Sampling (signal processing)

In signal processing, sampling is a way to change a smooth signal into separate pieces of information. It is like taking quick snapshots of something that is moving or changing fast. For example, when you record sound, the smooth sound wave is turned into a list of numbers. These numbers show what the sound was like at many tiny moments in time. Each number is called a sample.

A device that takes these snapshots is called a sampler. In the best case, a sampler catches the exact value of the signal each time, just like a perfect photo.

We can put these snapshots back together to recreate the original signal, if we take enough snapshots. This idea is linked to the Nyquist limit. It tells us how often to take samples for a good copy of the original. Using a special filter called a reconstruction filter, we can turn the list of numbers back into something that sounds or looks like the original signal.

Theory

Sampling is a way to take small pieces of information from a signal, like a sound wave, at regular times. Imagine recording your voice and saving just a tiny snapshot of the sound every moment. Each snapshot is called a "sample."

The time between each snapshot is called the sampling interval. How often we take these samples is the sampling frequency, measured in samples per second, sometimes called hertz. For example, 48 kHz means we take 48,000 samples every second. To bring these samples back into a smooth signal, we use special math methods. If the original signal changes very fast, we might need a special filter to prepare the signal before sampling.

Practical considerations

In real life, we turn a smooth signal into separate samples using a device called an analog-to-digital converter. This device has limits, so the results aren’t perfect and can have small differences called distortion.

Some types of distortion include:

Aliasing: This happens when fast parts of the signal get mixed up. It can be reduced with special filters.
Aperture error: This occurs because the sample is an average over a tiny time period.
Jitter: This is when the timing between samples isn’t perfect.
Noise: Small random changes can come from the equipment or the environment.
Slew rate limit error: Sometimes the device can’t change fast enough.
Quantization: This happens because the device can only use a limited number of levels to represent the signal.

While some methods can reduce certain types of distortion, they can’t remove all of them. At higher frequencies, some of these distortions become more noticeable.

Applications

Digital audio systems change sound waves into small pieces called samples. These samples help us store and play back music and other sounds on computers and devices. Audio is often sampled at rates like 44.1 kHz for CDs, so it can capture all the sounds humans can hear.

Video also uses sampling to turn what we see into digital data. For example, standard TV uses a certain number of pixels to make images, while high-definition TV uses even more pixels for clearer pictures. Sampling helps create videos by quickly taking many small pieces of information.

In medical imaging, sampling is used to create three-dimensional pictures from data. This helps doctors see inside the body without needing surgery.

Sampling rate	Use
5,512.5 Hz	Supported in Flash.
8,000 Hz	Telephone and encrypted walkie-talkie, wireless intercom and wireless microphone transmission; adequate for human speech but without sibilance (ess sounds like eff (/s/, /f/)).
11,025 Hz	One quarter the sampling rate of audio CDs; used for lower-quality PCM, MPEG audio and for audio analysis of subwoofer bandpasses.
16,000 Hz	Wideband frequency extension over standard telephone narrowband 8,000 Hz. Used in most modern VoIP and VVoIP communication products.
22,050 Hz	One half the sampling rate of audio CDs; used for lower-quality PCM and MPEG audio and for audio analysis of low-frequency energy. Suitable for digitizing early 20th century audio formats such as 78s and AM Radio.
32,000 Hz	miniDV digital video camcorder, video tapes with extra channels of audio (e.g. DVCAM with four channels of audio), DAT (LP mode), Germany's Digitales Satellitenradio, NICAM digital audio, used alongside analogue television sound in some countries. High-quality digital wireless microphones. Suitable for digitizing FM radio.
37,800 Hz	CD-XA audio
44,055.9 Hz	Used by digital audio locked to NTSC color video signals (3 samples per line, 245 lines per field, 59.94 fields per second = 29.97 frames per second).
44,100 Hz	Audio CD, also most commonly used with MPEG-1 audio (VCD, SVCD, MP3). Originally chosen by Sony because it could be recorded on modified video equipment running at either 25 frames per second (PAL) or 30 frame/s (using an NTSC monochrome video recorder) and cover the 20 kHz bandwidth thought necessary to match professional analog recording equipment of the time. A PCM adaptor would fit digital audio samples into the analog video channel of, for example, PAL video tapes using 3 samples per line, 588 lines per frame, 25 frames per second.
47,250 Hz	world's first commercial PCM sound recorder by Nippon Columbia (Denon)
48,000 Hz	The standard audio sampling rate used by professional digital video equipment such as tape recorders, video servers, vision mixers and so on. This rate was chosen because it could reconstruct frequencies up to 22 kHz and work with 29.97 frames per second NTSC video, as well as 25 frame/s, 30 frame/s and 24 frame/s systems. With 29.97 frame/s systems, it is necessary to handle 1601.6 audio samples per frame, delivering an integer number of audio samples only every fifth video frame. Also used for sound with consumer video formats like DV, digital TV, DVD, films, and many video streaming platforms such as YouTube and Netflix. The lossless audio such as FLAC can select either 44100 Hz, 48000 Hz or higher. The professional serial digital interface (SDI) and High-definition Serial Digital Interface (HD-SDI) used to connect broadcast television equipment together use this audio sampling frequency. Most professional audio gear uses 48 kHz sampling, including mixing consoles, and digital recording devices.
50,000 Hz	First commercial digital audio recorders from the late 70s from 3M and Soundstream.
50,400 Hz	Sampling rate used by the Mitsubishi X-80 digital audio recorder.
64,000 Hz	Uncommonly used, but supported by some hardware and software.
88,200 Hz	Sampling rate used by some professional recording equipment when the destination is CD (multiples of 44,100 Hz). Some pro audio gear uses (or is able to select) 88.2 kHz sampling, including mixers, EQs, compressors, reverb, crossovers, and recording devices.
96,000 Hz	DVD-Audio, some LPCM DVD tracks, BD-ROM (Blu-ray Disc) audio tracks, HD DVD (High-Definition DVD) audio tracks, and Hi-Res Audio. Some professional recording and production equipment is able to select 96 kHz sampling. This sampling frequency is twice the 48 kHz standard commonly used with audio on professional equipment.
176,400 Hz	Sampling rate used by HDCD recorders and other professional applications for CD production. Four times the frequency of 44.1 kHz.
192,000 Hz	DVD-Audio, some LPCM DVD tracks, BD-ROM (Blu-ray Disc) audio tracks, and HD DVD (High-Definition DVD) audio tracks, High-Definition audio recording devices, Hi-Res Audio, and audio editing software. This sampling frequency is four times the 48 kHz standard commonly used with audio on professional video equipment.
352,800 Hz	Digital eXtreme Definition, used for recording and editing Super Audio CDs, as 1-bit Direct Stream Digital (DSD) is not suited for editing. 8 times the frequency of 44.1 kHz.
384,000 Hz	Maximum sample rate available in common software.
2,822,400 Hz	SACD, 1-bit delta-sigma modulation process known as Direct Stream Digital, co-developed by Sony and Philips.
5,644,800 Hz	Double-Rate DSD, 1-bit Direct Stream Digital at 2× the rate of the SACD. Used in some professional DSD recorders.
11,289,600 Hz	Quad-Rate DSD, 1-bit Direct Stream Digital at 4× the rate of the SACD. Used in some uncommon professional DSD recorders.
22,579,200 Hz	Octuple-Rate DSD, 1-bit Direct Stream Digital at 8× the rate of the SACD. Used in rare experimental DSD recorders. Also known as DSD512.
45,158,400 Hz	Sexdecuple-Rate DSD, 1-bit Direct Stream Digital at 16× the rate of the SACD. Used in rare experimental DSD recorders. Also known as DSD1024.

Undersampling

Main article: Undersampling

When we try to record a fast signal but do it too slowly, it might appear to move more slowly than it really does. This is called undersampling. Sometimes, people do this on purpose because it can make things simpler. Even though the signal looks slower, it still gives us all the information we need about the original fast signal. Undersampling is also known as bandpass sampling, harmonic sampling, IF sampling, and direct IF to digital conversion.

Oversampling

Main article: Oversampling

Oversampling is a way to make digital tools better at changing sounds and signals. It turns smooth, continuous signals into step-by-step digital forms. By taking many small pieces of the original signal, it helps reduce mistakes when changing the digital signals back to smooth ones.

Complex sampling

Complex sampling, also called I/Q sampling, is a way to take two related signals at the same time and treat them as pairs of numbers. This helps in studying signals, especially in reducing how often we need to measure them.

When one signal is a special version of the other, called the Hilbert transform, we can work with these pairs more efficiently. This method lets us use half the number of samples to keep all the information, which is very useful in many technologies.