Tag: what should my sample rate be

Why upsampling? Part 2

For every doubling of the sampling frequency, the spectral density of the noise is reduced by half and the signal-to-noise ratio increases by 3 dB. Since the resolution limit for the pressure level is approximately 1 dB, these decibels are unlikely to have a noticeable effect on sound perception in the high-frequency region. Based on these numbers, it is absolutely impossible to draw tentative conclusions about the change in sound quality.

In order to relate the spectrum of quantization errors, sampling frequency and sound quality, in this article it is proposed to use a tonal signal as a music model, as is usual to evaluate the quality of sound paths. This approach relies heavily on materials published in the “Sound Engineer” magazine.

The results can be summarized as follows. Unlike analog audio, digital audio is the product of amplitude modulation. This is manifested in a rigid functional dependence of the quantization error spectrum of the frequency multiplicity factor of the audio signal F and the sampling frequency fs, represented as the ratio of prime numbers y and x (k = fs / F = y / x). The frequency spectrum of quantization errors is always discrete and is determined solely by the multiplicity factor; the components of this spectrum are also determined solely by the amplitude of the audio signal, expressed in quanta. This means that the mechanism for shaping the quantization error spectrum does not depend on the number of bits used. With an increase in the quantization bit depth, the spectrum does not change in shape and composition, but only changes in level by 6 dB with each additional digit. (There are situations where a change in bit depth leads to a change in spectrum, – Ed.) The auditory perception of the quantization error spectrum is largely determined by the frequency response of hearing, which, in turn, it depends largely on the sound pressure level.

The frequencies of digital sound are divided into multiples when x = 1 and submultiples when x> 1. At multiple frequencies, the spectrum of quantization errors is harmonic and the main pitch is the frequency of the audio signal. If y is an even number, then the spectrum contains only odd harmonics. If y is an odd number, then the odd and even harmonics of the audio signal are present in the spectrum.

At multiple sub-frequencies in the quantization error spectrum, the components appear below the frequency of the audio signal, down to zero, and the lower limit of the spectrum Fn (x) is determined by the formula x – Fn (x) = F / X. In this case, the frequency Fn (x) becomes the fundamental pitch of the sound for quantization errors, and all other components, including the frequency of the sound signal, are converted to its harmonics. If the number is even at the submultiple frequency yskr, then the spectrum contains only odd harmonics of the frequency Fn (x). If yskr is an odd number, then the spectrum contains odd and even harmonics of this frequency. Low-frequency components in the quantization error spectrum lead to the appearance of harmonics in the form of pitch or consonance. They are especially noticeable at high frequencies in the audio signal when there is no frequency masking effect.

To clarify, we will give an example of a quantization error spectrum at an audio signal level of minus 30 dB with 8-bit quantization. Let fs = 48 kHz and F = 12800 Hz, then the multiplicity factor k skr = y / x = 48000/12800 = 15/4 and therefore the lower cutoff frequency Fn (x) = F / x = 3200 Hz, and the spectrum consists of odd and even harmonics of this frequency.

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Figure 1. Quantization error spectra at submultiple frequency deviation

When the frequency of an audio signal deviates from a submultiple value by a small amount, sidebands appear around all harmonics of the spectrum, including zero (Fig. 1a), the number of spectrum components increases dramatically, and the limit bottom of the spectrum decreases, since the current value of x increases a lot.

Suppose, for example, that the frequency increment of the audio signal is 1 Hz, then the value of the multiplicity factor k = y / x = 48000/12801 = 16000/4267 and the lower limit frequency of the deviation spectrum becomes Fno = 12801/4267 = 3 Hz, and the interval between the components of the spectrum decreases to 6 Hz (Fig. 1b).

Why upsampling?

When it comes to improving digital sound quality, experts in this field agree on only one thing: with an increase in sample rate, sound quality improves dramatically.

Why upsampling?
When it comes to improving digital sound quality, experts in this field agree on only one thing: As the sample rate increases, the sound quality improves dramatically. Also, under the word “improvement”, everyone already understands something for himself. All the variety of opinions on this topic boils down to the following: the sound becomes clearer, softer, more natural, the low frequencies are perceived more clearly.

However, these nuances are only noticed by listeners trained with a good ear for music on specially selected sound material and using technically advanced equipment.

There are many hypotheses that explain why sound quality is improved by higher sampling. Many technicians are inclined to believe that this relationship is due to distortions that arise from filtering and interpolation during audio signal reconstruction.

On a modern technical level, high-quality interpolators may be practically impossible to implement, therefore, instead of improving them, manufacturers simply increase the sample rate. Maybe it’s not about them at all.

Another version, which many music lovers adhere to, is that at a low sampling frequency, for example 44100 Hz, digital sound is completely devoid of nuances of high sounds, the main frequencies of which are above 7 kHz. , and at lower frequencies there are very few harmonics for a high quality perception of music.

In fact, many musical instruments generate vibrations of up to 100 kHz. It is true that the fraction of energy that falls in the frequency band above 20 kHz is 0.01 to 2% for sounds of a harmonic nature and 0.02 to 68% for sounds created by a cymbal, triangle or striking the metal edge of a drum (hoop shot – editor’s note).

Even the frequency range of speech in hissing-hissing sounds extends up to 40 kHz. Supporters of this version are not ashamed that a person cannot perceive sounds with a frequency higher than 20 kHz. Ultrasound is assumed to be perceived bypassing the auditory system, for example through bone conduction.

Discussions that harmonics above 20 kHz make a significant contribution to sounding have culminated in the creation and widespread introduction of analog-to-digital converters using 96 kHz and 192 kHz sample rates; The sample rate is expected to increase to 384 kHz.

Based on modern knowledge of human perception of sound, it must be assumed that the relationship between digital sound quality and sampling frequency is due to the transformation of the quantization error spectrum in the audio frequency range.

In technical literature, this topic is considered only for a particular mathematical model, when music is represented by a signal with a uniform distribution in level and frequency. In this case, the quantization errors are converted to noise with a uniform spectral density from 0 Hz to the Nyquist frequency.

Relationship between sound quality and sample rate

The conversion of an analog signal to digital consists of two steps: sampling in time and quantization in amplitude.

Time sampling means that the signal is represented by a series of samples (samples) taken at regular intervals. For example, when we say that the sample rate is 44.1 kHz, this means that the signal is measured 44 100 times in one second.

The main problem in the first stage of converting an analog to digital signal (digitization) is the choice of the sampling frequency of the analog signal. The higher the frequency, the closer the digital signal is to the analog. However, in proportion to the increase in frequency, they increase:

The intensity of the digital data flow and the bandwidth of the interfaces are not unlimited, especially if several channels are recorded / played simultaneously;
The computational load on digital processors and their computing capabilities are also limited;
The amount of memory required to store the digital signal is increased.
Obviously, a compromise is needed. The choice of the sampling frequency affects the frequency range of the received digital sound and the maximum frequency of the analog signal, correctly represented in the digital one. It is believed that a person hears frequencies in the range of 20 to 20,000 Hz. According to the well-known Kotelnikov theorem, in order for an analog signal (continuous in time) to be accurately reconstructed from its samples, the sampling frequency must be at least twice the maximum audio frequency.

An audio frequency equal to half the sampling frequency is called the Nyquist frequency and is the maximum frequency that a given digital system can store and reproduce correctly. Therefore, if the actual analog signal that we are going to convert to digital format contains frequency components from 0 to 20 kHz, then the sampling frequency of that signal must be at least 40 kHz. The most common sample rates today are 44.1 kHz (CD) and 48 kHz (DAT).