# Digital signal

A digital signal (from English digit = number; represented by numbers) is a signal that is represented by discrete values ​​and describes its development over time. It can be formed from an analog signal which describes the time-continuous course of a physical variable . The conversion of an analog signal into a digital signal takes place through quantization and sampling , which takes place at defined times. Digital values ​​are usually coded as binary numbers . Their quantization is thus given in bits .

Digital signals play an important role in communications technology and digital signal processing . Examples of digital signals are the video signals that are transmitted with the DVI and HDMI standards or the audio signals with Dolby Digital and S / PDIF . Data transmission via DSL and WLAN , as well as the GSM and UMTS cellular standards , is based on digital signals, just like modern computer technology .

Continuous and discrete signals

## General

The conversion of an analog signal into a digital signal takes place in two steps, which can be carried out in any order:

• The sampling in order to convert a time-continuous signal into a time-discrete signal .
• The quantization to convert a continuous-value signal into a discrete-value signal.

Different definitions of terms are common, depending on the area of ​​application. The exact differentiation of what is meant by a digital signal usually results from the respective context: In signal theory , mathematical sequences are used as representation, which are clearly characterized by an "infinitely thin" pulse sequence in the time sequence. In contrast, in digital circuits, as is common in the field of digital technology , a mathematical sequence cannot be represented by physical parameters such as an electrical voltage: In this case, the digital signal is formed by a time-continuous curve, with the continuous curve only changing at certain points in time and the value is constant between the points in time.

The sampling and generation of the digital signal usually takes place at constant time intervals, but this is not absolutely necessary.

### Signal theory

Graphic representation of a digital signal in the form of red markings

A digital signal x [ n ] can be described as a sequence of numbers which originate from a delimited set of values. The index n represents the time variable standardized to the sampling rate - sampling usually takes place at constant time intervals T s . The reciprocal value is referred to as the sampling rate or the sampling frequency f s . In the figure opposite, the exemplary course of an analog signal is shown in gray and the resulting digital signal sequence in red with the values:

${\ displaystyle x [n] = \ {4,5,4,3,4,6,7,5,3,3,4,4,3 \}}$with the index n = 1, 2, ..., 13

shown. It is essential that the values ​​between the sampling times are not zero or include other values, but are not defined. The mapping to whole numbers is chosen arbitrarily.

In this case, the Nyquist-Shannon sampling theorem describes the effect that the sequence x [ n ] can only contain the complete information of the analog signal curve if its highest frequency components f a are less than half the sampling frequency f s :

${\ displaystyle f_ {a} <{\ frac {f_ {s}} {2}}}$

### Digital technology

Course of a digital signal in red, as is usual in digital technology

In digital technology, a digital signal is also formed with a sample-and-hold circuit of the 0th order and then represents a time-continuous course, which only changes in value at the individual sampling times.

The individual time-discrete samples of the sequence are the rectangular function folded . This creates a digital signal, as shown in red in the adjacent figure. This profile f (t) can be implemented, at least approximately, for example by a voltage profile physically in a digital circuit and in integrated circuits .

It should be noted that the convolution with the square-wave function results in a distortion of the frequency spectrum when converting to the original analog signal curve using a digital-to-analog converter (DAC), which must be compensated for by appropriate filters . The distortion corresponds to the sinc function , which is the Fourier transform of the rectangular function.

## example

When recording an audio CD , each channel (left / right) of the source signal is scanned 44,100 times per second. The sampling frequency in this case is 44.1  kHz . Higher frequency details of the recorded source signal than half the sampling rate, in this case approx. 22 kHz, are not recorded and are removed by anti-aliasing filters during recording .

The thus obtained individual samples ( "Samples") are continuously even in size, which means they can have any value. In order to be able to represent these values ​​in numerical form, they must first be fitted into a fixed value grid using quantization, a form of rounding. Finer changes between the value grid levels are not recorded or generate a change by a full level. This word length , resolution is therefore selected as finely as possible and includes 65,536 possible values ​​with a linear characteristic curve for the audio CD, i.e. That is, regardless of the signal size, a constant resolution is used in the value range. In the audio CD, the individual words per sample are encoded as binary numbers with 16 bits per word, the number of digits per word is also referred to as dynamics , and are then available for further processing, such as recording on the data carrier Available.

## application

### transmission

Only signals that are continuous in time can be transmitted. Furthermore, there are disturbances such as thermal noise and other inaccuracies in components , which practically exclude the transfer of exactly a certain value (represented by voltage, field strength, etc.). This means that every digital signal can only be transmitted in the form of an analog signal and then has to be digitized again on the receiver side .

In the simplest case, each possible value of the digital signal is simply assigned a value or range of values ​​to a physical quantity that is transmitted. In TTL technology, for example (with positive logic ) the binary digit "0" is represented by the voltage 0 ... 0.4 V and the "1" by 2.4 ... 5.0 V.

Such an analog signal, which has sudden, rapid changes in the course (such as, for example, the digital signal in circuit technology described above) has the undesirable property of having a very broad spectrum . This leads to interference in neighboring channels or channels with a large bandwidth must be used. Therefore, in today's digital transmission methods, the digital signal is folded with a continuous basic pulse with specific properties , e.g. B. the raised cosine filter . The result is then also an analog signal.

### Transmission and interference

A digital signal is less susceptible to interference during transmission, as the signal levels can still be assigned to the correct value with a certain tolerance. Every signal is always superimposed or disturbed by noise during transmission. If the noisy signal is digitized again, these disturbances disappear again due to the quantization . As long as the interference is not too great, the original signal is obtained again.

Therefore, digital signals are better suited than analog signals to be transmitted over long distances. Provide repeaters along the route that process the signal, i.e. H. digitize (correct errors if necessary) and then forward it (again in analogue), this removes the unwanted noise each time and only sends the useful signal on. A purely analog signal can also be amplified again and again, but here the noise is also amplified every time.

This makes it possible to transmit digital signals without loss and to copy again digitally transmitted or stored signals as often as required - and without any loss of quality.

At the end of the information processing chain, a conversion into an analog signal is usually necessary again for communication to humans, e.g. B. from the audio CD into electrical voltage and sound pressure.

### Differentiation from other discrete-value signals

A digital signal must be both time and value discrete. In terms of the circuit technology, the feature of the time discreteness is also satisfied when the signal only at discrete points in time change can be, but between them constant and is continuously present to the extent (time). There are also a number of discrete-value signals, but these are not digital signals.

For example, this is a pulse-width-modulated signal that consists of a square-wave signal of a fixed frequency with a continuously variable duty cycle . Likewise, a sequence of square-wave pulses, such as those produced when measuring the speed with a light barrier, is not a digital signal. Although this signal is value-discrete, even binary, it can change its value with the frequency of the pulses without being tied to a time cycle . Another example of the colloquial but incorrect use of the term is the designation "digital amplifier" for class D amplifiers that work by means of pulse width modulation.

## meaning

Due to the so-called digital revolution , the use of digital signals has increased dramatically. Most household appliances are now based either entirely or at least in large part on digital signals. Communication systems such as the Internet and mobile telephony are based on a digital signal network. The advantages over analog technology are more versatile processing options and the error-free storage capacity over a long period, for example on CD-ROMs .