To determine the spectra for various types of pulse modulation, we find the spectrum of the carrier itself. Let's take a pulse carrier with rectangular pulses (Fig. 3.10).
Rice. 3.10 Periodic square wave train
The sequence of such pulses can be represented by Fourier series.
,
(3.32)
where 
- complex amplitude of the k-th harmonic;
is a constant component.
Let's find the complex amplitudes for the specified limits (Fig. 3.10).
(3.33)
DC
(3.34)
Substitute (3.33) and (3.34) into (3.32) and after transformation we get:
(3.35)
It can be seen from the expression that the spectrum is line-shaped with an envelope that repeats the spectrum of a single pulse (Fig. 3.11). In other words, for pulses of the same shape, the lattice function is inscribed in a continuous S(jω).
R 
is. 3.11 Spectrum periodic sequence impulses
The constant component A 0 /2 has half the value. The distance between the components of the harmonics is equal to the fundamental frequency of the carrier ω 0 =2π/T. It follows that a change in the period T of the pulse repetition leads to a change in the density of discrete components, and a change in the duty cycle T/τ at a constant period (i.e., a change in τ) causes a narrowing or expansion of the envelope while maintaining its shape, leaving the distance between the lines of the discrete spectrum unchanged . At a sufficiently high density of these lines, when at least several lines of the spectrum (T>>τ) are placed between the nodes, the width of the spectrum ω of the pulsed carrier can be considered practically the same as for a single pulse. As τ approaches T, these spectra may turn out to be different in width. On Fig. 3.12 shows the deformation of the spectrum of a pulsed carrier with a change in T, and in Fig. 3.13 when changing τ for rectangular pulses.
R 
is. 3.12 Changing the nature of the spectrum of the carrier when changing
period T of the repetition of rectangular pulses.
With a constant amplitude of the pulses, according to expression (3.25), the envelope of the discrete spectrum increases in proportion to the increase in the area of the pulses (Fig. 3.13).
It should be noted that there is no periodic sequence in its pure form, since any sequence has a beginning and an end. The degree of approximation depends on the number of pulses in the sequence. Therefore, for a rigorous description of a pulsed carrier, the latter should be considered as a single pulse, which is a packet of elementary pulses of a certain shape. Such a signal has a continuous spectrum.
However, as the number of pulses in the sequence is accumulated, its spectrum is split up and deformed in such a way that it approaches more and more the lattice pattern.
Rice. 3.13 Changing the nature of the spectrum of the carrier when changing
pulse duration τ for rectangular pulses.
3.7 Signal spectra with pulse modulation
The spectra of all types of pulse modulations have a complex structure, and the conclusions are often too cumbersome. For this reason, we will consider the question of the spectral composition of pulse modulation signals, omitting in some cases too complex intermediate transformations. Such a consideration allows us to show the approach to the problem, outline the path of solution and analyze the final conclusions.
Let's find the spectrum with amplitude-pulse modulation (AIM). To simplify, we choose the modulating function f(t) containing one harmonic sint
Expanding this expression and replacing the product of sine with cosine
.
(3.36)
AND 
From (3.36) it can be seen that the signal spectrum contains the frequency of the modulating function and the highest harmonic components kω 0 ± with two lateral satellites. In this case, the highest harmonic components fit into the envelope of the spectrum of a single carrier pulse. On Fig. 3.14 shows the spectrum with amplitude-pulse modulation.
Rice. 3.14 Spectrum with amplitude-pulse modulation.
The spectrum width does not change during AIM, since the magnitude of the amplitudes that must be taken into account when determining the width depends only on the ratio τ /T, and this value is constant during AMI. If the sequence of pulses is modulated by a complex function from min to max , then not spectral lines appear in the spectrum after modulation, but frequency bands min ... max and kω 1 ±( min ... max)
Let us consider the features of the spectrum during pulse-phase modulation (PPM), which belongs to a variety of time-pulse modulation (PPM).
P 
For PIM - modulation (Fig. 3.15), the dotted line shows the change in the modulating function over time. The vertical dotted lines correspond to the position of the transition edges of the unmodulated pulse train. The figure shows that the position of the pulses (phase) changes with respect to the so-called clock points t k corresponding to the position on the time axis of the leading edges of the unmodulated pulse sequence. The shift of one of the pulses by the time ∆t k is shown in the figure.
Rice. 3.15 Illustration of PIM - modulation.
Rice. 3.16 Pulse position without modulation
and in the presence of modulation.
On fig. 3.16, the dotted line shows an unmodulated pulse located symmetrically about the clock point corresponding to the origin. During modulation, the pulse will shift by an amount 
, where t 1 corresponds to the new position of the leading edge, and t 2 to the new position of the trailing edge. We assume that the maximum momentum shift ∆t K corresponds to the value U(t) = 1.
If the modulating function changes sinusoidally, then for the modulated pulse, the times corresponding to the position of the leading and trailing edges will be:
(3.37)
(3.38)
In the last expression (3.38), the time value is (t-τ) since the trailing edge is shifted relative to the leading one by the value of the pulse duration.
To obtain the spectrum with PIM, it is necessary to substitute the value t 2 -t 1 instead of τ, since t 1 and t 2 are the current coordinates. You can reflect the offset of the centerline by replacing the time t with the time 
. As a result of substitution of these values in (3.35) we get:
(3.39)
Substituting the values t 1 and t 2 into expression (3.39) and after the transformation we obtain an expression that coincides with the spectrum at AIM, only near the component of the fundamental frequency and each higher harmonic, not one lower and one upper side spectral lines appeared, but bands of side harmonics with frequencies (kω 0 ±n).
An approximate view of the spectrum is shown in fig. 3.17. However, the lateral satellites decay rapidly, since they include Bessel functions.
R 
is. 3.17 Spectrum with phase-pulse modulation.
Spectra with PWM and PFM in their composition are the same as the spectrum with PIM - modulation.
Despite the fact that the nature of the spectrum during carrier modulation changes and depends on the type of modulation, its width remains the same as for a single pulse and is determined mainly by the pulse duration τ.
The transmission of measurement information in time division telemetry devices is often more preferable than transmission using frequency division channels, since time division does not require filters and, in addition, the bandwidth does not depend on the number of channels.
Depending on the type of modulation in the channels (primary) and the type of carrier frequency modulation (secondary), there are main types of telemetry devices with time division of channels: AIM-FM, PWM-FM, FIM-AM, FIM-FM, KIM-AM, KIM- World Cup
Time division channel systems are used to transmit measurement information from artificial satellites and spacecraft.
Name educational organization:
State budget professional educational institution Stavropol College of Communications named after Hero Soviet Union V.A. Petrov"
Year and place of work creation: 2016, cycle commission of natural and general professional disciplines.
Guidelines to fulfillment practical work in the discipline "Theory of telecommunications"
"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: solidify the knowledge gained in theoretical classes, develop skills in calculating the spectrum of a periodic sequence of rectangular pulses.
Literature: P.A. Ushakov "Chains and signals of telecommunication". 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 signal.
2.2 . Harmonics are called oscillations, the frequencies of which are an integer number of times greater than the frequency of the signal pulses.
2.3. The instantaneous value of the voltage of a periodic signal of the derivative form can be written as follows:
Where is the constant component equal to the average value of the signal for 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 sinusoidal voltage of the second harmonic;
Second harmonic frequency;
Amplitude of the second harmonic;
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 third harmonic oscillation;
2.4. The spectrum of a signal is a set of harmonic components with specific values of 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 constant component is equal to
where Um is the voltage amplitude of the PPTR
s - signal duty cycle (S - T/t);
T is the pulse repetition period;
t is the duration of the pulses;
The amplitudes of all harmonics are determined by the expression:
Umk = 2Um | sin kπ/s | /kπ
where k is the harmonic number;
2.5. Harmonic numbers whose amplitudes are equal to zero
where n is any integer 1,2,3…..
The number of the harmonic, the amplitude of which vanishes for the first time, is equal to the duty cycle of the PPPI
2.6. The interval between any neighboring spectral lines is equal to the frequency of the first harmonic or the pulse repetition rate.
2.7 Envelope of the amplitude spectrum of the signal (shown in Fig. 1 by a dotted line)
identifies groups of spectral lines called petals. According to fig. 1, each lobe of the spectrum envelope contains a number of lines equal to the duty cycle of the signal.
3 . Pwork order.
3.1. Get an individual task variant that matches the number in the group's journal list (see appendix).
3.2. Read the calculation example (see section 4)
4. Example
4.1. Let the repetition period of the FPPI T=.1 µs, pulse duration t=0.25 µs, pulse amplitude=10V.
4.2. Calculation and construction of the time diagram of the FRPP.
4.2.1 . To construct a timing diagram of the PPTR, it is necessary to know the pulse repetition period T, the amplitude and duration of the pulses t, which are known from the condition of the problem.
4.2.2. To build a time diagram of the FPPI, it is necessary to select the scales along the stress and time axes. The scales should correspond to the numbers 1,2 and 4 times 10 n - (where n=0,1,2,3...). The time axis should occupy approximately 3/4 of the sheet width and 2-3 signal periods should be placed on it. The vertical stress axis should be equal to 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 each period will have L 1 \u003d 5 cm. Because
Mt=T/Lt=1µs/5cm= 0.2µs/cm
The result obtained does not contradict the above conditions. It is convenient to take the scale Mu=2V/cm on the stress axis (see Fig. 2).
4.3. Calculation and construction of the spectral diagram.
4.3.1. The duty cycle of the FPPI is equal to
4.3.2. Since the duty cycle is S = 4, then 3 petals should be calculated, because 12 harmonics.
4.3.3. The frequencies of the harmonic components are equal
Where k is the number of the harmonic, l is the period of the FPPI.
4.3.4. The amplitudes of the APPI components are equal to
4.3.5. Mathematical model of voltage FPPI
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 must be shown on the frequency axis, it is convenient to take a scale along this axis Mf = 1 MHz / cm.
The stress axis is located vertically and should have a length of 4-5 cm. Since the largest stress must be shown from the stress axis
It is convenient to take the scale along this axis M=1V/cm.
4.3.7. The spectral diagram is shown in Fig. 3
The task:
T=0.75ms; τ=0.15ms 21.T=24µs; τ=8μs
T=1.5 µs; τ=0.25ms 22. T=6.4ms; τ=1.6ms
T=2.45ms; τ=0.35ms 23. T=7ms; τ=1.4ms
T=13.5µs; τ=4.5ms 24. T=5.4ms; τ=0.9ms
T=0.26ms; τ=0.65µs 25. T=17.5µs; τ=2.5µs
T=0.9ms; τ=150µs 26. T=1.4µs; τ=0.35µs
T=0.165ms; τ=55µs 27. T=5.4µs; τ=1.8ms
T=0.3ms; τ=75µs 28. T=2.1ms; τ=0.3ms
T=42.5µs; τ=8.5ms 29. T=3.5ms; τ=7ms
T=0.665ms; τ=95µs 30. T=27µs; τ=4.5µs
T=12.5µs; τ=2.5µs 31. T=4.2µs; τ=0.7µs
T=38µs; τ=9.5µs 32.T=28µs; τ=7μs
T=0.9µs; τ=0.3ms 33. T=0.3ms; τ=60µs
T=38.5µs; τ=5.5µs
T=0.21ms; τ=35ms
T=2.25ms; τ=0.45ms
T=39µs; τ=6.5µs
T=5.95ms; τ=0.85ms
T=48µs; τ=16μs
2. Spectrum of a periodic sequence of rectangular pulses
Consider a periodic sequence of rectangular pulses shown in fig. 5. This signal is characterized by the pulse duration, its amplitude and period. The voltage is plotted along the vertical axis.
Fig.5. Periodic sequence of rectangular pulses
Let's choose the beginning of counting in the middle of the pulse. Then the signal is decomposed only in cosines. Harmonic frequencies are n/T , where n- any integer. The amplitudes of the harmonics according to (1.2.) will be equal to:
because V(t)=E at , where is the pulse duration and V(t)=0 at , then
This formula can be conveniently written as:
(2.1.)
). This function is called the spectrum envelope. It should be borne in mind that physical meaning it only has frequencies where the corresponding harmonics exist. On fig. 6 shows the spectrum of a periodic sequence of rectangular pulses.
Fig.6. Periodic Sequence Spectrum
square pulses.
When constructing the envelope, we mean that - is
An oscillating function of frequency, and the denominator increases monotonically with increasing frequency. Therefore, a quasi-oscillating function with a gradual decrease is obtained. When the frequency tends to zero, both the numerator and the denominator tend to zero at the same time, their ratio tends to unity (the first classical limit). Zero values of the envelope occur at points where i.e.
Where mis an integer (exceptm
IN previous sections we considered the expansion of periodic signals in a Fourier series, and also studied some properties of the representation of periodic signals in a Fourier series. We said that periodic signals can be represented as a series of complex exponentials separated from each other by a frequency rad/s, where is the signal repetition period. As a result, we can treat the representation of a signal as a series of complex harmonics as a complex signal spectrum. The complex spectrum, in turn, can be divided into the amplitude and phase spectra of a periodic signal.In this section, we will consider the spectrum of a periodic sequence of rectangular pulses, as one of the most important signals used in practical applications.
Spectrum of a periodic sequence of rectangular pulses
Let the input signal be a periodic sequence of rectangular pulses of amplitude , duration of seconds followed by a period of seconds, as shown in Figure 1
Figure 1. Periodic sequence of rectangular pulses
The unit of signal amplitude depends on the physical process that describes the signal. It can be voltage, or current, or any other physical quantity with its own unit of measurement, which changes over time like . In this case, the units of measurement of the amplitudes of the spectrum , , will coincide with the units of measurement of the amplitude of the original signal.
Then the spectrum , , of this signal can be represented as:
The spectrum of a periodic sequence of rectangular pulses is a set of harmonics with an envelope of the form 
.
Properties of the spectrum of a periodic train of rectangular pulses
Let us consider some properties of the spectrum envelope of a periodic sequence of rectangular pulses.
The constant component of the envelope can be obtained as a limit:
To disclose the uncertainty, we use the L'Hopital rule:
Where is called the duty cycle of the pulses and sets the ratio of the pulse repetition period to the duration of a single pulse.
Thus, the value of the envelope at zero frequency is equal to the amplitude of the pulse divided by the duty cycle. With an increase in the duty cycle (ie, with a decrease in the pulse duration at a fixed repetition period), the value of the envelope at zero frequency decreases.
Using the duty cycle of the pulses, expression (1) can be rewritten as:
The zeros of the envelope of the spectrum of a train of rectangular pulses can be obtained from the equation:
The denominator vanishes only when , however, as we found out above 
, then the solution to the equation is
Then the envelope vanishes if
Figure 2 shows the envelope of the spectrum of a periodic sequence of rectangular pulses (dashed line) and the frequency relationships of the envelope and the discrete spectrum .
Figure 2. Spectrum of a periodic sequence of rectangular pulses
Also shown are the amplitude envelope , amplitude spectrum, as well as the phase envelope and phase spectrum .From Figure 2, you can see that the phase spectrum takes on values when the envelope has negative values. Note that and correspond to the same point of the complex plane equal to .
An example of the spectrum of a periodic train of rectangular pulses
Let the input signal be a periodic sequence of rectangular pulses of amplitude following with a period of a second and different duty cycles. Figure 3a shows the time oscillograms of these signals, their amplitude spectra (Figure 3b), as well as the continuous envelopes of the spectra (dashed line).
Figure 3. Spectrum of a periodic sequence of rectangular pulses at different duty cycles
a - time oscillograms; b - amplitude spectrum
As can be seen from Figure 3, as the signal duty cycle increases, the pulse duration decreases, the spectrum envelope expands and decreases in amplitude (dashed line). As a result, the number of spectrum harmonics increases within the main lobe.
Spectrum of a Time-Shifted Periodic Rectangular Pulse Train
Above, we studied in detail the spectrum of a periodic sequence of rectangular pulses for the case when the original signal was symmetrical with respect to . As a result, the spectrum of such a signal is real and is given by expression (1). Now we will consider what will happen to the spectrum of the signal if we shift the signal in time, as shown in Figure 4.
Figure 4. Timeshifted Periodic Rectangular Pulse Train
A shifted signal can be represented as a signal delayed by half the pulse duration 
. The spectrum of the shifted signal can be represented according to the property of the cyclic time shift as:
Thus, the spectrum of a periodic sequence of rectangular pulses shifted relative to zero is not a purely real function, but acquires an additional phase factor 
. The amplitude and phase spectra are shown in Figure 5.
Figure 5. Amplitude and phase spectra of a time-shifted periodic sequence of rectangular pulses
It follows from Figure 5 that the shift of a periodic signal in time does not change the amplitude spectrum of the signal, but adds a linear component to the phase spectrum of the signal.
conclusions
In this section, we have obtained an analytical expression for the spectrum of a periodic train of rectangular pulses.
We have considered the properties of the spectrum envelope of a periodic sequence of rectangular pulses and have given examples of spectra for different values of the duty cycle.
The spectrum was also considered with a time shift of a sequence of rectangular pulses and it was shown that the time shift changes the phase spectrum and does not affect the amplitude spectrum of the signal.
Moscow, Soviet Radio, 1977, 608 p.Dech, G. Guide to the practical application of the Laplace transform. Moscow, Nauka, 1965, 288 p.
Periodic and non-periodic signals other than sinusoidal waveforms are commonly referred to as pulse signals. The processes of generation, conversion, as well as the issues of practical application of pulsed signals are today related to many areas of electronics.
So, for example, not a single modern power supply can do without a rectangular pulse generator located on its printed circuit board, such as, for example, on the TL494 chip, which produces pulse sequences with parameters suitable for the current load.
Since pulse signals can have different shapes, they name different pulses in accordance with a geometric figure similar in shape: rectangular pulses, trapezoidal pulses, triangular pulses, sawtooth pulses, stepped pulses, and pulses of various other shapes. Meanwhile, the most commonly used rectangular pulses. Their parameters will be discussed in this article.
Of course, the term "rectangular pulse" is somewhat arbitrary. Due to the fact that nothing is ideal in nature, just as there are no ideally rectangular pulses. In fact, a real pulse, which is commonly called a rectangular one, can also have oscillatory spikes (shown as b1 and b2 in the figure), due to quite real capacitive and inductive factors.
These emissions may, of course, be absent, however, there are electrical and temporal parameters of the pulses, reflecting, among other things, "their non-ideal squareness."
A rectangular pulse has a certain polarity and operating level. Most often, the polarity of the pulse is positive, since the vast majority of digital microcircuits are powered by a positive, relative to the common wire, voltage, and therefore the instantaneous value of the voltage in the pulse is always greater than zero.
But there are, for example, comparators powered by a bipolar voltage, in such circuits you can find bipolar pulses. In general, microcircuits powered by a negative polarity voltage are not as widely used as microcircuits with a conventional positive supply.
In a sequence of pulses, the operating voltage of the pulse can take a low or high level, with one level replacing another over time. The low voltage level is indicated by U0, the high level by U1. The highest instantaneous value of the voltage in the pulse Ua or Um, relative to the initial level, is called pulse amplitude.
Developers of impulse devices often operate with active impulses high level, such as the one shown on the left. But sometimes it is practically expedient to use low-level pulses as active pulses, for which the initial state is a high voltage level. The low level pulse is shown in the figure on the right. Calling a low-level impulse a "negative impulse" is illiterate.
The voltage drop in a rectangular pulse is called a front, which is a fast (commensurate in time with the time of the transient process in the circuit) change in the electrical state.
The transition from a low level to a high level, that is, a positive edge, is called the rising edge or simply the edge of the pulse. The high-to-low transition, or falling edge, is called a cutoff, roll-off, or simply the trailing edge of an impulse.
The leading edge is denoted in the text by 0.1 or schematically by _|, and the trailing edge by 1.0 or schematically by |_.
Depending on the inertial characteristics of active elements, the transient process (drop) in a real device always takes some finite time. Therefore, the total duration of the pulse includes not only the times of existence of high and low levels, but also the times of the duration of the fronts (front and cut), which are denoted by Tf and Tav. In almost any particular circuit, rise and fall times can be seen with .
Since in reality it is not easy to single out the start and end moments of transient processes in drops very accurately, it is customary to consider the time interval during which the voltage changes from 0.1Ua to 0.9Ua (front) or from 0.9Ua to 0, as the duration of the drop, 1Ua (cut). So the steepness of the front Kf and the steepness of the cut Ks.r. are set according to the given boundary states, and are measured in volts per microsecond (V/µs). The duration of the pulse is directly called the time interval counted from the level of 0.5Ua.
When the processes of formation and generation of pulses are considered in general, the front and cutoff are taken as zero in duration, since these small time intervals are not critical for rough calculations.
These are impulses that follow each other in a certain order. If the pauses between pulses and the duration of the pulses in the sequence are equal to each other, then this is a periodic sequence. The pulse repetition period T is the sum of the pulse duration and the pause between pulses in the sequence. The pulse repetition frequency f is the reciprocal of the period.
Periodic sequences of rectangular pulses, in addition to the period T and frequency f, are characterized by a couple of additional parameters: the DC duty cycle and the duty cycle Q. The duty cycle is the ratio of the pulse duration time to its period.
The duty cycle is the ratio of the pulse period to the time of its duration. A periodic sequence of duty cycle Q=2, that is, one in which the pulse duration time is equal to the pause time between pulses or in which the duty cycle is DC=0.5, is called a meander.
