1. Short theoretical data
1.1. Discretization
Discretization is a process of sample discrete on time of instant values of an analog signal. To begin with it is useful to talk about a parity between frequencies of an analog and digital signal. We will consider an analog sinusoidal signal of a kind x (t) = Acos (ωt + ). Discretizing this signal during time moments t = nTS with the period of discretization TS, we will receive a discrete signal on time:
х [n] =A cos (ωnTS + ) =А cos (n + ), n = 0, 1,2..., (1)
Where = ωTS = 2f/fS - shows frequency of a digital signal in radians (in comparison with analog frequency ω which is measured in radians in a second).
Distinction between frequencies of analog and digital signals can be understood better if to consider cases when from various continuous signals on time the same discrete signal if frequency ωTS remains to the same turns out. Also various discrete signals can be received from same analog (or continuous) a signal at various frequency of discretization. In other words, frequency of a corresponding digital signal is defined both frequency of an analog signal, and frequency of discretization fS.
To understand the restrictions connected with process considered above of discretization, research of signals will help with frequency area. Let direct and return transforms of Fourier in analog (2) and discrete (3) representations are set:
(2)
(3)
When the analog signal with the maximum pass - band W (or with the maximum frequency fmах) was discretized with period TS = 1/fS, the corresponding frequency characteristic repeats everyone 2 a radian, or fS. In other words, transform of Fourier to digital representation becomes the periodic version of transform of Fourier in analog representation. There Fourier for discrete signals us the range of frequencies [0, fS/2] interests only.
Hence, to avoid occurrence of false signals or distortion of spectral structure of a discrete signal and thus to have possibility to restore frequency of an initial analog signal, it is necessary for us, that fS 2fmах. This parity is known as criterion of Nyquist according to which frequency of discretization should exceed, at least, in 2 times the maximum frequency of a signal. Usually be Fourier any action with a digital signal the filter for elimination of effects of imposing of the spectra, limiting the maximum frequency of an analog signal is used.
It is possible to show a problem of occurrence of false frequencies on a sinusoid example, discretized with too low frequency (a Figure 1).
Figure 1
In this drawing the sinusoid with frequency 1 kHz, discretized with frequency of discretization fS = 0.8 kHz which is less than frequency of Nyquist equal 2 kHz is represented. The signal put by a dashed line — a sinusoid with frequency of 200 Hz, passing through the same points of discretization. Thus, at frequency of discretization 0.8 kHz the signal on exit ADC (the analog - digital converter) will be identical to entrance signals with frequency both 1 kHz, and 200 Hz. On the other hand, the signal, discredited with the raised frequency, provides more exact description of a signal, than a signal, discredited with frequency of Nyquist.