filter is placed before the sampler (analogue-to-digital converter) to remove frequency components above the desired passband so that spectral replicas do not overlap. This filter is called an anti-aliasing filter.
• Choose f_s appropriately: Use a sampling frequency at or above the Nyquist rate for baseband signals. For signals occupying a narrow band at high frequency, use carefully chosen bandpass (or bandlimited) sampling strategies so that the band folds into baseband without overlap.
• Oversampling: Sampling at a rate higher than required reduces the burden on the anti-aliasing filter and simplifies subsequent digital processing.
• ADC and system design: Practical analogue-to-digital converters and acquisition systems must be designed so that input bandwidth and sampling rates are matched and unwanted high-frequency energy is suppressed.

Advantages and controlled use of aliasing

Although aliasing is usually undesirable, it can be exploited in controlled ways. The following points describe practical uses or perceived advantages in certain contexts.

• Undersampling can be used intentionally with band-pass (high-frequency, narrow-band) signals so that a high-frequency band is folded (aliased) into baseband; this is called bandpass sampling or harmonic sampling. It allows use of a lower sampling frequency while preserving the information in a band-limited signal if sampling frequency and filters are chosen correctly.
• In some transmission or conversion schemes the sampling/clock (carrier) frequency f_s may be used as the implicit carrier, and band filters can be used at receiver to select the desired aliased band. This can simplify RF down-conversion in hardware if done carefully.
• It is possible to use sampling at integer multiples or specific fractions of a base rate to place a desired band at a convenient frequency for processing; choosing these multiples correctly avoids overlap of spectral replicas.
• In certain sampling and modulation implementations the sampling pulses need not have zero average value; the pulse shape and its DC component affect the amplitudes of the replicas (coefficients C_k) but do not necessarily prevent useful aliasing-based schemes. Care must be taken because a non-zero average introduces additional spectral components that alter scaling of replicas and may require compensation.

When aliasing prevents reconstruction

If a signal is undersampled without the necessary constraints (for example, if the signal is not restricted to known narrow bands and no appropriate anti-alias filter is applied), then the original continuous-time signal cannot be uniquely reconstructed. Higher-frequency components are folded into lower frequencies in the sampled spectrum and information about the original frequencies is lost.

Summary

• Aliasing happens when spectral replicas produced by sampling overlap; it results from sampling at too low a rate or from inappropriate choice of sampling relative to signal band locations.
• The Nyquist condition f_s ≥ 2B guarantees perfect reconstruction for baseband band-limited signals; for other spectra, careful selection of f_s and pre-sampling filters is required.
• The stroboscopic effect is a clear visual demonstration of aliasing in time-domain sampling of motion.
• Aliasing is not always undesirable: bandpass sampling and other controlled undersampling schemes allow useful reduction of sampling rates when applied correctly, but they require careful design of sampling frequency and filtering.