The following methods of organizing radio channels (radio technologies) are currently known: FDMA, TDMA, CDMA, FH-CDMA. Combinations of these are possible (for example, FDMA/TDMA). The timing of the application of these technologies largely coincides with the stages of development of mobile communication systems. The first generation of mobile radiotelephone equipment used frequency division multiple access (FDMA) technology. FDMA radio technology has so far been successfully used in advanced equipment cellular communication first generation, as well as in simpler mobile radiotelephone systems with a non-cellular structure. As for the mobile communication standards of the first stage, the concept of standards was not used for the first radial systems, and the equipment differed according to the names of the systems (Altai, Volemot, Actionet, etc.). Cellular communication systems have begun to differ in standards. FDMA technology is the basis for such standards of first generation cellular communication systems as NMT-450, NMT-900, AMPS, TACS. Second generation cellular mobile communication systems made the transition to digital processing of transmitted voice messages, which began to use time division multiple access (TDMA) radio technology. As a result of the transition to TDMA: the noise immunity of the radio path has increased, its protection from eavesdropping has become better, etc. TDMA is used in systems with standards such as GSM, D-AMPS(the latter is often referred to simply as TDMA in the American version). Code division multiple access radio technology CDMA, or in the English version CDMA, has been actively introduced on public radiotelephone networks only in the last five years. This radio technology has its advantages, because in CDMA equipment: - the efficiency of using the radio frequency spectrum is 20 times higher compared to radio equipment of the AMPS standard (FDMA technology) and 3 times higher compared to GSM (TDMA technology); - significantly better quality, reliability and confidentiality of communications than in other 2nd generation TDMA systems; - it is possible to use small-sized low-power terminals with long term work; - at the same distance from the base station, the radiation power of CDMA subscriber terminals is lower by more than 5 times compared to the same indicator in standard networks based on other radio technologies; - it is possible to optimize the network topology when calculating coverage areas. CDMA technology was first implemented in cellular equipment of the IS-95 standard. In terms of their service capabilities, existing CDMA systems belong to second generation cellular communication systems. According to statistics from the National Telecommunications Institute (ETRI), the number of CDMA network subscribers is growing by 2,000 people every day. In terms of the growth rate of the number of subscribers, these networks surpass the networks of other existing cellular communication standards, outstripping the development of cellular networks even of such a popular standard as GSM. Currently, there are at least 30 million subscribers in CDMA networks. The global telecommunications community is inclined to believe that CDMA will occupy a leading position in future wireless access systems for subscriber lines (third generation personal communication systems). This conclusion was made due to the fact that CDMA technology is most capable of meeting the requirements for third-generation IMT-2000 equipment, in particular, for ensuring the exchange of information with high transmission rates. However, in future wireless access systems it is planned to use so-called broadband CDMA systems, where the frequency band per channel will be at least 5 MHz (in modern second-generation CDMA systems the band per channel is 1.23 MHz). In the last few years, tools have begun to appear wireless communication, which are based on Frequency Hopping Spread Spectrum (FH-CDMA) technology. This technology combines the specifics of TDMA, where each frequency is divided into several time slots, and CDMA, where each transmitter uses a specific sequence of noise-like signals. This technology has found its application in systems designed for organizing fixed-line communications.

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44. Representation of periodic signals in the form of Fourier series

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Periodic signals and Fourier series

A mathematical model of a process that repeats over time is a periodic signal with the following property:

Here T is the period of the signal.

The task is to find the spectral decomposition of such a signal.

Fourier series.

Let us set on the time interval considered in Chap. I orthonormal basis formed by harmonic functions with multiple frequencies;

Any function from this basis satisfies the periodicity condition (2.1). Therefore, by performing an orthogonal decomposition of the signal in this basis, i.e., by calculating the coefficients

we get the spectral decomposition

valid throughout the infinity of the time axis.

A series of the form (2.4) is called the Fourier series of a given signal. Let us introduce the fundamental frequency of the sequence that forms the periodic signal. Calculating the expansion coefficients using formula (2.3), we write the Fourier series for a periodic signal

with odds

(2.6)

So, in the general case, a periodic signal contains a time-independent constant component and an infinite set of harmonic oscillations, the so-called harmonics with frequencies that are multiples of the fundamental frequency of the sequence.

Each harmonic can be described by its amplitude and initial phase. To do this, the coefficients of the Fourier series should be written in the form

Substituting these expressions into (2.5), we obtain another, equivalent form of the Fourier series:

which sometimes turns out to be more convenient.

Spectral diagram of a periodic signal.

That's what they call it graphic image Fourier series coefficients for a specific signal. There are amplitude and phase spectral diagrams (Fig. 2.1).

Here, the horizontal axis represents the harmonic frequencies on a certain scale, and the vertical axis represents their amplitudes and initial phases.

Rice. 2.1. Spectral diagrams of some periodic signal: a - amplitude; b - phase

They are especially interested in the amplitude diagram, which allows one to judge the percentage of certain harmonics in the spectrum of a periodic signal.

Let's study a few specific examples.

Example 2.1. Fourier series of a periodic sequence of rectangular video pulses with known parameters, even relative to the point t = 0.

In radio engineering, the ratio is called the duty cycle of the sequence. Using formulas (2.6) we find

It is convenient to write the final formula of the Fourier series in the form

In Fig. Figure 2.2 shows the amplitude diagrams of the sequence under consideration in two extreme cases.

It is important to note that a sequence of short pulses, following each other quite rarely, has a rich spectral composition.

Rice. 2.2. Amplitude spectrum of a periodic sequence of rectangular video pulses: a - with a large duty cycle; b - with low duty cycle

Example 2.2. Fourier series of a periodic sequence of pulses formed by a harmonic signal of the form limited at the level (it is assumed that ).

Let us introduce a special parameter - the cutoff angle, determined from the relation where

In accordance with this, the value is equal to the duration of one pulse, expressed in angular measure:

The analytical recording of the pulse generating the sequence under consideration has the form

Constant sequence component

First harmonic amplitude factor

Similarly, the amplitudes of the harmonic components are calculated at

The results obtained are usually written like this:

where the so-called Berg functions:

Graphs of some Berg functions are shown in Fig. 2.3.

Rice. 2.3. Graphs of the first few Berg functions

Spectral density of signals. Direct and inverse Fourier transforms.

The signal is called periodic, if its form is repeated cyclically in time. A periodic signal in general form is written as follows:

Here is the period of the signal. Periodic signals can be either simple or complex.

For the mathematical representation of periodic signals with a period, this series is often used, in which harmonic (sine and cosine) oscillations of multiple frequencies are selected as basis functions:

Where . - the main angular frequency of the sequence of functions. For harmonic basis functions, from this series we obtain a Fourier series, which in the simplest case can be written in the following form:

where are the coefficients

From the Fourier series it is clear 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 Fourier series coefficients of this signal. There are amplitude and phase diagrams. To construct these diagrams, the values ​​of 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 only take positive values, the phases can take both positive and negative values ​​in the interval .

Spectral diagrams of a periodic signal:

a) - amplitude; b) - phase.

Signal spectrum- this 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 briefly - 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 a big role in telecommunications technology. Knowing the spectrum of the signal, 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 multi-channel systems with frequency division. 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 reduce interference.

To observe signal spectra, there are instruments called spectrum analyzers. They allow you to observe and measure the parameters of individual components of the spectrum of a periodic signal, as well as measure spectral density continuous signal.

Often, the mathematical description of even deterministic signals that are simple in structure and form is a difficult task. Therefore, an original technique is used, in which real complex signals are replaced (represented, approximated) by a set (weighted sum, i.e., a series) of mathematical models described by elementary functions. This provides an important tool for analyzing the passage of electrical signals through electronic circuits. In addition, the representation of the signal can also be used as an initial one in its description and analysis. In this case, you can significantly simplify the inverse problem - synthesis complex signals from a set of elementary functions.

Spectral representation of periodic signals by Fourier series

Generalized Fourier series.

The fundamental idea of ​​the spectral representation of signals (functions) dates back to more than 200 years ago and belongs to the physicist and mathematician J. B. Fourier.

Let us consider systems of elementary orthogonal functions, each of which is obtained from one initial one - the prototype function. This prototype function acts as a “building block”, and the desired approximation is found by appropriately combining identical blocks. Fourier showed that any complex function can be represented (approximated) as a finite or infinite sum of a series of multiple harmonic oscillations with certain amplitudes, frequencies and initial phases. This function can be, in particular, the current or voltage in the circuit. A sunbeam, decomposed by a prism into a spectrum of colors, is a physical analogue of the mathematical Fourier transforms (Fig. 2.7).

The light emerging from the prism is separated in space into individual pure colors, or frequencies. The spectrum has an average amplitude at each frequency. Thus, the function of intensity versus time was transformed into a function of amplitude versus frequency. A simple illustration of Fourier's reasoning is shown in Fig. 2.8. Periodic, rather complex in shape curve (Fig. 2.8, A) - this is the sum of two harmonics of different but multiple frequencies: single (Fig. 2.8, b) and doubled (Fig. 2.8, V).

Rice. 2.7.

Rice. 2.8.

A- complex oscillation; b,c- 1st and 2nd approximating signals

Using Fourier spectral analysis, a complex function is represented as a sum of harmonics, each of which has its own frequency, amplitude and initial phase. The Fourier transform defines functions representing the amplitude and phase of the harmonic components corresponding to a particular frequency, and the phase is the starting point of the sine wave.

The transformation can be obtained by two different mathematical methods, one of which is used when original function continuous, and the other when it is given by many individual discrete values.

If the function under study is obtained from values ​​​​with certain discrete intervals, then it can be divided into a successive series of sinusoidal functions with discrete frequencies - from the lowest, fundamental or main frequency, and then with frequencies doubled, tripled, etc. above the main one. This sum of components is called next to Fourier.

Orthogonal signals. In a convenient way The spectral description of a signal according to Fourier is its analytical representation using a system of orthogonal elementary functions of time. Let there be a Hilbert space of signals u0(t)y G/,(?), ..., u n (t) with finite energy, defined over a finite or infinite time interval (t v 1 2). On this segment we will define an infinite system (subset) of interconnected elementary functions of time and call it basic".

Where g = 1, 2, 3,....

Functions u(t) And v(t) are orthogonal on the interval (?, ? 2) if their scalar product, provided that none of these functions is identically zero.

In mathematics this is defined in the Hilbert space of signals orthogonal coordinate basis, i.e. system of orthogonal basis functions.

The property of orthogonality of functions (signals) is associated with the interval of their definition (Fig. 2.9). For example, two harmonic signals m,(?) = = sin(2nr/7’ 0) and u.,(t)= sin(4 nt/T Q)(i.e. with frequencies / 0 = 1/7’ 0 and 2/ 0, respectively) are orthogonal over any time interval whose duration is equal to an integer number of half-cycles T 0(Fig. 2.9, A). Therefore, in the first period the signals and ( (1) And u2(t) are orthogonal on the interval (0.7" 0 /2); but on the interval (O, ZG 0 /4) they are non-orthogonal. Pa Fig. 2.9, b the signals are orthogonal due to the different times of their appearance.

Rice. 2.9.

A- on the interval; b - due to different times of appearance Signal presentation u(t) elementary models is significantly simplified if a system of basis functions is chosen vff), having the property orthonormality. It is known from mathematics if for any pair of functions from the orthogonal system (2.7) the condition is satisfied

then the system of functions (2.7) orthonormal.

In mathematics, such a system of basis functions of the form (2.7) is called orthonormal basis.

Let, at a given time interval |r, t 2| an arbitrary signal is active u(t) and to represent it, the orthonormal system of functions (2.7) is used. Arbitrary Signal Design u(t) on the axis of the coordinate basis is called expansion into a generalized Fourier series. This expansion has the form

where c, are some constant coefficients.

To determine the coefficients from to generalized Fourier series, we choose one of the basis functions (2.7) v k (t) s any number To. Let's multiply both sides of expansion (2.9) by this function and integrate the result over time:

Due to the orthonormality of the basis of the chosen functions, on the right side of this equality all terms of the sum at i ^ To will go to zero. Only the only member of the sum with the number will remain non-zero i = To, That's why

Product of the form c k v k (t), included in the generalized Fourier series (2.9), is spectral component signal u(t), and a set of coefficients (projections of signal vectors on the coordinate axes) (с 0 , с,..., from to,..., s„) completely determines the analyzed signal ii(t) and it's called spectrum(from lat. spectrum- image).

The essence spectral representation (analysis) of the signal consists in determining the coefficients with i in accordance with formula (2.19).

The choice of a rational orthogonal system of coordinate basis of functions depends on the purpose of the research and is determined by the desire to maximize the simplification of the mathematical apparatus of analysis, transformation and data processing. The polynomials of Chebyshev, Hermite, Laguerre, Legendre, etc. are currently used as basis functions. The most widespread transformation of signals in the bases of harmonic functions: complex exponential exp(J 2ft) and real trigonometric sine-cosine functions related by Euler's formula e>x= cosx + y"sinx. This is explained by the fact that a harmonic oscillation theoretically completely retains its shape when passing through linear circuits with constant parameters, and only its amplitude and initial phase change. The symbolic method, well developed in circuit theory, is also widely used. The operation of representing deterministic signals in the form of a set of constant components ( constant component) and the sum of harmonic oscillations with multiple frequencies is usually called spectral decomposition. The fairly widespread use of the generalized Fourier series in signal theory is also associated with its very important property: with a chosen orthonormal system of functions vk(t) and a fixed number of terms in series (2.9), it provides the best representation of a given signal u(t). This property of Fourier series is widely known.

In the spectral representation of signals, orthonormal bases are most widely used. trigonometric functions. This is due to the following: harmonic oscillations are the easiest to generate; harmonic signals are invariant with respect to transformations carried out by stationary linear electrical circuits.

Let us evaluate the temporal and spectral representations analog signal(Fig. 2.10). In Fig. 2.10, A shows the timing diagram of a complex continuous signal, and Fig. 2.10, b - its spectral decomposition.

Let us consider the spectral representation of periodic signals as a sum of either harmonic functions or complex exponentials with frequencies forming an arithmetic progression.

Periodic they call the signal u„(?). repeating at regular intervals (Fig. 2.11):

where Г is the period of repetition or repetition of pulses; n = 0,1, 2,....

Rice. 2.11. Periodic signal

If T is the period of the signal u(t), then the periods will also be multiples of it: 2G, 3 T etc. A periodic sequence of pulses (they are called video pulses) is described by the expression

Rice. 2.10.

A- time diagram; b- amplitude spectrum

Here uQ(t)- the shape of a single pulse, characterized by amplitude (height) h = E, duration t„, period of follow-up T= 1/F(F - frequency), position of pulses in time relative to clock points, for example t = 0.

For spectral analysis of periodic signals, the orthogonal system (2.7) in the form of harmonic functions with multiple frequencies is convenient:

where co, = 2p/T- pulse repetition rate.

By calculating the integrals using formula (2.8), it is easy to verify the orthogonality of these functions on the interval [-Г/2, Г/2|. Any function satisfies the periodicity condition (2.11), since their frequencies are multiples. If system (2.12) is written as

then we obtain an orthonormal basis of harmonic functions.

Let's imagine a periodic signal, the most common in signal theory trigonometric(sine-cosine) shape Fourier series:

It is known from the mathematics course that expansion (2.11) exists, i.e. the series converges if the function (in this case the signal) u(t) on the interval [-7/2, 7/2] satisfies Dirichlet conditions(unlike Dirichlet’s theorem, they are often interpreted in a simplified way):

• there should be no discontinuities of the 2nd kind (with branches going to infinity);
• the function is bounded and has a finite number of discontinuities of the 1st kind (jumps);
• a function has a finite number of extrema (i.e. maxima and minima).

Formula (2.13) contains the following components of the analyzed signal:

Constant component

Amplitudes of cosine components

Amplitudes of sinusoidal components

The spectral component with frequency co, in communication theory is called first (basic) harmonic, and components with frequencies ISO, (n> 1) - higher harmonics periodic signal. The frequency step Aco between two adjacent sinusoids from the Fourier expansion is called frequency resolution spectrum

If the signal is an even function of time u(t) = u(-t), then in the trigonometric representation of the Fourier series (2.13) there are no sinusoidal coefficients b n, since in accordance with formula (2.16) they vanish. For signal u(t), described by an odd function of time, on the contrary, according to formula (2.15), the cosine coefficients are equal to zero a p(constant component a 0 is also absent), and the series contains components b p.

The limits of integration (from -7/2 to 7/2) do not have to be the same as in formulas (2.14)-(2.16). Integration can be performed over any time interval of width 7 - the result will not change. Specific limits are chosen for reasons of computational convenience; for example, it may be easier to integrate from O to 7 or from -7 to 0, etc.

Branch of mathematics that establishes the relationship between a function of time u(t) and spectral coefficients a p, b p, called harmonic analysis due to the connection of the function u(t) with sine and cosine terms of this sum. Further, spectral analysis is mainly limited to the framework of harmonic analysis, which finds exclusive application.

Often the use of the sine-cosine form of the Fourier series is not entirely convenient, since for each value of the summation index n(i.e. for each harmonic with frequency mOj) two terms appear in formula (2.13) - cosine and sine. From a mathematical point of view, it is more convenient to represent this formula by an equivalent Fourier series in real form/.

Where A 0 = a 0 / 2; A n = yja 2 n + b - amplitude; nth harmonics signal. Sometimes in relation (2.17) a “plus” sign is placed in front of sr L, then the initial phase of the harmonics is written as sr u = -arctg ( b n fa n).

In signal theory, the complex form of the Fourier series is widely used. It is obtained from the real form of the series by representing the cosine as a half-sum of complex exponentials using Euler’s formula:

By applying this transformation to the real form of the Fourier series (2.17), we obtain the sums of complex exponentials with positive and negative exponents:

And now we will interpret in formula (2.19) the exponents at frequency с, with a minus sign in the exponent, as members of a series with negative numbers. Within the same approach, the coefficient A 0 will become a member of the series with number zero. After simple transformations we arrive at complex form Fourier series

Complex amplitude n th harmonics.

Values S p by positive and negative numbers n are complex conjugate.

Note that the Fourier series (2.20) is an ensemble of complex exponentials exp(jn(o ( t) with frequencies forming an arithmetic progression.

Let us determine the connection between the coefficients of the trigonometric and complex forms of the Fourier series. It's obvious that

It can also be shown that the coefficients a p= 2C w coscp„; b n = 2C/I sincp, f .

If u(t) is an even function, the coefficients of series C will be real, what if u(t) - the function is odd, the coefficients of the series will become imaginary.

The spectral representation of a periodic signal by the complex form of the Fourier series (2.20) contains both positive and negative frequencies. But negative frequencies do not exist in nature, and this is a mathematical abstraction (the physical meaning of a negative frequency is rotation in the direction opposite to that which is taken to be positive). They appear as a consequence of the formal representation of harmonic vibrations in a complex form. When passing from the complex form of notation (2.20) to the real form (2.17), the negative frequency disappears.

The signal spectrum is visually judged by its graphical representation - the spectral diagram (Fig. 2.12). Distinguish amplitude-frequency And phase-frequency spectra. Set of harmonic amplitudes A p(Fig. 2.12, A) called amplitude spectrum, their phases (Fig. 2.12, b) Wed I - phase spectrum. Totality S p = |S p is complex amplitude spectrum(Fig. 2.12, V). On spectral diagrams, the abscissa axes indicate the current frequency, and the ordinate axes indicate either the real or complex amplitude or phase of the corresponding harmonic components of the analyzed signal.

Rice. 2.12.

A - amplitude; b - phase; V - amplitude spectrum of the complex Fourier series

The spectrum of a periodic signal is called ruled or discrete, since it consists of individual lines with a height equal to the amplitude A p harmonics Of all types of spectra, amplitude spectra are the most informative, since they allow one to estimate the quantitative content of certain harmonics in the frequency composition of the signal. In signal theory it has been proven that the amplitude spectrum is even frequency function, and phase - odd.

Note equidistance(equidistance from the origin of coordinates) of the complex spectrum of periodic signals: symmetrical (positive and negative) frequencies at which the spectral coefficients of the trigonometric Fourier series are located form an equidistant sequence (..., -zho v..., -2so p -so p 0, v 2so, ..., nco v...), containing frequency co = 0 and having a step co t = 2l/7’. The coefficients can take any value.

Example 2.1

Let us calculate the amplitude and phase spectra of a periodic sequence of rectangular pulses with amplitude?, duration m and repetition period T. The signal is an even function (Fig. 2.13).

Rice. 2.13.

Solution

It is known that an ideal rectangular video pulse is described by the following equation:

those. it is formed as the difference of two unit functions a(?) (inclusion functions), shifted in time by tn.

The sequence of rectangular pulses is a known sum of single pulses:

Since the given signal is an even function of time and during one period acts only on the interval [t and /2, t and /2], then according to formula (2.14)

Where q = T/ T".

Analyzing the resulting formula, you can see that the repetition period and pulse duration are included in it in the form of a ratio. This option q- the ratio of the period to the duration of the pulses is called duty cycle periodic sequence of pulses (in foreign literature, instead of duty cycle, the inverse value is used - duty cycle, from English, duty cycle, equal to m and /7); at q = 2 a sequence of rectangular pulses, when the durations of the pulses and the intervals between them become equal, is called meander(from the Greek paiav5poq - pattern, geometric ornament).

Due to the parity of the function describing the analyzed signal, in the Fourier series, along with the constant component, only cosine components (2.15) will be present:

On the right side of formula (2.22), the second factor has the form of an elementary function (sinx)/x. In mathematics, this function is denoted as sinc(x), and only for the value X= 0 it is equal to one (lim (sinx/x) =1), passes

through zero at points x = ±l, ±2l,... and decays with increasing argument x (Fig. 2.14). Finally, the trigonometric Fourier series (2.13), which approximates the given signal, is written in the form

Rice. 2.14. Graph of a function sinx/x

The sine function has a petal character. Speaking about the width of the lobes, it should be emphasized that for graphs of discrete spectra of periodic signals, two options for calibrating the horizontal axis are possible - in harmonic numbers and frequencies. For example, in Fig. 2.14 The ordinate axis is calibrated to correspond to frequencies. The width of the lobes, measured in the number of harmonics, is equal to the duty cycle of the sequence. This implies an important property of the spectrum of a sequence of rectangular pulses - it does not contain (have zero amplitudes) harmonics with numbers that are multiples of the duty cycle. With a pulse duty cycle of three, every third harmonic disappears. If the duty cycle were equal to two, then only odd harmonics of the fundamental frequency would remain in the spectrum.

From formula (2.22) and Fig. 2.14 it follows that the coefficients of a number of higher harmonics of the signal have a negative sign. This is due to the fact that the initial phase of these harmonics is equal to p. Therefore, formula (2.22) is usually presented in a modified form:

With this recording of the Fourier series, the amplitude values ​​of all higher harmonic components on the spectral diagram graph are positive (Fig. 2.15, A).

The amplitude spectrum of the signal largely depends on the ratio of the repetition period T and pulse duration t and, i.e. from duty cycle q. The frequency distance between adjacent harmonics is equal to the pulse repetition frequency with 1 = 2l/T. The width of the spectrum lobes, measured in frequency units, is equal to 2π/tn, i.e. is inversely proportional to the pulse duration. Note that for the same pulse duration m and with increasing non-

Rice. 2.15.

A- amplitude;b- phase

period of their repetition T the fundamental frequency co decreases and the spectrum becomes denser.

The same picture is observed if the pulse duration t is shortened and the period remains unchanged T. The amplitudes of all harmonics decrease. This is a manifestation of the general law (W. Heisenberg’s uncertainty principle - Uncertainty principle)’, The shorter the signal duration, the wider its spectrum.

The phases of the components are determined from the formula cp = arctg (bn/an). Since here the coefficients b„= 0, then

Where m = 0, 1, 2,....

Relation (2.24) shows that when calculating the phases of spectral components we are dealing with mathematical uncertainty. To reveal it, let us turn to formula (2.22), according to which the amplitudes of the harmonics periodically change sign in accordance with the change in the sign of the function sin(nco 1 x 1I /2). Changing the sign in formula (2.22) is equivalent to shifting the phase of this function by p. Therefore, when this function positive, harmonic phase (p u = 2 tp, and when negative - = (2t + 1 )To(Fig. 2.15, b). Note that although the amplitudes of the components in the spectrum of rectangular pulses decrease with increasing frequency (see Fig. 2.15, A), this decay is quite slow (amplitudes decrease in inverse proportion to frequency). To transmit such pulses without distortion, an infinite frequency band of the communication channel is required. For relatively subtle distortions, the limiting value of the frequency band should be many times greater than the inverse value of the pulse duration. However, all real channels have a finite bandwidth, which leads to distortions in the shape of the transmitted pulses.

Fourier series of arbitrary periodic signals can contain an infinitely large number of terms. When calculating the spectra of such signals, calculating the infinite sum of the Fourier series causes certain difficulties and is not always required, therefore we are limited to summing a finite number of terms (the series is “truncated”).

The accuracy of signal approximation depends on the number of summed components. Let's consider this using the example of approximation by the sum of the first eight harmonics of a sequence of rectangular pulses (Fig. 2.16). The signal has the form of a unipolar meander with a repetition period That amplitude E= 1 and pulse duration t and = T/2 (specified signal - even function - Fig. 2.16, A; duty cycle q= 2). The approximation is shown in Fig. 2.16, b, and the graphs show the number of summed harmonics. In the ongoing approximation of a given periodic signal (see Fig. 2.13) by the trigonometric series (2.13), the summation of the first and higher harmonics will be carried out only over odd coefficients Pu since if their values ​​and pulse duration are even, m and = T/2 = = tm/co, the value sin(mo,T H /2) = sin(wt/2) becomes zero.

The trigonometric form of the Fourier series (2.23) for a given signal has the form

Rice. 2.16.

A - given signal; 6 - intermediate stages of summation

For ease of presentation, the Fourier series (2.25) can be written simplified:

From formula (2.26) it is obvious that the harmonics that approximate the meander are odd, have alternating signs, and their amplitudes are inversely proportional to the numbers. Note that a sequence of rectangular pulses is poorly suited for representation by a Fourier series - the approximation contains ripples and jumps, and the sum of any number of harmonic components with any amplitudes will always be a continuous function. Therefore, the behavior of the Fourier series in the vicinity of discontinuities is of particular interest. From the graphs in Fig. 2.16, b it is easy to see how, with an increase in the number of summed harmonics, the resulting function increasingly approaches the shape of the original signal u(t) everywhere except the points of its break. In the vicinity of the discontinuity points, the summation of the Fourier series gives a slope, and the slope of the resulting function increases with the number of summed harmonics. At the very point of discontinuity (let us denote it as t = t 0) Fourier series u(t 0) converges to half the sum of the right and left limits:

In the sections of the approximated curve adjacent to the discontinuity, the sum of the series gives noticeable pulsations, and in Fig. 2.16 it is clear that the amplitude of the main surge of these pulsations does not decrease with increasing number of summed harmonics - it only compresses horizontally, approaching the break point.

At n-? at the break points the ejection amplitude remains constant,

and its width will be infinitely narrow. Both the relative amplitude of the pulsations (relative to the amplitude of the jump) and the relative attenuation do not change; Only the pulsation frequency changes, which is determined by the frequency of the last summed harmonics. This is due to the convergence of the Fourier series. Let's take a classic example: will you ever reach the wall if you walk half the remaining distance with each step? The first step will lead to the halfway mark, the second will lead to the three-quarters mark, and after the fifth step you will have covered almost 97% of the way. You are almost there, but no matter how many steps forward you take, you will never reach it in a strict mathematical sense. You can only prove mathematically that in the end you will be able to approach any given distance, no matter how small. This proof would be equivalent to demonstrating that the sum of the numbers is 1/2,1/4,1/8,1/16, etc. tends to unity. This phenomenon, inherent in all Fourier series for signals with discontinuities of the 1st kind (for example, jumps, as on the fronts of rectangular pulses), is called Gibbs effect*. In this case, the value of the first (largest) amplitude surge in the approximated curve is about 9% of the jump level (see Fig. 2.16, n = 4).

The Gibbs effect leads to an irremovable error in the approximation of periodic pulse signals with discontinuities of the 1st kind. The effect occurs when there is a sharp violation of the monotony of functions. At horse races, the effect is maximum; in all other cases, the amplitude of the pulsations depends on the nature of the violation of monotony. For a number of practical applications, the Gibbs effect causes certain problems. For example, in sound reproduction systems this phenomenon is called “ringing” or “blinking”. Moreover, each sharp consonant or other sudden sound may be accompanied by a short sound that is unpleasant to the ear.

The Fourier series can be applied not only to periodic signals, but also to signals of finite duration. In this case, the time is specified

nal interval for which the Fourier series is constructed, and at other times the signal is considered equal to zero. To calculate the coefficients of a series, this approach means periodic continuation signal outside the considered interval.

Note that nature (for example, human hearing) uses the principle of harmonic signal analysis. A person performs a virtual Fourier transform whenever he hears a sound: the ear automatically performs this, representing the sound as a spectrum of successive loudness values ​​for tones of different pitches. The human brain turns this information into perceived sound.

Harmonic synthesis. In signal theory, along with harmonic analysis of signals, they widely use harmonic synthesis- obtaining specified oscillations of a complex shape by summing a number of harmonic components of their spectrum. Essentially, the synthesis of a periodic sequence of rectangular pulses by the sum of a number of harmonics was carried out above. In practice, these operations are performed on a computer, as shown in Fig. 2.16, b.

• Jean Baptiste Joseph Fourier (J.B.J. Fourier; 1768-1830) - French mathematician and physicist.
• Josiah Gibbs (J. Gibbs, 1839-1903) - American physicist and mathematician, one of the founders of chemical thermodynamics and statistical physics.

Forms of recording the Fourier series. The signal is called periodic, if its shape repeats cyclically in time Periodic signal u(t) in general it is written like this:

u(t)=u(t+mT), m=0, ±1,±2,…

Here is the T-period of the signal. Periodic signals can be either simple or complex.

For the mathematical representation of periodic signals with a period T series (2.2) is often used, in which harmonic (sine and cosine) oscillations of multiple frequencies are chosen as basis functions

y 0 (t)=1; y 1 (t)=sinw 1 t; y 2 (t)=cosw 1 t;

y 3 (t)=sin2w 1 t; y 4 (t)=cos2w 1 t; ..., (2.3)

where w 1 =2p/T is the main angular frequency of the sequence

functions. For harmonic basis functions, from series (2.2) we obtain the Fourier series (Jean Fourier - French mathematician and physicist of the 19th century).

Harmonic functions of the form (2.3) in the Fourier series have the following advantages: 1) simple mathematical description; 2) invariance to linear transformations, i.e. if there is a harmonic oscillation at the input of a linear circuit, then at its output there will also be a harmonic oscillation, differing from the input only in amplitude and initial phase; 3) like a signal, harmonic functions are periodic and have infinite duration; 4) the technique for generating harmonic functions is quite simple.

It is known from a mathematics course that in order to expand a periodic signal into a series in harmonic functions (2.3), the Dirichlet conditions must be met. But all real periodic signals satisfy these conditions and can be represented in the form of a Fourier series, which can be written in one of the following forms:

u(t)=A 0 /2+ (A’ mn cosnw 1 t+A” mn nw 1 t), (2.4)

where are the coefficients

A mn ”= (2.5)

u(t)=A 0 /2+ (2.6)

A mn = (2.7)

or in complex form

u(t)= (2.8)

Cn= (2.9)

From (2.4) - (2.9) it follows that in the general case, the periodic signal u(t) contains a constant component A 0 /2 and a set of harmonic oscillations of the fundamental frequency w 1 =2pf 1 and its harmonics with frequencies w n =nw 1, n=2 ,3,4,… Each of the harmonic

Fourier series oscillations are characterized by amplitude and initial phase y n .nn

Spectral diagram and spectrum of a periodic signal. If any signal is presented as a sum of harmonic oscillations with different frequencies, then it is said that spectral decomposition signal.

Spectral diagram signal is usually called a graphical representation of the coefficients of the Fourier series of this signal. There are amplitude and phase diagrams. In Fig. 2.6, on a certain scale, the values ​​of harmonic frequencies are plotted along the horizontal axis, and their amplitudes A mn and phases y n are plotted along the vertical axis. Moreover, the harmonic amplitudes can take only positive values, the phases can take both positive and negative values ​​in the interval -p£y n £p

Signal spectrum- this is a set of harmonic components with specific values ​​of frequencies, amplitudes and initial phases, which together form a signal. IN technical applications in practice, spectral diagrams are called more briefly - amplitude spectrum, phase spectrum. Most often people are interested in the amplitude spectral diagram. It can be used to estimate the percentage of harmonics in the spectrum.

Example 2.3. Expand a periodic sequence of rectangular video pulses into a Fourier series With known parameters (U m , T, t z), even "Relative to point t=0. Construct a spectral diagram of amplitudes and phases at U m =2B, T=20ms, S=T/t and =2 and 8.

A given periodic signal on an interval of one period can be written as

To represent this signal, we will use the Fourier series form V form (2.4). Since the signal is even, only cosine components will remain in the expansion.

Rice. 2.6. Spectral diagrams of a periodic signal:

a - amplitude; b- phase

The integral of an odd function over a period is equal to zero. Using formulas (2.5) we find the coefficients

allowing us to write the Fourier series:

To construct spectral diagrams for specific numerical data, we set i=0, 1, 2, 3, ... and calculate the harmonic coefficients. The results of calculating the first eight components of the spectrum are summarized in table. 2.1. In a series (2.4) A" mn =0 and according to (2.7) A mn =|A’ mn |, the main frequency f 1 =1/T= 1/20-10 -3 =50 Hz, w 1 =2pf 1 =2p*50=314 rad/s. The amplitude spectrum in Fig.

2.7 is built for these n, at which And mn more than 5% of the maximum value.

From the given example 2.3 it follows that with increasing duty cycle, the number of spectral components increases and their amplitudes decrease. Such a signal is said to have a rich spectrum. It should be noted that for many practically used signals there is no need to calculate the amplitudes and phases of harmonics using the previously given formulas.

Table 2.1. Amplitudes of the Fourier series components of a periodic sequence of rectangular pulses

Rice. 2.7. Spectral diagrams of a periodic pulse sequence: A- with duty cycle S-2; - b-with duty cycle S=8

In mathematical reference books there are tables of expansions of signals in a Fourier series. One of these tables is given in the Appendix (Table A.2).

The question often arises: how many spectral components (harmonics) should be taken to represent a real signal in a Fourier series? After all, the series is, strictly speaking, endless. A definite answer cannot be given here. It all depends on the shape of the signal and the accuracy of its representation by the Fourier series. Smoother signal change - less harmonics required. If the signal has jumps (discontinuities), then it is necessary to sum larger number harmonics to achieve the same error. However, in many cases, for example in telegraphy, it is believed that three harmonics are sufficient for transmitting rectangular pulses with steep fronts.

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 representation is studied in sufficient detail in other courses, so we recall only the basic relationships and definitions.

As noted above, any periodic function u(t) , for which the equality holds u(t)=u(t+T) , Where T=1/F=2p/W , can be represented in a Fourier series:

Each term of this series can be expanded using the cosine formula for the difference of 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 average values ​​of the product of the function u(t) and harmonic vibration with frequency nw over an interval of duration T . We already know (section 2.5) that these are cross-correlation functions that determine the extent of their connection. Therefore, the coefficients A n And Bn show us "how much" sine or cosine waves with frequency nW contained in this function u(t) , expandable in a Fourier series.

So we can represent the periodic function u(t) as a sum of harmonic oscillations, where the numbers Cn are amplitudes, and the numbers φn - phases. Usually in literature is called the amplitude spectrum, and - spectrum of phases. Often only the amplitude spectrum is considered, which is depicted as lines located at points nW on the frequency axis and having a height corresponding to the number Cn . However, it should be remembered that in order to obtain a one-to-one correspondence between the time function u(t) and its spectrum must use both the amplitude spectrum and the phase spectrum. This can be seen from this simple example. The signals will have the same amplitude spectrum, but completely different type temporary functions.

Not only a periodic function can have a discrete spectrum. For example, a signal: is not periodic, but has a discrete spectrum consisting of two spectral lines. Also, a signal consisting of a sequence of radio pulses (pulses with high-frequency filling), for which the repetition period is constant, but the initial phase of the high-frequency filling varies from pulse to pulse according to some law, will not be strictly periodic. Such signals are called almost periodic. As we will see later, they also have a discrete spectrum. We will study the physical nature of the spectra of such signals in the same way as periodic ones.