To simplify the methods for solving circuit analysis problems, signals are represented as a sum of certain functions.
This process is substantiated by the concept of a generalized Fourier series. It has been proven in mathematics that any function that satisfies the Dirichlet conditions can be represented as a series:
To determine, we multiply the left and right parts of the series by and take the integral of the left and right parts:
for the interval in which the orthogonality conditions are satisfied.
It can be seen that. We got an expression for the generalized Fourier series:
We single out a specific type of function for expanding the signal into a series. As such a function, we choose an orthogonal system of functions:
To determine the series, we calculate the value:
Thus, we get:
Graphically, this series is represented as two graphs of the amplitude harmonic components.
The resulting expression can be represented as:
We got the second form of recording the trigonometric Fourier series. Graphically, this series is presented in the form of two graphs - amplitude and phase spectra.
Let's find the complex form of the Fourier series, for this we use the Euler formulas:
Graphically, the spectrum in this form is represented on the frequency axis in the range.
Obviously, the spectrum of a periodic signal, expressed in complex or amplitude form, is discrete. This means that the spectrum contains components with frequencies
Spectral characteristics of a non-periodic signal
Since a single signal is considered as a non-periodic signal in radio engineering, to find its spectrum, we represent the signal as a periodic signal with a period. Let's use the transformation of the Fourier series for the given period. Get for:
An analysis of the obtained expression shows that at , the amplitudes of the components become infinitely small and they are located continuously on the frequency axis. Then, in order to get out of this situation, we use the concept of spectral density:
We substitute the resulting expression into the complex Fourier series, we get:
Finally we get:
Here is the spectral density, and the expression itself is the direct Fourier transform. To determine the signal from its spectrum, the inverse Fourier transform is used:
Properties of the Fourier Transform
From the formulas of direct and reverse transformations Fourier, it is obvious that if the signal changes, then its spectrum will also change. The following properties set the dependency of the spectrum of the changed signal on the spectrum of the signal before the changes.
1) Linearity property of the Fourier transform
We found that the spectrum of the sum of the signals is equal to the sum of their spectra.
2) The spectrum of the signal shifted in time
It was found that when the signal is shifted, the amplitude spectrum does not change, but only the phase spectrum changes by the value
3) Changing the time scale
that is, when the signal expands (narrows) several times, the spectrum of this signal narrows (expands).
4) Displacement spectrum
5) The spectrum of the derivative of the signal
Take the derivative of the left and right sides of the inverse Fourier transform.
We see that the spectrum of the derivative of the signal is equal to the spectrum of the original signal multiplied by, that is, the amplitude spectrum changes and the phase spectrum changes by.
6) Signal integral spectrum
Take the integral of the left and right sides of the inverse Fourier transform.
We see that the spectrum of the derivative of the signal is equal to the spectrum of the original signal divided by,
7) Spectrum of the product of two signals
Thus, the spectrum of the product of two signals is equal to the convolution of their spectra multiplied by the coefficient
8) Duality property
Thus, if a spectrum corresponds to some signal, then a signal in shape coinciding with the above spectrum corresponds to a spectrum in shape coinciding with the above signal.
General remarks
Among the various systems of orthogonal functions that can be used as bases for representing radio signals, an exceptional place is occupied by harmonic (sine and cosine) functions. The importance of harmonic signals for radio engineering is due to a number of reasons.
In radio engineering, one has to deal with electrical signals that are associated with transmitted messages. accepted way coding.
We can say that an electrical signal is a physical (electrical) process that carries information. The amount of information that can be transmitted using a certain signal depends on its main parameters: duration, frequency band, power, and some other characteristics. Importance also has a level of interference in the communication channel: the lower this level, the more information can be transmitted using a signal with a given power. Before talking about the information capabilities of a signal, it is necessary to familiarize yourself with its main characteristics. It is advisable to consider separately deterministic and random signals.
Any signal is called deterministic, the instantaneous value of which at any time can be predicted with a probability of one.
Examples of deterministic signals are pulses or bursts of pulses whose shape, magnitude and position in time are known, as well as a continuous signal with given amplitude and phase relationships within its spectrum. Deterministic signals can be divided into periodic and non-periodic.
A periodic signal is any signal for which the condition
where period T is a finite segment, and k is any integer.
The simplest periodic deterministic signal is harmonic oscillation. Strictly harmonic oscillation is called monochromatic. This term, borrowed from optics, emphasizes that the spectrum of a harmonic oscillation consists of a single spectral line. For real signals that have a beginning and an end, the spectrum is inevitably blurred. Therefore, strictly monochromatic oscillations do not exist in nature. In the future, a harmonic and monochromatic signal will conditionally mean an oscillation. Any complex periodic signal, as is known, can be represented as a sum of harmonic oscillations with frequencies that are multiples of the fundamental frequency w = 2*Pi/T. The main characteristic of a complex periodic signal is its spectral function, which contains information about the amplitudes and phases of individual harmonics.
A non-periodic deterministic signal is any deterministic signal for which the condition s(t)s(t+kT) is satisfied.
As a rule, a non-periodic signal is limited in time. Examples of such signals are the already mentioned pulses, bursts of pulses, “scraps” of harmonic oscillations, etc. Non-periodic signals are of primary interest, since they are predominantly used in practice.
The main characteristic of a non-periodic, as well as a periodic signal, is its spectral function;
Random signals include signals whose values are not known in advance and can only be predicted with a certain probability less than one. Such functions are, for example, electrical voltage corresponding to speech, music, a sequence of characters of a telegraph code when transmitting a non-repeating text. Random signals also include a sequence of radio pulses at the input of the radar receiver, when the amplitudes of the pulses and the phases of their high-frequency filling fluctuate due to changes in propagation conditions, the position of the target, and some other reasons. Many other examples of random signals can be given. Essentially, any signal that carries information should be considered random. The listed deterministic signals, "completely known", no longer contain information. In the following, such signals will often be referred to as "oscillations".
A statistical approach is used to characterize and analyze random signals. The main characteristics of random signals are:
a) the law of probability distribution.
b) spectral distribution of signal power.
Based on the first characteristic, one can find the relative residence time of the signal value in a certain range of levels, the ratio of the maximum values to the root mean square, and a number of others. important parameters signal. The second characteristic gives only the frequency distribution medium power signal. More detailed information regarding the individual components of the spectrum - about their amplitudes and phases - the spectral characteristic of a random process does not give.
Along with useful random signals in theory and practice, one has to deal with random interference - noise. As mentioned above, the noise level is the main factor limiting the information transfer rate for a given signal.
SAINT PETERSBURG STATE UNIVERSITY
FACULTY OF PHYSICS
DIRECTION
"APPLIED MATHEMATICS AND PHYSICS"
Methods of determination
spectral characteristics
electrical signals
Saint Petersburg
Introduction ................................................ ................................................. ................................... 3
The real form of the Fourier series.................................................... ................................................. 3
The complex form of the Fourier series .......................................................... ................................................. .. 4
Spectrum of a periodic function ............................................... ................................................. 5
Fourier transform .................................................................. ................................................. ............... 6
Properties of the Fourier Transform .............................................................. ................................................. 7
Discrete signal spectrum .............................................................. ................................................. ...... 9
Discrete Fourier Transform .............................................................. ............................................ 12
Spreading of the spectrum .................................................. ................................................. ................... 14
Laboratory setup and measurements....................................................................... ................... 15
Tasks................................................. ................................................. ...................................... 17
Appendix 1. Segment of a sinusoid .............................................. ............................................. 18
Literature................................................. ................................................. ................................ 19
Introduction
This work is the first in a series laboratory work in the educational laboratory "Methods of processing and transmission of information" (MOPI) of the Faculty of Physics, St. Petersburg State University. The laboratory is carried out in the second year and supports the course of lectures "Physical foundations of methods for processing and transmitting information". By this time, the course has already been taken by students, the laboratory is designed to consolidate and expand knowledge in this area.
The concept of the signal spectrum is necessary for the development of information transmission devices, it is used for indirect measurement of other physical quantities, and simply for calculating an electrical circuit. Knowing the spectrum of the signal allows you to better understand its nature, and it is no coincidence that the cycle of laboratory work begins with this work.
The work will have both computational and experimental character. The experimental part of the work contains an important innovative element - the use of digital signal processing, digitized using a data acquisition system. In addition, the entire computational part of the work, as well as the processing of experimental results, is performed on the basis of the modern mathematical package MATLAB and its additional library - Signal Processing Toolbox. The possibilities inherent in them for mathematical modeling of various types of signals and data processing are used.
It is assumed that the reader is familiar with the basic workings of this package. Calculation programs and various additions will be referred to the Applications for work.
Real form of the Fourier series
Consider a periodic function with a period equal to: 
, where is any integer. Under certain conditions, this function can be represented as a sum, finite or infinite, of harmonic functions of the form 
, whose period coincides with the period of the original function , where https://pandia.ru/text/78/330/images/image007_33.gif" width="19 height=24" height="24"> is a constant..gif" width="15" height="17 src=">. Thus, we will solve the problem of expanding a periodic function into a trigonometric series:
(1)
A separate term of this sum https://pandia.ru/text/78/330/images/image011_19.gif" width="28" height="23">. Our task is to select such coefficients and , for which row (1) will converge to the given function https://pandia.ru/text/78/330/images/image013_18.gif" width="301 height=53" height="53"> (2)
where the new coefficients are expressed as https://pandia.ru/text/78/330/images/image015_16.gif" width="105" height="24 src=">.gif" width="273" height="117 "> (3)
It can be proved that the trigonometric series will converge uniformly to the function https://pandia.ru/text/78/330/images/image019_13.gif" width="48 height=53" height="53">.gif" width= "28" height="23 src="> can be approximated with a certain accuracy by a trigonometric order polynomial N, that is, a finite number of terms.
Complex form of the Fourier series
Another, complex form of the trigonometric series is obtained by writing the sines and cosines in (2) through complex exponents:
(4)
The coefficients of the real and complex forms are interconnected by the relations:
(5)
Using formulas (5), from (3) we obtain expressions for the coefficients of the complex form of the trigonometric series. These coefficients can be written for any number k in the following way
(6)
A trigonometric series in complex form converges uniformly to the function if the series and converge. This will be true if the original function satisfies the Dirichlet conditions.
Spectrum of a periodic function
Let us introduce the concept of the spectrum of a periodic function. It is based on the possibility of representing the signal either as a real Fourier series (1) or as a complex series (4). This means that the real coefficients and , or the complex coefficients carry full information about periodic with a known period https://pandia.ru/text/78/330/images/image012_20.gif" width="21" height="24"> and is called the real signal spectrum..gif" width="69" height="41 src=">). Therefore, the set is called the amplitude spectrum..gif" width="20" height="24">. Unlike the real spectrum, the complex spectrum is defined for both positive and negative frequencies. Below we will show that the moduli of these coefficients determine the amplitudes harmonics and therefore can be called the amplitude spectrum, and the arguments (phase spectrum) determine the initial phases of the harmonics..gif" width="61 height=29" height="29">. This relationship implies the property of parity for the amplitude complex spectrum and oddness for the phase one.
Let us see how the real and complex spectra are related. We write series (4) as
Terms with negative numbers can be expressed in terms of terms with positive numbers, since and 
. Then only the sum with positive numbers remains
After summing exponents with the same numbers https://pandia.ru/text/78/330/images/image035_4.gif" width="237" height="53"> (9)
Comparing series (1) and (9), we obtain the desired relationship between the real and complex spectra: and .
Since the spectrum of a periodic signal consists of individual harmonics, it is called discrete or line. Harmonic frequencies are inversely proportional to the period https://pandia.ru/text/78/330/images/image011_19.gif" width="28" height="23"> is a continuously differentiable absolutely integrable function on the entire axis: . A non-periodic signal can be considered as periodic, but with an infinitely large period. Having made the limit transition from a finite to an infinitely large signal period in formulas (6) and (4), we obtain formulas for the direct Fourier transform:
(10)
and vice versa:
(11)
The function https://pandia.ru/text/78/330/images/image011_19.gif" width="28" height="23 src=">. Thus, the spectrum of a non-periodic signal is continuous (in contrast to the line spectrum of a periodic signal), it is defined on the entire frequency axis.
Properties of the Fourier Transform
Consider the basic properties of the Fourier transform.
Linearity. Let us consider functions and having spectra and :
Then the spectrum of their linear combination will be:
Time delay..gif" width="28" height="23 src=">
(14)
Let's calculate the spectrum of the signal shifted in time: https://pandia.ru/text/78/330/images/image050_1.gif" width="59" height="21">, then
We got that the signal delay is for the time https://pandia.ru/text/78/330/images/image055_1.gif" width="41" height="25">.
Scale change. We assume that the spectrum is known https://pandia.ru/text/78/330/images/image011_19.gif" width="28" height="23 src=">.gif" width="36" height="23 ">. We introduce a new variable , make a substitution integration variable https://pandia.ru/text/78/330/images/image059_1.gif" width="312" height="61"> (16)
Multiplication by https://pandia.ru/text/78/330/images/image041_3.gif" width="40 height=23" height="23"> signal . Find the spectrum of this signal multiplied by .
Thus, multiplying the signal by https://pandia.ru/text/78/330/images/image062_1.gif" width="23" height="24">.
Derivative spectrum. In this case, the key point is the absolute integrability of the function. From the fact that the integral of the modulus of a function must be bounded, it follows that at infinity the function must tend to zero. The integral of the derivative of the function is taken in parts, the resulting non-integral terms are equal to zero, since the function tends to zero at infinity.
(18)
Spectrum of the integral. Let's find the spectrum of the signal https://pandia.ru/text/78/330/images/image065_1.gif" width="81" height="57">, that is, the signal does not have a constant component. This requirement is necessary so that the non-integral terms were equal to zero when the integral is taken by parts.
(19)
Convolution theorem. It is known that https://pandia.ru/text/78/330/images/image067_1.gif" width="37" height="23 src="> feature spectra and https://pandia.ru/text/78 /330/images/image069_1.gif" width="153" height="57"> through and . To do this, in the Fourier integral of the convolution of one of the functions, we will replace the variable , then in the exponent, you can make a replacement 181"> (20)
The Fourier transform of the convolution of two signals gives the product of the spectra of these signals.
Production of signals. It is known that https://pandia.ru/text/78/330/images/image067_1.gif" width="37" height="23 src="> are feature spectra and https://pandia.ru/text/ 78/330/images/image073_1.gif" width="53" height="23"> through spectra and ..gif" width="409" height="123"> (21)
The spectrum of the product of signals is the convolution of the spectra of these signals.
Discrete signal spectrum
Special attention it is worth paying attention to discrete signals, since it is precisely such signals that are used in digital processing. discrete signal unlike continuous, it is a sequence of numbers corresponding to the values of a continuous signal at certain points in time. Conventionally, a discrete signal can be considered as a continuous signal, which at certain moments of time takes on some values, and at other times it is equal to zero. (Fig. 1).
https://pandia.ru/text/78/330/images/image078_1.gif" width="87" height="24"> (22)
Rectangular pulses have a duration https://pandia.ru/text/78/330/images/image079_1.gif" width="19 height=24" height="24">:
(23)
The pulse amplitude is chosen so that the integral of the pulse over the period is . In this case, the clock pulses are dimensionless. We expand the sequence of such impulses into a trigonometric series:
(24)
To get instant signal readings https://pandia.ru/text/78/330/images/image082_1.gif" width="44" height="19">. all will be equal to 1.
(25)
Exactly the same form has the expansion in the Fourier series of the function:
(26)
Coefficients of expansion into a trigonometric series of the clock signal:
(27)
Then the discrete signal will look like:
When calculating the Fourier transform of a discrete signal, we swap the operation of summation and integration, and then use the property δ - Functions:
The spectrum of a discrete signal is a periodic function. Consider the exponential in the individual term as a function of frequency..gif" width="45" height="19">, and this, accordingly, will be the repetition period of the entire spectrum. That is the spectrum of a discrete signal has a repetition period equal to the quantization frequency 
.
Let's get another idea. Due to the fact that it is a product of functions and , the spectrum of a discrete signal is calculated as a convolution of the spectra of a continuous signal https://pandia.ru/text/78/330/images/image094_1.gif" width="37" height="23"> .
(30)
Let us calculate using (25). Since it is a periodic function, its spectrum is discrete.
So convolution (30)
https://pandia.ru/text/78/330/images/image099_1.gif" width="39" height="23 src=">.
The very fact that qualitative changes occur in the signal spectrum as a result of sampling suggests that the original signal can be distorted, since it is completely determined by its spectrum. However, on the other hand, the periodic repetition of the same spectrum in itself does not introduce anything new into the spectrum, therefore, under certain conditions, knowing the signal values at individual points in time, you can find what value this signal took at any other point in time, that is, get original continuous signal. This is the meaning of the Kotelnikov theorem, which imposes a condition on the choice of the quantization frequency in accordance with the maximum frequency in the signal spectrum.
If this condition is violated, then after the digitization of the signal, a periodically repeating spectrum will be superimposed (Fig. 2). The spectrum resulting from the overlay will correspond to another signal.
Rice. 2. Spectra overlap.
Discrete Fourier Transform
In the previous section, it was said that when the condition of the Kotelnikov theorem is fulfilled, the samples of a discrete signal store all information about the original continuous signal, and hence about its spectrum. Therefore, the signal spectrum can also be found from its discrete readings, which provides ample opportunities for signal analysis in digital processing. Previously, it was shown that the spectrum of a periodic signal is discrete, that is, the signal can be decomposed into certain harmonics. A discrete signal has a periodic spectrum. A discrete periodic signal will have a discrete periodic spectrum. A discrete signal is represented as a sequence of signal values at fixed times ..gif" width="19" height="19 src=">, that is, it is performed for any. Usually, the discrete Fourier transform of a signal specified by samples as a vector of elements, calculated by the formula:
(33)
Inverse Fourier transform according to the formula:
(34)
Comparing (33) with (4) we get, that the complex amplitude of the harmonic with the number https://pandia.ru/text/78/330/images/image110_1.gif" width="69" height="43 src="> and corresponds to the frequency 
or, which is the same 
, where the quantization frequency is in hertz: https://pandia.ru/text/78/330/images/image114_0.gif" width="53" height="41 src="> is the quantization period, the period is considered equal to the duration of the recorded fragment signal.
In MATLAB, the discrete Fourier transform is performed using the fft (Fast Fourier Transform) command, which performs calculations using a special fast transform algorithm. Command syntax:
y = fft(x, n, dim)
x is a vector with signal samples;
y - vector with the result of the transformation ;
n is an optional parameter that specifies the number of signal samples used to perform the transformation. In this case, the vector y will consist of n elements;
dim is an optional parameter that specifies the number of the dimension by which the transformation is performed. Used when x contains multiple signals, each in a column or row, as indicated by dim.
A similar interface has a command that performs the inverse transformation:
x = ift(n, n, dim)
The fft command returns an array in which the harmonic amplitudes correspond to harmonic frequencies in the range https://pandia.ru/text/78/330/images/image117_0.gif" width="80" height="48 src=">, more familiar In general, if all values of the vector x are real, which is typical for any measured physical quantity, then, as shown above (9), only harmonics in the frequency range have a value https://pandia.ru/text/78/330 /images/image104_1.gif" width="20" height="24 src="> is exactly one signal period. That is, in this case, the recorded segment of the periodic signal must be periodically continued, while the repetition period must be the duration of the entire signal recording. If the duration of the recording is different from the period of the signal that was recorded, then with the periodic repetition of the recording of the signal, the signal shape will be distorted, respectively, and its spectrum.
For example, a sinusoidal signal with a period was recorded, and the duration of the recording is , and , where is an integer. Then, with periodic repetition of the signal recording (Fig. 3), discontinuities of the first kind will appear, since the signal values at the beginning and end of the recording are different.
https://pandia.ru/text/78/330/images/image054_1.gif" width="13" height="15">. The segment of the recorded signal can also be interpreted as the original signal convolved with a rectangular pulse that determines the segment of time Then, according to the properties of the Fourier transform, the spectrum of the recorded signal will be the product of the original spectrum with the spectrum of a rectangular pulse (Fig. 4).
https://pandia.ru/text/78/330/images/image123.jpg" width="562" height="229 src=">
Rice. 5. Laboratory installation.
Consider each block of this scheme in more detail.
1. The source of analog model signals is the Model Signal Generator. The following devices can be used as it (at the choice of the teacher):
· Standard laboratory signal generator of various shapes (sinusoidal and rectangular pulses);
Digital generator assembled on a digital-to-analog converter (DAC) of the L-Card device ;
· With the help of MATLAB, the signals can be played back on the computer's sound card.
Using MATLAB, it became possible to reproduce signals of almost any shape, the spectrum of which is in the audio range, the possibilities are limited only by the characteristics of the sound card, namely the quantization frequency, frequency response and the maximum possible voltage value. Sound cards designed primarily for sound reproduction have frequency response, which allows you to reproduce the signal in the frequency range from approximately 100 Hz to 20 kHz. These boundaries are determined by the internal device of the sound card, usually filters are used there that limit the signal spectrum in this range. Another feature of the sound card is that most of them can only work with certain sampling frequencies: 8000Hz, 11025Hz, 22050Hz and 44100Hz. Output voltage for different sound cards may differ, but usually the maximum possible value is about 1V. Sound card advantage:
They are in almost any computer;
Supported by many programs, including MATLAB and Simulink.
Flaws:
For different boards, the characteristics can vary greatly;
How measuring device they do not have an accuracy class;
Lack of internal protection circuits (galvanic or optical isolation), which can lead to failure.
2. Analog signals taken from the output of any of the generators listed above are visually controlled on the screen of a cathode-ray oscilloscope. Such control is necessary to observe the shape of the generated signals and set their parameters - amplitude, duration, repetition period, etc.
3. The next element of the experimental setup is a low-pass filter (LPF). This is an analog device that is commonly used in such circuits. Its purpose is to limit the spectrum of the studied signals from above in order to satisfy the conditions of the Kotelnikov theorem. The maximum quantization frequency of the L-Card is 125 kHz, then, from the Kotelnikov theorem, to restore the signal without distortion, the signal spectrum should not exceed fgr:
As instructed by the teacher, you should solder the simplest low-pass filter. Its scheme is shown in Fig. 6.
https://pandia.ru/text/78/330/images/image126_0.gif" width="85" height="41"> (36)
4. Analog-to-digital converter (ADC) - a device for converting analog signals into digital implementations that can be processed on a computer. Our laboratory uses L-Card type L-761 and L-783 ADCs located directly in system unit computer.
Tasks
1. Analytically calculate the spectral functions of periodic signals of a simple form given by the teacher (rectangular video pulse, triangular pulse, exponential pulse, etc.). Construct graphs of the amplitude and phase spectrum of these signals.
2. Perform Fourier analysis of the listed signals in MATLAB using the Fast Fourier Transform (FFT). Construct the corresponding graphs of the amplitude and phase spectra in the region of positive and negative frequencies (using the functions fft, fftshift, stem, after looking at them in the documentation). The amplitudes of harmonics and their frequencies on the graphs must correspond to their values in a given signal. Pay special attention to the influence of the ratio of pulse duration and signal recording time on the signal spectrum, explain the result. For each type of signal in the same coordinates plot the amplitude spectra found analytically (task 1) and numerically calculated.
3. Using the FFT command, find and compare the spectra of segments of a sinusoid consisting of an integer and a non-integer number of periods.
4. Conduct a spectral analysis of a segment of a sinusoid consisting of several periods. See how the spectrum changes depending on the number of periods.
5. Using the L-Graph digital oscilloscope, observe the signal distortion as a result of violation of the Kotelnikov theorem. To do this, connect an analog harmonic signal generator to the L-Card, set the quantization frequency, for example, 20 kHz, and, smoothly changing the generator frequency in the range from 1 kHz to 20 kHz, observe the frequency of the digitized signal, explain the observed effects.
6. Set the quantization frequency to 100kHz, the frequency of the harmonic signal generator to 10kHz, and the amplitude to 1V. Record a segment of a harmonic signal with a duration of 0.01 s and plot its amplitude spectrum in MATLAB. At the same time, the frequencies and amplitudes on the graph should correspond to those that actually exist.
7. Using the results obtained in the first task, approximate a rectangular impulse with a finite number of terms of the trigonometric series. Compare on the same graph the original impulse and the approximated two first harmonics, the first ten harmonics.
Appendix 1. A segment of a sinusoid
To complete one of the tasks, you will need to write a program to calculate the spectrum of a sinusoid, an example of such a program is given below. At the beginning of the program, parameters are defined that specify the duration of the signal in periods and the number of periods. By changing these parameters, you can get various options segment of a sinusoid.
clear, clc, close all
f0 = 1000; % sine frequency
N1 = 20; % duration of the entire track in periods
N2 = 10; % number of readings per period
N3 = 2; % number of periods
N = N1*N2; % number of samples in the entire record
fs = f0*N2; % sample rate
% create a signal
t = (0:(N-1))/fs; % time
x(1:N2*N3) = sin(2*pi*(0:(N2*N3-1))/N2);
% calculate range
X = fftshift(abs(fft(x))/N);
f = (ceil(N/2)-N:ceil(N/2)-1)*fs/N;
subplot(2,1,1), plot(t, x,"k"), xlabel("t, c"), ylabel("x(t)")
subplot(2,1,2), stem(f, X,"k."), xlabel("f, Hz"), ylabel("|X|")
Literature
1. Budylin and Fourier integrals. St. Petersburg State University. 2002.
2., Romanov transformations in MATLAB. SPb. 2007
3. Smirnov of Higher Mathematics (vol.
The shape of the frequency response is nothing more than a spectral image of a damped sinusoidal signal. In addition, as is known, the amplitude-frequency flow characteristic of a single electric oscillatory circuit.
The relationship between the shape of the amplitude-frequency characteristic of certain devices and the properties of the signal is studied in the fundamentals of theoretical electrical engineering and theoretical radio engineering. Briefly, what we should now be interested in from this is the following.
The amplitude-frequency characteristic of the oscillatory circuit coincides in outline with the image frequency spectrum signal that occurs when shock excitation of this oscillatory circuit. To illustrate this point, Fig. 1-3 is shown, which shows a damped sinusoid, which occurs when a shock is applied to an oscillatory circuit. This signal is given in time O m ( A) and spectral ( b) image.
Rice. 1-3
According to the section of mathematics called spectral-temporal transformations, the spectral and temporal images of the same time-varying process are, as it were, synonyms, they are equivalent and identical to each other. This can be compared to translating the same concept from one language into another. Anyone familiar with this branch of mathematics will say that Figures 1-3 A and 1-3 b are equivalent to each other. In addition, the spectral image of this signal obtained by shock excitation of the oscillatory system (oscillatory circuit) is simultaneously geometrically similar to the amplitude-frequency characteristic of this very circuit.
It is easy to see that the graph ( b) in Fig.1-3 is geometrically similar to the graph 3 in Fig.1-1. That is, seeing that as a result of measurements a graph was obtained 3 , I immediately treated it not just as an amplitude-frequency characteristic of sound attenuation in the roof rocks, but also as evidence of the presence of an oscillatory system in the rock mass.
On the one hand, the presence of oscillatory systems in rocks lying in the roof of an underground working did not raise any questions for me, because it is impossible to obtain a sinusoidal (or, in other words, harmonic) signal in other ways. On the other hand, I have never heard of the presence of oscillatory systems in the earth's thickness.
To begin with, let us recall the definition of an oscillatory system. An oscillatory system is an object that reacts to a shock (impulse) action with a damped harmonic signal. Or, in other words, it is an object that has a mechanism for converting an impulse (impact) into a sinusoid.
The parameters of a damped sinusoidal signal are the frequency f 0 and quality factor Q , the value of which is inversely proportional to the attenuation coefficient. As can be seen from Fig. 1-3, both of these parameters can be determined from both the temporal and the spectral image of this signal.
Spectral-temporal transformations are an independent section of mathematics, and one of the conclusions that we must draw from the knowledge of this section, as well as from the shape of the amplitude-frequency characteristic of the rock mass sound conductivity shown in Fig. 1-1 (curve 3), is that that, in terms of acoustic properties, the studied rock mass showed the property of an oscillatory system.
This conclusion is quite obvious to anyone who is familiar with spectral-temporal transformations, but is categorically unacceptable to those who are professionally involved in acoustics of solid media, seismic exploration, or geophysics in general. It so happened that in the course of training students of these specialties this material is not given.
As is known, in seismic exploration it is considered that the only mechanism that determines the shape of a seismic signal is the propagation of the field of elastic oscillations according to the laws of geometric optics, its reflection from the boundaries lying in the earth's thickness and interference between the individual components of the signal. It is believed that the shape of seismic signals is due to the nature of the interference between many small echo signals, that is, reflections from many small boundaries lying in the mountain range. In addition, it is believed that with the help of interference, a signal of any shape can be obtained.
Yes, this is all true, but the fact of the matter is that a harmonic (including harmonic damped) signal is an exception. It is impossible to get it by interference.
A sinusoid is an elementary information brick that cannot be decomposed into simpler components, because a signal in nature does not exist easier than a sinusoid. That is why, by the way, the Fourier series is a collection of precisely sinusoidal terms. Being an elementary, indivisible information element, a sinusoid cannot be obtained by adding (interference) any other, even simpler components.
You can get a harmonic signal in one and only way - namely, by influencing the oscillatory system. When shock (impulse) impact on the oscillatory system occurs damped sinusoid, and periodic or noise exposure - undamped sinusoid. And therefore, having seen that the amplitude-frequency characteristic of an object is geometrically similar to the spectral image of a harmonic damped signal, it is no longer possible to treat this object otherwise than as an oscillatory system.
Before taking my first measurements in a mine, I, like all other people working in the field of acoustics of solid media and seismic exploration, was convinced that there are no oscillatory systems in the rock mass and cannot be. However, having discovered such an amplitude-frequency characteristic of attenuation, I simply had no right to remain with this opinion.
Carrying out measurements similar to those described above is very laborious, and the processing of the results of these measurements takes a long time. Therefore, when I saw that the nature of the sound conductivity of the rock mass is an oscillatory system, I realized that another measurement scheme should be used, which is used in the study of oscillatory systems, and which we use to this day. According to this scheme, the source of the probing signal is an impulse (impact) impact on the rock mass, and the receiver is a seismic receiver, specially designed for spectral seismic measurements. The scheme of indication and processing of the seismic signal makes it possible to observe it both in temporal and spectral form.
Applying this measurement scheme at the same point of the underground working as in our first measurement, we made sure that when the rock mass of the roof is impacted, the signal that occurs in this case really has the form of a damped sinusoid, similar to that shown in Fig. 1 -3 a, and its spectral image is similar to the graph shown in Fig. 1-3 b.
Most often it happens that the seismic signal contains not one, but several harmonic components. However, no matter how many harmonic components, they all arise solely due to the presence of an appropriate number of oscillatory systems.
Multiple studies of seismic signals obtained in a variety of conditions - both in underground workings, and on the earth's surface, and in the conditions of a sedimentary cover, and in the study of crystalline basement rocks - have shown that in all possible cases there are no signals received not as a result of the presence of oscillatory systems, but as a result of interference processes.
Fourier images - complex coefficients of the Fourier series F(j w k) periodic signal (1) and spectral density F(j w) non-periodic signal (2) - have a number of common properties.
1. Linearity . Integrals (1) And (2) carry out linear transformation functions f(t). Therefore, the Fourier image of a linear combination of functions is equal to a similar linear combination of their images. If f(t) = a 1 f 1 (t) + a 2 f 2 (t), That F(j w) = a 1 F 1 (j w) + a 2 F 2 (j w), where F 1 (j w) and F 2 (j w) - Fourier images of signals f 1 (t) And f 2 (t), respectively.
2. Delay (changing the origin of time for periodic functions) . Consider the signal f 2 (t), delayed for a while t 0 relative to signal f 1 (t) which has the same form: f 2 (t) = f 1 (t – t 0). If the signal f 1 has a picture F 1 (j w), then the Fourier image of the signal f 2 equals F 2 (j w) == . Multiplying and dividing by , we group the terms as follows:
Since the last integral is F 1 (j w), then F 2 (j w) = e -j w t 0 F 1 (j w) . Thus, when the signal is delayed for a time t 0 (changing the origin of time), the modulus of its spectral density does not change, and the argument decreases by w t 0 proportional to the delay time. Therefore, the amplitudes of the signal spectrum do not depend on the origin, and the initial phases with a delay of t 0 decrease by w t 0 .
3. Symmetry . For valid f(t) image F(j w) has conjugate symmetry: F(– j w) = . If f(t) is an even function, then Im F(j w) = 0; for the odd function Re F(j w) = 0. Module | F(j w)| and the real part of Re F(j w) - even frequency functions, argument arg F(j w) and Im F(j w) - odd.
4. Differentiation . From the direct transformation formula, integrating by parts, we obtain the connection of the image of the derivative of the signal f(t) with the image of the signal itself
For an absolutely integrable function f(t) the non-integral term is equal to zero, and, therefore, at , and the last integral represents the Fourier image of the original signal F(j w) . Therefore, the Fourier image of the derivative df/dt is related to the image of the signal itself by the relation j w F(j w) - when differentiating a signal, its Fourier image is multiplied by j w. The same relation is valid for the coefficients F(j w k), which are determined by integration within finite limits from – T/2 to + T/2. Indeed, the product within the appropriate limits
Since, due to the periodicity of the function f(T/2) = f(– T/2), a = = = (– 1) k, then in this case the term outside the integral vanishes, and the formula
where the arrow symbolically denotes the operation of the direct Fourier transform. This relation can also be generalized to multiple differentiation: for n-th derivative we have: d n f/dt n (j w) n F(j w).
The obtained formulas allow us to find the Fourier image of the derivatives of a function from its known spectrum. It is also convenient to use these formulas in cases where, as a result of differentiation, we arrive at a function whose Fourier image is calculated more simply. So if f(t) is a piecewise linear function, then its derivative df/dt is a piecewise constant, and for it the direct transformation integral can be found elementarily. To obtain the spectral characteristics of the integral of the function f(t) its image should be divided into j w.
5. The duality of time and frequency . Comparison of the integrals of the direct and inverse Fourier transforms leads to the conclusion about their peculiar symmetry, which becomes more obvious if the formula for the inverse transformation is rewritten, transferring the factor 2p to the left side of the equation:
For signal f(t), which is an even function of time f(– t) = f(t) when the spectral density F(j w) - real value F(j w) = F(w), both integrals can be rewritten in the trigonometric form of the cosine Fourier transform:
With mutual replacement t and w the integrals of direct and inverse transformations transform into each other. From this it follows that if F(w) represents the spectral density of an even function of time f(t), then the function 2p f(w) is the spectral density of the signal F(t). For odd functions f(t) [f(t) = – f(t)] spectral density F(j w) purely imaginary [ F(j w) = jF(w)]. Fourier integrals in this case are reduced to the form of sine transforms, from which it follows that if the spectral density jF(w) corresponds to an odd function f(t), then the value j 2p f(w) represents the spectral density of the signal F(t). Thus, the graphs of the time dependence of the signals of these classes and its spectral density are dual to each other.
Integral (1)
Integral (2)
In radio engineering, the spectral and temporal representation of signals is widely used. Although signals are random processes by their nature, however, individual implementations of a random process and some special (for example, measuring) signals can be considered deterministic (that is, known) functions. The latter are usually divided into periodic and non-periodic, although strictly periodic signals do not exist. A signal is called periodic if it satisfies the condition
on a time interval , where T is a constant value, called a period, and k is any integer.
The simplest example of a periodic signal is a harmonic oscillation (or harmonic for short).
where is the amplitude, = is the frequency, is the circular frequency, is the initial phase of the harmonic.
The importance of the concept of harmonics for the theory and practice of radio engineering is explained by a number of reasons:
- harmonic signals retain their shape and frequency when passing through stationary linear electrical circuits(for example, filters), changing only the amplitude and phase;
- harmonic signals are quite simply generated (for example, using LC oscillators).
A non-periodic signal is a signal that is non-zero over a finite time interval. A non-periodic signal can be considered as periodic, but with an infinitely large period. One of the main characteristics of a non-periodic signal is its spectrum. The signal spectrum is a function that shows the dependence of the intensity of various harmonics in the signal composition on the frequency of these harmonics. The spectrum of a periodic signal is the dependence of the coefficients of the Fourier series on the frequency of the harmonics to which these coefficients correspond. For a non-periodic signal, the spectrum is the direct Fourier transform of the signal. So, the spectrum of a periodic signal is a discrete spectrum (a discrete function of frequency), while a non-periodic signal is characterized by a continuous spectrum (continuous) spectrum.
Let us pay attention to the fact that discrete and continuous spectra have different dimensions. The discrete spectrum has the same dimension as the signal, while the dimension of the continuous spectrum is equal to the ratio of the signal dimension to the frequency dimension. If, for example, the signal is represented by an electrical voltage, then the discrete spectrum will be measured in volts [V] and the continuous spectrum in volts per hertz [V/Hz]. Therefore, the term "spectral density" is also used for the continuous spectrum.
Consider first the spectral representation of periodic signals. It is known from the course of mathematics that any periodic function that satisfies the Dirichlet conditions (one of the necessary conditions is the condition that the energy be finite) can be represented by a Fourier series in trigonometric form:
where determines the average value of the signal over the period and is called the constant component. The frequency is called the fundamental frequency of the signal (the frequency of the first harmonic), and the multiples of it are called the higher harmonics. Expression (3) can be represented as:
The inverse relationships for the coefficients a and b have the form
Figure 1 shows a typical view of the graph of the amplitude spectrum of a periodic signal for the trigonometric form of series (6):
Using an expression (Euler's formula).
instead of (6), we can write the complex form of the Fourier series:
where the coefficient is called the complex amplitudes of the harmonics, the values of which, as follows from (4) and the Euler formula, are determined by the expression:
Comparing (6) and (9), we note that when using the complex form of the Fourier series, negative values of k allow us to speak of components with "negative frequencies". However, the appearance of negative frequencies is of a formal nature and is associated with the use of a complex notation to represent a real signal.
Then instead of (9) we get:
has the dimension [amplitude / hertz] and shows the signal amplitude per band of 1 Hertz. Therefore, this continuous frequency function S(jw) is called the spectral density of complex amplitudes or simply the spectral density. We note one important circumstance. Comparing expressions (10) and (11), we notice that for w=kwo they differ only by a constant factor, and
those. complex amplitudes of a periodic function with period T can be determined from the spectral characteristic of a non-periodic function of the same form, given in the interval . The above is also true with respect to the modulus of the spectral density:
It follows from this relationship that the envelope of the continuous amplitude spectrum of a non-periodic signal and the envelope of the amplitudes of the line spectrum of a periodic signal coincide in shape and differ only in scale. Let us now calculate the energy of the non-periodic signal. Multiplying both parts of inequality (14) by s(t) and integrating in infinite limits, we get:
where S(jw) and S(-jw) are complex conjugate quantities. Because
This expression is called Parseval's equality for a non-periodic signal. It determines the total energy of the signal. It follows that there is nothing more than the signal energy per 1 Hz of the frequency band around the frequency w. Therefore, the function is sometimes called the spectral energy density of the signal s(t). We now present, without proof, several theorems on spectra expressing the main properties of the Fourier transform.
Loading...