Very often we hear such definitions as a “digital” or “discrete” signal, what is its difference from the “analog” one?
The essence of the difference is that the analog signal is continuous in time (blue line), while the digital signal consists of a limited set of coordinates (red dots). If everything is reduced to coordinates, then any segment of an analog signal consists of an infinite number of coordinates.
For a digital signal, the coordinates along the horizontal axis are located at regular intervals, in accordance with the sampling frequency. In the common Audio-CD format, this is 44100 dots per second. Vertical accuracy of the coordinate height corresponds to the bit depth of the digital signal, for 8 bits it is 256 levels, for 16 bits \u003d 65536 and for 24 bits \u003d 16777216 levels. The higher the bit depth (the number of levels), the closer the vertical coordinates to the original wave.
Analog sources are: vinyl and audio cassettes. Digital sources are: CD-Audio, DVD-Audio, SA-CD (DSD) and files in WAVE and DSD formats (including derivatives of APE, Flac, Mp3, Ogg, etc.).
Advantages and disadvantages of an analog signal
The advantage of an analog signal is that it is in the analog form that we perceive sound with our ears. And although our auditory system translates the perceived sound stream into a digital form and transmits it to the brain in this form, science and technology have not yet reached the possibility of connecting players and other sound sources directly in this form. Similar studies are now being actively conducted for people with disabilities, and we enjoy exclusively analog sound.The disadvantage of an analog signal is the ability to store, transmit and replicate the signal. When recording onto tape or vinyl, the signal quality will depend on the properties of the tape or vinyl. Over time, the tape is demagnetized and the quality of the recorded signal deteriorates. Each reading gradually destroys the medium, and rewriting introduces additional distortions, where the next medium (tape or vinyl), reading, recording and transmitting devices add additional deviations.
To make a copy of the analog signal, it is the same as to copy it again to copy a photograph.
Advantages and disadvantages of a digital signal
The advantages of a digital signal include accuracy when copying and transmitting an audio stream, where the original is no different from a copy.The main disadvantage can be considered that the signal in digital form is an intermediate stage and the accuracy of the final analog signal will depend on how detailed and accurately the sound wave describes the coordinates. It is logical that the more points there are and the more accurate the coordinates, the more accurate the wave will be. But so far there is no consensus on how many coordinates and accuracy of the data is sufficient to say that the digital representation of the signal is enough to accurately restore the analog signal, indistinguishable from the original by our ears.
If you operate with data volumes, then the capacity of a conventional analog audio cassette is only about 700-1.1 MB, while a regular CD-ROM can hold 700 MB. This gives an idea of \u200b\u200bthe need for large-capacity media. And this gives rise to a separate war of compromises with different requirements for the number of describing points and for the accuracy of coordinates.
To date, it is considered quite sufficient to represent a sound wave with a sampling frequency of 44.1 kHz and a bit depth of 16 bits. At a sampling frequency of 44.1 kHz, a signal with a frequency of up to 22 kHz can be restored. As psychoacoustic studies show, a further increase in the sampling rate is not noticeable, but an increase in bit depth gives a subjective improvement.
How DACs Build a Wave
DAC is a digital-to-analog converter, an element that converts digital sound to analog. We will examine superficially basic principles. If the comments show an interest in considering a number of points in more detail, a separate material will be released.Multi-bit DACs
Very often, a wave is presented in the form of steps, which is due to the architecture of the first generation of R-2R multi-bit DACs, operating similarly to a relay switch.
At the input of the DAC, the next vertical coordinate value is received, and at each cycle it switches the current (voltage) level to the appropriate level until the next change.
Although it is believed that a person’s ear hears no higher than 20 kHz, and according to the Nyquist theory, a signal can be restored to 22 kHz, the quality of this signal after recovery remains a question. In the high-frequency region, the shape of the resulting “stepped” wave is usually far from the original. The easiest way out of the situation is to increase the sampling rate during recording, but this leads to a significant and undesirable increase in the file size.
An alternative is to artificially increase the sampling rate when playing in the DAC, adding intermediate values. Those. we represent the path of a continuous wave (gray dashed line) smoothly connecting the original coordinates (red dots) and add intermediate points on this line (dark purple).
With an increase in the sampling frequency, it is usually necessary to increase the bit depth so that the coordinates are closer to the approximated wave.
Thanks to the intermediate coordinates, it is possible to reduce the “steps” and build a wave closer to the original.
When you see the function of increasing the frequency from 44.1 to 192 kHz in the player or an external DAC, this is a function of adding intermediate coordinates, not restoring or creating sound in the region above 20 kHz.
Initially, these were separate SRC chips before the DAC, which then migrated directly to the DAC chips themselves. Today you can find solutions where such a microcircuit is added to modern DACs, this is done in order to provide an alternative to the built-in algorithms in the DAC and sometimes get even better sound (such as this is done in the Hidizs AP100).
The main refusal in the industry from multi-bit DACs occurred due to the impossibility of further technological development of quality indicators with current production technologies and higher cost against “pulse” DACs with comparable characteristics. Nevertheless, in Hi-End products, preference is often given to old multi-bit DACs, rather than new solutions with technically better characteristics.
Pulse DAC
At the end of the 70s, an alternative version of DACs based on the "pulse" architecture - "delta-sigma" became widespread. The technology of pulsed DACs has made possible the emergence of super-fast keys and allowed the use of a high carrier frequency.
The signal amplitude is the average value of the pulse amplitudes (pulses of equal amplitude are shown in green, and the resulting sound wave in white).
For example, a sequence of eight cycles of five pulses will give the average amplitude (1 + 1 + 1 + 0 + 0 + 1 + 1 + 0) / 8 \u003d 0.625. The higher the carrier frequency, the more pulses fall under the smoothing and a more accurate amplitude value is obtained. This allowed us to present the sound stream in a single-bit form with a wide dynamic range.
Averaging can be done with a conventional analog filter, and if such a set of pulses is applied directly to the speaker, then we will get sound at the output, and ultra high frequencies will not be reproduced due to the high inertness of the emitter. According to this principle, PWM amplifiers operate in class D, where the energy density of the pulses is created not by their quantity, but by the duration of each pulse (which is easier to implement but impossible to describe with a simple binary code).
A multi-bit DAC can be thought of as a printer capable of applying color with pantone inks. Delta-Sigma is an inkjet printer with a limited set of colors, but due to the possibility of applying very small dots (compared to an antler printer), due to the different density of dots per unit surface, it gives more shades.
In the image, we usually do not see individual points due to the low resolution of the eye, but only the average tone. Similarly, the ear does not hear impulses separately.
Ultimately, with current technologies in pulsed DACs, you can get a wave close to that which theoretically should be obtained by approximating the intermediate coordinates.
It should be noted that after the appearance of the delta-sigma DAC, the relevance of drawing a “digital wave” by steps has disappeared. so modern DACs do not build the steps of a wave. To build a properly discrete signal with points connected by a smooth line.
Are Pulse DACs Ideal?
But in practice, not everything is cloudless, and there are a number of problems and limitations.
Because Since the overwhelming number of records is stored in a multi-bit signal, conversion to a pulse signal on a bit-to-bit basis requires an excessively high carrier frequency, which modern DACs do not support.
The main function of modern pulsed DACs is to convert a multi-bit signal into a single-bit signal with a relatively low carrier frequency with data thinning. Basically, it is these algorithms that determine the final sound quality of pulsed DACs.
To reduce the problem of high carrier frequency, the audio stream is divided into several single-bit streams, where each stream is responsible for its discharge group, which is equivalent to a multiple increase in the carrier frequency from the number of streams. Such DACs are called multi-bit delta sigma.
Today, pulsed DACs got a second wind in general-purpose high-speed microcircuits in NAD and Chord products due to the ability to flexibly program conversion algorithms.
DSD format
After the widespread distribution of delta-sigma DACs, the appearance of a binary recording format directly for delta-sigma encoding was quite logical. This format is called DSD (Direct Stream Digital).The format was not widely used for several reasons. Editing files in this format was unnecessarily limited: you can not mix streams, adjust the volume and apply equalization. And this means that without loss of quality, you can only archive analog recordings and produce two-microphone recordings of live performances without further processing. In a word - money doesn’t really make money.
In the fight against piracy, SA-CD format disks were not supported (and still are not supported) by computers, which does not allow making copies of them. No copies - no wide audience. It was possible to play DSD audio content only from a separate SA-CD player from a company disc. If there is a SPDIF standard for PCM format for digital data transmission from a source to a separate DAC, then there is no standard for DSD format and the first pirated copies of SA-CD discs were digitized from the analog outputs of SA-CD players (although the situation seems dumb, some recordings were released only on SA-CD, or the same recording on Audio-CD was specially made poorly to promote SA-CD).
A turning point occurred with the release of SONY game consoles, where the SA-CD disk was automatically copied to the console hard disk before playback. This was used by fans of the DSD format. The advent of pirated recordings stimulated the market for the release of separate DACs for playback of the DSD stream. Most external DACs with DSD support today support USB data transfer using the DoP format as a separate digital signal encoding via SPDIF.
The carrier frequencies for DSD are relatively small, 2.8 and 5.6 MHz, but this audio stream does not require any thinning conversions and is quite competitive with high-resolution formats such as DVD-Audio.
When asked which is better, DSP or PCM there is no definite answer. Everything rests on the quality of the implementation of a specific DAC and the talent of a sound engineer when recording the final file.
General conclusion
An analog sound is what we hear and perceive as the surrounding world through our eyes. Digital sound is a set of coordinates describing a sound wave, and which we cannot directly hear without conversion to an analog signal.An analog signal recorded directly to an audio cassette or vinyl cannot be overwritten without loss of quality, while a wave in digital representation can be copied bit to bit.
Digital recording formats are a constant compromise between the amount of coordinate accuracy versus file size and any digital signal is only an approximation to the original analog signal. However, at the same time, a different level of technology for recording and reproducing a digital signal and storing media for an analog signal gives more advantages to the digital representation of the signal, similar to a digital camera versus a film camera.
Signal information - physical process having for a person or technical device informationalvalue. It can be continuous (analog) or discrete
The term “signal” is very often identified with the concepts of “data” and “information”. Indeed, these concepts are interrelated and do not exist one without the other, but belong to different categories.
Signalis an information function that carries a message about the physical properties, state or behavior of any physical system, object or environment, and the purpose of signal processing can be considered to be the extraction of certain information information that is displayed in these signals (briefly - useful or target information) and transformation this information in a form convenient for perception and further use.
Information is transmitted in the form of signals. A signal is a physical process that carries information. The signal can be sound, light, in the form of mail, etc.
A signal is a material medium of information that is transmitted from a source to a consumer. It can be discrete and continuous (analog)
Analog signal- a data signal in which each of the representing parameters is described by a function of time and a continuous set of possible values.
Analog signals are described by continuous functions of time, so an analog signal is sometimes called a continuous signal. Analog signals are opposed to discrete (quantized, digital).
Examples of continuous spaces and corresponding physical quantities: (direct: electrical voltage; circle: position of the rotor, wheel, gear, analogue clock hands, or phase of the carrier signal; segment: position of the piston, control lever, liquid thermometer or electrical signal, limited in amplitude multidimensional spaces: color, quadrature modulated signal.)
The properties of analog signals are largely the opposite of the properties of quantized or digitalsignals.
The absence of clearly distinct discrete signal levels makes it impossible to use the concept of information in its description in the form in which it is understood in digital technologies. The "amount of information" contained in a single count will be limited only by the dynamic range of the measuring instrument.
Lack of redundancy. From the continuity of the value space, it follows that any interference introduced into the signal is indistinguishable from the signal itself and, therefore, the original amplitude cannot be restored. In fact, filtering is possible, for example, by frequency methods, if any additional information about the properties of this signal (in particular, the frequency band) is known.
Application:
Analog signals are often used to represent continuously changing physical quantities. For example, an analog electric signal taken from a thermocouple carries information about a change in temperature, a signal from a microphone indicates rapid changes in pressure in a sound wave, etc.
Discrete signalis composed of a countable set (i.e., such a set whose elements can be counted) of elements (they say - information elements). For example, a brick signal is discrete. It consists of the following two elements (this is the syntactic characteristic of this signal): a red circle and a white rectangle inside a circle located horizontally in the center. It is in the form of a discrete signal that the information that the reader is now mastering is presented. The following elements can be distinguished: sections (for example, “Information”), subsections (for example, “Properties”), paragraphs, sentences, individual phrases, words and individual characters (letters, numbers, punctuation marks, etc.). This example shows that depending on the pragmatics of the signal, different information elements can be distinguished. In fact, for a person studying computer science in this text, larger information elements, such as sections, subsections, individual paragraphs, are important. They allow him to navigate easier in the structure of the material, better to assimilate it and prepare for the exam. For those who prepared this methodological material, in addition to the indicated information elements, smaller ones, for example, individual sentences, with the help of which this or that idea is presented and which implement one or another way of material accessibility, are also important. The set of the smallest elements of a discrete signal is called the alphabet, and the discrete signal itself is also called message.
Discretization is the conversion of a continuous signal to a discrete (digital) one.
The difference between discrete and continuous presentation of information is clearly visible on the example of a watch. In an electronic watch with a digital dial, information is presented discretely - in numbers, each of which is clearly different from each other. In a mechanical watch with a dial, information is presented continuously - by the positions of two hands, and two different positions of the hands are not always clearly distinguishable (especially if there are no minute divisions on the dial).
Continuous signal- is reflected by a certain physical quantity that varies in a given time interval, for example, by timbre or sound power. In the form of a continuous signal, real information is presented for those student students who attend lectures on computer science and through sound waves (in other words, the lecturer's voice), which are continuous, perceive the material.
As we will see later, a discrete signal lends itself better to transformations, therefore it has advantages over a continuous one. At the same time, in technical systems and in real processes, a continuous signal prevails. This forces us to develop ways to convert a continuous signal to a discrete one. \\
To convert a continuous signal to a discrete one, a procedure called quantization.
A digital signal is a data signal in which each of the representing parameters is described by a discrete time function and a finite set of possible values.
A discrete digital signal is more difficult to transmit over long distances than an analog signal, so it is pre-modulated on the side of the transmitter, and demodulated on the side of the receiver of information. The use of digital verification and recovery algorithms for digital information in digital systems can significantly increase the reliability of information transfer.
Comment. It should be borne in mind that a real digital signal by its physical nature is analog. Due to noise and changes in the parameters of transmission lines, it has fluctuations in amplitude, phase / frequency (jitter), and polarization. But this analog signal (pulse and discrete) is endowed with the properties of a number. As a result, it is possible to use numerical methods for its processing (computer processing).
Today we’ll try to understand what are analog and digital signals? Their advantages and disadvantages. We will not throw various scientific terms and definitions, but try to understand the situation on the fingers.
What is an analog signal?
An analog signal is based on the analogy of an electric signal (current and voltage values) to the value of the original signal (pixel color, frequency and amplitude of sound, etc.). Those. certain current and voltage values \u200b\u200bcorrespond to the transmission of a specific pixel color or sound signal.
I will give an example on an analog video signal.
The voltage on the wire 5 volts corresponds to blue, 6 volts to green, 7 volts to red.
In order for the red, blue and green stripes to appear on the screen, you need to alternately apply 5, 6, 7 volts to the cable. The faster we change voltages, the finer the strips we get on our monitor. Having reduced the interval between voltage changes to a minimum, we will no longer receive strips, but colored dots alternating one after another.
An important feature of the analog signal is the fact that it is transmitted strictly from the transmitter to the receiver (for example, from the antenna to the TV), there is no feedback. Therefore, if interference interferes with signal transmission (for example, instead of six volts, four will come), the pixel color will be distorted and a ripple will appear on the screen.
The analog signal is continuous.
What is a digital signal?
Data is also transmitted using an electric signal, but the values \u200b\u200bof these signals are only two and they correspond to 0 and 1. That is, a sequence of zeros and ones is transmitted over the wires. Something like this: 01010001001, etc. In order for the receiving device (for example, a TV) not to get confused in the transmitted data, the numbers are transmitted in batches. This happens like this: 10100010 10101010 10100000 10111110. Each such pack carries some kind of information, for example, the color of a pixel. An important feature of a digital signal is that the transmitting and receiving devices can communicate with each other and correct one after another errors that may occur during transmission.
Examples of digital and analog signal transmission
For a digital signal, transmission takes place like this:
- Hindrance: AAAAAAAAAAAAAA!
- TV: Which one? I can not hear!
- VCR: Green!
- TV: Yeah, got it! I draw green.
- TV: Please confirm that the color is red.
- VCR: confirm.
- TV: Ok! I draw.
Transmission for analog signal:
- VCR: Hey TV, pixel color with coordinates 120x300 - green.
- Hindrance: AAAAAAAAAAAAAA!
- TV: Which one? I can not hear! Damn, I’ll draw blue.
- VCR: The next color is red!
- Hindrance: BAM! BOOM!
- TV: Like red! I draw.
- VCR: Shovel!
- Hindrance: PShShShShShSh!
- TV set: ?!. Need to draw something ?! Let there be a shovel!
Advantages and disadvantages of digital and analog signals
From the foregoing, we can conclude that, ceteris paribus, the quality of information transmission using a digit will be higher than with the analog representation of the signal. At the same time, with good noise immunity, the two technologies can compete on equal terms.
With these words, John began his gospel, describing times that went beyond our era. We begin this article no less pathetic, and with all seriousness we declare that in the broadcasting business “there was a signal in the beginning”.
In television, as in all electronics, the signal is the basis. Speaking of it, we mean electromagnetic oscillations, which propagate in the air with the help of a transmitting antenna and cause current fluctuations in the receiving antenna. The ether wave can be represented both in continuous and in pulsed form, which significantly affects the final result - the quality of TV reception.
What is analog television? This is television, familiar to everyone, which the parents of our parents found. It is broadcast in an unencoded way, its basis is an analog signal, and it receives its usual analogue television, familiar to us from childhood,. Currently, in many countries, the process of digitizing an analog signal, and therefore, on-air television, is ongoing. In some European countries, this process has already been completed and terrestrial analogue TV is turned off. There are reasons for this article that this article suggests.
Differences between a digital signal and an analog
For most people, the distinction between analog and digital can be completely implicit. Nevertheless, their difference is significant and is not just in the quality of the broadcast.
An analog signal is the received data that we see, hear and perceive as the world that surrounds us. This method of generating, processing, transmitting and recording signals is traditional and still very common. Data is converted into electromagnetic waves, reflecting the frequency and intensity of phenomena according to the principle of full compliance.
A digital signal is a set of coordinates describing an electromagnetic wave, which is not inaccessible to perception directly, without decoding, because is a sequence of electromagnetic pulses. Speaking of discreteness and continuity of signals, they mean respectively “accepting values \u200b\u200bfrom a finite set” and “accepting values \u200b\u200bfrom an infinitely many”. 
An example of discreteness can be school grades, which take values \u200b\u200bfrom a set of 1,2,3,4,5. In fact, a digital video signal is often created by digitizing an analog signal.
Departing from theory, in fact, the following key differences between analog and digital signals can be distinguished:
- analogue television is vulnerable to interference, introducing noise into it, while a digital pulse is either completely blocked by interference and is absent, or is received in its original form.
- any device whose operation is based on the same principle as broadcasting a transmitter can receive and read an analog signal. A digital wave is intended for a certain “target”, and therefore, is resistant to interception, because securely encoded.
Image quality
The quality of the picture on the TV that analog TV provides is largely due to the TV standard. The frame that carries analogue broadcasting includes 625 lines with an aspect ratio of 4 × 3. Thus, the old picture tube shows an image of television lines, while a digital image is composed of pixels.
With poor reception and interference, the TV will “sniff” and hiss, not giving the viewer the image and sound. In an attempt to improve this situation, at one time, it was realized.
Other features
Despite the rapid development of electronic technology and the advantages of a digital signal over analog, there are still areas in which analog technology is indispensable, such as professional sound processing. But, although the original record can be no worse than a “figure”, after editing and copying it will inevitably be noisy.
Here is a set of basic operations that can be performed with an analog stream:
- strengthening and weakening;
- modulation aimed at reducing its susceptibility to interference, and demodulation;
- frequency filtering and processing;
- multiplication, summation and logarithm;
- processing and changing the parameters of its physical quantities.
Features of analog and digital television
The narrow-minded opinion about the collapse of on-air TV and the transition to broadcasting technologies of the future is somewhat unfair, just because viewers are replacing concepts: broadcast and analog TV. After all, the term “broadcast” is understood to mean any television broadcast on a terrestrial radio channel.
Both “analogue” and “figure” are varieties of broadcast TV. Despite the fact that analog television is different from digital, their general broadcasting principle is identical - the television tower broadcasts channels and guarantees a high-quality signal only in a limited radius. At the same time, the digital coverage radius is shorter than the range of the unencoded stream, which means that the repeaters should be installed closer to each other.
But the opinion that the “figure” will bypass the “analogue” in the long run is true. Viewers in many countries have already become "eyewitnesses" of converting an analog signal into a digital one and enjoy watching TV programs in HD quality.
Features of terrestrial television
The existing broadcast television system uses analog signals to transmit a television product. They propagate through waves with a high level of vibration, reaching terrestrial antennas. In order to increase the area of \u200b\u200bbroadcast coverage, repeaters are installed. Their function is to concentrate and amplify the signal, transmitting it to remote receivers. The signals are transmitted at a fixed frequency, so each channel corresponds to its own frequency and is fixed in the numbering order on the TV.
Advantages and Disadvantages of Digital Broadcasting
Information transmitted using a digital code contains virtually no errors or distortions. A device that digitizes an original signal is called an analog-to-digital converter (ADC).
For coding pulses, a system of ones and zeros is used. To read and convert the binary decimal code, a device called a digital-to-analog converter ”(DAC) is built into the receiver. Neither for the ADC, nor for the DAC does not exist half values, for example, 1.4 or 0.8.
This method of encryption and data transfer gave us a new TV format, which has many advantages:
- a change in the strength or length of the pulse does not affect its recognition by the decoder;
- uniform broadcast coverage;
- unlike analog broadcasting, reflections from obstacles of the transformed ether add up and improve reception;
- broadcast frequencies are used more efficiently;
- reception on an analogue TV is possible.
Differencedigital television from analog
The difference between analog and digital broadcasting is most easily noticed by presenting the final characteristics of both technologies in a table.
| Digital tv | Analog tv |
| The resolution of the digital image is 1280 × 720, which gives a total of 921600 pixels. In the case of the 1080i scan format, the image resolution is 1920 × 1080, which gives an impressive result: more than 2 million 70 thousand pixels. | The maximum resolution of the analog “picture” is approximately 720 × 480, which gives a total of more than 340,000 pixels. |
| Sound | |
| Audio, like video, is transmitted without distortion. Many programs are accompanied by a stereo surround signal. | Sound quality varies. |
| Receiver | |
| The cost of a TV adapted for digital reception is several times higher than the price of a conventional TV. | Analog TV has a reasonable price. |
| TV channels | |
| Watching digital channels gives the viewer an extensive choice: a large number and thematic focus of TV channels. | Number of programs up to 100. |
| Other | |
| Reception of programs on one TV. Additional services, such as “private broadcasting”, “virtual cinema”, “storage of programs”, etc. | The ability to connect more receivers and simultaneously view multiple programs. |
| Total | |
| The new television brings with it excellent picture and sound quality, the ability to create a multimedia home station for playing, working and learning. However, the high cost of adapted TVs and the slow introduction of TV encoding technology in the Russian market so far leave it behind the existing television. | Good old TV is inferior to digital in image and sound quality. Nevertheless, the price of receivers and the ability to distribute the signal to a larger number of TVs (the ability to watch several programs at the same time) is a significant plus. |
Antenna Sensitivity for TV
There is no universal recipe for choosing the perfect antenna, but there are mandatory requirements that must be met in order for it to receive analog and digital signals. With increasing distance from the broadcasting object, these requirements increase. In particular, to the sensitivity of the receiver - its ability to pick up television signals that are weak in intensity. Often, they become the cause of the fuzzy image. This problem is solved with the help of which significantly increases the sensitivity of the antenna and removes the question: how to connect it to digital television? The same TV, and the same antenna, only an on-air digital tuner will appear near the TV.
What is the antenna pattern
In addition to the sensitivity of the antenna, there is a parameter that determines the extent to which it is able to focus energy. It is called directional gain or directivity, and is the ratio of the radiation density in a given direction to the average radiation density. 
A graphical interpretation of this characteristic is an antenna pattern. At its core, it is a three-dimensional figure, but for the convenience of work it is expressed in two planes located perpendicular to each other. Having such a flat diagram at hand and comparing it with a map of the area, you can plan the area for receiving an analog video signal by an antenna. Also from this graph, a number of useful practical characteristics of the television antenna can be extracted, such as the intensity of the side and back radiation and the coefficient of protective action.
Which signal is better
It should be recognized that, despite the many improvements implemented in the field of analog presentation of information, this method of translation retained its shortcomings. Among them - distortion during transmission and noise during playback.
Also, the need to convert an analog signal to digital is caused by the unsuitability of the existing recording method for storing information in a semiconductor memory.
Unfortunately, the existing TV has practically no obvious advantages over digital, excluding the ability to receive a signal from a conventional TV antenna, and share it between TVs.
Lecture number 1
“Analog, discrete, and digital signals.”
The two most fundamental concepts in this course are signal and system concepts.
Under the signal This refers to a physical process (for example, a voltage changing over time) that displays some information or a message. Mathematically, a signal is described by a function of a certain type.
One-dimensional signals are described by a real or complex function defined on the interval of the real axis (usually the time axis). An example of a one-dimensional signal is an electric current in a microphone wire that carries information about the perceived sound.
Signal x (t ) is called bounded if there is a positive numberA such that for anyt.
Signal powerx (t ) is called the quantity
,(1.1)
If then they say that the signalx (t ) has limited energy. Energy limited signals have the property
If the signal has limited energy, then it is limited.
Signal strengthx (t ) is called the quantity
,(1.2)
If, then they say that the signalx (t ) has limited power. Signals with limited power can take non-zero values \u200b\u200bfor an arbitrarily long time.
In the real nature, signals with unlimited energy and power do not exist. Most signals existing in real nature are analogue.
Analog signals
are described by a continuous (or piecewise-continuous) function, moreover, the function itself and the argumentt can take any values \u200b\u200bat some intervals 
. In fig. 1.1 a presents an example of an analog signal that varies in time according to the law, where. Another example of an analog signal, shown in Figure 1.1b, changes in time according to the law.
An important example of an analog signal is the signal described by the so-called "Unit function"described by the expression
(1.3),
where 
.
The graph of the unit function is shown in Fig. 1.2.
 |
Function 1 (t ) can be considered as the limit of the family of continuous functions 1 (a, t ) when changing the parameter of this familya.
(1.4).
Chart Family 1 (a, t ) at various valuesa presented in Fig.1.3.
 |
In this case, function 1 (t ) can be written as
(1.5).
Denote the derivative of 1 (a, t) as d(a, t).
(1.6).
Chart Familyd(a, t ) is presented in Fig. 1.4.
 |
Area under the curved(a, t ) does not depend ona and always equal to 1. Indeed
(1.7).
Function
(1.8)
called dirac impulse function ord - function. Values d - the functionsequal to zero at all points exceptt \u003d 0. At t \u003d 0 d-function is equal to infinity, but so that the area under the curved- function is equal to 1. Figure 1.5 shows a graph of the functiond(t) and d(t - t).
 |
Note some propertiesd- features:
1. (1.9).
This follows from the fact that only at t \u003d t.
2. (1.10) .
In the integral, infinite limits can be replaced by finite ones, but so that the argument of the functiond(t - t) vanishes within these limits.
(1.11).
3. Conversion Laplaced-the functions
(1.12).
AT in particular, whent=0
(1.13).
4. Fourier Transformd- functions. For p \u003d j v from 1.13 we get
(1.14)
At t=0
(1.15),
those. spectrum d- function equals 1.
Analog signalf (t ) is called periodic if there is a real numberT such that f (t + T) \u003d f (t) for any t. Moreover, T called the period of the signal. An example of a periodic signal is the signal shown in Fig.1.2a, moreoverT \u003d 1 / f . Another example of a periodic signal is the sequenced- functions described by the equation
(1.16)
schedule which is presented in Fig. 1.6.
 |
Discrete signals differ from analog ones in that their values \u200b\u200bare known only at discrete time instants. Discrete signals are described by lattice functions - sequences -x d(nT), where T \u003d const - interval (period) of sampling,n \u003d 0,1,2, .... Function itselfx d(nT) at discrete moments can take arbitrary values \u200b\u200bon a certain interval. These function values \u200b\u200bare called function samples or samples. Another designation of the lattice functionx ( nT) is x (n) or x n. In fig. 1.7a and 1.7b show examples of lattice functions and. Sequencex (n ) can be finite or infinite, depending on the interval of definition of the function.
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The process of converting an analog signal to a discrete one is called time sampling. Mathematically, the process of time sampling can be described as modulation by the input analog signal of the sequenced- functions d T (t)
(1.17)
The process of recovering an analog signal from a discrete one is called temporary extrapolation.
For discrete sequences, the concepts of energy and power are also introduced. Energy sequencex (n ) is called the quantity
,(1.18)
Power sequencex (n ) is called the quantity
,(1.19)
For discrete sequences, the same patterns regarding power and energy limitations are preserved as for continuous signals.
Periodic called sequencex ( nT) satisfying the conditionx ( nT) \u003d x ( nT+ mNT), where m and N - whole numbers. WhereinN called the period of the sequence. It is sufficient to specify a periodic sequence on a period interval, for example, at.
Digital signals represent discrete signals, which at discrete moments of time can take only a finite series of discrete values \u200b\u200b- quantization levels. The process of converting a discrete signal to digital is called quantization by level.Digital signals are described by quantized lattice functions.x c(nT) Examples of digital signals are presented in fig. 1.8a and 1.8b.
The relationship between the lattice functionx d(nT) and the quantized lattice functionx c(nT) is determined by the nonlinear quantization functionx c(nT)= F k(x d(nT)). Each of the quantization levels is encoded by a number. Usually binary encoding is used for these purposes, so that quantized samplesx c(nT) are encoded with binary numbers withn discharges. Number of quantization levelsN and the smallest number of bitsm which can be used to encode all these levels are related by the relation
,(1.20)
where int(x ) Is the smallest integer not less thanx.
Thus, quantization of discrete signals consists in representing the signal samplex d(nT) using a binary number containingm discharges. As a result of quantization, the count is presented with an error called a quantization error
.(1.21)
Quantization Step Q determined by the weight of the least significant bit of the resulting number
.(1.22)
The main methods of quantization are truncation and rounding.
Truncation to m -bit binary number consists in discarding all the least significant bits of the number exceptn seniors. In this case, the truncation error
. For positive numbers using any coding method 
. For negative numbers, when using the direct code, the truncation error is non-negative, and when using the additional code, this error is non-positive. Thus, in all cases, the absolute value of the truncation error does not exceed the quantization step:
.(1.23)
The graph of the truncation function of the additional code is shown in Fig. 1.9, and the direct code is shown in Fig. 1.10.
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Rounding differs from truncation in that, in addition to dropping the least significant bits of the number,m - th (younger non-discarded) discharge of a number. Its modification consists in the fact that it either remains unchanged or increases by one, depending on whether the discarded part of the number of the quantity is more or less. Rounding can be practically done by adding a unit to (m +1) - the category of numbers with the subsequent truncation of the resulting number ton discharges. The rounding error for all encoding methods is within 
and therefore
.(1.24)
The graph of the rounding function is shown in Fig. 1.11.
Consideration and use of various signals implies the possibility of measuring the value of these signals at given points in time. Naturally, the question arises of the reliability (or, conversely, uncertainty) of measuring the value of signals. Deals with these issues information theory, the founder of which is C. Shannon. The basic idea of \u200b\u200binformation theory is that information can be handled in much the same way as physical quantities like mass and energy.
We usually characterize the accuracy of measurements with the numerical values \u200b\u200bobtained during the measurement or the estimated errors. In this case, the concepts of absolute and relative errors are used. If the measuring device has a measuring range fromx 1 to x 2 , with absolute error± Dindependent of the current valuex measured quantity then having received the measurement result in the form x n we record its likex n± Dand characterize the relative error.
The consideration of these same actions from the point of view of information theory is of a slightly different nature, characterized in that all these concepts are given a probabilistic, statistical meaning, and the result of the measurement is interpreted as a reduction in the region of uncertainty of the measured quantity. In information theory, the fact that a measuring device has a measuring range from x 1 to x 2 meansthat when using this device, readings can only be obtained fromx 1 to x 2 . In other words, the probability of obtaining samples smallerx 1 or large x 2 , is 0. The probability of obtaining readings somewhere in the range fromx 1 to x 2 is 1.
Assuming that all measurement results ranging from x 1 to x 2 are equally probable, i.e. the probability distribution density for different values \u200b\u200bof the measured value along the entire scale of the device is the same, from the point of view of information theory, our knowledge of the value of the measured value before the measurement can be represented by a graph of the probability density distribution p (x).
Since the full probability of getting a readout somewhere in the range fromx 1 to x 2 equal to 1, then an area equal to 1 should be enclosed under the curve, which means that
(1.25).
After the measurement, we get the reading of the device equal tox n. However, due to the error of the device, equal to± D, we cannot say that the measured value is exactly equalx n. Therefore, we write the result in the formx n± D. This means that the actual measured valuex lies somewhere betweenx n- Dbefore x n+ D. From the point of view of information theory, the result of our measurement consists only in the fact that the region of uncertainty was reduced to a value of 2D and characterized by much higher probability density
(1.26).
Obtaining any information about the value of interest to us is, therefore, to reduce the uncertainty of its value.
As a characteristic of the uncertainty of the value of some random variable, K. Shannon introduced the concept of entropy valuesx which is calculated as
(1.27).
The units of measurement of entropy depend on the choice of the base of the logarithm in the above expressions. When using decimal logarithms, entropy is measured in the so-called. decimal units or babies . In the case of using binary logarithms, the entropy is expressed in binary units or bits.
In most cases, the uncertainty of knowledge about the value of the signal is determined by the action of interference or noise. The disinformation effect of noise during signal transmission is determined by the entropy of noise as a random variable. If the noise in the probabilistic sense does not depend on the transmitted signal, then regardless of the signal statistics, a certain amount of entropy can be attributed to the noise, which characterizes its disinformation effect. Moreover, the analysis of the system can be carried out separately for noise and signal, which greatly simplifies the solution of this problem.
Shannon's theorem on the amount of information. If a signal with entropy is applied to the input of the information transmission channel H( x), and the noise in the channel has entropy H (D ) , then the amount of information at the output of the channel is defined as
(1.28).
If in addition to the main signal transmission channel there is an additional channel, then to correct errors arising from noise with entropyH ( D), through this channel it is necessary to transmit an additional amount of information, not less than
(1.29).
This data can be encoded so that it will be possible to correct all errors caused by noise, with the exception of an arbitrarily small fraction of these errors.
In our case, for a uniformly distributed random variable, the entropy is defined as
(1.30),
and the remaining or conditional entropy measurement result after receiving the countx n is equal to
(1.31).
Hence, the amount of information received equal to the difference between the original and the remaining entropy is
(1.32).
When analyzing systems with digital signals, quantization errors are considered as a stationary random process with a uniform probability distribution over the range of quantization error distribution. In fig. 1.12a, b, and c, the probability densities of quantization errors are given when rounding an additional code, direct code, and truncation, respectively.
Quantization is obviously a nonlinear operation. However, the analysis uses the linear signal quantization model shown in Fig. 1.13.
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(1.33)
