Bit depth of the old tape recorder

Sharp WQ-T238


The history of this work lasts from open spaces of the modern Internet in nostalgia according to the magnetic record of the last century. Came across somehow at himself in a cushion a pile of the old cartridges which lay without need it is a lot of years. It was decided to get the old Sharp who did not bad to maintain the shape, to wipe heads with cologne and to include “memoirs”. Yes, I hung up at several o’clock, driving to and fro old records. Nostalgia created a rare question in we wash an inquisitive brain – you have an Internet where all music is collected, and under it someone opens a mouth for what to you this old material? Well … yes, in global network it is possible to see big sets of tape recorders. Well … yes, got into a wide area network, it appears, the whole museums of tape recorders at people. Huge collections, worthy the former audio of shops are collected. Sparkling on shelves of Sony’s, Panasonic’s and Sharp’s filled with deep-voices tunes space around “collector”. Likely and this field of activity is not so useless, I thought, besides rare models can be brought with confidence to the level of the work of art, such top of a design thought of the past. Laudatory odes of “analog recording” can be seen in posts and at the forums devoted to tape recorders. Thereby, the market of tape recorders continues to live. In the past I took part in developments of the similar equipment in the twilight of the magnetophonobuilding, and the circuitry of these products is not bad familiar to me. But there is a strong feeling that it is just nostalgia, and we are made a fool, pushing the pipes of gramophone rubbed to gloss and lulling memories of the light passed. Anyway, I decided to find out how analog recording really is analog? To whom formulas and details of technical character are not interesting, wind in the end of work at once – on an output.



Let’s provide amplitude as any function on time, then function of amplitude:

(1)   \begin{equation*} A=f(x) \end{equation*}

Let’s consider any changing signal (Fig. 1), and we will allocate from it the site of time of ∆t which has the dimension of time of quantization of the digital signal, we will not penetrate now into the identity of this site since it can be any quantum of the signal. Let’s consider this site of ∆t in more detail. If ∆t is so a little that the piece cut on the function can be taken for linear, then the behavior of function (1) can take only three forms:

Figure 1. Schedule of amplitudes

    1. increase;
    2. decrease;
    3. horizontal site.

The relation of tendencies of features of two neighboring places of function cannot exceed some constant because frequency range is limited from above to technical capabilities. Restriction of frequency range from below gives us also impossibility of long existence of the place (3). In effect tendency of feature – function derivative in point (A, t) our schedule. The biggest tendency of feature is defined by the greatest value of frequency of way of record reproduction, we will call it \omega_ {max}. For this path of record reproduction the minimum amplitude of signal corresponds to Boolean function of single jump of signal:

(2)   \begin{equation*} \begin{matrix} |A_{min}|& = & \left\{ \begin{matrix} 1, & \mbox{if } A = A_{min} \\ 0, & \mbox{if } A \le A_{min} } \end{matrix} \right. \end{matrix} \end{equation*}

The module of amplitude indicates tolerance of positive and negative half waves. Then, the maximum amplitude we can present the sums of the minimum jumps in the form:

(3)   \begin{equation*} |A_{max}|=n|A_{min}| \end{equation*}

Values of shift of the phase of (\varphi) the pica of the signal of {(A} _ {max}) in the point (π/2), two close frequencies, differ with emergence of the single signal (bit) at smaller of two close frequencies. And as the written-down signal amplitude {(A} _ {max}) same on the condition, function of single amplitude of the written-down sine signal for higher frequency, lags behind on corner (\varphi). Amplitude of single jump of the signal:

(4)   \begin{equation*} A_{(1)}=A_{max}\sin{\left(\omega_1t\right)=A_{max}\ sin(\omega_{lim}t)=\ logical(1)}. \end{equation*}

This formula is very remarkable what amplitude we would not try to write down at the limiting frequency, at the exit we will receive only one value equal to the minimum bit of the signal as limiting frequency is initially accepted to minimum possible signal amplitude at the exit. Amplitude with the maximum frequency which is higher than limit does not form record of the signal on the tape and is respectively taken for logical zero:

(5)   \begin{equation*} A_{(0)}=A_{max}\ sin(\omega_0t)=A_{max}\ sin(\omega_{max}t)=0\ =logical(0) \end{equation*}

Let’s define the delta of frequencies as the module of the difference maximum at which amplitude of record of the signal does not differ from noise of the sound path and the tape and limit, showing stable bit:

(6)   \begin{equation*} \mathrm{\Delta\omega}=|\omega_0-\omega_1| \end{equation*}

We determine amplitude of the bit step:

(7)   \begin{equation*} A_{(1)}=A_{max}\ sin(\Delta\omega\ t) \end{equation*}

(8)   \begin{equation*} A_{(1)}=A_{max} sin (\omega_0t-\omega_1t) \end{equation*}

We find the maximum frequency:

(9)   \begin{equation*} \omega_0t=\omega_1t+asin\frac{\left|A_{min}\right|}{\left|A_{max}\right|} \end{equation*}

(10)   \begin{equation*} f_{max\ }=\frac{\omega_1}{2\pi}+\frac{asin\frac{\left|A_{min}\right|}{\left|A_{max}\right|}}{2\pi t} \end{equation*}

From the schedule of noise and the maximum amplitude of this Sharp we receive the maximum frequency of the path of record reproduction:

(11)   \begin{equation*} f_{max\ } =   \frac{9}{0,0005} + \frac{arcsin{(0,002/0,9)}}{0,001\pi }\approx 18kHz \end{equation*}

Figure 2. Schedule of noise component



It should be noted that we do not become attached to standards, and look for practical result on the basis of which describe digital conversion including at the minimum levels of the signal where the frequency of record is greatest possible. By the way, to whom it is interesting, the augend of the formula (10) represents fluctuation of the frequency of one bit of the analog signal. Also, we will note that we operate with amplitude in conventional units (c.u). The loudness level does not change at measurements, and as loading serves pure nominal resistance. Sampling rate of one channel of this Sharp, so that the upper harmonics did not influence, has to be higher, at least twice greatest possible, and the sound path is provided with the low pass filter which is present at circuitry of this tape recorder:

(12)   \begin{equation*} f_{sampling} = 2 f_{lim} = 2 \cdot 18kHz = 36kHz \end{equation*}


Figure 3. Maximum amplitude of record reproduction


Dynamic range is bounded above overload capacity of the tape. Write amplifiers are supplied with limiters or regulators which prevent the tape overload. Determine the “bit” of the analog path:

(13)   \begin{equation*} N_{bit} =log_2 \left (  \frac{2|A_{max} - A_{min}|}{|A_{min}|} \right ) \approx log_2 \left ( 2 \frac{|A_{max}|}{|A_{min}|} \right ) \end{equation*}

The two is the scope of amplitudes. Compander squelch can expand the dynamic range, but not the bit depth of the system. It depends on circuitry of devices. Only the amplitude of the A_{min}. bit step is increased. By the way, characteristic distortions of systems of noise reduction are the consequence of increase in amplitude of the bit step. And so a moment of truth:

(14)   \begin{equation*} N_{bit} =log_2 \left ( 2 \frac{0,9}{0,002} \right ) = 10 bit! \end{equation*}

Output: Even with considerable indulgences, the real bit depth of the cassette tape recorder makes 10 bits, with a sampling rate of 36 kHz. About any exceeding quality, in comparison with digit 16 – 24 bits, out of the question! The magnetic record – the technology which left is more interesting to collectors of “brilliant devices”, and the tonal quality is subjective emotional appraisal. It is possible to draw a conclusion on inability of a tape to transfer signals with the bit depth exceeding signal-to-noise ratio level also. Fluctuation of limiting frequency of record of a signal leads to a failure of legibility of a phonogram – to deterioration in its transparency in

comparison with the digital medium. Trial records of a reference phonogram in high resolution on the cartridge and in worsened (10 bits 36 kHz) a digital format were

made for confidence. When listening the amazing proximity of quality of sounding of these phonograms was observed.

Thank you all for your attention, visit our portal, there will be many more interesting things.

Copyright © Aleksei Tarasov (Bit depth of the old tape recorder) 2019

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