1.2 Spectral characteristics of signals
Signals used in radio engineering have a rather complex structure. Describing such signals mathematically is difficult. Therefore, to simplify the procedure for analyzing signals and passing them through radio engineering circuits, a technique is used that provides for the decomposition of complex signals into a set of idealized mathematical models described by elementary functions.
Harmonic spectral analysis of periodic signals involves an expansion in a Fourier series in trigonometric functions - sines and cosines. These functions describe harmonic oscillations that retain their shape during the conversion by linear devices (only the amplitude and phase change), which makes it possible to use the theory of oscillatory systems to analyze the properties of radio engineering circuits.
The Fourier series can be represented as
Practical application has another form of writing the Fourier series
where is the amplitude spectrum;
- phase spectrum.
Complex form of the Fourier series
The above formulas are used to obtain the spectral response of a periodic signal. Fourier transforms are used to obtain the spectrum of a non-periodic signal.
Direct Fourier Transform
Inverse Fourier Transform
Expressions (1.5), (1.6) are the main relations for obtaining spectral characteristics.
1.3 Properties of the Fourier transform
The formulas of the direct and inverse Fourier transforms make it possible to determine its spectral density S (jω) from the signal s (t) and, if necessary, to determine the signal s (t) from the known spectral density S (jω). To indicate this correspondence between the signal and its spectrum, the symbol s (t) ↔ S (jω) is used.
Using the properties of Fourier transforms, you can determine the spectrum of the modified signal by transforming the spectrum of the original signal.
Basic properties:
1. Linearity
s 1 (t) ↔ S 1 (jω)
s n (t) ↔ S n (jω)
_____________________
We use the direct Fourier transform
Final Result
Conclusion: the direct Fourier transform is a linear operation, has the properties of homogeneity and additivity. Therefore, the spectrum of the sum of the signals is equal to the sum of the spectra.
2. Spectrum of the time-shifted signal
s (t ± t 0) ↔ S c (jω)
Final Result
Conclusion: a shift of the signal in time by an amount of ± t 0 leads to a change in the phase characteristic of the spectrum by an amount of ± ωt 0. The amplitude spectrum does not change.
3. Changing the scale in time
s (αt) ↔ S m (jω)
Final Result
Conclusion: when the signal is compressed (expanded) in time by a certain number, its spectrum is expanded (compressed) by the same amount along the frequency axis with a proportional decrease (increase) in the amplitudes of its components.
4. Spectrum of the derivative
ds (t) / dt↔ S п (jω).
To determine the spectrum of the signal derivative, we take the time derivative of the right and left sides of the inverse Fourier transform:
Final Result
Conclusion: the spectrum of the signal derivative is equal to the spectrum of the original signal multiplied by jω. In this case, the amplitude spectrum changes in proportion to the change in frequency, and a constant component is added to the phase characteristic of the original signal, equal to π / 2 at ω> 0 and equal to -π / 2 at ω
5. Spectrum of the integral
Take the integral of the right and left sides of the inverse Fourier transform
Comparing the result with the inverse Fourier transform, we obtain
Final Result
Conclusion: the spectrum of a signal equal to the integral of the original signal is equal to the spectrum of the original signal divided by jω. In this case, the amplitude spectrum changes in inverse proportion to the change in frequency, and a constant component is added to the phase characteristic of the original signal, equal to π / 2 at ω 0.
6. Spectrum of the product of two signals
s 1 (t) ↔ S 1 (jω)
s 2 (t) ↔ S 2 (jω)
s 1 (t) s 2 (t) ↔ S pr (jω).
Find the spectrum of the product of two signals using the inverse Fourier transform
Final Result
Conclusion: The spectrum of the product of two signals is equal to the convolution of their spectra, multiplied by a factor of 1 / (2π).
In the course of calculating the signal spectra, the linearity and integral properties of the signal will be used.
1 .4 Classification and properties of radio circuits
In the theoretical foundations of radio engineering, an important place is occupied by methods of analysis and synthesis of various radio engineering circuits. In this case, a radio engineering circuit is understood as a set of passive and active elements connected in a certain way, ensuring the passage and functional transformation of signals. Passive elements are resistors, capacitors, inductors and means for connecting them. Active elements are transistors, vacuum tubes, power supplies and other elements capable of generating energy, increasing the signal power. If there is a need to emphasize the functionality of a circuit, the term device is used instead of the term circuit. Radio engineering circuits used to convert signals are very diverse in their composition, structure and characteristics. In the process of their development and analytical research, various mathematical models are used that meet the requirements of adequacy and simplicity. In the general case, any radio engineering circuit can be described by a formalized relationship that determines the transformation of the input signal x (t) into the output y (t), which can be symbolically represented as
where T is an operator indicating the rule according to which the transformation of the input signal is carried out.
Thus, a set of operator T and two sets X = (), Y = () of signals at the input and output of the circuit can serve as a mathematical model of a radio engineering circuit so that
By the type of conversion of input signals into outputs, i.e. by the type of operator T, the radio circuits are classified.
1. An electronic circuit is linear if the operator T is such that the circuit satisfies the conditions of additivity and homogeneity.
It is characteristic that the linear transformation of a signal of any shape is not accompanied by the appearance of harmonic components with new frequencies in the spectrum of the output signal, i.e. linear transformation does not lead to an enrichment of the signal spectrum.
2. A radio engineering circuit is nonlinear if the operator T does not ensure the fulfillment of the additivity and homogeneity conditions. The functioning of such circuits is described by nonlinear differential equations, i.e. equations, at least one coefficient of which is a function of the input signal or its derivatives. Non-linear circuits do not satisfy the superposition principle. When analyzing the passage of signals through a non-linear circuit, the result is defined as the response to the signal itself. It cannot be decomposed into simpler signals. At the same time, nonlinear circuits have a very important property - to enrich the signal spectrum. This means that with nonlinear transformations in the spectrum of the output signal, harmonic components appear with frequencies that were not in the spectrum of the input signal. The appearance of components with frequencies equal to the combination of frequencies of harmonic components of the spectrum of the input signal is also possible. This property of nonlinear circuits has led to their use for solving a wide class of problems associated with the generation and conversion of signals. Structurally, linear circuits contain only linear elements, which include non-linear elements operating in a linear mode (on the linear sections of their characteristics). Linear circuits are linear amplifiers, filters, long lines, delay lines, etc. Nonlinear circuits contain one or more nonlinear elements. Nonlinear circuits include generators, detectors, modulators, multipliers and frequency converters, limiters, etc.
To simplify methods for solving problems of network analysis, signals are presented as the sum of certain functions.
This process is substantiated by the concept of a generalized Fourier series. In mathematics, it has been proven that any function satisfying the Dirichlet conditions can be represented as a series:
To determine, we multiply the left and right sides of the series by and take the integral of the left and right sides:
for the interval in which the orthogonality conditions are satisfied.
It can be seen that. Received an expression for the generalized Fourier series:
Let's select a specific kind of function for the signal series expansion. As such a function, we choose the orthogonal system of functions:
To determine the series, we calculate the value:
Thus, we get:
Graphically, this series is presented in the form of two graphs of amplitude harmonic components.
The resulting expression can be represented as:
Received 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 us find the complex form of the Fourier series, for this we use the Euler formulas:
The spectrum in this form is graphically represented on the frequency axis in the range.
It is obvious that 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 periodic with a period. Let's use the transformation of the Fourier series for this period. We get for:
Analysis of the obtained expression shows that at the amplitudes of the components become infinitesimal and they are located continuously on the frequency axis. Then, to get out of this situation, we use the concept of spectral density:
Substituting the resulting expression in the complex Fourier series, we get:
We finally 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:
Fourier transform properties
From the formulas of the direct and inverse Fourier transforms, it is obvious that if the signal changes, then its spectrum will also change. The following properties establish the dependence of the spectrum of the altered signal, from the spectrum of the signal before the changes.
1) The property of linearity of the Fourier transform
We got that the spectrum of the sum of the signals is equal to the sum of their spectra.
2) Spectrum of the time-shifted signal
We found that when the signal is shifted, the amplitude spectrum does not change, but only the phase spectrum changes by an amount
3) Change the time scale
that is, when expanding (narrowing) the signal 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) Property of duality
Thus, if a spectrum corresponds to a signal, then a signal in shape that coincides with the above spectrum corresponds to a spectrum in shape that coincides with the above signal.
The shape of the amplitude-frequency characteristic is nothing more than a spectral image of the decaying sinusoidal signal. In addition, as is known, the amplitude-frequency transmission characteristic of a single electric oscillatory circuit has a similar shape.
The relationship between the shape of the frequency response of certain devices and the properties of the signal is studied in the foundations of theoretical electrical engineering and theoretical radio engineering. In short, what should interest us now from this is as follows.
The amplitude-frequency characteristic of the oscillatory circuit in its outlines coincides with the image of the frequency spectrum of the signal that arises during shock excitation of this oscillatory circuit. To illustrate this point, Fig. 1-3 is shown, which shows a damped sinusoid that occurs when an impact 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 branch 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 to another. Anyone familiar with this section of mathematics will tell you that Figures 1-3 a and 1-3 b are equivalent to each other. In addition, the spectral image of this signal obtained upon shock excitation of the oscillatory system (oscillatory circuit) is simultaneously geometrically similar to the amplitude-frequency characteristic of this circuit itself.
It is easy to see that the graph ( b) in Figure 1-3 is geometrically similar to the graph 3 See Figure 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 rocks of the roof, 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 mine 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 before.
To begin with, let us recall the definition of an oscillatory system. An oscillatory system is an object that reacts to a shock (impulse) effect with a damped harmonic signal. Or, in other words, it is an object with a mechanism for converting an impulse (shock) into a sinusoid.
The parameters of the decaying sinusoidal signal are the frequency f 0 and quality factor Q , the value of which is inversely proportional to the damping coefficient. As can be seen from Fig. 1-3, both of these parameters can be determined from both the temporal and spectral images of this signal.
Spectral-temporal transformations are an independent branch of mathematics, and one of the conclusions that we must draw from the knowledge of this section, as well as from the form of the amplitude-frequency characteristic of the sound conductivity of the rock mass, shown in Fig. 1-1 (curve 3), is that , that the acoustic properties of the rock mass under study showed the property of an oscillatory system.
This conclusion is quite obvious to anyone who is familiar with spectral-temporal transformations, but it is categorically unacceptable for those who are professionally engaged in solid media acoustics, seismic exploration, or geophysics in general. It so happened that this material is not given in the course of teaching students of these specialties.
As you know, in seismic exploration it is generally accepted that the only mechanism that determines the shape of the seismic signal is the propagation of the elastic vibration field according to the laws of geometric optics, its reflection from the boundaries lying in the earth's thickness and the interference between individual signal components. It is believed that the shape of the seismic signals is due to the nature of the interference between many small echoes, that is, reflections from many small boundaries occurring in the mountain range. In addition, it is believed that any waveform can be obtained using interference.
Yes, this is all true, but the fact of the matter is that a harmonic (including harmonic decaying) 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 simpler 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 acting on the oscillatory system. With a shock (impulse) impact on the oscillatory system, a damped sinusoid arises, and with a periodic or noise impact, an undamped sinusoid appears. And therefore, having seen that the amplitude-frequency characteristic of a certain object is geometrically similar to the spectral image of a harmonic damped signal, it is no longer possible to relate to this object otherwise than to an oscillatory system.
Before taking my first measurements in the mine, I, like all other people working in the field of solid media acoustics 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 processing the results of these measurements is time-consuming. Therefore, having seen that the rock mass is an oscillatory system in terms of sound conductivity, I realized that a different 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 sounding signal is an impulse (shock) impact on the rock mass, and the receiver is a seismic receiver specially designed for carrying out spectral seismic measurements. The scheme of indication and processing of the seismic signal makes it possible to observe it both in time and in spectral form.
Applying this measurement scheme at the same point of the underground workings as in our first measurement, we made sure that when the rock mass of the roof is hit by a shock, the signal arising 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, the seismic signal contains not one, but several harmonic components. However, no matter how many harmonic components there are, they all arise exclusively 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 rocks of the crystalline basement - have shown that in all possible cases, signals received not as a result of the presence of oscillatory systems, and as a result of interference processes, does not exist.
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) perform a linear transformation of the function 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), then 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 (change the time reference for periodic functions) . Consider the signal f 2 (t), delayed for a time 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 an image F 1 (j w), then the Fourier image of the signal f 2 equals F 2 (j w) = = . After 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 (change in the origin of time), the modulus of its spectral density does not change, and the argument decreases by the value 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 an odd function Re F(j w) = 0. Module | F(j w) | and the real part Re F(j w) - even functions of frequency, argument arg F(j w) and Im F(j w) are odd.
4. Differentiation . From the direct transformation formula, integrating by parts, we obtain the connection between the image of the signal derivative f(t) with the image of the signal itself
For an absolutely integrable function f(t) outside the 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 associated with the image of the signal itself by the ratio j w F(j w) - when differentiating the 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 relationship is generalized to multiple differentiation: for n-th derivative we have: d n f/dt n (j w) n F(j w).
The formulas obtained make it possible to find the Fourier image of the derivatives of a function from its known spectrum. It is also convenient to apply these formulas in cases when, as a result of differentiation, we arrive at a function, the Fourier image of which is calculated more simply. So if f(t) is a piecewise linear function, then its derivative df/dt- piecewise constant, and for it the integral of the direct transformation is found elementary. To obtain the spectral characteristics of the integral of the function f(t) its image should be divided by 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 inverse transformation formula is rewritten by transferring the factor 2p to the left side of the equality:
For signal f(t), which is an even function of time f(– t) = f(t) when the spectral density F(j w) is a real value F(j w) = F(w), both integrals can be rewritten in trigonometric form by the Fourier cosine transform:
When interchangeable t and w, the integrals of the direct and inverse transformations transform into each other. Hence 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)]. In this case, the Fourier integrals 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 quantity j 2p f(w) represents the spectral density of the signal F(t). Thus, the graphs of the time dependence of the signals of the indicated classes and its spectral density are dual to each other.
Integral (1)
Integral (2)
Spectral and temporal representation of signals is widely used in radio engineering. Although signals are inherently random processes, however, individual implementations of a random process and some special (for example, measurement) 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, called a period, and k is any integer.
The simplest example of a periodic signal is a harmonic wave (or harmonic for short).
where is the amplitude, = is the frequency, is the angular 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 can be easily generated (eg with LC auto-generators).
A non-periodic signal is a signal that is nonzero at a finite time interval. A non-periodic signal can be considered as a periodic one, 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, 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 (discrete function of frequency), while a non-periodic signal is characterized by a continuous spectrum (continuous) spectrum.
Note that the 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.
Let us first consider the spectral representation of periodic signals. It is known from the course of mathematics that any periodic function satisfying the Dirichlet conditions (one of the necessary conditions is the condition that the energy is 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 its multiples are called the higher harmonics. Expression (3) can be represented as:
The inverse dependences for the coefficients a and b have the form
Figure 1 shows a typical graph of the spectrum of the amplitudes of a periodic signal for the trigonometric form of the 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 writing the Fourier series, negative values of k allow us to speak of components with "negative frequencies". However, the appearance of negative frequencies has a formal character and is associated with the use of a complex notation to represent a valid signal.
Then instead of (9) we get:
has the dimension [amplitude / hertz] and shows the signal amplitude per 1 Hertz bandwidth. Therefore, this continuous function of the frequency S (jw) is called the spectral density of the complex amplitudes or simply the spectral density. Let us note one important circumstance. Comparing expressions (10) and (11), we notice that for w = kwo they differ only by a constant factor, and
those. the complex amplitudes of a periodic function with a period T can be determined from the spectral characteristic of a non-periodic function of the same shape, specified in the interval. The same is true for the spectral density modulus:
It follows from this relationship that the envelope of the continuous amplitude spectrum of the non-periodic signal and the envelope of the amplitudes of the line spectrum of the periodic signal coincide in shape and differ only in scale. Let us now calculate the energy of the non-periodic signal. Multiplying both sides of inequality (14) by s (t) and integrating in infinite limits, we obtain:
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 from this 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 that express the basic properties of the Fourier transform.
The spectral characteristics are used to estimate the internal composition (spectrum) of the signal. For this, the signal x (t) represent in the form of a generalized Fourier series, expanding it in terms of the system of basis functions T k (t)
where C to - constant coefficients reflecting the contribution of the function H ^ (?) to the formation of signal values at the considered time interval.
The ability to represent a complex signal x (t) in the form of a sum of simple signals "RDO turns out to be especially important for linear dynamic systems. In such systems, superposition principle, i.e. their reaction to the sum of influences (signals) is equal to the sum of reactions to each of the influences separately. Therefore, knowing the reaction of a linear system to a simple signal, it is possible, by summing up the results, to determine its reaction to any other complex signal.
Selecting functions Y k (t) subject to the requirements of maximum signal approximation accuracy x (t) series (7.21) with the minimum number of terms of this series and, if possible, reducing the computational difficulties arising in determining the coefficients of the series C.
As basis functions, the most widely used are real trigonometric functions
and complex exponential functions
The classical spectral analysis of signals is based on them. At the same time, it is possible to use other systems of basis functions (functions of Taylor, Walsh, Laguerre, Hermite, Legendre, Chebyshev, Kotelnikov, etc. 121), which in a number of cases allows, taking into account the specifics of the approximated function x (t), reduce the number of terms in series (7.21) while maintaining the specified approximation error.
In recent years, a new, very promising system of basic functions has appeared, called wavelets. Unlike harmonic functions, they are able, by changing their shape and properties, to adapt to the local features of the approaching signal. As a result, it becomes possible to simply represent complex signals (including those with local jumps and discontinuities) by sets of wavelets of one type or another.
When using trigonometric basis functions (7.22), series (7.21) takes the form of the classical trigonometric Fourier series
where Q = 2p / T is the frequency of the fundamental harmonic of the series (G is the signal period); k = 1, 2, 3, ... is an integer; ak, bk are real numbers (Fourier coefficients) calculated according to the formulas
In these formulas, as before (see (7.20)), t 0 - an arbitrary number that can be chosen for reasons of convenience in calculating integrals (7.25), since the values of these integrals of the quantity t 0 do not depend; x T (t) - basic signal pulse (see fig. 7.3, v).
Coefficient a 0 determines the doubled average (over the period) signal value, the remaining coefficients a k> b k (k= 1, 2, 3, ...) - contribution To-th harmonic of the Fourier series (7.24) in the formation of instantaneous signal values X(?).
The trigonometric Fourier series (7.24) can be written in two other forms: in the form of an expansion in sines
and in the form of an expansion in cosines
where A 0/2 = a 0/2 - constant component of the signal; A k - amplitude k-and harmonics of the series, calculated by the formula
The initial phases of these harmonics are calculated from the relations
The set of amplitudes of the harmonic components of a periodic signal (A to) °? = ( called amplitude spectrum this signal. The set of the initial phases of these components (φ / ^) ^ = 1 - phase spectrum signal.
Using the 5-Dirac function 8 (?), Both spectra can be represented lattice functions frequency
tp the amplitude and phase spectra of the periodic signal are discrete spectra. This distinguishes the periodic signal from other continuous-spectrum signals.
Thus, a periodic signal can be represented as a sum of harmonics (7.24). In this case, the frequency of each harmonic component of the Fourier series is a multiple of the fundamental frequency λ2, which depends on the signal period T.
The more such harmonics, the smaller the error in the approximation of the function x (t) the finite sum of the Fourier series (7.24). An exception are the points of discontinuity of the function x (i). In the vicinity of such points, the so-called Gibbs phenomenon| 2 |. According to this phenomenon, in the vicinity of the discontinuity points, the finite sums of the Fourier series
form oscillating "tails", the height of which does not decrease with an increase in the number of taken into account harmonics of the Fourier series N - it is about 9% of the value of the jump of the function x (t) at the break point.
To calculate the amplitude and initial phase of the kth harmonic of a periodic signal, instead of formulas (7.28) and (7.29), one can use the formulas
where X t = X t (p) = L (x T (t)) index T variable X - Laplace image of the basic signal pulse, determined by the formula (see Appendix 2)
i - imaginary unit; & = 0,1,2, ... is a positive integer. The use of these formulas eliminates the need to calculate the integrals (7.25), which greatly simplifies the calculations. Let's show an example of such a calculation.
Example 7.1
Determine the amplitude spectrum of a periodic signal Solution
In fig. 7.3, a, a graph of such a signal is shown. It can be seen that the signal has a period T= me. Consequently, the frequency of the fundamental harmonic of the corresponding Fourier series (7.24) is equal to Q = 2p / T = 2 s -1. Taking t 0 = 0, x T (t) = sin? (for 0 t 
Rice. 73.
a - waveform; b - amplitude spectrum of the signal
Hence, A 0/2 = 2 / n, A k= 4 / i (4 & 2 - 1), SCH= l, where k= 1,2,3, i.e. expansion of the function | sin (?) | in the trigonometric Fourier series has the form
Note: here we have taken f / = l (a ns y k = 0) due to the use of the minus sign in front of the sum of the harmonics of the series.
In fig. 7.3, b shows the amplitude spectrum of the signal under consideration. The value of the amplitude of the? -Th harmonic of the series And to represented by a vertical segment of the corresponding length, at the base of which the harmonic number is indicated.
It should be borne in mind that the amplitudes And to some harmonics of the Fourier series may be zero. In addition, a monotonic decrease in the amplitudes of these harmonics with an increase in the harmonic number is optional, as is the case in Fig. 7.3, b.
However, in all cases the condition lim And to= 0, which follows from the requirement
convergence of the Fourier series.
Let's solve the problem using formulas (7.32). To do this, we first find the Laplace image of the basic signal pulse x T (t)
Substituting here p = ikQ = 2ik(where i- imaginary unit, k= 1, 2, 3, ...), we get that coincides with the previous results.
In technical applications, a complex form of notation of the Fourier series is often used
In this case, complex exponential functions (7.23) are used as basis functions. Therefore, the coefficients C n rows (7.36) become complex... They are calculated by the formula
where, as in formula (7.6), the index variable P can be either a positive or negative integer.
When using the complex form of the Fourier series (7.36) amplitude spectrum periodic signal x (t) call the set of absolute values of the complex Fourier coefficients C n
a spectrum of phases- many main arguments of these coefficients
Many quantities (WITH%) ^> = _ is called power spectrum periodic signal, and the set of complex numbers (C n - spectral sequence periodic signal. It is these three characteristics (amplitude spectrum, phase spectrum and power spectrum) that relate to the main spectral characteristics of a periodic signal.
In contrast to the amplitude and phase spectra of a periodic signal, represented in the form of a trigonometric Fourier series (7.24), the spectra of the same signal, plotted using complex Fourier coefficients (7.37), turn out to be double-sided. This is a consequence of the presence in (7.36) of "negative frequencies" on.(for negative values P). The latter, of course, do not exist in reality. They only reflect the representation of the exponential harmonic function used in the formation of the complex Fourier series f ~ t in the form of a unit vector rotating clockwise with an angular velocity ω.
If there is a Laplace image of the basic pulse of a periodic signal X T (p) = L (x T (t)), then the amplitude spectrum and the phase spectrum of the periodic signal can be calculated by the formulas
Algorithms of the so-called fast Fourier transform, thanks to which it is possible to reduce the calculation time of the Fourier coefficients so much that the spectra of signals during their processing are obtained practically in real time.
In conclusion, we note the three most important properties of the spectral characteristics of a periodic signal.
- 1. If x (t) - is an even function, then the imaginary components of all complex Fourier coefficients Im (C w) are equal to zero and, on the contrary, if this function is odd, then the real components of all complex Fourier coefficients Re (C „) are equal to zero.
- 2. At the break point of the first kind t = t r functions x (t) Fourier series sum S (t) is equal to the half-sum of the limiting values of the function as the argument approaches the discontinuity point t = t r left and right, i.e.
Note: if the function values x (€) at the ends + D) of the basic impulse x T (t) are not equal to each other, then with a periodic continuation of the pulse, these points become points of discontinuity of the first kind.
3. The powers of the periodic signal in the time and frequency domains are equal to each other, i.e.
This ratio expresses Parseval's theorem.
The presence in the formula (7.36) of "negative frequencies" nQ.(for yy
