Analog-to-digital signal conversion for beginners

Introduction


The topics of analog-to-digital and digital-to-analog conversions are quite important in the course of electronics, since most devices that interact with a computer have analog input / output, and the computer can only process digital signals. In this article I want to share with you the very basics of such transformations.

Signal types

Before you understand the transformations themselves, you need to know what signals exist. And there are 3 types of them:
  • Analog
  • Discrete
  • Digital

Analogue signals are continuous in time; they are determined at all time instants.
Discrete - these are signals represented by a sequence of samples, i.e. signal values ​​at discrete time instants.
Digital signals are discrete in time (or in space) and quantized in level. Computational procedures in a computer are performed precisely in digital signals.

In order for the computer to be able to process the signal, it is necessary to convert the signal from analog to digital form.
After processing, the inverse conversion is performed, since most household devices are controlled by analog signals.

The block diagram of digital signal processing in general is as follows:
Structural scheme


Analog to Digital Signal Conversion


Analog-to-digital signal conversion includes two stages:
  1. Signal discretization (in time or space)
  2. Level Quantization

At the sampling stage, signal samples with a certain sampling period (T) are taken.
The sampling frequency can be determined by the formula. image

The process of obtaining a sample of the input signal must occupy a very small part of the sampling period in order to reduce the dynamic conversion errors due to a change in the signal during the sampling time.

The sampling rate is selected from the Kotelnikov theorem . It states that in order for the continuous signal to be restored arbitrarily accurately from the samples of the signal, the sampling frequency must be at least two times higher than the upper frequency of the spectrum of the sampled signal.

Any signal has its own spectral representation. Any representation of a signal is a representation in the form of a sum (or integral) of harmonic components (sinusoids and cosine waves), various frequencies taken with specific weighting factors (having a certain amplitude)
For periodic signals, this is the sum, for non-periodic signals it is an integral.
The transition to the signal spectrum is carried out using the direct Fourier transform.

Consider the transition to the spectral representation in the form of a periodic function:

image


image


As is known, a periodic function satisfying the Dirichlet condition can be represented by a number of harmonic functions.

image


According to Euler’s formula, any expression can be represented as image
image- the frequency of the first harmonic

image- the frequency of the nth harmonic

image- the circular frequency of the nth harmonic

image- the complex amplitude of the harmonic, where imageis the phase spectrum.

The set of amplitudes of the harmonics of the Fourier series is called the amplitude spectrum, and the set of their phases is called the phase spectrum.

image


Spectrum example:
image


For a non-periodic function image, imagethen it imageis replaced by a continuously changing frequency => the sum is replaced by the integral.

image


Direct Fourier Transform for a Non-Periodic Signal

image


Thus, the spectrum of a non-periodic function is represented by the sum of an infinite number of harmonic oscillations, whose frequencies are located infinitely close to each other.

Level quantization

image


The number of quantization levels is determined by the formula image
n - the number of bits
N - quantization level

The choice of the number of quantization levels of signals is based on a compromise approach that takes into account the need for a sufficiently accurate representation of the signal, which requires a large number of quantization levels, and on the other hand, the number of quantization levels to be less, so that the bit depth of the code is minimal.

On this I will end my article so as not to overload the reader with unnecessary information. Good luck in your endeavors!

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