D / A converters
Let’s move on to DAC: digital to analog converters. This complex subject is always covered with a veil of secrecy and peppered with audiophile mysticism.
Additionally, there is a lot of speculation from opposing camps around digital-to-analog converters: marketers, audiophiles, and skeptics. Let’s find out what the problem is.
Multibit DAC
In the beginning when the audio CD format first appeared, PCM was converted to an analog signal using multi-bit DACs. They were built on the basis of a resistive matrix with constant impedance, the so-called R-2R matrix.
Simplified multi-bit DAC circuit
Simplified multi-bit DAC circuit
Multi-bit DACs work like this: the PCM stream is split into two channels, left and right, and converted from serial to parallel, for example by shift registers. Data from the right channel is written to the buffer of one register and data from the left channel is written to the buffer of the other. Data is transmitted simultaneously through parallel ports with a certain sample rate (most often 44.1 kHz), as in the picture below, only the parallel outputs are not eight, but sixteen, because the bit width it is 16 bit. Depending on the position in the frame, the high and low bits will encounter different resistances along the path of the electric current, since the number of resistors connected in series will be different. The older the bit, the greater its importance.
Multi-bit, or multi-bit, DACs require very high-quality components and precise resistance adjustment, because any inaccuracies in component ratings add up. This leads to serious deviations from the original waveform and creates a multi-digit error in quantization.
There is no PCM manipulation in multi-bit DACs from the eighties. The multibits are connected directly to the I2S bus and reproduce PCM as is: the data from the right channel (16 bits) arrived, they waited for the data from the second channel (16 bits), they sent both channels to the resistive matrix, and so on with a 44.1 kHz frequency.
In the eighties, the frequency and the bit depth were determined by the CDDA format, which became almost a reference implementation of Kotelnikov’s theorem. With some reservations, this is how the later MP3 can be characterized. Only from the DVD Audio format has the approach to digitizing and sound reproduction been revised.
This is how the first simpler DACs worked, then they began to use converters with a more complex device. Circuitry was modernized, component quality was improved, and multi-bit DAC oversampling technology was also used. Oversampling is the oversampling of a digital stream with upsampling and quantization bit depth to reduce quantization noise.
To explain why oversampling is used, it is necessary to talk about the application of Kotelnikov’s theorem in practice. Not everything here is as optimistic as it seems in the world of mathematics; it is not about anything “precisely”, as it is written in the theorem.
Kotelnikov’s theorem
“Any function F (t), consisting of frequencies from 0 to 1, can be transmitted continuously with any precision using numbers that occur in 1 / (2f 1) seconds”
Consequences of Kotelnikov’s theorem:
Any analog signal can be reconstructed with any precision from its discrete samples taken with a frequency f> 2fc, where fc is the maximum frequency that is limited by the spectrum of the real signal;
If the maximum frequency in the signal is equal to or greater than half the sampling frequency (aliasing), then there is no way to recover the signal from discrete to analog without distortion.
If you are interested in the details, you can consult the main source – the work “On the bandwidth of” ether “and cable in telecommunications” by V. A. Kotelnikov (PDF).
Difficulties with Kotelnikov’s theorem
Kotelnikov’s theorem is often taken too literally and elevated to the absolute. How many articles by staunch skeptics I have read about the wonderful MP3 and CDDA formats and the crazy audiophiles who sell their unnecessary DVD-Audio and DSD to everyone! Of course, your main argument is Kotelnikov’s theorem.
To begin with, the Nyquist frequency, in practice, is not sufficient to transmit an accurate waveform. Due to imperfect conditions, noises and distortions inevitably appear: quantization noise when recording an audio signal, rounding noise during processing and playback, and more.
