Spectral (harmonic) analysis of signals. Harmonic analysis of periodic signals Mathematical recording of harmonic oscillations

Mathematical notation of harmonic vibrations. Amplitude and phase spectra of a periodic signal. Spectrum of a periodic sequence of rectangular pulses. Internal integral, which is a function of frequency. Spectra of non-periodic signals.

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Spectral (harmonic) signal analysis

Literature

spectral harmonic signal oscillation

Harmonic analysis is a branch of mathematics that studies the possibilities of representing functions in the form of trigonometric series and integrals. The main concept in harmonic analysis is harmonic oscillation, which can be written mathematically as follows:

where Um, f0, 0, and 0 are the amplitude, frequency, angular frequency, and initial phase of the oscillation, respectively.

In harmonic analysis we introduce nth concept harmonics of a periodic oscillation of frequency u0, which again means a harmonic oscillation with a frequency n times higher than the frequency of the main harmonic oscillation.

The next important concept is the signal spectrum. The spectrum of a signal is understood as the totality of its harmonic components. The introduction of the concept of signal spectrum led to the use in technical applications of the name spectral analysis for harmonic analysis of signals.

1. Spectral analysis of periodic signals

As is known, any signal S(t), described by a periodic function of time that satisfies the Dirichlet conditions (models of real signals satisfy them), can be represented as a sum of harmonic oscillations, called a Fourier series:

where is the average value of the signal over the period or the constant component of the signal;

Fourier series coefficients;

Fundamental frequency (first harmonic frequency); n=1,2,3,…

The set of values An and n (or when expanded in sinusoidal functions n) is called the spectrum of a periodic function. The harmonic amplitudes An characterize the amplitude spectrum, and the initial phases n (or "n) characterize the phase spectrum.

Thus, the spectrum of a periodic signal is represented as a constant component and an infinite number of harmonic oscillations (sine or cosine) with corresponding amplitudes and initial phases. All harmonic frequencies are multiples of the fundamental frequency. This means that if a periodic signal follows a frequency of, for example, 1 kHz, then its spectrum can only contain frequencies of 0 kHz, 1 kHz, 2 kHz, etc. The spectrum of such a periodic signal cannot contain, for example, frequencies of 1.5 kHz or 1.2 kHz.

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In Fig. 1. The amplitude and phase spectra of a certain periodic signal are shown. Each harmonic component is depicted as vertical segments, the lengths of which (on a certain scale) are equal to its amplitude and phase. As you can see, the spectrum of a periodic signal is discrete or, as they say, lined.

In order to simplify calculations, instead of the trigonometric form of writing the Fourier series, they often use a complex form of writing it, the coefficients of which combine the coefficients An and n:

The set of complex amplitudes n is called the complex spectrum of a periodic signal.

The calculation of signal spectra in the complex domain is much simpler, since there is no need to consider separately the coefficients and the trigonometric form of writing the Fourier series.

2. Spectrum of a periodic sequence rectangular pulses

Before considering the spectrum of a periodic sequence of rectangular pulses, let us consider the parameters of these pulses.

The parameters of a single pulse are amplitude, pulse duration, rise time, fall duration, and flat top fall (cleavage).

The pulse amplitude Um is measured in volts.

Pulse duration is measured at the base, at levels of 0.1Um or 0.5Um. In the latter case, the pulse duration is called active. The pulse duration is measured in units of time.

The duration of the front tf and fall tс is measured either at the level 0 - Um, or at the level (0.1-0.9) Um. In the latter case, the duration of the front and decline is called active.

Flat top cleavage is characterized by cleavage coefficient? = ?u/Um,

where?u is the chip value; Um - pulse amplitude.

The parameters of the pulse series are the repetition period T, repetition frequency f, duty cycle Q, duty cycle, average voltage values Uav and average power value Pav.

Repetition period T = ti +tp, where T is the period, ti is the pulse duration,
tп - pause duration. T, tи, and tп are measured in units of time.

The repetition frequency f = 1/T is measured in hertz, etc.

The duty cycle Q = T/ti is a dimensionless quantity.

Fill factor = ti/T is a dimensionless quantity.

Average voltage

Let's move on to considering the amplitude and phase spectra of the signal in the form of a periodic sequence of rectangular pulses with duration and amplitude Um, following with a period T (Fig. 2).

Let us consider the case when the middle of the pulse is the beginning of the time count. Then the signal over the period is described by the expression

Complex amplitudes of harmonic components.

The function is sign-alternating and changes its sign to the opposite when the argument n1 changes by the amount?

where k is the serial number of the interval on the frequency scale, counted from zero frequency.

Thus, the harmonic amplitudes, including the DC component, are determined by the expression:

and the phases - by the expression =1, 2,3,...

The function characterizes the change in the amplitude spectrum of the signal depending on frequency. It vanishes for values of its argument that are multiples. It follows that harmonics with number n = , where
= 1,2,3,...will have zero amplitudes, i.e. absent from the spectrum.

As you know, the ratio is called the duty cycle of the pulse sequence. Thus, in the spectrum of the sequence under consideration there will be no harmonics whose numbers are multiples of the duty cycle.

If the beginning of the time count is associated with the beginning of the pulse, then the amplitude spectrum will remain unchanged, and the phases of the harmonics, in accordance with the property of the Fourier transform, will receive an additional phase shift nп1ф/2. As a result

Expressions for the trigonometric form of recording the Fourier series when counting time from the middle and beginning of the pulse, respectively, have the form:

In Fig. 3. The amplitude and phase spectra of the considered sequence of rectangular pulses with a duty cycle of two are shown.

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The phase spectra are shown respectively when counting time from the middle and beginning of the pulse. The dotted lines in the amplitude spectra characterize the behavior of the modulus of the spectral density of a single pulse.

The expression for the amplitudes and phases of harmonics can be easily obtained in a form convenient for calculations. So, when counting time from the middle of the pulse for a duty cycle equal to two, we have

N = 1,3,5,7, …,

3. Spectra of some periodic signals

Table 1 shows the amplitude and phase spectra, as well as trigonometric forms of recording Fourier series of some of the most frequently encountered periodic signals in practice.

Signals No. 1 and No. 2 are sequences of rectangular pulses with a duty cycle of 2 and zero constant component and differ only in the beginning of the time count. Please note that the amplitude spectra of these signals are the same, but the phase spectra are different.

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Signals No. 3 and No. 4 are sequences of rectangular pulses with

duty cycle, respectively, 3 and 3/2 and zero constant component. The amplitude spectra of these signals are the same. Please note that for signal No. 3, each of the intervals Дш = 2р/ф contains two harmonics, and for signal No. 4, each of the intervals Дш1 = 2р/2ф contains only one harmonic. The conclusion about the coincidence of the amplitude spectra of these signals can also be made based on the fact that when signal No. 3 is shifted by T/2, it is inverse (i.e., has the opposite sign) with respect to signal 4.

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Signal No. 5 is a sequence of symmetrical triangular pulses with a zero constant component. When choosing the time reference, as shown in the figure in Table 3.1, all harmonics have zero initial phases.

Signal No. 6 is a sequence of so-called sawtooth pulses with a zero constant component.

Signals No. 7 and No. 8 are sequences of pulses that approximate with good accuracy, respectively, the signals obtained from full-wave and half-wave rectification of sinusoidal signals.

The dotted lines on the amplitude spectra of signals No. 1 - No. 8 depict spectral densities that characterize the behavior of the modulus of the spectral density of single pulses forming sequences.

Signal No. 9 is an oscillation with a frequency u0, modulated in amplitude by an oscillation with a frequency u. Such a signal is called an amplitude-modulated oscillation. The coefficient m is called the amplitude modulation coefficient:

where DU is the amplitude of the change in the envelope of the amplitude-modulated oscillation.

4. Spectra of non-periodic signals

Let a non-periodic signal be described by a function S(t) specified over a finite time interval t1< t < t2, которая удовлетворяет условиям Дирихле и абсолютно интегрируема, т.е.

The latter physically means that the signal has finite energy.

Let us assume that the signal S(t) is converted by repeating it with an arbitrary period T > t2-t1 into a periodic signal S1(t). For this signal, the Fourier series expansion is applicable:

The coefficients An in this case will be smaller, the larger the interval T chosen as the period. As T tends to infinity, in the limit we obtain infinitesimal amplitudes of the harmonic components. The number of harmonic components included in the Fourier series will be infinitely large, since as T tends to infinity, the fundamental frequency of the signal u = 2p/T tends to zero. In other words, the distance between harmonics, equal to the fundamental frequency, becomes infinitesimal, and the spectrum becomes continuous.

As a result, at T, the signal S1(t) turns into the signal S(t), frequency 1 decreases to d, and n1 turns into the current frequency. Replacing summation by integration, we obtain

The internal integral, which is a function of frequency, is called the complex spectral density or spectral characteristic () of the signal S(t):

In the general case, when the limits t1 and t2 are not specified

Thus, the time and frequency representations of non-periodic signals are related to each other by a pair of Fourier transforms.

Complex spectral density can be presented in the following forms:

() = S()e-j()=A() + jB(),

where A() = B() =

() = arctan.

The function S() is called the spectral density of amplitudes of a non-periodic signal, and the function () is called the spectral density of phases.

Unlike the spectrum of a periodic signal, the spectrum of a non-periodic signal is continuous (continuous). Dimension S() - amplitude/frequency, () - phase/frequency. At each specific frequency, the amplitude of the corresponding component is zero. Therefore, we can only talk about amplitude harmonic components, the frequencies of which are contained in a small but finite frequency range, + d.

We emphasize that the connection between the time and frequency representation of the signal, given by Fourier transforms, exists only for spectral density.

Literature

Kasatkin A.S. Electrical engineering: textbook. for universities / A.S. Kasatkin, M.V. Nemtsov. - 11th ed., erased. ; Grif MO. - M.: Academy, 2007. - 539 p.

Kasatkin A.S. Electrical engineering: textbook. for universities / A.S. Kasatkin, M.V. Nemtsov. - 9th ed., erased. ; Grif MO. - M.: Academia, 2005. - 639 p.

Nemtsov M.V. Electrical engineering: textbook. allowance for the environment. textbook institutions / M.V. Nemtsov, I.I. Svetlakova. - Grif MO. - Rostov n/d: Phoenix, 2004. - 572 p.

Moskalenko V.V. "Automated electric drive". Textbook for universities. M.: Energoatomizdat, 1986.

"Electrical Engineering", ed. V.S. Pantyushina, M.: Higher School, 1976.

"General Electrical Engineering" ed. A.T. Blazhkina, L.: Energy, 1979.

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Transcript

1 Topic 3 HARMONIC ANALYSIS OF NON-PERIODIC SIGNALS Direct and inverse Fourier transforms Spectral characteristics of a signal Amplitude-frequency and phase-frequency spectra Spectral characteristics of the simplest signals Properties of the Fourier transform Energy distribution in the spectrum of a non-periodic signal 3 Fourier transform Harmonic analysis can be extended to non-periodic signals Consider a signal which is defined by some function (t) on the interval [t, t] and is equal to zero outside this interval (this signal is shown in Fig. 3 with a solid line) We will assume that this function satisfies the Dirichlet conditions and is absolutely integrable Fig. 3 Periodic function formed by repeating (t ) Let's take an arbitrary period of time of duration T, completely including the interval [t, t], and form a periodic function n (t) (t k T) k in which the function (t) is repeated after the interval T (a fragment of this function is shown in Fig. 3) Obviously , that (t) lm (t) (3) T The periodic function n (t) can be written as a Fourier series in complex form where n n () j t t c e, (3) T j t (33) c (t) e d t Substituting ( 33) in (3) and replacing T, we obtain T T t j j t p () [ [ () ] (34) t e d e t 8

2 To obtain the spectral representation of the signal (t), we substitute (34) into (3) and let T go to infinity. At T, the angular frequency T turns into an infinitesimal frequency increment d, the frequency of the th component of the series into the current frequency, and the summation operation can be replaced by the integration operation As a result, we obtain t (t) j t e [ () j e d ] d (35) t Taking into account that the values of t and t are not defined, for the internal integral in (35) we introduce the notation X (j) (t) e d t j t (36) The function X (j) is called the spectral characteristic of the signal () Expression (35) taking into account (36) takes the form (t) j t X (j) e d 9 t (37) Formulas (36) and (37) form a pair of transformations Fourier and establish a one-to-one correspondence between the representation (t) of the signal in the time domain and its representation X (j) in the frequency domain. Formula (36) is called the direct Fourier transform, and the function X (j) is the spectral characteristic of the signal (t). Formula (37) allows carry out an inverse transformation and calculate the instantaneous value of the signal (t), if its spectral characteristic X (j) is known. Symbolically, these transformations are written in the form X (j) [ (t)], (t) [ X (j)] Spectral characteristic X ( j) signal (t) in the general case is a complex function of frequency. Applying the well-known Euler formula, it can be written in the following form j t X (j) (t) e d t (t) c o s t d t j (t) s t d t Real part a () j b () X () e j () a () (t) c o s t d t of the spectral characteristic is an even function of frequency, and the imaginary part b () (t) s t d t (38) is an odd function of frequency. It follows that the modulus of the spectral characteristic X () X (j) a () b ()

3 is an even function of frequency, and the argument of the spectral characteristic () a rg X (j) is an odd function of frequency. Graphically, the spectral characteristic X (j) of the signal (t) in the general case can be represented as a hodograph on the complex plane (Figure 3, a) However more often, amplitude-frequency X () and phase-frequency () spectral characteristics are plotted (Figure 3, b, c) Considering the symmetry of the spectral characteristics at positive and negative frequency values, they are, as a rule, constructed only at positive frequency values Fig. 3 Spectral characteristics of the signal : a hodograph, b amplitude, c phase Formula (37) of the inverse Fourier transform using the Euler formula and expression (38) can be transformed to the following form: (t) [ a () c o s t b () s t ] d (39) 3 Spectral characteristics the simplest non-periodic signals Spectral characteristics of a single rectangular pulse A rectangular pulse with a reference point aligned with its middle (Figure 33, a) is described by the expression t D p p and t, (t) D re c t p and t and t Using formula (36 ), we find j j j t D s () (3) j X (j) D e d t (e e) D The spectral characteristic of a rectangular pulse at the selected reference point is a real function (Fig. 33, b) The maximum value of X (j) is achieved at It can be calculated according to L'Hopital's rule: X () D The spectral characteristic vanishes at the values of the argument (where any (positive or negative) integer is 3),

4 Fig33 Spectral characteristics of a rectangular pulse (a): b general; in amplitude; g phase As the pulse duration increases, the distance between the zeros of the function X (j) decreases, that is, the spectrum narrows. The value of X () increases. When the pulse duration decreases, on the contrary, the distance between the zeros of the function X (j) increases, which indicates an expansion of the spectrum, and the value of X () decreases. The amplitude spectral characteristic X () of a rectangular pulse is shown in Fig. 33, c. When constructing the phase spectral characteristic () (Fig. 33, d), each change in the sign of the function X (j) is taken into account by the phase increment by Spectral characteristic of the delta function The delta function (Dirac function) is defined as follows: p p and t, (t) p p and t The function satisfies the condition (t) d t, which means that the pulse area is equal to unity. In practice, it is impossible to obtain a signal described by such a function. However, the delta function is a very convenient mathematical model. Figure 34, a shows a graphical representation of the delta function in the form of a vertical segment ending with an arrow. The length of this segment is taken to be proportional to the area of the delta pulse. Let's find the spectral characteristic of the delta function. To do this, take a rectangular pulse described by the function v (t) (Fig. 34, b) The pulse duration is equal, and the amplitude is Therefore, the pulse area is equal to unity We will reduce the pulse duration to zero, while its amplitude will tend to infinity Therefore, (t) lm v (t) 3

5 Fig34 To determine the spectral characteristics of the delta function: a delta function; b rectangular pulse; c spectral characteristic The spectral characteristic of a rectangular pulse is determined by the expression (3) Hence, taking into account that A, we obtain the spectral characteristic of the delta function s () X (j) lm Thus, the delta pulse has a uniform spectrum at all frequencies (Fig. 34, c) Spectral characteristic of an exponential signal Consider the signal described by the function t (t) A e (t) with a positive real value of the parameter (Fig. 35, a) The spectral characteristic of the exponential signal is equal to A X (j) A e e d t e j j t j t A (j) t The hodograph of the spectral characteristic is shown in Fig. 35, b The amplitude and phase spectra are determined respectively by the expressions: X () X (j) A () arg X (j) arctg (), Fig35 To determine the spectral characteristics of an exponential pulse: a exponential pulse; b spectral characteristic Spectral characteristic of a step signal Consider a signal described by a step function (t) A (t) (3) 3

6 The step function (t) is not an absolutely integrable function, therefore the direct Fourier transform formula cannot be used. However, function (3) can be represented as the limit of an exponential function: (t) A lm e t In this case, the spectral characteristic X (j) can be defined as the limit spectral characteristic of an exponential signal at: A X (j) lm A lm ja lm j When the first term on the right side of this expression is equal to zero at all frequencies except where it goes to infinity Let's find the area d d a rc tg () The limit of the second term - Therefore, the limit of the first term is () which is obvious. Therefore, we finally obtain X (j) () j 33 Basic properties of the Fourier transform There is a one-to-one correspondence between the signal (t) and its spectrum X (j) To solve practical problems, it is necessary to know the relationship between changes in the signal and the corresponding changes in the spectral characteristic Let us consider the most important transformations of signals and the corresponding changes in the spectral characteristic Linearity of the Fourier transform If signals (t), (t) are Fourier transformable and their spectral characteristics are respectively functions X (j), X (j) and if, quantities, independent of t and, then the following equalities are valid: (t) X (j), X (j) (t) Thus, a linear combination of signals corresponds to a linear combination of the spectral characteristics of these signals Spectral characteristic of the derivative If the function (t) describing the signal , and its derivative y (t) d d t are Fourier transformable and (t) has a spectral characteristic X (j), then the spectral characteristic of the derivative d (t) Y (j) j X (j) dt (3) 33

7 Thus, differentiating a signal with respect to time is equivalent to a simple algebraic operation of multiplying the spectral characteristic by a factor j. Therefore, it is customary to say that the imaginary number j is a differentiation operator operating in the frequency domain. Formula (3) is generalized to the case of the spectrum of the th-order derivative. It is easy to show that if the derivative y (t) d (t) d t is absolutely integrable in the interval (,), then Y (j) (j) X (j) Spectral characteristic of the integral If the function (t) describing the signal is Fourier transformable, has a spectral characteristic () t, then the spectral characteristic of the integral y (t) () d is equal to X j and (t) d t t X (j) Y (j) () d j Thus, the factor (j) is an integration operator in the frequency domain This property extends and on the integrals of multiplicity Spectral characteristic of a shifted signal Let there be a signal (t) (Fig. 36, a) of an arbitrary shape, existing on the interval [t, t] and having a spectral characteristic X (j) Consider the same signal, but arising a time later and therefore, described by the function (t) (t) This function is defined on the interval [t, t] (Fig. 36, b) Fig. 36 Initial (a) and “delayed” (b) signals If the signal (t) is Fourier transformable and has a spectral characteristic X (j), then the spectral characteristic of the “delayed” signal (t) is equal to j X (j) (t) e X (j) In the case of the “advanced” signal (t) (t) we will have 34

8 j X (j) (t e X (j) Shift of the spectral characteristic If the function (t) is Fourier transformable and has a spectral characteristic X (j), then j a t e (t) X [ j (a)], where a is any real non-negative number Compression and stretching of signals Let a signal (t) and its spectral characteristic X (j) be given. Let us subject this function to a change in the time scale, forming a new function (t) (k t), where k is some real number. Figure 37 shows, for example, graphs of the signal , described by the function for values of Ф k 5 ; 5 kt, (33) (t) e c o s k t Fig. 37 Signal graphs (33): a k ; b k ; c k 5 It is easy to see that at k the signal is “compressed” (Fig. 37). , b), and at k the “stretching” of the signal (Fig. 37, c) It can be shown that the spectral characteristic of the signal (t) is determined by the expression X (j) (k t) X (j) k k From this expression it follows that when the signal is compressed on the time axis by k times, its spectrum expands by the same number on the frequency axis. The modulus of the spectral characteristic decreases by k times. When the signal is stretched in time, that is, at k, the spectrum narrows and the modulus of the spectral characteristic increases. Spectral characteristic of the product of signals Let there are two signals that are described by the functions (t) and (t) We form a signal If the signals () t and () t are Fourier transformable and their spectral characteristics are, respectively () y(t) is determined by the expression y (t) (t) ( t) X j and () X j, then the spectral characteristic of the signal is 35

9 Parseval’s theorem If the functions () Y (j) F (t) (t) X [ j ()] X (j) d t and () t are Fourier transformable and their spectral characteristics are respectively equal to () X j and () converge absolutely, then the equality X j is true, and the integrals X (j) d, X (j) d (t) (t) d t X (j) X (j) d (34) Formula (34) allows you to find the integral in infinite limits from the product of two functions by performing the corresponding operations with the spectral characteristics of the functions. After simple transformations, formula (34) can be written in real form (t) (t) d t X (j) X (j) c o s[ () ()] d If (t ) (t) (t), then X (j) X (j) X (j) and from (34) we obtain an equality called the Parseval formula: (t) d t X (j) d X ( j) d Invertibility of the Fourier transform It is easy to notice that the formulas for the direct transform and the inverse Fourier transform j t X (j) (t) e d t j t (t) X (j) e d are very similar to each other. For this reason, all “pairs” of transformations have close mirror images images Let's show this with an example As shown above, a rectangular pulse described by the function (t) has a spectral characteristic D p p and t, p p and t and t s () X (j) D On the other hand, if we subject the signal to a direct Fourier transform 36

10 we obtain D s (t) y(t) t D pr and, Y (j) pr and and 34 Energy distribution in the spectrum of a non-periodic signal Practical spectrum width The value E (t) d t is called the signal energy Exactly this energy is released in a resistor with a resistance of Ohm if a voltage (t) is applied to its terminals. Using the Parseval formula, the signal energy can be expressed through its spectral characteristic: (35) E (t) d t X (j) d X (j) d Relation ( 35) allows you to determine the signal energy by integrating the square of the modulus of the spectral characteristic over the entire frequency range. In addition, this relationship shows how the signal energy is distributed over various frequency components. It is clear from it that energy falls on an infinitely small frequency range. Therefore, the function d E 37 X (j) d N () X (j) can be called the spectral characteristic of the signal energy (t) It characterizes the distribution of the signal energy over its harmonic components In the process of solving practical problems of analysis and synthesis of signals using the Fourier transform, it is necessary to limit the frequency interval in which spectral characteristic This frequency interval [, ], called the practical spectrum width, contains components that are essential for this study. When determining the practical width of the signal spectrum from a given intensity of harmonic components, the amplitude spectral characteristic is used. The value of the harmonic amplitude is chosen from the condition that when ical components do not exceed a given value From an energy point of view, the practical width of the spectrum of a non-periodic signal is estimated by the frequency range within which the overwhelming majority of the signal energy is concentrated. In accordance with formula (35), the signal energy concentrated in the frequency band from to

11 pr E X j d () Depending on the requirements for the share of useful energy used by the signal and the practical spectrum width is selected Example Given is a rectangular pulse described by the function The signal energy is equal to (t) D p p and t, p p and t and t E (t) d t D d t D The spectral characteristic of a rectangular pulse is found above: s () X (j) D (36) Let D, Then according to (36) E Integrating the squared modulus of the spectral characteristic in the frequency interval [, ] gives an estimate of the pulse energy E Test questions Specify the main fundamental difference between the spectra of periodic and non-periodic signals Explain physical meaning amplitude and phase spectra of a non-periodic signal 3 Explain what happens to the spectrum of a non-periodic signal when the polarity of the latter changes to the opposite 4 How are the spectra of a single pulse and a periodic sequence of the same pulses related? 5 How will the amplitude and phase spectra of the signal change when it is differentiated (integrated)? 6 Explain what is the relationship between the amplitude and phase spectra of a given signal and a signal delayed by an amount 7 Explain how the spectral characteristic (39) of a rectangular pulse will change if the pulse duration 8 Show that the principle of superposition is valid for the Fourier transform 9 What is the physical meaning Parseval's equality? What does the concept of practical spectrum width mean and why is it introduced? 38

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When decomposing a periodic signal s(t) in the Fourier series of trigonometric functions we take as the orthogonal system

The orthogonality interval in both cases coincides with the period
functions s(t).

The system of functions (1.18) leads to the trigonometric form of the Fourier series, and the system (1.19) leads to the complex form. There is a simple connection between these two forms.

Let us first use the orthogonal system (1.19). Then the Fourier series must be written in the form

Set of coefficients With n the Fourier series in the basis of trigonometric functions is called frequency spectrum periodic signal. Series coefficients (1.20 ) With n are easily determined using the formulas given in the previous paragraph.

From formula (1.16) it follows that

. (1.21)

Thus, regardless of n norm
. Using formula (1.9), we obtain

. (1.22)

Expressions (1.21) and (1.22) take into account that the functions
corresponds to the complex conjugate function

Odds With n in the general case they are complex quantities. Substituting in (1.22)

Cosine (real) and sine (imaginary) parts of the coefficient With n are determined by the formulas

,
. (1.24)

It is often convenient to write coefficients in the form

, (1.25)

,
. (1.26), (1.27)

Module is a function even with respect to n, and the argument showing that is even,a odd functions n.

The general expression (1.20) can be reduced to the form

. (1.28)

Now it is easy to move on to the trigonometric form of the Fourier series. Having isolated from the series (1.28) a pair of terms corresponding to any given value |n| , for example |n|=2, and, taking into account the relations
,
we obtain for the sum of these terms

From this it is clear that when passing to the trigonometric form, series (1.28) must be written as follows:

. (1.30)

The meaning of doubling Fourier coefficients c n in the trigonometric series at n > 1 becomes clear from considering the vector diagram (Fig. 1.3), corresponding to (1.29) for |n|=2. Real function
is obtained as the sum of projections onto the horizontal axis OB two vectors of length | With n| , rotating with angular frequency
in mutually opposite directions. A vector rotating counterclockwise corresponds to a positive frequency, and a vector rotating clockwise corresponds to a negative frequency. . After the transition to trigonometric form, the concept of “negative frequency” loses its meaning. Coefficient c Q does not double, since in the spectrum of a periodic signal the component with zero frequency does not have a “understudy”.

Instead of expression (1.30), the following form of notation is often found in mathematical and radio engineering literature:

and
.

Rice. 1.3. Representation of harmonic vibration in the form of two complex

components: with positive and negative frequencies

From a comparison of expressions (1.31) and (1.30) it is clear that the amplitude n th harmonics A n is related to the coefficient |c n | series (1.28) by the relation

, A
,
.

Thus, for all positive values n (including n = 0)

,
. (1.32)

If the signal is a function that is even with respect to t, i.e. s(t)= s(-t), In the trigonometric notation of the series, only cosine terms remain, since the coefficients b n in accordance with formula (1.32) vanish. For odd relatively t functions s(t) , on the contrary, the coefficients go to zero A n and the series consists only of sinusoidal terms.

Two characteristics - amplitude and phase, i.e., the modules and arguments of the complex coefficients of the Fourier series, completely determine the structure of the frequency spectrum of a periodic oscillation. A visual representation of the “width” of the spectrum is given by a graphical representation of the amplitude spectrum. As an example in Fig. 1.4.a, the spectrum of coefficients is constructed | With n |, and in Fig. 1.4, b - amplitude spectrum A n = 2|s n| for the same periodic oscillation. For a comprehensive characterization of the spectrum, such constructions must be supplemented by specifying the initial phases of individual harmonics.

Rice. 1.4. Coefficients of complex (a) and trigonometric (b) Fourier series of a periodic function of time

The spectrum of a periodic function is called linear or discrete, since it consists of separate lines corresponding to discrete frequencies, etc.

The use of Fourier series for harmonic analysis of complex periodic oscillations in combination with the superposition principle is an effective means for studying the influence of linear circuits on the passage of signals. It should, however, be noted that determining the signal at the output of a circuit from the sum of harmonics with given amplitudes and phases is not an easy task, especially if rapid convergence of the Fourier series representing the input signal is not ensured. The most common signals in radio engineering do not meet this condition, and to satisfactorily reproduce the waveforms it is usually necessary to sum large number harmonics

Mathematical notation of harmonic vibrations. Amplitude and phase spectra of a periodic signal. Spectrum of a periodic sequence of rectangular pulses. Internal integral, which is a function of frequency. Spectra of non-periodic signals.

Test

Option No. 4

Spectral (harmonic) signal analysis

Literature

spectral harmonic signal oscillation

where Um, f0, 0, and 0 are the amplitude, frequency, angular frequency, and initial phase of the oscillation, respectively.

In harmonic analysis, the concept of the nth harmonic of a periodic oscillation of frequency u0 is introduced, which is again understood as a harmonic oscillation with a frequency n times higher than the frequency of the main harmonic oscillation.

1. Spectral analysis of periodic signals

where is the average value of the signal over the period or the constant component of the signal;

Fourier series coefficients;

Fundamental frequency (first harmonic frequency); n=1,2,3,…

The set of complex amplitudes n is called the complex spectrum of a periodic signal.

The calculation of signal spectra in the complex domain is much simpler, since there is no need to consider separately the coefficients and the trigonometric form of writing the Fourier series.

2. Spectrum of a periodic sequence of rectangular pulses

Before considering the spectrum of a periodic sequence of rectangular pulses, let us consider the parameters of these pulses.

The parameters of a single pulse are amplitude, pulse duration, rise time, fall duration, and flat top fall (cleavage).

The pulse amplitude Um is measured in volts.

Pulse duration is measured at the base, at levels of 0.1Um or 0.5Um. In the latter case, the pulse duration is called active. The pulse duration is measured in units of time.

The duration of the front tf and fall tс is measured either at the level 0 - Um, or at the level (0.1-0.9) Um. In the latter case, the duration of the front and decline is called active.

Flat top cleavage is characterized by cleavage coefficient? = ?u/Um,

where?u is the chip value; Um - pulse amplitude.

The parameters of the pulse series are the repetition period T, repetition frequency f, duty cycle Q, duty cycle, average voltage values Uav and average power value Pav.

Repetition period T = ti +tp, where T is the period, ti is the pulse duration,
tп - pause duration. T, tи, and tп are measured in units of time.

The repetition frequency f = 1/T is measured in hertz, etc.

The duty cycle Q = T/ti is a dimensionless quantity.

Fill factor = ti/T is a dimensionless quantity.

Average voltage

Let's move on to considering the amplitude and phase spectra of the signal in the form of a periodic sequence of rectangular pulses with duration and amplitude Um, following with a period T (Fig. 2).

Let us consider the case when the middle of the pulse is the beginning of the time count. Then the signal over the period is described by the expression

Complex amplitudes of harmonic components.

The function is sign-alternating and changes its sign to the opposite when the argument n1 changes by the amount?

where k is the serial number of the interval on the frequency scale, counted from zero frequency.

Thus, the harmonic amplitudes, including the DC component, are determined by the expression:

and the phases - by the expression =1, 2,3,...

Expressions for the trigonometric form of recording the Fourier series when counting time from the middle and beginning of the pulse, respectively, have the form:

In Fig. 3. The amplitude and phase spectra of the considered sequence of rectangular pulses with a duty cycle of two are shown.

N = 1,3,5,7, …,

3. Spectra of some periodic signals

Table 1 shows the amplitude and phase spectra, as well as trigonometric forms of recording Fourier series of some of the most frequently encountered periodic signals in practice.

Signals No. 3 and No. 4 are sequences of rectangular pulses with

Signal No. 6 is a sequence of so-called sawtooth pulses with a zero constant component.

Signals No. 7 and No. 8 are sequences of pulses that approximate with good accuracy, respectively, the signals obtained from full-wave and half-wave rectification of sinusoidal signals.

The dotted lines on the amplitude spectra of signals No. 1 - No. 8 depict spectral densities that characterize the behavior of the modulus of the spectral density of single pulses forming sequences.

where DU is the amplitude of the change in the envelope of the amplitude-modulated oscillation.

4. Spectra of non-periodic signals

The latter physically means that the signal has finite energy.

Let us assume that the signal S(t) is converted by repeating it with an arbitrary period T > t2-t1 into a periodic signal S1(t). For this signal, the Fourier series expansion is applicable:

As a result, at T, the signal S1(t) turns into the signal S(t), frequency 1 decreases to d, and n1 turns into the current frequency. Replacing summation by integration, we obtain

The internal integral, which is a function of frequency, is called the complex spectral density or spectral characteristic () of the signal S(t):

In the general case, when the limits t1 and t2 are not specified

Thus, the time and frequency representations of non-periodic signals are related to each other by a pair of Fourier transforms.

Complex spectral density can be presented in the following forms:

() = S()e-j()=A() + jB(),

where A() = B() =

() = arctan.

The function S() is called the spectral density of amplitudes of a non-periodic signal, and the function () is called the spectral density of phases.

We emphasize that the connection between the time and frequency representation of the signal, given by Fourier transforms, exists only for spectral density.

Literature

Kasatkin A.S. Electrical engineering: textbook. for universities / A.S. Kasatkin, M.V. Nemtsov. - 11th ed., erased. ; Grif MO. - M.: Academy, 2007. - 539 p.

Kasatkin A.S. Electrical engineering: textbook. for universities / A.S. Kasatkin, M.V. Nemtsov. - 9th ed., erased. ; Grif MO. - M.: Academia, 2005. - 639 p.

Nemtsov M.V. Electrical engineering: textbook. allowance for the environment. textbook institutions / M.V. Nemtsov, I.I. Svetlakova. - Grif MO. - Rostov n/d: Phoenix, 2004. - 572 p.

Moskalenko V.V. "Automated electric drive". Textbook for universities. M.: Energoatomizdat, 1986.

"Electrical Engineering", ed. V.S. Pantyushina, M.: Higher School, 1976.

"General Electrical Engineering" ed. A.T. Blazhkina, L.: Energy, 1979.

Spectral (harmonic) analysis of signals. Harmonic analysis of periodic signals Mathematical recording of harmonic oscillations

Mathematical notation of harmonic vibrations. Amplitude and phase spectra of a periodic signal. Spectrum of a periodic sequence of rectangular pulses. Internal integral, which is a function of frequency. Spectra of non-periodic signals.

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Mathematical notation of harmonic vibrations. Amplitude and phase spectra of a periodic signal. Spectrum of a periodic sequence of rectangular pulses. Internal integral, which is a function of frequency. Spectra of non-periodic signals.

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