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. One day I came across a stack of old compact cassettes that had lain unnecessarily for many years. It was decided to get, old Sharp, forgotten by time, well preserved his appearance, wipe the rollers with heads and turn on the “memories”.

Yes, I hung for several hours, driving back and forth time-worn tapes. The flood of nostalgia inquiringly formed one trivial question in my inquisitive brain – do you have the Internet, where all the music is collected, and besides, someone opens its mouth wide for it, why do you need this old trash? 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 “Birch” and “Commission”. Sparkling on the shelves of Sonya, Panasonic and Sharpa, 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 area 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 they make a fool of us, “vaping” “tubes from gramophones” rubbed with a triple cologne, giving our trembling fibers nostalgic notes of bygone years. One way or another, I decided to find out how much the analogue recording is really “analogue”? Further, he 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:

(1)

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

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. tilt;

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 call it as (ω_max). At this frequency, the minimum signal amplitude corresponds to the Boolean function of a single jump:

(2)

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)

The amplitude function of the sinusoidal signal in the limit of the recording frequency range:

(4)

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

.

The output will be only one value:

(5)

We do not 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 (А_max), and only technical instability leads to fluctuations in the output amplitude. We use a noise component that extends to the limit of resolution. We choose the region of values of the peaks of the noise signal with a minimum of Δt. We explain that the noise signal is removed from the erased tape. We find the minimum possible amplitude as the minimum change in the level of the noise component. It is assumed that in the limit of the maximum frequency, the amplitude of the useful component decreases to the level (A_min). Find φ:

(6)

We get the maximum possible recording frequency of one 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 is written to 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 conclude, because the first term (q.8) is remarkable, it reflects the frequency difference at which an unstable bit of information appears. In this device, it is approximately 2.8 Hz. You need to add the fluctuation frequency 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 appear by ear as a floating sound, and the difference is> 5 Hz, like a broken sound. towards which we experience nostalgic moods and prejudiced expectations. In fact, the ear is incapable of intelligibility for nearby signal peaks due to the uncertainty of their appearance. Conclusion, substituting f. 6:

(7)

The maximum possible signal recording frequency at the A_min level:

(8)

(9)

From the graph of noise and the maximum amplitude of this “Sharpe” at t = 0.0005c we get the maximum frequency:

(10)

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

(11)

The maximum amplitude of the recording-playback path (315 Hz) 

 

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

(12)

Two in brackets is the range of amplitudes. And I intentionally do not take into account the noise component, since compressor silencers can expand the dynamic range, which depends on the circuitry of the devices, but this does not affect the “bit”, since the resolution is determined by the physical environment. By the way, why when using compressor systems there are special rainbow distortions, they stretch the signal, and these transitions become audible with an unstable mechanism and incorrect settings. Therefore, they needed accuracy in setting the levels and stability of the mechanisms. These systems, known as Dolby (A; B; C; S; X), were distinguished by the degree of compression-expansion, frequency range and levels. Accordingly, the higher the compression-expansion ratio of the signal, the more high-quality mechanism was necessary. To reduce detonation, complex belt mechanisms have been developed that provide only the stability of the movement of the mechanical carrier. A little tip. If you still decide to purchase a similar tape recorder, select the Dolby system. This system was not installed on poor mechanics, with rare exceptions. And so, back to our sheep. The moment of truth:

(14)

Conclusion: even with significant leniency the actual bit depth of a cassette recorder is about 10 bits with a sampling frequency of 36 kHz per channel. You can’t talk about any superior quality compared to modern digital recording! Hence the consequence. Magnetic recording is an outdated technology that is interesting to collectors and connoisseurs of “copper pipes”, and sound quality is a subjective assessment.

Thank you all for your attention.

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