Bit depth of the old tape recorder

The history of the creation of this work leaves the sites of the modern Internet in nostalgia for magnetic recording technologies of the end of the last century. Somehow I stumbled upon a stack of old compact cassettes that had lain unnecessarily for many years. It was decided to get old Sharp, forgotten by time, who had kept his appearance well, wipe the rollers with heads and turn on the “memories”.

Yes, I hung for several hours, driving back and forth time-worn tapes. The stream of nostalgia questioningly formed one trivial question in my inquisitive brain: do you have the Internet, where all the music is collected, why do you need this old gramophone? Well … yes, I had to delve into the global network, it turns out, entire museums of tape recorders in people. Huge collections worthy of the former electronics stores. Sparkling on the shelves of Sony, Panasonic and Sharp, they again make sounds of bass songs, filling the space with nostalgia for the level of the spring of the 80s.  It is likely that this field of activity is not so useless, I thought, in addition, some rare models can be confidently elevated to the rank of a work of art, a sort of peak of design thought of the past. Laudatory odes for analogue recording can be seen in various posts and on forums dedicated to tape recorders. Even the number of sales has increased. However, I had a persistent feeling that this was just an echo of bygone years, and we were fooled, giving our quivering fibers nostalgic notes of bygone years. One way or another, I decided to find out how much the analogue recording is really “analogue”? Further, my sat down at the «desk» in the evening and that’s what happened. A bit of my extreme physics.

We represent the amplitude as an arbitrary function in time:

On an arbitrary signal, cut off the portion ∆t, which has the dimension of time. We will not delve into the individuality of it, as this is any part of the signal.

arbitrary function

We consider the ∆t region in more detail. If ∆t is so small that the segment cut off by the function can be taken as linear, then the behavior of the function (form 1) can exist only in two forms:

  1. incline;

  2. horizontal section.

The angle of inclination cannot exceed a certain constant, because the frequency range of the recording-playback path is limited from above by the technical capabilities of the apparatus and tape. The limitation of the frequency range from below makes it impossible for the long-term existence of section (2). In fact, the slope of the characteristic is a derivative of the function at the point (A, t) of our graph. The maximum possible slope of the characteristic is determined by the maximum possible value of the frequency of the recording-playback path, we will call it as (ω_max). At a given frequency, the minimum signal amplitude corresponds to the Boolean function of a single jump:

The amplitude modulus indicates the tolerance of the positive and negative half-waves.

Then, we can represent the maximum amplitude as the sum of the minimum jumps:

(3)

Amplitude function of a sinusoidal signal in the limit of the recording frequency range:

(4)

This formula is very remarkable, no matter what amplitude we try to write:

the output will be only one value:

(5)

We don’t know what the true frequency and dynamic range of the tape recorder is, but we are sure that the amplitude at the points (Kπ / 2, where K = 1;3 …) at φ = 0 is (A_max), and only instability of a technical nature leads to fluctuations in the output amplitude. We use the noise component, which extends up to the resolution limit. We will choose the region of the peak values of the noise signal with a minimum ∆t. Let’s explain, remove the noise signal from the erased tape. We find the minimum possible amplitude as the minimum change in the level of the noise component. We assume that in the limit of the maximum frequency, the amplitude of the useful component decreases to the level (A_min). Find φ:

(6)

We derive the maximum possible recording frequency of a single signal from frequency fluctuations. Let us explain this hooliganism, (ω_limit) is the frequency of the maximum possible signal, a signal with an amplitude of one jump, at which (А_min) = 0, that is, at this frequency the signal recorded on the tape, whatever it is, starts to disappear. (ω_max) is the frequency at which (A_min) = 1 (the logical appearance of the minimum possible signal, in accordance with (f.2). Since the amplitude of the minimum signal (A_min) is not stable, the frequencies (ω_max) ≅ (ω_lim) But we can deduce, because the first term (f.8) is remarkable, it reflects the frequency difference at which an unstable bit of information appears. In this device it is equal to about 2.8 Hz. You need to add frequency fluctuation to the detonation of the tape , although there is a signal-to-noise parameter, but it can be technically reduced. Typical distortions from the detonation of the mechanism in the region from 2 to 5 Hz are manifested by ear as a floating sound, and the difference is > 5 Hz, like a torn sound. to which we experience nostalgic moods and prejudiced expectations. In fact, the ear fails to intelligibility for nearby peaks of the signal due to the uncertainty of their appearance. Conclusion, substituting f.6 subtracting all angles:

(7)

(8)

The maximum possible signal recording frequency, at A_min level:

(9)

noise tape

From the graph of noise and the maximum amplitude given by Sharp, at t = 0.0005c, we obtain the maximum frequency:

(10)

It should be noted that we are not tied to the standards, but are looking for a practical result, on the basis of which we describe the digital conversion, including at the minimum signal levels, where the recording frequency is the highest possible. We also note that we operate with amplitude in arbitrary units. The volume level does not change during measurements, and the resistance is the load. The sampling frequency of one channel, so that the higher harmonics do not affect, must be higher than the second harmonic of the maximum possible frequency, and the sound path is provided with a low-pass filter in the region of the first ω_max:

(11)

maximum amplitude

The dynamic range is limited from above by the tape overload capacity, but usually recording amplifiers have limiters or knobs that prevent tape overloading. Define the «bit» of the analog path, as:

(12)

The two in brackets is the amplitude range. And I deliberately do not take into account the noise component, since compader noise suppressors can expand the dynamic range, which depends on the circuitry of the devices, but this does not affect the “bitness”, since the resolution is determined by the physical medium. By the way, why there are specific iridescent distortions when using compader systems, they stretch the signal, and these transitions become audible with an unstable mechanism and incorrectly configured parameters. Therefore, they required the accuracy of setting levels and the stability of the mechanisms. These systems, the famous Dolby (A; B; C; S; X), differed in the degree of compression-expansion, the frequency range and levels. Accordingly, the higher the compression-expansion ratio of the signal, the more high-quality mechanism was necessary. To reduce detonation, sophisticated tape drives with closed paths were produced to only ensure the stability of the movement of the mechanical carrier. A little tip. If you still decide to purchase a similar tape recorder, then choose with the Dolby system. This system was not installed on poor mechanics, with rare exceptions. And so, back to our sheep. The moment of truth:

(13)

Conclusion: Even with significant indulgences, the real «bitness» of a cassette recorder is about 10 bits, with a sampling frequency of 36 kHz per channel. About any superior quality, in comparison with the modern «digital», there can be no question! Hence the consequence. Magnetic recording is a technology of the last century is more interesting to collectors and connoisseurs of «copper pipes» and sound quality is a subjective assessment.

Thank you for your attention, I apologize for my English, this resource will be much more interesting.

©Alexey Tarasov