Sampling (signal processing)

A common example is the conversion of a sound wave to a sequence of "samples".

[A] A sampler is a subsystem or operation that extracts samples from a continuous signal.

A theoretical ideal sampler produces samples equivalent to the instantaneous value of the continuous signal at the desired points.

The Whittaker–Shannon interpolation formula is mathematically equivalent to an ideal low-pass filter whose input is a sequence of Dirac delta functions that are modulated (multiplied) by the sample values.

The fidelity of a theoretical reconstruction is a common measure of the effectiveness of sampling.

contains frequency components whose cycle length (period) is less than 2 sample intervals (see Aliasing).

[4] In practice, the continuous signal is sampled using an analog-to-digital converter (ADC), a device with various physical limitations.

This results in deviations from the theoretically perfect reconstruction, collectively referred to as distortion.

Various types of distortion can occur, including: Although the use of oversampling can completely eliminate aperture error and aliasing by shifting them out of the passband, this technique cannot be practically used above a few GHz, and may be prohibitively expensive at much lower frequencies.

Furthermore, while oversampling can reduce quantization error and non-linearity, it cannot eliminate these entirely.

Jitter, noise, and quantization are often analyzed by modeling them as random errors added to the sample values.

Integration and zero-order hold effects can be analyzed as a form of low-pass filtering.

Early professional audio equipment manufacturers chose sampling rates in the region of 40 to 50 kHz for this reason.

There has been an industry trend towards sampling rates well beyond the basic requirements: such as 96 kHz and even 192 kHz[8] Even though ultrasonic frequencies are inaudible to humans, recording and mixing at higher sampling rates is effective in eliminating the distortion that can be caused by foldback aliasing.

Conversely, ultrasonic sounds may interact with and modulate the audible part of the frequency spectrum (intermodulation distortion), degrading the fidelity.

[9] One advantage of higher sampling rates is that they can relax the low-pass filter design requirements for ADCs and DACs, but with modern oversampling delta-sigma-converters this advantage is less important.

Thermal noise limits the true number of bits that can be used in quantization.

Few analog systems have signal to noise ratios (SNR) exceeding 120 dB.

However, digital signal processing operations can have very high dynamic range, consequently it is common to perform mixing and mastering operations at 32-bit precision and then convert to 16- or 24-bit for distribution.

The image sampling frequency is the repetition rate of the sensor integration period.

Since the integration period may be significantly shorter than the time between repetitions, the sampling frequency can be different from the inverse of the sample time: Video digital-to-analog converters operate in the megahertz range (from ~3 MHz for low quality composite video scalers in early game consoles, to 250 MHz or more for the highest-resolution VGA output).

When analog video is converted to digital video, a different sampling process occurs, this time at the pixel frequency, corresponding to a spatial sampling rate along scan lines.

The sampling rates and resolutions in both spatial directions can be measured in units of lines per picture height.

Spatial aliasing of high-frequency luma or chroma video components shows up as a moiré pattern.

Volume rendering is common in medical imaging, X-ray computed tomography (CT/CAT), magnetic resonance imaging (MRI), positron emission tomography (PET) are some examples.

That is often done purposefully in such a way that the lowest-frequency alias satisfies the Nyquist criterion, because the bandpass signal is still uniquely represented and recoverable.

[23] Oversampling is used in most modern analog-to-digital converters to reduce the distortion introduced by practical digital-to-analog converters, such as a zero-order hold instead of idealizations like the Whittaker–Shannon interpolation formula.

, is called an analytic signal, whose Fourier transform is zero for all negative values of frequency.

In that case, the Nyquist rate for a waveform with no frequencies ≥ B can be reduced to just B (complex samples/sec), instead of

For instance, the equivalent baseband waveform can be created without explicitly computing