entranceRepresentation of periodic signals by a Fourier series. Spectral representation of deterministic signals
In the last century, Ivan Bernoulli, Leonard Euler, and then Jean-Baptiste Fourier were the first to use the representation of periodic functions by trigonometric series. This view is studied in sufficient detail in other courses, so we only recall the basic relationships and definitions.
As noted above, any periodic function u (t) for which the equality u (t) = u (t + T) , where T = 1 / F = 2p / W , can be represented by a Fourier series:
Each term in this series can be expanded using the cosine formula for the difference between two angles and represented as two terms:
,
where: A n = C n cosφ n, B n = C n sinφ n
, so 
, a 
Odds A n and In n are determined by Euler's formulas:
; 
.
At n = 0 :
a B 0 = 0.
Odds A n and In n , are the mean values of the product of the function u (t) and harmonic oscillations with frequency nw on an interval of duration T ... We already know (Section 2.5) that these are cross-correlation functions that determine the measure of their relationship. Therefore, the coefficients A n and B n show us "how many" sinusoids or cosines with frequency nW contained in this function u (t) , expanded in a Fourier series.
Thus, we can represent the periodic function u (t)
as a sum of harmonic vibrations, where the numbers C n
are the amplitudes, and the numbers φ n
- phases. Usually in literature 
is called the spectrum of amplitudes, and 
- the spectrum of phases. Often only the spectrum of amplitudes is considered, which is depicted as lines located at points nW
on the frequency axis and having a height corresponding to the number C n
... However, it should be remembered that in order to obtain a one-to-one correspondence between the temporal function u (t)
and its spectrum, it is necessary to use both the amplitude spectrum and the phase spectrum. This can be seen from such a simple example. The signals will have the same amplitude spectrum, but completely different types of temporal functions.
A discrete spectrum can have not only a periodic function. For example, signal: is not periodic, but has a discrete spectrum consisting of two spectral lines. Also, there will not be a strictly periodic signal consisting of a sequence of radio pulses (pulses with high-frequency filling), in which the repetition period is constant, but the initial phase of the high-frequency filling changes from pulse to pulse according to some law. Such signals are called almost periodic. As we will see later, they also have a discrete spectrum. Investigation of the physical nature of the spectra of such signals, we will carry out in the same way as for periodic ones.
A periodic signal of any shape with a period T can be represented as a sum
harmonic oscillations with different amplitudes and initial phases, the frequencies of which are multiples of the fundamental frequency. The harmonic of this frequency is called the fundamental or first, the rest - the higher harmonics.
Trigonometric form of the Fourier series:
,
where 
- constant component;
- the amplitudes of the cosine components;
- the amplitudes of the sinusoidal components.
Even signal ( 
) has only cosine, and odd ( 
- only sinusoidal terms.
The equivalent trigonometric form of the Fourier series is more convenient:
,
where 
- constant component;
- the amplitude of the n-th harmonic of the signal. The aggregate of the amplitudes of the harmonic components is called the amplitude spectrum;
- the initial phase of the n-th harmonic of the signal. The set of phases of harmonic components is called the phase spectrum.
Spectrum of a periodic sequence of rectangular pulses. Dependence of the spectrum on the pulse repetition period and their duration. Spectrum width. Fourier series pppi
Let us calculate the amplitude and phase spectra of the AEFI having an amplitude 
, duration 
, following period 
and located symmetrically about the origin (signal is an even function).
Figure 5.1 - AEFI timing diagram.
A signal on an interval of one period can be recorded:
Calculations:
,
The Fourier series for the PPPI is:
Figure 5.2 - Amplitude spectral diagram of the AEFI.
Figure 5.3 - Phase spectral diagram of the AEFI.
The AEFI spectrum is linear (discrete) (represented by a set of individual spectral lines), harmonic (spectral lines are at the same distance from each other ω 1), decreasing (the amplitudes of harmonics decrease with increasing number), has a petal structure (the width of each lobe is 2π / τ), unlimited (the frequency interval in which the spectral lines are located is infinite);
At integer duty cycle, the frequency components with frequencies that are multiples of the duty cycle are absent in the spectrum (their frequencies coincide with the zeros of the envelope of the amplitude spectrum);
With increasing duty cycle, the amplitudes of all harmonic components decrease. Moreover, if it is associated with an increase in the repetition period T, then the spectrum becomes denser (ω 1 decreases), with a decrease in the pulse duration τ - the width of each petal becomes larger;
The frequency range containing 95% of the signal energy (equal to the width of the first two lobes of the envelope) is taken as the width of the AEFI spectrum:
or 
;
All harmonics located in one envelope lobe have the same phase, equal to either 0 or π.
Using Fourier Transform for Spectrum Analysis of Non-Periodic Signals. Spectrum of a single rectangular pulse. Integral Fourier Transforms
Communication signals are always limited in time and therefore are not periodic. Among non-periodic signals, single impulses (SS) are of the greatest interest. OI can be considered as a limiting case of a periodic sequence of pulses (PPI) with a duration 
with an infinitely large period of their repetition 
.
Figure 6.1 - PPI and OI.
A non-periodic signal can be represented as the sum of an infinitely large number of oscillations infinitely close in frequency with vanishingly small amplitudes. The OI spectrum is continuous and is introduced by the Fourier integrals:
-
(1) - direct Fourier transform. Allows you to analytically find the spectral function for a given signal shape;
-
(2) - inverse Fourier transform. Allows you to analytically find the shape for a given spectral function of the signal.
Complex form of integral Fourier transform(2) gives a two-sided spectral representation (having negative frequencies) of a non-periodic signal 
as a sum of harmonic vibrations 
with infinitesimal complex amplitudes 
whose frequencies continuously fill the entire frequency axis.
The complex spectral density of a signal is a complex function of frequency, simultaneously carrying information about both the amplitude and the phase of elementary harmonics.
The modulus of the spectral density is called the spectral density of the amplitudes. It can be considered as the frequency response of the continuous spectrum of a non-periodic signal.
Spectral density argument 
called the spectral density of the phases. It can be considered as the phase-frequency characteristic of the continuous spectrum of a non-periodic signal.
Let's transform the formula (2):
Trigonometric form of integral Fourier transform gives a one-way spectral representation (having no negative frequencies) of a non-periodic signal:
.
Coursework in Mathematical Analysis
Topic: Calculation of partial sums and spectral characteristics of the Fourier series for an explicit function
signal spectrum fourier function
1.Model of the physical process
Solution of a problem with theoretical calculations
An example of solving the problem
An example of solving a problem in the Matlab R2009a environment
Bibliography
1.Model of the physical process
Mathematical model a radio signal can serve as some function of time f(t) . This function can be real or complex, one-dimensional or multidimensional, deterministic or random (noisy signals). In radio engineering, the same mathematical model describes with equal success the current, voltage, electric field strength, etc.
Consider real one-dimensional deterministic signals
Sets of functions (signals) are usually considered as linear functional normed spaces, in which the following concepts and axioms are introduced:
) all the axioms of the linear space are satisfied;
) the dot product of two real signals is defined as follows:
) two signals are called orthogonal if their dot product is equal to zero;
) the system of orthogonal signals forms an infinite-dimensional coordinate basis, which can be used to decompose any periodic signal belonging to the linear space;
Among the various systems of orthogonal functions that can be used to decompose the signal, the most common is the system of harmonic (sinusoidal and cosine) functions:
The representation of a certain periodic signal as a sum of harmonic oscillations with different frequencies is called the spectral representation of the signal. The individual harmonic components of the signal form its spectrum. From a mathematical point of view, the spectral representation is equivalent to the expansion of a periodic function (signal) in a Fourier series.
The significance of the spectral decomposition of functions in radio engineering is due to a number of reasons:
) simplicity of studying the properties of the signal, because harmonic functions are well understood;
) the ability to generate an arbitrary signal, because the technique for generating harmonic signals is quite simple;
) ease of transmission and reception of a signal over the radio channel, tk. harmonic oscillation is the only function of time that retains its shape when passing through any linear circuit. The signal at the output of the circuit remains harmonic with the same frequency, only the amplitude and the initial phase of the oscillation change;
) the decomposition of the signal into sines and cosines allows the use of a symbolic method developed for analyzing the transmission of harmonic oscillations through linear circuits.
As a model of the physical process, consider the electrocardiogram of the heart.
2.Solution of the problem with theoretical calculations
Objective 1:
Let us describe, with the help of Fourier series, a periodically repeating impulse in the area of the electrocardiogram, the so-called QRS complex.
The QRS complex can be defined by the following piecewise linear function
Where
This function can be continued periodically with a period T = 2l.
Fourier series of functions:
Definition 1: The function is called piecewise continuous on the segment [a, b] if it is continuous at all points of this segment, except for a finite number of points at which its finite one-sided limits exist.
Definition 2: The function is called piecewise smooth on some segment if it and its derivative are piecewise continuous.
Theorem 1 (Dirichlet test): Fourier series of a piecewise-smooth function on an interval f (x) converges at each point of continuity to the value of the function at this point and to the value at each point of discontinuity.
Our function satisfies the conditions of the theorem.
For a given function, we obtain the following coefficients of the Fourier series:
Complex form of the Fourier series
To represent the series in complex form, we use Euler's formulas:
Let us introduce the notation:
Then the series can be rewritten as
In addition, the coefficients of the complex Fourier series can be obtained directly by calculating them by the formula
We write in complex form the Fourier series of a given function
Spectral characteristics of the series
Expression 
in the Fourier series is called nth harmonic. It is known that
where or
, 
Aggregates are named accordingly amplitude and phase spectrum periodic function.
Spectra are graphically depicted as length segments drawn perpendicular to the axis on which the value is plotted n= 1,2 ... or.
A graphical representation of the corresponding spectrum is called an amplitude or phase diagram. In practice, the amplitude spectrum is most often used.
.An example of solving the problem
Task 2: Consider a specific example of a problem for the selected model of a physical process.
We extend this function to the entire number axis, we obtain the periodic function f(x)
with period T = 2 l= 18 (Fig. 1.).
Rice. 1. Graph of a periodically continued function
Let's calculate the Fourier coefficients of the given function.
Let's write down the partial sums of the series:
Rice. 2. Plots of partial sums of the Fourier series
With growth n plots of partial sums at points of continuity approach the plot of a function f(x)
... At break points, the values of the partial sums approach 
.
Let's build the amplitude and phase diagrams.
given a quarter.
table
4. An example of solving a problem in the Matlab R2009a environment Objective 3: As an example, consider the entire PR and QT intervals. Rice For this function, build graphs of partial sums, as well as amplitude and phase diagrams. Let's take specific values of the parameters for our task: A script for building the required graphs and charts. The script allows you to solve a number of similar problems by choosing the parameters and coordinates of the points Q, R, S. % CALCULATION OF PARTIAL SUMS AND SPECTRAL CHARACTERISTICS OF THE FOURIER SERIES FOR EXPRESS % Spectral analysis. L I1 I2 Q R S I3 I4 I5 P T w v a b c d q r Qy Ry Sy nCase = 18; = 6; I2 = 10; Q = 11; Qy = -2; R = 12; Ry = 17; S = 13; Sy = -4; I3 = 15; I4 = 20; I5 = 26; = 2; T = 3; ExprNum = 9; = 250; = 30; = 0; flag == 0 = 1; (k<15) k = menu ("Changing parameters", ... sprintf ("Parameter1 P =% g", P), ... ("Parameter2 I1 =% g", I1), ... ("Parameter3 I2 =% g", I2), ... ("Parameter4 Qx =% g ", Q), ... (" Parameter5 Qy =% g ", Qy), ... (" Parameter6 Rx =% g ", R), ... (" Parameter7 Ry =% g ", Ry), ... ("Parameter8 Sx =% g", S), ... ("Parameter9 Sy =% g", Sy), ... ("Parameter10 I3 =% g", I3), .. . ("Parameter11 I4 =% g", I4), ... ("Parameter12 T =% g", T), ... ("Parameter13 I5 =% g", I5), ... ("Parameter13 Ns =% g ", Ns), ... "Continue"); k == 1, = input (); endk == 2, = input (); endk == 3, = input (); endk == 4, = input (); endk == 5, = input (); endk == 6, = input (); endk == 7, = input (); "New Sx value ="]); endk == 9, = input (); endk == 10, = input (); endk == 11, = input (); endk == 12, = input (); endk == 13, = input () endk == 14, = input () % Application of parameters = Qy / (Q-I2); v = Qy * I2 / (I2-Q); = (Ry-Qy) / (RQ); = (Qy * RQ * Ry) / (RQ); = (Sy-Ry) / (SR); = (Ry * SR * Sy) / (SR); = Sy / (S-I3); = I3 * Sy / (I3-S); = 2 * L / N; = 0: Ts: 2 * L; = length (t ); = zeros (1, Dim); = floor (I1 * N / 2 / L) +1; = floor ((I2-I1) * N / 2 / L) +1; = floor ((Q-I2) * N / 2 / L) +1; = floor ((RQ) * N / 2 / L) +1; = floor ((SR) * N / 2 / L) +1; = floor ((I3-S) * N / 2 / L) +1; = floor ((I4-I3) * N / 2 / L) +1; = floor ((I5-I4) * N / 2 / L) +1; = floor (( 2 * L-I4) * N / 2 / L) +1; i = 1: u1 (i) = P * sin (pi * t (i) / I1); i = u1: u2 (i) = 0; i = (u2 + u1) :( u3 + u2 + u1) (i) = w * t (i) + v; i = (u3 + u2 + u1): (u4 + u3 + u2 + u1) (i) = a * t (i) + b; i = (u4 + u3 + u2 + u1): (u5 + u4 + u3 + u2 + u1) (i) = c * t (i) + d; i = (u5 + u4 + u3 + u2 + u1): (u6 + u5 + u4 + u3 + u2 + u1) (i) = q * t (i) + r; i = (u6 + u5 + u4 + u3 + u2 + u1 ): (u7 + u6 + u5 + u4 + u3 + u2 + u1) (i) = 0; i = (u7 + u6 + u5 + u4 + u3 + u2 + u1): (u8 + u7 + u6 + u5 + u4 + u3 + u2 + u1) (i) = T * sin (pi * (t (i) -I4) / (I5-I4)); (t, y, "LineWidth", 2), grid, set ( gca, "FontName", "Arial Cyr", "FontSize", 16); title ("Process chart"); xlabel ("Time (s)"); ylabel ("Y (t)"); % Partial sum plot n n = 0; j = 1: ExprNum = j; j1 = quad (@f, 0, I1); 2 = a0 + quad (@f, I1, I2); 3 = a0 + quad (@f, I2, Q ); 4 = a0 + quad (@f, Q, R); 5 = a0 + quad (@f, R, S); 6 = a0 + quad (@f, S, I3); 7 = a0 + quad ( @f, I3, I4); 8 = a0 + quad (@f, I4, I5); 9 = a0 + quad (@f, I5, 2 * L); = a0 / L; = zeros (1, Ns) ; = zeros (1, Ns); i = 1: Ns = i; j = 1: ExprNum = j; j1 (i) = quad (@f, 0, I1); (i) = quad (@g, 0 , I1); 2 (i) = an (i) + quad (@f, I1, I2); (i) = bn (i) + quad (@g, I1, I2); 3 (i) = an ( i) + quad (@f, I2, Q); (i) = bn (i) + quad (@g, I2, Q); 4 (i) = an (i) + quad (@f, Q, R ); (i) = bn (i) + quad (@g, Q, R); 5 (i) = an (i) + quad (@f, R, S); (i) = bn (i) + quad (@g, R, S); 6 (i) = an (i) + quad (@f, S, I3); (i) = bn (i) + quad (@g, S, I3); 7 (i) = an (i) + quad (@f, I3, I4); (i) = bn (i) + quad (@g, I3, I4); 8 (i) = an (i) + quad ( @f, I4, I5); (i) = bn (i) + quad (@g, I4, I5); 9 (i) = an (i) + quad (@f, I5, 2 * L); ( i) = bn (i) + quad (@g, I5, 2 * L); (i) = an (i) / L; (i) = bn (i) / L; = t; = zeros (1, length (x)); = fn + a0 / 2; i = 1: Ns = i; = fn + an (i) * cos (n * pi * x / L) + bn (i) * sin (n * pi * x / L); (t, y, x, fn, "LineWidth", 2), grid, set (gca, "FontName", "Arial Cyr", "FontSize", 16); title ("Signal and partial sum graph"); xlabel ("Time (s)"); ylabel (sprintf ("Sn (t)")); % Plotting an amplitude diagram = zeros (1, Ns); wn = pi / L; = wn: wn: wn * Ns; i = 1: Ns (i) = sqrt (an (i). ^ 2 + bn (i). ^ 2); (Gn, A, ". "), grid, set (gca," FontName "," Arial Cyr "," FontSize ", 16); (" Amplitude diagram of the signal "); xlabel ("n"); ylabel ("An"); % Construction of the phase diagram of the signal = zeros (1, Ns); for i = 1: Ns (an (i)> 0) (i) = atan (bn (i) / an (i)); ((an (i)<0)&&(bn(i))>0) (i) = atan (bn (i) / an (i)) + pi; ((an (i)<0)&&(bn(i))<0)(i)=pi-atan(bn(i)/an(i));((an(i)==0)&&(bn(i))>0) (i) = pi / 2; ((an (i) == 0) && (bn (i))<0)(i)=-pi/2;(Gn,Fi,"."),
grid, set(gca,"FontName","Arial Cyr","FontSize",16);("Фазовая диаграмма сигнала");
xlabel("n"); ylabel("Fi");Figure 1;Figure 2;Figure 3;Figure 4;=0;=input("Закончить
работу-<3>, continue - 1. Fikhtengolts, G.M. The course of differential and integral calculus: in 3 volumes, Moscow, 1997.3 volumes. Vodnev, V.T., Naumovich, A.F., Naumovich, N.F., Basic mathematical formulas. Minsk, 1998 Kharkevich A.A. Spectra and Analysis. Moscow, 1958 Lazarev, Yu. F., Beginnings of programming in the MatLAB environment. Kiev 2003. Demidovich, B.P. Collection of problems and exercises in mathematical analysis, M., 1988.
Listliterature
To check if the program is working correctly, let's form an array of samples as the sum of two sinusoids sin (10 * 2 * pi * x) + 0.5 * sin (5 * 2 * pi * x) and slip it into the program. The program drew the following:
Fig. 1 The graph of the time function of the signal
Fig. 2 Signal spectrum graph
The spectrum graph has two sticks (harmonics) of 5 Hz with an amplitude of 0.5 V and 10 Hz - with an amplitude of 1 V, everything is as in the formula of the original signal. Everything is fine, well done programmer! The program is working correctly.
This means that if we feed a real signal from a mixture of two sinusoids to the ADC input, then we will get a similar spectrum, consisting of two harmonics.
Total, our real measured signal, lasting 5 sec, digitized ADC, that is, presented discrete counts, has discrete non-periodic spectrum.
From a mathematical point of view, how many mistakes are there in this phrase?
Now the bosses decided we decided that 5 seconds is too long, let's measure the signal in 0.5 seconds.
Fig. 3 Graph of the function sin (10 * 2 * pi * x) + 0.5 * sin (5 * 2 * pi * x) at a measurement period of 0.5 sec
Fig. 4 Function spectrum
Something seems to be wrong! The 10 Hz harmonic is drawn normally, and instead of the 5 Hz stick, some incomprehensible harmonics appeared. We look on the Internet, what and how ...
In, they say that zeros must be added to the end of the sample and the spectrum will be drawn normal.
Fig. 5 We finished off zeros up to 5 sec
Fig. 6 Received the spectrum
Still not what it was at 5 seconds. We'll have to deal with the theory. Go to Wikipedia- source of knowledge.
2. Continuous function and its representation by the Fourier series
Mathematically, our signal with a duration of T seconds is a function f (x) defined on the interval (0, T) (X in this case is time). Such a function can always be represented as a sum of harmonic functions (sinusoids or cosines) of the form:
(1), where:
K - number of trigonometric function (number of harmonic component, number of harmonic)
T - the segment where the function is defined (signal duration)
Ak is the amplitude of the kth harmonic component,
θk is the initial phase of the kth harmonic component
What does it mean to "represent a function as the sum of a series"? This means that by adding at each point the values of the harmonic components of the Fourier series, we get the value of our function at this point.
(More strictly, the root-mean-square deviation of the series from the function f (x) will tend to zero, but despite the root-mean-square convergence, the Fourier series of the function, generally speaking, is not obliged to converge to it pointwise. See https://ru.wikipedia.org/ wiki / Fourier_Row.)
This series can also be written as:
(2),
where, k-th complex amplitude.
The relationship between the coefficients (1) and (3) is expressed by the following formulas:
Note that all these three representations of the Fourier series are completely equivalent. Sometimes, when working with Fourier series, it is more convenient to use exponents of the imaginary argument instead of sines and cosines, that is, to use the Fourier transform in complex form. But it is convenient for us to use formula (1), where the Fourier series is presented as a sum of cosine waves with the corresponding amplitudes and phases. In any case, it is incorrect to say that the result of the Fourier transform of a real signal will be the complex amplitudes of the harmonics. As the Wiki correctly says, "The Fourier transform (ℱ) is an operation that assigns one function to a real variable to another function, also a real variable."
Total:
The mathematical basis for spectral analysis of signals is the Fourier transform.
The Fourier transform allows you to represent the continuous function f (x) (signal), defined on the segment (0, T) as the sum of an infinite number (infinite series) of trigonometric functions (sinusoids and \ or cosines) with certain amplitudes and phases, also considered on the segment (0, T). Such a series is called a Fourier series.
Let's note some more points, the understanding of which is required for the correct application of the Fourier transform to signal analysis. If we consider the Fourier series (the sum of sinusoids) on the entire X axis, then we can see that outside the segment (0, T), the function represented by the Fourier series will periodically repeat our function.
For example, in the graph in Fig. 7, the original function is defined on the segment (-T \ 2, + T \ 2), and the Fourier series represents a periodic function defined on the entire x-axis.
This is because sinusoids themselves are periodic functions, and accordingly, their sum will be a periodic function.
Fig. 7 Representation of a non-periodic original function by the Fourier series
In this way:
Our original function is continuous, non-periodic, defined on some segment of length T.
The spectrum of this function is discrete, that is, it is presented in the form of an infinite series of harmonic components - the Fourier series.
In fact, the Fourier series defines a certain periodic function that coincides with ours on the segment (0, T), but for us this periodicity is not essential.
The periods of the harmonic components are multiples of the value of the segment (0, T), on which the original function f (x) is defined. In other words, the periods of the harmonics are multiples of the duration of the signal measurement. For example, the period of the first harmonic of the Fourier series is equal to the interval T, on which the function f (x) is defined. The period of the second harmonic of the Fourier series is equal to the interval T / 2. And so on (see fig. 8).
Fig. 8 Periods (frequencies) of harmonic components of the Fourier series (here T = 2π)
Accordingly, the frequencies of the harmonic components are multiples of 1 / T. That is, the frequencies of the harmonic components Fk are equal to Fk = k \ T, where k ranges from 0 to ∞, for example k = 0 F0 = 0; k = 1 F1 = 1 \ T; k = 2 F2 = 2 \ T; k = 3 F3 = 3 \ T; ... Fk = k \ T (at zero frequency - constant component).
Let our original function be a signal recorded for T = 1 sec. Then the period of the first harmonic will be equal to the duration of our signal T1 = T = 1 sec and the frequency of the harmonic is 1 Hz. The second harmonic period will be equal to the signal duration divided by 2 (T2 = T / 2 = 0.5 sec) and the frequency is 2 Hz. For the third harmonic, T3 = T / 3 sec and the frequency is 3 Hz. Etc.
The step between harmonics in this case is 1 Hz.
Thus, a signal with a duration of 1 sec can be decomposed into harmonic components (to obtain a spectrum) with a frequency resolution of 1 Hz.
To increase the resolution by 2 times to 0.5 Hz, it is necessary to increase the measurement duration by 2 times - up to 2 sec. A signal with a duration of 10 seconds can be decomposed into harmonic components (to obtain a spectrum) with a frequency resolution of 0.1 Hz. There is no other way to increase the frequency resolution.
There is a way to artificially increase the signal duration by adding zeros to the sample array. But it does not increase the real frequency resolution.
3. Discrete signals and discrete Fourier transform
With the development of digital technology, the methods of storing measurement data (signals) have also changed. If earlier the signal could be recorded on a tape recorder and stored on tape in analog form, now the signals are digitized and stored in files in the computer's memory as a set of numbers (counts).A typical scheme for measuring and digitizing a signal is as follows.
Fig. 9 Diagram of the measuring channel
The signal from the measuring transducer arrives at the ADC for a period of time T. The samples of the signal (sample) obtained during the time T are transferred to the computer and stored in memory.
Fig. 10 Digitized signal - N samples obtained during time T
What are the requirements for the signal digitization parameters? A device that converts an input analog signal into a discrete code (digital signal) is called an analog-to-digital converter (ADC) (Wiki).
One of the main parameters of the ADC is the maximum sampling rate (or sampling rate, English sample rate) - the sampling rate of a continuous signal in time during its sampling. Measured in hertz. ((Wiki))
According to the Kotelnikov theorem, if a continuous signal has a spectrum limited by the frequency Fmax, then it can be completely and unambiguously reconstructed from its discrete samples taken at time intervals 
, i.e. with a frequency of Fd ≥ 2 * Fmax, where Fd is the sampling frequency; Fmax is the maximum frequency of the signal spectrum. In other words, the signal sampling frequency (ADC sampling frequency) must be at least 2 times higher than the maximum frequency of the signal that we want to measure.
And what will happen if we take samples with a lower frequency than is required by the Kotelnikov theorem?
In this case, the effect of "aliasing" (aka stroboscopic effect, moiré effect) occurs, in which a high-frequency signal, after digitization, turns into a low-frequency signal, which in fact does not exist. In fig. 11 high frequency red sine wave is a real signal. The blue sinusoid of a lower frequency is a dummy signal that arises due to the fact that during the sampling time it manages to pass more than half a period of the high-frequency signal.
Rice. 11. The appearance of a false signal of low frequency with insufficiently high sampling rate
To avoid the effect of aliasing, a special anti-aliasing filter is installed in front of the ADC - a low-pass filter (low-pass filter), which passes frequencies below half the sampling frequency of the ADC, and cuts higher frequencies.
In order to calculate the spectrum of the signal from its discrete samples, the discrete Fourier transform (DFT) is used. Note again that the spectrum of the discrete signal is "by definition" limited by the frequency Fmax, less than half of the sampling frequency Fd. Therefore, the spectrum of a discrete signal can be represented by the sum of a finite number of harmonics, in contrast to the infinite sum for the Fourier series of a continuous signal, the spectrum of which can be unlimited. According to the Kotelnikov theorem, the maximum frequency of a harmonic should be such that it has at least two counts, so the number of harmonics is equal to half the number of samples of a discrete signal. That is, if there are N samples in the sample, then the number of harmonics in the spectrum will be equal to N / 2.
Consider now the discrete Fourier transform (DFT).
Comparing with the Fourier series
We see that they coincide, except that the time in the DFT is discrete and the number of harmonics is limited to N / 2, which is half the number of counts.
DFT formulas are written in dimensionless integer variables k, s, where k are the numbers of signal samples, s are the numbers of spectral components.
The value of s shows the number of total harmonic oscillations at the period T (the duration of the signal measurement). Discrete Fourier transform is used to find the amplitudes and phases of harmonics numerically, i.e. "on the computer"
Going back to the results at the beginning. As mentioned above, when expanding a non-periodic function (our signal) in a Fourier series, the resulting Fourier series actually corresponds to a periodic function with a period T. (Fig. 12).
Fig. 12 Periodic function f (x) with a period T0, with a measurement period T> T0
As can be seen in Fig. 12, the function f (x) is periodic with a period T0. However, due to the fact that the duration of the measuring sample T does not coincide with the period of the function T0, the function obtained as a Fourier series has a discontinuity at the point T. As a result, the spectrum of this function will contain a large number of high-frequency harmonics. If the duration of the measurement sample T coincided with the period of the function T0, then in the spectrum obtained after the Fourier transform, only the first harmonic (sinusoid with a period equal to the duration of the sample) would be present, since the function f (x) is a sinusoid.
In other words, the DFT program “does not know” that our signal is a “piece of a sinusoid,” but tries to represent a periodic function as a series, which has a discontinuity due to the inconsistency of individual pieces of a sinusoid.
As a result, harmonics appear in the spectrum, which should summarize the shape of the function, including this discontinuity.
Thus, in order to obtain a "correct" spectrum of a signal, which is the sum of several sinusoids with different periods, it is necessary that an integer number of periods of each sinusoid fit into the signal measurement period. In practice, this condition can be met for a sufficiently long signal measurement duration.
Fig. 13 An example of a function and spectrum of the signal of the kinematic error of the gearbox
With a shorter duration, the picture will look "worse":
Fig. 14 Example of rotor vibration signal function and spectrum
In practice, it can be difficult to understand where are the "real components" and where are the "artifacts" caused by the multiple periods of the components and the duration of the signal sampling or "jumps and breaks" in the waveform. Of course, the words "real components" and "artifacts" are not in vain taken in quotes. The presence of many harmonics on the spectrum graph does not mean that our signal in reality "consists" of them. It is like thinking that the number 7 "consists" of the numbers 3 and 4. The number 7 can be represented as the sum of the numbers 3 and 4 - this is correct.
So our signal ... or rather not even "our signal", but a periodic function composed by repeating our signal (sample) can be represented as a sum of harmonics (sinusoids) with certain amplitudes and phases. But in many cases that are important for practice (see the figures above), it is really possible to associate the harmonics obtained in the spectrum with real processes that have a cyclical nature and make a significant contribution to the signal shape.
Some results
1. The real measured signal, duration T sec, digitized by the ADC, that is, represented by a set of discrete samples (N pieces), has a discrete non-periodic spectrum, represented by a set of harmonics (N / 2 pieces).2. The signal is represented by a set of real values and its spectrum is represented by a set of real values. Harmonic frequencies are positive. The fact that mathematicians find it more convenient to represent the spectrum in a complex form using negative frequencies does not mean that "this is correct" and "this should always be done."
3. The signal measured at the time interval T is determined only at the time interval T. What was before we started measuring the signal, and what will happen after that - this is unknown to science. And in our case, it is not interesting. The DFT of a time-limited signal gives its “true” spectrum, in the sense that, under certain conditions, it allows the amplitude and frequency of its components to be calculated.
Used materials and other useful materials.
The signal is called periodic if its shape is cyclically repeated in time. A periodic signal is generally written as follows:
Here is the signal period. Periodic signals can be simple or complex.
For the mathematical representation of periodic signals with a period, this series is often used, in which harmonic (sinusoidal and cosine) oscillations of multiple frequencies are selected as basis functions:
where . is the fundamental angular frequency of the sequence of functions. With harmonic basis functions, from this series we obtain a Fourier series, which in the simplest case can be written in the following form:
where the coefficients
From the Fourier series it can be seen that, in the general case, a periodic signal contains a constant component and a set of harmonic oscillations of the fundamental frequency and its harmonics with frequencies. Each harmonic oscillation of the Fourier series is characterized by an amplitude and an initial phase.
Spectral diagram and spectrum of a periodic signal.
If any signal is presented as a sum of harmonic oscillations with different frequencies, then this means that spectral decomposition signal.
Spectral diagram signal is a graphical representation of the coefficients of the Fourier series of this signal. There are amplitude and phase diagrams. To construct these diagrams, the harmonic frequencies are plotted on a certain scale along the horizontal axis, and their amplitudes and phases are plotted along the vertical axis. Moreover, the amplitudes of the harmonics can take only positive values, the phases - both positive and negative values in the interval.
Spectral diagrams of a periodic signal:
a) - amplitude; b) - phase.
Signal spectrum is a set of harmonic components with specific values of frequencies, amplitudes and initial phases, which together form a signal. In practice, spectral diagrams are called more concisely - amplitude spectrum, phase spectrum... The greatest interest is shown in the amplitude spectral diagram. It can be used to estimate the percentage of harmonics in the spectrum.
Spectral characteristics play an important role in telecommunication technology. Knowing the signal spectrum, you can correctly calculate and set the bandwidth of amplifiers, filters, cables and other nodes of communication channels. Knowledge of signal spectra is necessary for building multichannel systems with frequency division multiplexing. Without knowledge of the interference spectrum, it is difficult to take measures to suppress it.
From this we can conclude that the spectrum must be known in order to carry out undistorted signal transmission over the communication channel, to ensure signal separation and to attenuate interference.
To observe the spectra of signals, there are devices called spectrum analyzers... They allow observing and measuring the parameters of individual components of the spectrum of a periodic signal, as well as measuring the spectral density of a continuous signal.