In electronic equipment for various applications, periodic sequences of rectangular pulses are widely used. In this case, the ratio of the pulse duration τ and the oscillation period T can be very different. For example, vibrations that produce clock generatorsspecifying the "pace" of computers are characterized by comparable values \u200b\u200bof τ and T, and the pulses used in radar can be hundreds of times shorter than the period. Attitude T/ τ is called duty cycle, and the reciprocal (τ / T) - fill factor.
Figure: 6. Sequence of rectangular pulses (a) and coefficients of the Fourier series (b)
Consider a sequence of rectangular pulses with an amplitude AND, duration τ and following with a period T (fig. 6, and). Let's choose the time origin as shown in the figure, that is, so that the pulse is symmetrical about the zero mark, and calculate the coefficients of the Fourier series (1). Since the function s(t) with this position of the axes turns out to be even, all b n are equal to zero, and for a n we get:
The Fourier series for a sequence of rectangular pulses takes the form:
(6)
The values \u200b\u200bof the coefficients of the Fourier series, calculated by formulas (5), are shown in the spectral diagram shown in Fig. 6, b.
Odds a n can be associated with a function 
... Indeed, they will be proportional (with a factor 
) to the values \u200b\u200bof the function 
with arguments corresponding to harmonic frequencies. This can be seen if expression (5) is rewritten as follows:
(7)
So a function like 
is an envelope for coefficients Fourier expansions sequence of rectangular pulses (see Fig. 6, b). Position of envelope zeros on the frequency axis f can be found from the condition 
or 
where. The first time the envelope vanishes at the frequency f\u003d 1 / τ (or ω \u003d 2π / τ). Further, the zeros of the envelope are repeated at f \u003d 2 / τ, 3 / τ, etc. These frequencies can coincide (with integer duty cycles) with the frequencies of any spectrum harmonics, and these frequency components will disappear from the Fourier series. If the duty cycle is an integer, the period T exactly multiples of the pulse duration. Then, between the two zeros of the envelope, there will be spectrum harmonics in the amount q-
1.
How the parameters of pulses are related in time and frequency representations is illustrated in Table. 2.With increasing period T harmonics on the spectral diagram approach each other (the spectrum becomes "thicker"). However, a change in only the period does not lead to a change in the shape of the amplitude spectrum envelope. The evolution of the envelope (shift of its zeros) depends on the pulse duration. Shown here is the evolution of the amplitude spectral diagrams for rectangular pulse trains with varying pulse durations and periods. The ordinate axes of the spectral diagrams show the relative values \u200b\u200bof the amplitudes of the harmonics: 
They are calculated using the formulas:
(8)
Table 2. Oscillograms and spectrograms of rectangular pulse trains
2.5. Spectra of chaotic (noise) oscillations
Chaotic swing s(t) - this is random process... Each of its implementation under unchanged conditions is not repeated, it is unique. In electronics, chaotic vibrations are associated with noises - fluctuations of currents and voltages, changing randomly due to the random movement of charge carriers. In this context, chaotic and noise fluctuations are considered synonymous.
Figure: 7. Block diagram for measuring the mean square noise voltage
Noise swing can be described in frequency representation: a certain spectral characteristic is assigned to it, and for a random process it is continuous. The theoretical foundations of the spectral decomposition of chaotic oscillations are presented in. Without plunging into a rigorous theory, we will explain the methodology for the experimental study of statistical parameters noise voltage s(t) according to the scheme shown in Fig. 8.
R 
is. 8. Scheme for measuring the spectral density of the noise voltage intensity
Let's pass the noise voltage s(t) through a filter that releases vibration energy in a narrow band 
near frequency f... Subject to the condition 
<<
f the oscillation at the filter output will resemble a sinusoid with a frequency f... However, the amplitude and phase of this sinusoid are subject to chaotic changes. With decreasing filter bandwidth 
the shape of the output waveform is increasingly approaching a sinusoid. Its amplitude decreases, but the ratio of the mean square of the voltage passed through the filter ( 
), to the bandwidth 
remains finite and with a successive decrease in the band tends to a certain limit W(f):
Limit value W(f) are called spectral intensity densityprocess s(t).
It is equal to the average intensity of the harmonic components per unit interval of the frequency axis. When measuring W(f) use a narrow-band tunable filter that can be tuned to any frequency within a given measurement range. The noise voltage passed through the filter is subjected to square-law detection and averaged (integrated). The result is a mean square: 
... Further along the known filter band 
calculate W(f).
Full process intensity - middle square 
- are found by integrating the spectral components of the noise over all frequencies:
(10)
To prepare for work, you should study this manual in full. More detailed information on the topic of laboratory work can be found in the chapter "Frequency Spectra of Electrical Oscillations, Spectral Analysis" of the book.
Name of the educational organization:
State budgetary professional educational institution "Stavropol College of Communications named after Hero of the Soviet Union V.A. Petrov "
Year and place of work creation: 2016, cyclic commission of natural and general professional disciplines.
Methodical instructions for the implementation of practical work in the discipline "Telecommunication Theory"
"Calculation and construction of the spectrum of a periodic sequence of rectangular pulses"
for students 2 course of specialties:
11.02.11 Communication networks and switching systems
11.02.09 Multichannel telecommunication systems
full-time education
Objective: to consolidate the knowledge gained in theoretical classes, to develop skills for calculating the spectrum of a periodic sequence of rectangular pulses.
Literature: P.A. Ushakov "Telecommunication circuits and signals". M .: Publishing Center "Academy", 2010, pp. 24-27.
1. Equipment:
1.Personal computer
2. Description of practical work
2. Theoretical material
2.1. A periodic signal of an arbitrary shape can be represented as a sum of harmonic oscillations with different frequencies, this is called a spectral decomposition of a signal.
2.2 ... Harmonics are vibrations whose frequencies are an integer number of times higher than the signal pulse repetition rate.
2.3. The instantaneous voltage value of the periodic signal of the derivative waveform can be written as follows:
Where is the constant component equal to the average value of the signal over the period;
Instantaneous value of the sinusoidal voltage of the first harmonic;
Harmonic frequency equal to the pulse repetition rate;
Amplitude of the first harmonic;
The initial phase of the oscillation of the first harmonic;
Instantaneous value of the second harmonic sinusoidal voltage;
Second harmonic frequency;
Second harmonic amplitude;
The initial phase of the second harmonic oscillation;
Instantaneous value of the sinusoidal voltage of the third harmonic;
Third harmonic frequency;
Amplitude of the third harmonic;
The initial phase of the oscillation of the third harmonic;
2.4. The signal spectrum is a collection of harmonic components with specific values \u200b\u200bof frequencies, amplitudes and initial phases that form the sum of the signal. In practice, the amplitude diagram is most often used
If the signal is a periodic sequence of rectangular pulses, then the dc component is
where Um is the amplitude of the PPPI voltage
s - signal duty cycle (S - T / t);
T is the pulse repetition period;
t is the pulse duration;
The amplitudes of all harmonics are determined by the expression:
Umk \u003d 2Um | sin kπ / s | / kπ
where k is the number of the harmonic;
2.5. Harmonic numbers with amplitudes equal to zero
where n is any integer 1,2,3 ... ..
The number of the harmonic, the amplitude of which turns to zero for the first time, is equal to the duty cycle of the AIR
2.6. The spacing between any adjacent spectral lines is equal to the first harmonic frequency or pulse repetition rate.
2.7 The envelope of the amplitude spectrum of the signal (in Fig. 1 shown by the dotted line)
highlights groups of spectral lines called petals. According to fig. 1 each lobe of the spectrum envelope contains the number of lines equal to the signal duty cycle.
3 ... Pwork order.
3.1. Get a variant of an individual task that corresponds to the number in the list of the group's journal (see appendix).
3.2. Read an example of calculation (see section 4)
4. Example
4.1. Let the repetition period of the PPPI T \u003d .1 μs, the pulse duration t \u003d 0.25 μs, the pulse amplitude \u003d 10V.
4.2. Calculation and construction of the AEFI timing diagram.
4.2.1 ... To construct a time diagram of the PPPI, it is necessary to know the pulse repetition period T, the amplitude and duration of the pulses t, which are known from the problem statement.
4.2.2. To plot the time diagram of the AEFI, it is necessary to select the scales along the stress and time axes. The scales should correspond to the numbers 1,2 and 4 multiplied by 10 n - (where n \u003d 0,1,2,3 ...). The time axis should occupy about 3/4 of the width of the sheet and 2-3 signal periods should be placed on it. The vertical axis of stresses should be 5-10 cm. With a sheet width of 20 cm, the length of the time axis should be approximately 15 cm. It is convenient to place 3 periods on 15 cm, while L 1 \u003d 5 cm will fall on each period. Because
Mt \u003d T / Lt \u003d 1μs / 5cm \u003d 0.2 μs / cm
The result obtained does not contradict the above conditions. On the stress axis, it is convenient to take the scale Мu \u003d 2V / cm (see Figure 2).
4.3. Calculation and construction of the spectral diagram.
4.3.1. The LOI ratio is
4.3.2. Since the duty cycle is S \u003d 4, then 3 petals should be calculated, since 12 harmonics.
4.3.3. The frequencies of the harmonic components are equal
Where k is the harmonic number, l is the PPPI period.
4.3.4. The amplitudes of the PPPI components are
4.3.5. Mathematical model of LOI voltage
4.3.6 Choice of scales.
The frequency axis is located horizontally and with a sheet width of 20 cm should have a length of about 15 cm. Since the highest frequency of 12 MHz should be shown on the frequency axis, it is convenient to take the scale along this axis Mf \u003d 1 MHz / cm.
The stress axis is located vertically and should have a length of 4-5 cm. Since the stress axis must show the greatest stress
It is convenient to take the scale along this axis M \u003d 1V / cm.
4.3.7 The spectral diagram is shown in Fig. 3
The task:
T \u003d 0.75ms; τ \u003d 0.15ms 21.T \u003d 24μs; τ \u003d 8μs
T \u003d 1.5 μs; τ \u003d 0.25μs 22. T \u003d 6.4ms; τ \u003d 1.6ms
T \u003d 2.45ms; τ \u003d 0.35ms 23. T \u003d 7ms; τ \u003d 1.4ms
T \u003d 13.5μs; τ \u003d 4.5μs 24. T \u003d 5.4ms; τ \u003d 0.9ms
T \u003d 0.26ms; τ \u003d 0.65μs 25. T \u003d 17.5μs; τ \u003d 2.5μs
T \u003d 0.9ms; τ \u003d 150μs 26. T \u003d 1.4μs; τ \u003d 0.35μs
T \u003d 0.165ms; τ \u003d 55μs 27. T \u003d 5.4μs; τ \u003d 1.8μs
T \u003d 0.3ms; τ \u003d 75μs 28. T \u003d 2.1ms; τ \u003d 0.3ms
T \u003d 42.5μs; τ \u003d 8.5μs 29. T \u003d 3.5ms; τ \u003d 7ms
T \u003d 0.665ms; τ \u003d 95μs 30. T \u003d 27μs; τ \u003d 4.5μs
T \u003d 12.5μs; τ \u003d 2.5 μs 31. T \u003d 4.2 μs; τ \u003d 0.7μs
T \u003d 38μs; τ \u003d 9.5μs 32.T \u003d 28μs; τ \u003d 7μs
T \u003d 0.9μs; τ \u003d 0.3μs 33. T \u003d 0.3ms; τ \u003d 60μs
T \u003d 38.5μs; τ \u003d 5.5μs
T \u003d 0.21ms; τ \u003d 35ms
T \u003d 2.25ms; τ \u003d 0.45ms
T \u003d 39μs; τ \u003d 6.5μs
T \u003d 5.95ms; τ \u003d 0.85ms
T \u003d 48μs; τ \u003d 16μs
From the output of the message source, signals are received that carry information, as well as clock signals used to synchronize the operation of the transmitter and receiver of the transmission system. Information signals are in the form of non-periodic, and clock signals are in the form of a periodic sequence of pulses.
For a correct assessment of the possibility of transmitting such pulses through communication channels, let us determine their spectral composition. A periodic signal in the form of pulses of any shape can be expanded in a Fourier series according to (7).
For transmission over air and cable communication lines, signals of various shapes are used. The choice of one form or another depends on the nature of the transmitted messages, the frequency spectrum of the signals, the frequency and time parameters of the signals. Signals similar in shape to rectangular pulses are widely used in the technology of transmitting discrete messages.
We calculate the spectrum, i.e. the set of amplitudes of constant and
harmonic components of periodic rectangular pulses (Figure 4, a) with duration and period. Since the signal is an even function of time, in expression (3) all even harmonic components vanish ( 
\u003d 0), and the odd components take on the values:
(10)
The constant component is
(11)
For 1: 1 signal (CW dots) Figure 4a:
,
.
(12)
Moduli of the amplitudes of the spectral components of a sequence of rectangular pulses with a period 
are shown in Fig. 4, b. The abscissa shows the main pulse repetition rate 
() and frequencies of odd harmonic components 
,
etc. The envelope of the spectrum changes according to the law.
With an increase in the period, in comparison with the pulse duration, the number of harmonic components in the spectral composition of the periodic signal increases. For example, for a signal with a period (Figure 4, c), we obtain that the DC component is equal to and
In the frequency band from zero to frequency, there are five harmonic components (Figure 4, d), while there is only one.
With a further increase in the pulse repetition period, the number of harmonic components becomes more and more. In the limiting case when 
the signal becomes a non-periodic function of time, the number of its harmonic components in the frequency band from zero to frequency increases to infinity; they will be located at infinitely close frequency distances; the spectrum of the non-periodic signal becomes continuous.
Figure 4
2.4 Single pulse spectrum
A single video pulse is set (Figure 5):
Figure 5
The Fourier series method allows for a deep and fruitful generalization, which allows one to obtain the spectral characteristics of non-periodic signals. To do this, mentally supplement a single pulse with the same pulses, periodically following at a certain time interval, and we obtain the previously studied periodic sequence:
Let's imagine a single pulse as the sum of periodic pulses with a large period.
,
(14)
where are integers.
For periodic oscillation
.
(15)
In order to return to a single impulse, let us direct the repetition period to infinity:. In this case, it is obvious:
,
(16)
We denote
.
(17)
Quantity is the spectral characteristic (function) of a single pulse (direct Fourier transform). It depends only on the temporal description of the impulse and, in general, is complex:
, (18) where 
; (19)
; (20)
,
where 
- the module of the spectral function (amplitude-frequency characteristic of the pulse);
- phase angle, phase-frequency characteristic of the pulse.
We find for a single pulse by formula (8) using the spectral function:
.
If, we get:
.
(21)
The resulting expression is called the inverse Fourier transform.
The Fourier integral defines momentum as an infinite sum of infinitesimal harmonic components located at all frequencies.
On this basis, one speaks of a continuous (continuous) spectrum possessed by a single pulse.
The total energy of the pulse (the energy released at the active resistance Ohm) is
(22)
Changing the order of integration, we get
.
The inner integral is the spectral function of the momentum taken with the argument -, i.e. is a complex conjugate quantity: 
Consequently
Modulus squared (the product of two conjugate complex numbers is equal to the modulus squared).
In this case, it is conventionally said that the pulse spectrum is two-sided, i.e. is located in the frequency band from to.
The above relation (23), which establishes the relationship between the pulse energy (at a resistance of 1 Ohm) and the modulus of its spectral function, is known as Parseval's equality.
It claims that the energy contained in a pulse is equal to the sum of the energies of all its spectrum components. Parseval's equality characterizes an important property of signals. If some electoral system passes only part of the signal spectrum, attenuating its other components, then this means that part of the signal energy is lost.
Since the square of the modulus is an even function of the variable of integration, then by doubling the value of the integral, integration can be introduced in the range from 0 to:
.
(24)
It is said that the pulse spectrum is located in the frequency band from 0 to and is called one-sided.
The integrand in (23) is called the energy spectrum (spectral energy density) of the pulse
It characterizes the distribution of energy over frequency, and its value at a frequency is equal to the pulse energy per frequency band equal to 1 Hz. Therefore, the pulse energy is the result of the integration of the signal energy spectrum over the entire frequency range separately; in other words, the energy is equal to the area enclosed between the curve representing the signal energy spectrum and the abscissa axis.
To estimate the distribution of energy over the spectrum, use the relative integral function of the distribution of energy (energy characteristic)
,
(25)
where 
is the pulse energy in a given frequency band from 0 to, which characterizes the fraction of the pulse energy concentrated in the frequency range from 0 to.
For single impulses of various shapes, the following regularities are fulfilled:
From the output of the message source, signals are received that carry information, as well as clock signals used to synchronize the operation of the transmitter and receiver of the transmission system. Information signals are in the form of non-periodic, and clock signals are in the form of a periodic sequence of pulses.
For a correct assessment of the possibility of transmitting such pulses through communication channels, let us determine their spectral composition. A periodic signal in the form of pulses of any shape can be expanded in a Fourier series according to (7).
For transmission over air and cable communication lines, signals of various shapes are used. The choice of one form or another depends on the nature of the transmitted messages, the frequency spectrum of the signals, the frequency and time parameters of the signals. Signals similar in shape to rectangular pulses are widely used in the technology of transmitting discrete messages.
We calculate the spectrum, i.e. the set of amplitudes of constant and
harmonic components of periodic rectangular pulses (Figure 4, a) with duration and period. Since the signal is an even function of time, in expression (3) all even harmonic components vanish ( 
\u003d 0), and the odd components take on the values:
(10)
The constant component is
(11)
For 1: 1 signal (CW dots) Figure 4a:
,
.
(12)
Moduli of the amplitudes of the spectral components of a sequence of rectangular pulses with a period 
are shown in Fig. 4, b. The abscissa shows the main pulse repetition rate 
() and frequencies of odd harmonic components 
,
etc. The envelope of the spectrum changes according to the law.
With an increase in the period, in comparison with the pulse duration, the number of harmonic components in the spectral composition of the periodic signal increases. For example, for a signal with a period (Figure 4, c), we obtain that the DC component is equal to and
In the frequency band from zero to frequency, there are five harmonic components (Figure 4, d), while there is only one.
With a further increase in the pulse repetition period, the number of harmonic components becomes more and more. In the limiting case when 
the signal becomes a non-periodic function of time, the number of its harmonic components in the frequency band from zero to frequency increases to infinity; they will be located at infinitely close frequency distances; the spectrum of the non-periodic signal becomes continuous.
Figure 4
2.4 Single pulse spectrum
A single video pulse is set (Figure 5):
Figure 5
The Fourier series method allows for a deep and fruitful generalization, which allows one to obtain the spectral characteristics of non-periodic signals. To do this, mentally supplement a single pulse with the same pulses, periodically following at a certain time interval, and we obtain the previously studied periodic sequence:
Let's imagine a single pulse as the sum of periodic pulses with a large period.
,
(14)
where are integers.
For periodic oscillation
.
(15)
In order to return to a single impulse, let us direct the repetition period to infinity:. In this case, it is obvious:
,
(16)
We denote
.
(17)
Quantity is the spectral characteristic (function) of a single pulse (direct Fourier transform). It depends only on the temporal description of the impulse and, in general, is complex:
, (18) where 
; (19)
; (20)
,
where 
- the module of the spectral function (amplitude-frequency characteristic of the pulse);
- phase angle, phase-frequency characteristic of the pulse.
We find for a single pulse by formula (8) using the spectral function:
.
If, we get:
.
(21)
The resulting expression is called the inverse Fourier transform.
The Fourier integral defines momentum as an infinite sum of infinitesimal harmonic components located at all frequencies.
On this basis, one speaks of a continuous (continuous) spectrum possessed by a single pulse.
The total energy of the pulse (the energy released at the active resistance Ohm) is
(22)
Changing the order of integration, we get
.
The inner integral is the spectral function of the momentum taken with the argument -, i.e. is a complex conjugate quantity: 
Consequently
Modulus squared (the product of two conjugate complex numbers is equal to the modulus squared).
In this case, it is conventionally said that the pulse spectrum is two-sided, i.e. is located in the frequency band from to.
The above relation (23), which establishes the relationship between the pulse energy (at a resistance of 1 Ohm) and the modulus of its spectral function, is known as Parseval's equality.
It claims that the energy contained in a pulse is equal to the sum of the energies of all its spectrum components. Parseval's equality characterizes an important property of signals. If some electoral system passes only part of the signal spectrum, attenuating its other components, then this means that part of the signal energy is lost.
Since the square of the modulus is an even function of the variable of integration, then by doubling the value of the integral, integration can be introduced in the range from 0 to:
.
(24)
It is said that the pulse spectrum is located in the frequency band from 0 to and is called one-sided.
The integrand in (23) is called the energy spectrum (spectral energy density) of the pulse
It characterizes the distribution of energy over frequency, and its value at a frequency is equal to the pulse energy per frequency band equal to 1 Hz. Therefore, the pulse energy is the result of the integration of the signal energy spectrum over the entire frequency range separately; in other words, the energy is equal to the area enclosed between the curve representing the signal energy spectrum and the abscissa axis.
To estimate the distribution of energy over the spectrum, use the relative integral function of the distribution of energy (energy characteristic)
,
(25)
where 
is the pulse energy in a given frequency band from 0 to, which characterizes the fraction of the pulse energy concentrated in the frequency range from 0 to.
For single impulses of various shapes, the following regularities are fulfilled:
SIGNALS
Consider a few examples of periodic oscillations often used in various radio engineering devices.
1. RECTANGULAR VIBRATION (FIG. 2.3)
Such an oscillation, often called a meander, is especially widely used in measurement technology.
When choosing the start of timing in accordance with Fig. 2.3, and the function is odd, and Fig. 2.3, b - even. Applying formulas (2.24), we find for an odd function (Fig. 2.3, a) with s (t) \u003d e (t):
Figure: 2.3. Periodic rectangular oscillation (meander)
Figure: 2.4. The coefficients of the complex (a) and trigonometric (b) Fourier series of the oscillation shown in Fig. 2.3
Considering that, we get
The initial phases in accordance with (2.27) are equal for all harmonics.
Let's write the Fourier series in trigonometric form
The spectrum of the coefficients of the complex Fourier series is shown in Fig. 2.4, a, and the trigonometric series - in Fig. 2.4, b (at).
When counting the time from the middle of the pulse (Fig. 2.3, b), the function is even with respect to t and for it
Graphs of the 1st harmonics and their sums are shown in Fig. 2.5, a. In fig. 2.5, b, this sum is supplemented by the 5th harmonic, and in Fig. 2.5, c - 7th.
With an increase in the number of summed harmonics, the sum of the series approaches the function everywhere, except for the points of discontinuity of the function, where an outlier is formed. When the value of this outlier is equal, i.e. the sum of the series differs from the given function by 18%. This convergence defect in mathematics is called the Gibbs phenomenon.
Figure: 2.5. Summation of the 1st and 3rd harmonics (a), 1st, 3rd and 5th harmonics (b), 1st, 3rd, 5th and 7th harmonics (c) of the oscillation shown in Fig. 2.3
Figure: 2.6 Periodic oscillation of the sawtooth shape
Figure: 2.7. The sum of the first five harmonics of the oscillation shown in Fig. 2.6
Despite the fact that, in the case under consideration, the Fourier series does not converge to the function being expanded at the points of its discontinuity, the series converges on average, since at the outliers are infinitely narrow and do not make any contribution to the integral (2.13).
2. SAW VIBRATION (FIG. 2.6)
Functions like these are often dealt with in oscilloscope scanners. Since this function is odd, the Fourier series for it contains only sinusoidal terms. Using formulas (2.24) - (2.31), it is easy to determine the coefficients of the Fourier series. Omitting these calculations, we write the final expression for the series
As you can see, the amplitudes of the harmonics decrease according to the law, where. In fig. 2.7 shows a graph of the sum of the first five harmonics (enlarged).
3. SEQUENCE OF UNIPOLAR TRIANGULAR PULSES (FIG. 2.8)
The Fourier series for this function is as follows:
Figure: 2.8. Sum of the first three harmonics of a periodic function
Figure: 2.9. Periodic sequence of rectangular pulses with high duty cycle
In fig. 2.8 shows the sum of the first three terms of this series. In this case, we note a more rapid decrease in the amplitudes of the harmonics than in the previous examples. This is due to the absence of breaks (jumps) in the function.
4. SEQUENCE OF UNIPOLAR RECTANGULAR PULSES (FIG 2.9)
Applying formula (2.32), we find the average value (constant component)
and harmonic distortion
