Sampling

(from Lat. discretio - “distinguish”, “recognize”) - in the general case - the representation of a continuous function by a discrete set of its values for different sets of arguments. For variable function $f(x)$ {\ displaystyle f (x)} $f (x)$ - presentation of her set $n$ ${\ displaystyle n}$ $n$ its meanings $f(x_{0}),f(x_{1}),...f(x_{n-1})$ ${\ displaystyle f (x_ {0}), f (x_ {1}), ... f (x_ {n-1})}$ ${\ displaystyle f (x_ {0}), f (x_ {1}), ... f (x_ {n-1})}$ on a given discrete set of argument values $x_{0},x_{1},...x_{n-1}$ ${\ displaystyle x_ {0}, x_ {1}, ... x_ {n-1}}$ ${\ displaystyle x_ {0}, x_ {1}, ... x_ {n-1}}$ .

In signal processing, the representation of an analog continuous signal $S(t)$ ${\ displaystyle S (t)}$ $S (t)$ the set of its values, this set is called samples $S(t_{0}),S(t_{1}),...S(t_{n-1})$ ${\ displaystyle S (t_ {0}), S (t_ {1}), ... S (t_ {n-1})}$ ${\ displaystyle S (t_ {0}), S (t_ {1}), ... S (t_ {n-1})}$ taken at time points $t_{0},t_{1},...t_{n-1}$ ${\ displaystyle t_ {0}, t_ {1}, ... t_ {n-1}}$ ${\ displaystyle t_ {0}, t_ {1}, ... t_ {n-1}}$ .

In the general case, the period of time from one sample to the next may differ for each pair of neighboring samples, but usually when processing a signal, the samples follow a fixed and constant period of time. This gap is then called the sampling period or sampling interval and is usually indicated by the letter $T$ ${\ displaystyle T}$ $T$ . Value inverse to the sampling period $F_{s}=1/T$ ${\ displaystyle F_ {s} = 1 / T}$ ${\ displaystyle F_ {s} = 1 / T}$ called the sampling rate or sampling rate ^[1] .

Examples of an analog signal are audio or video signals, signals from various measuring sensors , etc. For subsequent digital processing, continuous analog signals must first be sampled and quantized by level using analog-to-digital converters .

The inverse process of obtaining a continuous analog signal given by a discrete set of its samples is called recovery . Recovery is carried out by digital-to-analog converters .

Content

1 Theory
2 Application
3 See also
4 See also
5 notes
6 Literature
7 References

Theory

Discretization by a comb of Dirac

In mathematical terms, discretization is the multiplication of a continuous function $s(t)$ ${\ displaystyle s (t)}$ to a function called the Dirac crest $\Delta _{T}(t)\ {\stackrel {\mathrm {def} }{=}}\ \sum _{k=-\infty }^{\infty }\delta (t-kT)$ ${\ displaystyle \ Delta _ {T} (t) \ {\ stackrel {\ mathrm {def}} {=}} \ \ sum _ {k = - \ infty} ^ {\ infty} \ delta (t-kT) }$ Where $T$ ${\ displaystyle T}$ - constant - sampling period and $\delta (t)$ ${\ displaystyle \ delta (t)}$ - Dirac delta function :

s_{\mathrm {a} }(t)=s(t)\cdot \sum _{n=-\infty }^{\infty }\delta (t-nT).

{\ displaystyle s _ {\ mathrm {a}} (t) = s (t) \ cdot \ sum _ {n = - \ infty} ^ {\ infty} \ delta (t-nT).}

Fourier transform of a discrete function $s_{\mathrm {a} }(t)$ ${\ displaystyle s _ {\ mathrm {a}} (t)}$ gives its spectrum $S_{\mathrm {a} }(f)$ ${\ displaystyle S _ {\ mathrm {a}} (f)}$ . According to Kotelnikov’s theorem, if the spectrum $S_{\mathrm {a} }(f)$ ${\ displaystyle S _ {\ mathrm {a}} (f)}$ the original function is bounded, i.e. the spectral density is zero above a certain frequency $f_{max}$ ${\ displaystyle f_ {max}}$ , then the original function is uniquely recoverable from the totality of its samples taken with a sampling frequency $1/T\geq 2f_{max}$ ${\ displaystyle 1 / T \ geq 2f_ {max}}$ .

For absolutely accurate recovery, it is necessary to apply to the input of an ideal low-pass filter a sequence of infinitely short pulses each with an area equal to the sample value.

It is practically impossible to ideally accurately restore real signals from samples, since, firstly, there are no signals with a limited spectrum, because real signals are limited in time, which necessarily gives a spectrum of infinite width. Secondly, physically we cannot realize the ideal low-pass filter ( sinc filter ), thirdly, infinitely short pulses with a finite area are impossible.

Application

All signals in nature are essentially analog. For digital signal processing, storage and transmission in digital form, the analog signals are pre-digitized. Digitization includes sampling and quantization by level, performed using the ADC. After digital processing, transmission, storage of digital data encoding a signal, it is often necessary to invert the digital image of the signal into an analog signal. For example, playing audio from a CD.

Sampling is also used in analog pulse modulation systems.

In practice, the reconstruction of an analog signal from a set of samples is performed with one degree or another degree of accuracy, and the restoration accuracy is higher, the higher the sampling frequency and the number of quantization levels of each sample. But the higher the sampling rate and the number of quantization levels, the more resources are required for processing, storage, transmission of digitized data. Therefore, the sampling rate and bit depth of the ADC is practically chosen on the basis of a reasonable compromise.

For example, with digital voice transmission, for good speech intelligibility, a sampling frequency of 8 kHz is sufficient, high-quality reproduction of music from CDs in the modern standard is performed with a sampling frequency of 48 kHz, which ensures high-quality sound reproduction in the entire audible band of 20 Hz - 20 kHz.

Digitization of a television video signal with a frequency band of 6 MHz is performed with a sampling frequency of more than 10 MHz ^[2] .

Notes

↑
The conversion of a continuous information set of analog signals into a discrete set is called level sampling or quantization (cf. "Time Quantization"). Level quantization is widely used in digital machines. When quantizing by level, various values of the value are displayed $x$ ${\ displaystyle x}$ to a discrete region consisting of quantities ${\overset {-}{x}}$ ${\ displaystyle {\ overset {-} {x}}}$ quantization level.
- Samofalov K.G., Romankevich A.M., Valuisky V.N., Kanevsky Yu.S., Pinevich M.M. 1.3 Information discretization // Applied theory of digital automata. - K .: Vishcha school, 1987 .-- 375 p.
↑ Dictionary of Cybernetics, p. 168 / Edited by V. S. Mikhalevich. - 2nd edition - K .: 1989. - 751 p., ISBN 5-88500-008-5

Literature

Samofalov K.G., Romankevich A.M., Valuisky V.N., Kanevsky Yu.S., Pinevich M.M. Applied theory of digital automata. - K .: Vishcha school, 1987 .-- 375 p.

Links

Signal quantization - an article from the Great Soviet Encyclopedia .
Sampling of analog signals Interactive presentation of time sampling. Institute of Telecommunications, University of Stuttgart

[1] 
The conversion of a continuous information set of analog signals into a discrete set is called level sampling or quantization (cf. "Time Quantization"). Level quantization is widely used in digital machines. When quantizing by level, various values of the value are displayed $x$ ${\ displaystyle x}$ to a discrete region consisting of quantities ${\overset {-}{x}}$ ${\ displaystyle {\ overset {-} {x}}}$ quantization level.
- Samofalov K.G., Romankevich A.M., Valuisky V.N., Kanevsky Yu.S., Pinevich M.M. 1.3 Information discretization // Applied theory of digital automata. - K .: Vishcha school, 1987 .-- 375 p.

[2] Dictionary of Cybernetics, p. 168 / Edited by V. S. Mikhalevich. - 2nd edition - K .: 1989. - 751 p., ISBN 5-88500-008-5