Literature: [L.1], p. 40
As an example, let us present the Fourier series expansion of a periodic sequence of rectangular pulses with amplitude, duration and repetition period symmetric about zero, i.e.
, (2.10)
Here 
The expansion of such a signal in a Fourier series gives
, (2.11)
where is the duty cycle.
To simplify the notation, you can enter the notation
, (2.12)
Then (2.11) can be written as follows
, (2.13)
In fig. 2.3 shows a sequence of rectangular pulses. The spectrum of a sequence, like any other periodic signal, is discrete (linear).
The envelope of the spectrum (Fig. 2.3, b) is proportional to 
... The distance along the frequency axis between two adjacent components of the spectrum is equal, and between two zero values \u200b\u200b(width of the spectrum lobe) -. The number of harmonic components within one lobe, including the zero value on the right in the figure, is, where the sign means rounding to the nearest integer, less (if the duty cycle is a fractional number), or (with an integer value of the duty cycle). With an increase in the period, the fundamental frequency 
decreases, the spectral components on the diagram approach each other, the amplitudes of the harmonics also decrease. In this case, the shape of the envelope is preserved.
When solving practical problems of spectral analysis, instead of angular frequencies, cyclic frequencies are used 
, measured in Hertz. Obviously, the distance between adjacent harmonics on the diagram will be, and the width of one lobe of the spectrum will be. These values \u200b\u200bare shown in the diagram in parentheses.
In practical radio engineering, in most cases, instead of a spectral representation (Fig. 2.3, b), spectral diagrams of the amplitude and phase spectra are used. The amplitude spectrum of a sequence of rectangular pulses is shown in Fig. 2.3, c.
Obviously, the envelope of the amplitude spectrum is proportional to 
.
As for the phase spectrum (Fig. 2.3, d), it is believed that the initial phases of the harmonic components change abruptly by the value -π when changing the sign of the envelope sinc kπ / q... The initial phases of the harmonics of the first lobe are assumed to be zero. Then the initial phases of the second lobe harmonics will be φ = -π , third petal φ \u003d -2π etc.
Let's consider another representation of the signal by the Fourier series. For this we use Euler's formula
.
In accordance with this formula, the kth component (2.9) of the signal expansion in the Fourier series can be represented as follows
; 
. (2.15)
Here the quantities and are complex and represent the complex amplitudes of the spectrum components. Then the series
Fourier (2.8) taking into account (2.14) takes the following form
, (2.16)
, (2.17)
It is not hard to verify that the expansion (2.16) is carried out in basis functions 
, which are also orthogonal on the interval 
, i.e.
Expression (2.16) is complex form Fourier series, which extends to negative frequencies. The quantities and 
, where means the complex conjugate of the quantity, are called complex amplitudes spectrum. Because is a complex quantity, it follows from (2.15) that
and 
.
Then the set makes up the amplitude, and the set makes up the phase spectrum of the signal.
In fig. 2.4 shows the spectral diagram of the spectrum of the above sequence of rectangular pulses, represented by a complex Fourier series
The spectrum also has a linear character, but in contrast to the previously considered spectra, it is determined both in the region of positive and in the region of negative frequencies. Since it is an even function of the argument, the spectral diagram is symmetric about zero.
Based on (2.15), it is possible to establish a correspondence between both the coefficients and the expansion (2.3). Because
and 
,
then as a result we get
. (2.18)
Expressions (2.5) and (2.18) allow us to find values \u200b\u200bin practical calculations.
Let us give a geometric interpretation of the complex form of the Fourier series. Let's select the k-th component of the signal spectrum. In complex form, the kth component is described by the formula
where and are defined by expressions (2.15).
In the complex plane, each of the terms in (2.19) is depicted as vectors of length 
rotated by an angle and relative to the real axis and rotating in opposite directions with frequency (Fig. 2.5).
Obviously, the sum of these vectors gives a vector located on the real axis, the length of which is. But this vector corresponds to the harmonic component
As for the projections of vectors on the imaginary axis, these projections have equal length, but opposite directions and add up to zero. This means that the signals presented in complex form (2.16) are in fact real signals. In other words, the complex form of the Fourier series is mathematical abstraction, which is very convenient for solving a number of problems of spectral analysis. Therefore, sometimes the spectrum determined by the trigonometric Fourier series is called physical spectrum, and the complex form of the Fourier series is mathematical spectrum.
And in conclusion, let us consider the issue of energy and power distribution in the spectrum of a periodic signal. For this we use Parseval's equality (1.42). When the signal is expanded into a trigonometric Fourier series, expression (1.42) takes the form
.
Constant energy
,
and the energy of the kth harmonic
.
Then the signal energy
. (2.20)
Because average signal strength
,
then in view of (2.18)
. (2.21)
When the signal is expanded into a complex Fourier series, expression (1.42) has the form
,
where 
is the energy of the kth harmonic.
Signal energy in this case
,
and its average power
.
From the above expressions it follows that the energy or average power of the k-th spectral component of the mathematical spectrum is half the energy or power of the corresponding spectral component of the physical spectrum. This is due to the fact that the physical spectrum is equally divided between and the mathematical spectrum.
| -τ and / 2 |
| τ and / 2 |
| T |
| t |
| U 0 |
| S (t) |
Task number 1, group RI - 210701
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:
To determine the spectra for various types of pulse modulation, we find the spectrum of the carrier itself. Take a pulse carrier with rectangular pulses (Fig. 3.10).
Figure: 3.10 Periodic sequence of rectangular pulses
The sequence of such pulses can be represented by Fourier series.
,
(3.32)
where 
- complex amplitude of the k-th harmonic;
- constant component.
Find the complex amplitudes for the specified limits (Fig. 3.10).
(3.33)
Constant component
(3.34)
Substitute (3.33) and (3.34) into (3.32) and after transformation we obtain:
(3.35)
It can be seen from the expression that the spectrum is linear with an envelope repeating the spectrum of a single pulse (Fig. 3.11). In other words, for pulses of the same shape, the lattice function fits into the continuous S (jω).
R 
is. 3.11 Spectrum of a periodic pulse train
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 \u003d 2π / T. It follows that a change in the pulse repetition period T leads to a change in the density of discrete components, and a change in the duty cycle T / τ with a constant period (i.e. change in τ) causes a narrowing or expansion of the envelope while maintaining its shape, leaving unchanged the distance between the lines of the discrete spectrum ... With a sufficiently high density of these lines, when at least several spectral lines are placed between the nodes (T \u003e\u003e τ), 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 be different in width. In Fig. 3.12 shows the deformation of the spectrum of the impulse carrier with a change in T, and Fig. 3.13 when changing τ for rectangular pulses.
R 
is. 3.12 Changing the nature of the spectrum of the carrier when changing
repetition period T of rectangular pulses.
With a constant pulse amplitude, according to expression (3.25), the envelope of the discrete spectrum increases in proportion to the increase in the area of \u200b\u200bthe 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 pulse 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 accumulates, its spectrum is fragmented and deformed in such a way that it increasingly approaches the lattice spectrum.
Figure: 3.13 Changing the nature of the carrier spectrum when changing
pulse duration τ for rectangular pulses.
3.7 Spectra of Pulse Modulated Signals
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. This consideration allows you to show an approach to the problem, outline a solution path and analyze the final conclusions.
Let's find the spectrum with pulse-amplitude modulation (AMM). For simplicity, we choose the modulating function f (t) containing one harmonic sint
Expanding this expression and replacing the product of sine by cosine
.
(3.36)
AND 
it is seen from (3.36) that the signal spectrum contains the frequency of the modulating function and the highest harmonic components kω 0 ± with two side satellites. In this case, the highest harmonic components fit into the envelope of the spectrum of a single pulse of the carrier. In Fig. 3.14 shows the spectrum with pulse amplitude modulation.
Figure: 3.14 Spectrum at amplitude-pulse modulation.
The width of the spectrum does not change with 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 with AMI. If a sequence of pulses is modulated by a complex function from min to max, then after modulation, not spectral lines appear in the spectrum, but frequency bands min… max and kω 1 ± ( min… max)
Consider the features of the spectrum for pulse-phase modulation (PPM), which belongs to a type of pulse-time modulation (PIM).
P 
at PPM - modulation (Fig. 3.15), the dashed line shows the change in the modulating function in time. Vertical dashed lines represent the position of the transition edges of the unmodulated pulse train. The figure shows that the position of the pulses (phase) changes relative to the so-called clock points t k, corresponding to the position on the time axis of the leading edges of the unmodulated pulse train. The displacement of one of the pulses by the time ∆t k is shown in the figure.
Figure: 3.15 Illustration of PPM - modulation.
Figure: 3.16 Pulse position without modulation
and in the presence of modulation.
In 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 will assume that the maximum pulse displacement ∆t K corresponds to the value U (t) \u003d 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 equal to (t-τ) since the trailing edge is displaced relative to the leading edge by the amount of the pulse duration.
To obtain a spectrum at FIM, it is necessary to substitute the value t 2 -t 1 instead of τ, since t 1 and t 2 are the current coordinates. The offset of the center line can be reflected by replacing time t with time 
... As a result of substitution of these values \u200b\u200bin (3.35), we obtain:
(3.39)
Substituting the values \u200b\u200bof t 1 and t 2 into expression (3.39) and after the transformation, we obtain an expression that coincides with the spectrum during AMI, only near the component of the fundamental frequency and each higher harmonic appeared not one lower and one upper side spectral lines, but bands of side harmonics with frequencies (kω 0 ± n).
An approximate spectrum is shown in Fig. 3.17. However, the lateral satellites quickly decrease, since they include Bessel functions.
R 
is. 3.17 Spectrum with phase-pulse modulation.
The spectra for PWM and PFM in their composition are the same as the spectrum for PPM - modulation.
Despite the fact that the nature of the spectrum changes during modulation of the carrier and depends on the type of modulation, its width remains the same as for a single pulse and is mainly determined by the pulse duration τ.
The transmission of measurement information in time division telemetry devices is often preferable to transmission using frequency division multiplexing, 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 modulation of the carrier frequency (secondary), there are the 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 multiplexing systems are used to transmit measurement information from artificial satellites and spacecraft.
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 55 μs 27. T \u003d 5.4 μs; 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 2.5μs 31. T \u003d 4.2μs; 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
Periodic sequence of rectangular video pulses is a modulating function for forming a periodic sequence of rectangular radio pulses (RPPVI), which are sounding signals for detecting and measuring the coordinates of moving targets. Therefore, based on the spectrum of the modulating function (PPPVI), it is relatively simple and quick to determine the spectrum of the probing signal (PPPRI). When the sounding signal is reflected from a moving target, the frequencies of the spectrum of the harmonics of the carrier oscillation change (Doppler effect). As a result, it is possible to distinguish a useful signal reflected from a moving target against the background of interfering (interference) vibrations reflected from stationary objects (local objects) or sedentary objects (meteorological formation, flocks of birds, etc.).
PPPVI (Fig. 1.42) is a set of single rectangular video pulses following each other at regular intervals. Analytical signal expression.
where is the pulse amplitude; - pulse duration; - pulse repetition period; - pulse repetition rate,; - duty cycle.
The Fourier series is used to calculate the spectral composition of a periodic pulse train. With the known spectra of single pulses forming a periodic sequence, one can use the relationship between the spectral density of pulses and the complex amplitudes of the series:
For a single rectangular video pulse, the spectral density is described by the formula
Using the relationship between the spectral density of a single pulse and the complex amplitudes of the series, we find
where \u003d 0; ± 1; ± 2; ...
The amplitude-frequency spectrum (Fig. 1.43) will be represented by a set of components:
in this case, positive values \u200b\u200bcorrespond to zero initial phases, and negative values \u200b\u200bcorrespond to initial phases equal to.
Thus, the analytical expression for PPPVI will be equal to
From the analysis of the graphs shown in Figure 1.43 it follows:
· Spectrum PPPVI discrete consisting of separate harmonics with frequency.
· The ASF envelope changes according to the law.
· The maximum value of the envelope at is, the value of the constant component.
· The initial phases of harmonics within the odd lobes are equal to 0, within the even ones.
· The number of harmonics within each lobe is equal.
Signal spectrum width at 90% signal energy
· Signal base, so the signal is simple.
If you change the duration of the pulses, or the frequency of their repetition F (period), then the parameters of the spectrum and its ASF will change.
Figure 1.43 shows an example of the change in the signal and its frequency response when the pulse duration is doubled.
Periodic sequences of rectangular video pulses and their frequency response parameters, T,. and, Tare shown in Figure 1.44.
From the analysis of the given graphs it follows:
1. For PPPVI with pulse duration:
Duty cycle q\u003d 4, therefore, 3 harmonics are concentrated within each lobe;
· Frequency of the k-th harmonic;
· Width of the signal spectrum at the level of 90% of the energy;
The constant component is
2. For PPPVI with pulse duration:
Duty cycle q \u003d2, therefore, within each lobe there is 1 harmonic;
· The frequency of the k-th harmonic remained unchanged;
· The width of the signal spectrum at the level of 90% of its energy has decreased by 2 times;
· The constant component has doubled.
Thus, it can be concluded that with an increase in the pulse duration, the ASF is “compressed” along the ordinate (the width of the signal spectrum decreases), while the amplitudes of the spectral components increase. Harmonic frequencies do not change.
Figure 1.44. an example of the change in the signal and its AFC is presented with an increase in the repetition period by 4 times (decrease in the repetition frequency by 4 times).
c) the spectrum width of the signal at 90% of its energy has not changed;
d) the constant component has decreased by 4 times.
Thus, we can conclude that with an increase in the repetition period (a decrease in the repetition frequency, “compression” occurs) of the ASF along the frequency axis (the amplitudes of harmonics decrease with an increase in their number within each lobe). In this case, the width of the signal spectrum does not change. A further decrease in the repetition frequency (increase in the repetition period) will lead (at) to a decrease in the amplitudes of the harmonics to infinitely small values. In this case, the signal will turn into a single one, respectively, the spectrum will become continuous.