Sampling (signal processing)

In signal processing, sampling is a way to change a continuous flow of information, like a smooth sound wave, into a set of separate numbers or "samples." Each sample captures the value of the signal at a specific moment in time. This process is very important in many modern technologies because it allows complex, continuous signals to be stored, processed, and transmitted using computers and other digital devices.

A device or operation called a sampler takes these snapshots from the continuous signal. In an ideal situation, each sample perfectly matches the value of the original signal at that exact point in time. This idea helps us understand how digital audio, images, and many other types of data can be created from real-world signals.

From these samples, the original signal can often be brought back to life, as long as the samples were taken at a fast enough rate. This maximum rate is known as the Nyquist limit, and when we pass the samples through a special reconstruction filter, we can get a close approximation of the original continuous signal. This ability to convert between continuous and discrete forms is the foundation of many technologies we use every day, from music players to mobile phones, making sampling a key concept in the world of sound wave and digital communication.

Theory

See also: Nyquist–Shannon sampling theorem

Sampling is a way to turn a continuous signal, like a sound wave, into a series of numbers by measuring it at regular time intervals. These intervals are called the sampling interval or sampling period. The number of these measurements taken each second is called the sampling frequency or sampling rate, and it’s measured in samples per second, sometimes called hertz. For example, 48 kHz means 48,000 samples are taken every second.

When we want to turn these samples back into a continuous signal, we use special math called interpolation. To avoid mistakes in this process, signals are often passed through a low-pass filter before sampling. This helps make sure that the reconstructed signal matches the original as closely as possible.

Practical considerations

In real life, devices called analog-to-digital converters (ADCs) are used to take samples of a continuous signal. Because of physical limits, these devices can't perfectly copy the original signal, leading to what we call distortion.

Some types of distortion include aliasing, where higher frequencies appear as lower ones; aperture error, which happens because the sample is an average over time rather than a single instant; jitter, a shift in timing; and noise, random variations in the signal. Other distortions can come from the ADC’s limited ability to change quickly (slew rate limit error) or from quantization, where the signal is rounded to a fixed number of levels. While some methods like oversampling can reduce certain errors, they have their own limits and costs.

Applications

Digital audio systems use a method called pulse-code modulation (PCM) to turn sound waves into a series of numbers, or samples. These samples capture the sound at specific moments, allowing computers to store and play back audio. For recording music or other sounds that humans can hear — from 20 to 20,000 Hz — common sampling rates are 44.1 kHz (used for CDs), 48 kHz, 88.2 kHz, or 96 kHz. Higher sampling rates, like 96 kHz or even 192 kHz, help reduce distortion but do not add more usable sound for human ears.

Video also uses sampling to turn moving images into digital data. Standard-definition TV uses either 720 by 480 pixels (as in US NTSC systems) or 720 by 576 pixels (as in UK PAL systems). High-definition TV uses formats like 720p, 1080i, and 1080p, which have more pixels and clearer images. In digital video, the sampling rate is linked to how many frames are shown each second, with common rates being 50 Hz for PAL and about 59.94 Hz for NTSC.

In 3D sampling, a grid of tiny cubes called voxels is used to create three-dimensional images from medical scans like X-ray computed tomography (CT), magnetic resonance imaging (MRI), and positron emission tomography (PET), as well as in other fields like seismic studies.

Undersampling

Main article: Undersampling

When a signal is sampled slower than a certain rate, called its Nyquist rate, the samples can look like they come from a lower-frequency signal. This is called undersampling. Sometimes, this is done on purpose so that the lower-frequency version still represents the original signal correctly and can be recovered. Undersampling is also known as bandpass sampling, harmonic sampling, IF sampling, and direct IF to digital conversion.

Oversampling

Main article: Oversampling

Oversampling is a technique used in modern devices that change continuous signals into digital ones. It helps reduce errors that can happen when turning digital signals back into continuous ones. This makes the final output clearer and more accurate.

Complex sampling

Complex sampling, also known as I/Q sampling, involves taking two related waveforms at the same time. These pairs of samples are then treated like complex numbers. This method helps in reducing the number of samples needed to represent a signal accurately.

When one waveform is the Hilbert transform of the other, it creates what is called an analytic signal. This special signal allows the sampling rate to be lowered without losing information. Instead of needing twice as many samples per second, only half as many complex samples are required to capture the same information. This technique is useful in many areas of signal processing.

Main article: Complex number

Main articles: Hilbert transform, Analytic signal, Nyquist rate, Equivalent baseband waveform