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ANALOG-TO-DIGITAL CONVERTER FROM A HISTORICAL POINT OF VIEW
Adzhemov Artem Sergeevich,
MTUCI, Moscow, Russia Khromoy Boris Petrovich, MTUCI, Moscow, Russia, oiris@mtuci.ru
Keywords: analog signaling methods, signal-to-noise, musical acoustics.
The transition from analog signaling methods to digital is the revolution in communications technology. Analog-to-digital converter (ADC) is available in each mobile phone. This was possible as result in modern achievements of micro-circuitry. The logical question, when a person has mastered the processes implemented in the ADC and started to apply? In the beginning we list these processes. They are called discretization, quantization, encoding. These terms appeared recently. However, this is not about the terms and what they mean. First consider the parameters of modern ADC. The discretization is carried out according to the linear law and on a logarithmic quantization. This allows to improve the signal-to-noise. The logarithmic law of quantization is provided different ways and often used reception companding.
Consider now the formation of sound signals in science called "musical acoustics". Engineers in the case usually consider the music is not as science and as art. However, it is not. When building a musical work is used a certain number of fixed frequency sounds, constituting the so-called scale. The scale is a category invented by nature and not a man. There is a natural scale on the basis of which was created scale currently in, t. e. musical scale. The natural scale is formed by the vibration of the string. This string was originally bowstring very first bow, which when shot ranged uttering a sound set, the ratio of the frequencies which are used at present time. Known string hesitate forms standing waves. The signalman is well known the length of the standing wave should be a part of the length of the oscillating body.
Summing up it should be recognized the quantization and coding developed and improved on a large historical interval and only the terms of explaining these processes appeared relatively recently.
For citation:
Adzhemov A.S., Khromoy B.P. Analog-to-digital converter from a historical point of view. T-Comm. 2015. Vol 9. No.4, pp. 83-87.
The transition from analog signaling methods to digital is the revolution in communications technology. Analog-to-digital converter (ADC) is available in each mobile phone. This was possible as result in modern achievements of micro-circuitry. The logical question, when a person has mastered the processes implemented in the ADC and started to apply? In the beginning we list these processes. They are called discretization, quantization, encoding. These terms appeared recently. However, this is not about the terms and what they mean.
First consider the parameters of modern ADC. The discretization is carried out according to the linear law and on a logarithmic quantization. This allows to improve the signal-to-noise. The logarithmic law of quantization is provided different ways and often used reception companding.
When considering this question most often used normalized characteristics which are obtained by replacing the relative values inpute for size "x" and "y", that is:
x and
U„
While the conversation equation take the form:
lg I N
" ' lg !
(I)
There should be conditions -1 < x < +1 H -I <y <+l.
In accordance with the recommendation ITU-T G7II = 255. The practical implementation of the law of conversion is made by the method of approximation the expression I straight line segments. In accordance with the recommendation G7I I when implementing compression positive value of the input signal divided into 7 equal parts. For the same number of parts divided negative phase amplitude characteristics. However, the initial plots are approximated by straight line which is beginning of coordinates. There fore, the total number of linear segments not 14 and 13. The number of quantization steps and their distribution by segments for positive values of the signal in accordance with the recommendation G7I I presented in the table I.
Table I
.№ segments 1 2 3 4 5 6 7
(Number interval) X (quantity interval) 32x2 16x4 16x8 16x16 16x32 16x64 16x128
Total number interval at the end of segment. 64 128 256 512 1024 2048 4096
As can be seen table I, except first segment, the number of quantization steps in the segments 2-7 similar and equal 16. Speed increases the quantization step. So in segment I the quantization step is 2 conditional unit, and in segment 7-128 unit.
The total number of steps quantization steps doubled by the end of each segment and at the end of the last (seventh)
segment equal to 4096. The same number of quantization steps provided for negative values of the signal.
So, from a consideration of the quantization process quantization derives the following conclusions,
1. Quantization is performed on the logarithmic law.
2. The quantization characteristic of the quantizer is divided into segments.
3. Except the first division of each segment is 16 equal portions (intervals).
4. By increasing the number of segment per unit of quantization step double.
It is obvious that such conversion speech signal could appear only after the person string instruments (prototypes of the modern harp), but on keyboard instruments. It is known, that the prototype of the modern body was invented in the second century BC in Alexandria.
Consider now the formation of sound signals in science called "musical acoustics". Engineers in the case usually consider the music is not as science and as art. However, it is not. When building a musical work is used a certain number of fixed frequency sounds, constituting the so-called scale. The scale is a category invented by nature and not a man. There is a natural scale on the basis of which was created scale currently in, t. e. musical scale.
The natural scale is formed by the vibration of the string. This string was originally bowstring very first bow, which when shot ranged uttering a sound set, the ratio of the frequencies which are used at present time. Known string hesitate forms standing waves. The signalman is well known the length of the standing wave should be a part of the length of the oscillating body. Thus the string oscillates not only as a whole, but parts forming harmonic components of a certain set of frequencies. A string forms a first harmonic half string second harmonic the third part of the third harmonic, and so on. Much less vibrating part of the string in many times higher than the oscillation frequency. The natural scale for strings oscillating with a frequency of 24 Hz presented in table 2.
Table 2
1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 i/10 l/ll 1/12 1/13 1/14 1/15 1/16
24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 384
H* Hz Hz Hz Hz Hz Hz Hz Hz Hz Hz Hz Hz Hz Hz Hz
The top row of the table specifies the fraction of the total length vibrating strings and the lower frequency corresponding harmonics. In the bottom row of the selected frequency which borders octave. It should be noted that the same ratio obtained if ranges are not a solid body but also the air in a certain volume.
The table shows that the natural scale is logarithmic structure. !f the first harmonic has a frequency of 24 Hz, the second 48 Hz. The first and second harmonic octave form. Known octave has a cutoff frequency which differs in two times. The second octave is formed by frequency 48 Hz and 96 Hz. Between the boundary values appeared intermediate
frequency 72 Hz. Third octave (frequency of 96-192 Hz) already contains three intermediate frequency, and the fourth seven frequencies.
Given the natural scale of the seemingly many stringed instrument needs to be set in accordance with table 2 correlations. Indeed many-stringed instrument the contractor can extract the sounds with multiple strings. The combination of sounds in a certain ratio may cause dissonance. On the other hand adding strings done to increase sound palette.
Note that according to table 2 frequency intervals between adjacent sounds 24 Hz are large enough, it Is known that at low frequencies man distinguishes between two tones differing by I Hz. in the high frequency range of 5 Hz. In the range in which you can customize musical instruments {16-4700 Hz) people can distinguish about 1500 different sounds!
Communicator can be surprised by the range of 16- 4700 Hz. It is known that audio broadcasting I St class the upper frequency range is set almost equal to 15000 Hz. Why if the instruments are configured in a much more narrow range? The point is that if the string is tuned to the frequency 4700 Hz, that radiated its third harmonic is 4700 x 3 = = 14100 Hz. This frequency is no need to configure anything, it will be created automatically, and send it should because it sounds in the hall and the listener perceives her.
So in the range of 16 4700 Hz person is able to distinguish about 1500 different sounds. But how many sounds should be used in the construction of musical instruments? It is natural that the number of configurable elements of a musical instrument should be less. Example to serve the violin has four strings. However, it cannot be taken as a basis. The fact that smoothly changing the position of the finger, holding the string to the fingerboard violinist as smoothly changes the frequency of the sound.
In other words, it configures the instrument to a specific frequency of sound during the execution of works. Otherwise performs its task performer on the harp or piano. He can't adapt and change during execution. The tool is configured before the concert and allows you to play a certain set of fixed frequencies. A similar situation among the performers on wind instruments.
What is the number of fixed frequency sounds used in practice? You can get that calculates the number of white and black keys at the piano. There are only 88 pieces, and piano covers almost the entire frequency range! But the violin? Although the violinist can remove about 400 frequency in practice it uses only about 50 even speaking solo and performing complex work. To depart from this rule, he can not playing in ensemble with other instruments.
So the number of frequency selected about 88. Note what happened the quantization process frequency. Same in practice is the choice discrete frequencies? To do this, contact history. Musicians and they were already professionals in ancient Egypt, empirically realized the necessity of changing the natural order. The fact that developing your technique games they have started to use chords. Known chord called simultaneous use of at least three sounds. Practice has shown that pleasant sounding chord it turns out if the fre-
quencies are related as 4: 5: 6. In the natural scale as it is obvious from table 2, there is only one such combination is 192:240:288. And musicians wanted what would this combination can be obtained from any frequency scale. This natural scale was upgraded. Modernization came to change the numerical values of the two frequencies and the destruction of one frequency.
To solve this problem it was decided to introduce additional sounds whose frequencies are divided least intervals of whole tones in half. For this purpose we have introduced five additional frequencies. Recall that at the piano each octave contains five black key that reflect this idea.
Moving forward with the introduction of additional sounds was made a long time ago long before the piano. How much? Almost 2500 years? What? Pythagoras. Typically, the name of Pythagoras connect with math. However, few people know that Pythagoras was a great musician.
Pythagoras experimented with using a device which was called the monochord figure i. Monos - one, chorde -string. The string is firmly attached to the box using still stands. One stand was rolling.
To determine the frequency of additional sounds reproduced played a musical instrument Pythagoras suggested a number of presents in the first row of table 2
Figure I
Table 3
(2/3)s (2/3)* (2/3)5 (2/3)' 2/3 1 3/2 (3/2)' (3/2)! (3/2)' (3/2)' (3/2)'
46,4 69.6 103.4 155.13 232,7 349 523.5 785 1176 1764 2646 3969
Hi Hi H z Hz H z Hi Hz Hz Hi Hi Hi Hi
sol flat re la mi si fa do Sol re la mi si
counter flat flat flat fiat 1 2 2 3 3 4 4
octave most most small small octive octave octave octave octave octave octave
octave octave octave octave
You can see in the middle of the first is the number I. It corresponds to a particular source frequency sound which can be chosen arbitrarily. Digit located on the right and left from the original I show how many times should increase {or decrease) the source frequency to obtain sound series.
As to the right are fraction 3/2 large I, built in varying degrees of frequency located on the right form a monotone sequence. On the contrary to the left of I are fraction 2/3, constructed in the same degree. Since 2/3 less I, the sequence on the left hand side is a decreasing.
The result of the calculation in the example shown in the second row of table 3. Per unit adopted frequency 349 Hz. The result is a sequence of discrete frequencies from 46,4 to 2646 Hz.
The source frequency 349 Hz was chosen not by chance. Engineers signalers are accustomed to the fact that many frequency settings standardized. In musical acoustics are the
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same standard, in the size at 88 frequency at which tune the strings of the piano. For example, in this standard for piano mode L-120 frequency 349 Hz corresponds to the note "fa" of the first octave. She indicated under I the second line Table 3. Note that strings are adjusted with high precision to tenths of Hz. The following frequency 523,5 calculated using the Pythagoras number.
Refer again to the above standard L-120. Do you have this in it, this figure? Yes there is it is equal to 523,25 Hz. According to the standard of this figure corresponds note "do", but already the second octave. Later in the standard you can find all frequencies calculated on a range of Pythagoras and presented in the second row Table 3, Frequency I 176 Hz according to the standard corresponds to the note "re" third octave, frequency 1764 Hz corresponds to the note "la" - third octave frequency 2646 Hz corresponds note "mi" fourth octave frequency 3969 Hz note "si".
Thus the number of Pythagorean allowed us to calculate standard frequency reproducible 7 white the keys of a piano "do", "re", "mi", "fa", "sol", "!a", "si". However belong to these frequencies different octaves, but it is not significant. Recall that during the time of Pythagoras there was no frequency and units Hz, because Hertz was born much later. His number Pythagor invented to tune musical instruments experienced by based on the length of the string.
The main difference musicians from the usua! communications engineers responsible for configuring frequency generator with electronic frequency, is the presence of a developed ear for music allows configuring one string is the presence of advanced music such as the note "fa" I octave, is to tune the second string to double the frequency that will be the note "fa" 2 octave. In other words, in modern parlance Pythagoras in addition to the mathematical study of the scale was invented the algorithm allows to configure musical instrument are not lost value to our time because in addition to the frequency tuning of the piano strings accepted by the white keys, allows you to customize the way the strings accepted the five black keys on the frequency of which is given in the second row of the table 3 to the left of the main frequency 349 Hz. It notes "sol flat", "re, mi, la, si flat". The term flat is inserted to that would not give for the black keys individual names.
Key sol flat is between adjacent keys reproducing "fa" and "sol", and a few left 'sol". If the source key to take "fa", the same black key to the right and will be called "fa sharp". Thus Pythagoras took care of setting up all the piano strings the piano was invented after approximately 1300 years.
Summarizing it should be noted that in musical acoustics is used quantization frequency it is based on dividing the total range on the octave, the position of the beginning and end of the octave in the frequency range obeys a logarithmic law with base 2, as the end frequency of each octave exceeds the initial frequency twice. Octave in evenly tempered the system is divided into 12 equal parts. The number of octaves used in music frequency range - 9, at the piano full octaves - 9.
Now compare the main characteristics of the quantization process in modern ADC and musical acoustics.
I. Quantization in both cases is on a logarithmic base 2.
2. The quantization characteristic of the quantizer or musical structure is divided into segments (or octave).
3. In ADC (except the first) the division of each segment is carried out on 16 equal (to the absolute value) of the parts (of the intervals). In musical acoustics into 12 equal intervals.
4. In electric ADC with increasing numbers segment per unit, the sizes of the intervals are doubled. In tempered musical system increase the octave number leads to an increase in the frequency the interval between adjacent notes in the same way twice.
To compare the quantization process in ADC and in musical acoustics should be noted and the difference. In the ADC quanthouse signal levels and in musical acoustics - frequency. However, this does not matter because we are talking about the comparison of processes and not devices. However, it should be noted objectives with which is the quantization process according to the logarithmic law. In the first case, the ADC is improved attitude the signal noise in the second musical acoustics is achieved harmony of sounds, as it is possible to avoid dissonances that in the performance of the work accomplished harmony sounds. Who in the performance of this coincidence is not accidental. The latest research in neuroscience (Norman Weinberger. University of California) showed the relationship of the considered phenomena with the processing of audio information with the processes in auditory apparatus person and in the brain.
Because in ADC except quantization processes are car* ried the processes of sampling and coding will consider how these processes are performed in music. A piece of music is performed within a certain the time interval. Composer composing music does not install the time of execution of works and sets the tempo. The rate is usually set using a metronome which was invented by the Vienna mechanic Maelzel (Figure 2).
The metronome the instrument is equipped with a clockwork mechanism, which accurately counts the duration of the intervals. Who exactly he looks tike a wooden pyramid which removed one panel. Bottom-anchored pendulum with a sliding weight and the pyramid dial with numerals. If you move the balance weight relative to the pendulum, in accordance with the fact about what number scale to install the pendulum swings faster or slower and clicks notes right bar beats. The notes indicate the tempo in the form of inscriptions M.M. =108. This means that marks the 108 quarter notes in minute. The metronome is a kind of historical analog master oscillator in modern ADC.
Discretization is that the whole time allowed for execution of works is divided into equal intervals -quanta. In this case, the metronome is similar to the clock generator cycles, posted in ADC and specify the frequency of sampling.
At this stage, there is virtually no difference in the process the sample rate of the ADC In music. In both cases, the discretization is performed on uniform law.
But this is only the first stage. The fact that in music the beat divided into shares. The number of beats per measure, can to be different. So for example if you divide the time spent on one into four parts that the share will be j its duration and
т
will be called "one-quarter". If we divide the quarter two will get "one-eighth" and so next. Further, the division will give "one sixteenth", one "thirty-second" and so next.
Three pieces
Mazurka
M. Glinka (1804 - 1857)
Figure 2
Besides composing music composer can in addition to complete cycle using three quarter share (waltz), two quarter share {march) and so next. Thus two adjacent cycles the number of notes may be various, however, using various percentages of the composer must choose the number to get the total playing time of each beat was the same. Thus, in the applied music primary sampling on uniform law and organized in a certain way and somehow organized secondary the discretization and it should be recognized that the sampling in this case is much harder than modern electronic ADC.
Harder than finally you should contact the encoding process encoding. Of course during the times of ancient Greece did not exist Hamming code and hardly used the term "code". However, musical works sought to preserve and for this it was necessary to record. It is natural that the sequence of sounds volume the duration of the tempo is very difficult to describe with words, therefore, for the preservation of works used code, which is now called "notes". Of course it is possible to doubt that the notes are code. However, refer to figure 3 which shows a fragment of music notes from the works of M. Glinka. It is obvious that the person did not receive music education the meaning of the signs in figure 3 is not clear. This is natural because each character is an element of the code. But if this is the code for decryption, you must have the key! There are even two keys are presented in figure 4.
Figure 3
Figure 4
The upper sign is called the treble clef, and the lower the bass clef. These notes and keys are included for the execution of works for piano. For other instruments the music used other keys.
Completing a comparative overview of the processes performed in modern ADC and in music it should be stated they inherently have a lot in common although developed in different historical epochs. The common characteristic for the processes due to the fact that in both cases the receiver of information is the human hearing apparatus with its characteristic logarithmic perception the volume and frequency. This has influenced the processes of quantization.
It should be noted and differences, when performing quantization of the signal in the electronic ADC can be used by all levels formed in the chip. When composing a musical work, the composer cannot use all the notes of the octave. Of the 12 notes he can only use 7 notes selected by a predetermined algorithm, called tonality. This is due to the peculiarity of the perception of melody the human ear. The alternation of sounds is perceived as a melody if they are not more than seven. Therefore, for compositions in any tonality a composer can use from 88 key piano about 50, The number of combinations used tones 24, that corresponds to the number of tonality: 12 major and 12 minor.
The second difference is the discretized. In ADC it once and runs on uniform law. The music on uniform sampling clock superimposed secondary sampling when the beat of a fixed duration split into multiple dayperiod. Thus, these processes the music is also much more difficult that is quite natural because they have evolved for a long period of time especially when you consider according to preserved ancient drawings mini orchestras already existed in the 5th century up to N. E.
Summing up it should be recognized the quantization quantization and coding developed and improved on a large historical interval and only the terms of explaining these processes appeared relatively recently.
References
J, Recommendation ITU-TG.7I L
2. Ghazaryan S. In the world of musical instruments. Moscow. Education, 1989.
ИСТОРИЧЕСКИЕ АСПЕКТЫ АНАЛОГО-ЦИФРОВОГО ПРЕОБРАЗОВАНИЯ
Аджемов А.С., Хромой Б.П., ФГОБУ ВПО МТУСИ, Россия
Переход от аналоговых методов передачи сигналов к цифровым методам совершенно справедливо считается революцией в технике связи. Микросхема, осуществляющая преобразование аналогового звукового сигнала в цифровой (АЦП), имеется в каждом мобильном телефоне. Нет сомнения в том, что это стало возможным лишь в результате современного развития микросхемотехники. Если отвлечься от устройства, осуществляющего нужное преобразование сигналов, и обратиться к самим процессам, то вполне закономерен вопрос: "А когда человек освоил эти процессы и начал применять их на практике?". Сделаны выводы, что квантование, дискретизация и кодирование, развивались и совершенствовались на большом историческом интервале, и только термины, поясняющие эти процессы, появились сравнительно недавно.
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