2. Spectrum of a periodic sequence of rectangular pulses
Consider the periodic sequence of rectangular pulses shown in Fig. 5. This signal is characterized by the pulse duration, its amplitude and period. The stress is plotted along the vertical axis.
Fig.5. Periodic sequence of rectangular pulses
We choose the starting point in the middle of the pulse. Then the signal is expanded only in cosines. The harmonic frequencies are n/T, where n:
- any integer. The harmonic amplitudes according to (1.2.) will be equal because=V(t) E because at , where is the pulse duration and
=0 at , then
(2.1.)
(function
). This function is called the spectrum envelope. It should be borne in mind that it has a physical meaning only at frequencies where corresponding harmonics exist. In Fig. Figure 6 shows the spectrum of a periodic sequence of rectangular pulses.
Fig.6. Spectrum of a periodic sequence
rectangular 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. As the frequency tends to zero, both the numerator and the denominator tend to zero, and their ratio tends to unity (the first classical limit).Zero values of the envelope occur at points where i.e.Therefore, a quasi-oscillating function with a gradual decrease is obtained. As the frequency tends to zero, both the numerator and the denominator tend to zero, and their ratio tends to unity (the first classical limit).
Where m:
– an integer (except Name educational organization State budgetary professional educational institution
"Stavropol College of Communications named after Hero Soviet Union
V.A. Petrova" Year and place of creation of the work:
“Calculation and construction of the spectrum of a periodic sequence of rectangular pulses”
for students 2 courses of specialties:
02/11/11 Communication networks and switching systems
02/11/09 Multichannel telecommunication systems
full-time education
Goal of the work: consolidate the knowledge acquired in theoretical classes, develop skills in 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 arbitrary shape can be represented as a sum harmonic vibrations with different frequencies, this is called spectral decomposition of the signal.
2.2 . Harmonics are vibrations whose frequencies are an integer number of times greater than the pulse repetition rate of the signal.
2.3. The instantaneous voltage value of a periodic derivative waveform can be written as follows:
Where is the constant component equal to the average signal value over the period;
Instantaneous value of the first harmonic sinusoidal voltage;
Harmonic frequency equal to the pulse repetition frequency;
Amplitude of the first harmonic;
The initial phase of the first harmonic oscillation;
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 third harmonic sinusoidal voltage;
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 PPIP
s - signal duty cycle (S - T/t);
T - pulse repetition period;
t - pulse duration;
The amplitudes of all harmonics are determined by the expression:
Umk = 2Um | sin kπ/s | / kπ
where k is the harmonic number;
2.5. Numbers of harmonics whose amplitudes are zero
where n is any integer 1,2,3…..
The number of the harmonic whose amplitude goes to zero for the first time is equal to the duty cycle of the PPIP
2.6. The interval between any adjacent spectral lines is equal to the frequency of the first harmonic or pulse repetition frequency.
2.7 Envelope of the amplitude spectrum of the signal (shown in Fig. 1 by a dotted line)
identifies groups of spectral lines called lobes. According to Fig. 1, each lobe of the spectrum envelope contains a number of lines equal to the signal duty cycle.
3 . Pwork order.
3.1. Receive an individual task option that matches the number in the group journal list (see appendix).
3.2. Read the calculation example (see section 4)
4. Example
4.1. Let the pulse repetition period T=.1 µs, pulse duration t=0.25 µs, pulse amplitude = 10V.
4.2. Calculation and construction of AEFI time diagram.
4.2.1 . To construct a time diagram of the PPIP, it is necessary to know the pulse repetition period T, the amplitude and duration of the pulses t, which are known from the problem conditions.
4.2.2. To construct a time diagram of SAI, it is necessary to select scales along the stress and time axes. The scales should correspond to the numbers 1,2 and 4, multiplied by 10 n - (where n=0,1,2,3...). The time axis should occupy approximately 3/4 of the width of the sheet 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, and for each period there will be L 1 = 5 cm. Because
Mt=T/Lt=1μs/5cm= 0.2 μs/cm
The obtained result does not contradict the above conditions. On the stress axis it is convenient to take the scale Mu = 2V/cm (see Fig. 2).
4.3.Calculation and construction of a spectral diagram.
4.3.1.The duty cycle of the FITR 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 harmonic number, l is the SAI period.
4.3.4. The amplitudes of the AEFI components are equal
4.3.5. Mathematical model of voltage SAI
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 needs to be shown on the frequency axis, it is convenient to take the 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
Exercise:
T=0.75ms; τ=0.15ms 21.T=24μs; τ=8μs
T=1.5 µs; τ=0.25μs 22. T=6.4ms; τ=1.6ms
T=2.45ms; τ=0.35ms 23. T=7ms; τ=1.4ms
T=13.5μs; τ=4.5μs 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=1.8μs
T=0.3ms; τ=75μs 28. T=2.1ms; τ=0.3ms
T=42.5μs; τ=8.5μs 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=0.3μs 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
Let us consider a periodic sequence of rectangular pulses with a period T, pulse duration and maximum value 
. 
Let us find the series expansion of such a signal by choosing the origin of coordinates as shown in Fig. 15. In this case, the function is symmetrical about the ordinate axis, i.e. all coefficients of sinusoidal components 
.
-
0
=0, and only the coefficients need to be calculated
T t 
(28)
constant component 
The constant component is the average value over the period, i.e. this is the area of the impulse 
, divided by the entire period, i.e.
, i.e. the same thing that happened with a strict formal calculation (28). 
Let us remember that the frequency of the first harmonic 1 = 
, where T is the period of the rectangular signal. 
Distance between harmonics= 1. If the harmonic number n turns out to be such that the argument of the sine, where
(29)
. 
The harmonic number at which its amplitude vanishes for the first time is called "first zero"=
and denote it with the letter N, emphasizing the special properties of this harmonic: on the other hand, the duty cycle S of pulses is the ratio of the period T to the pulse duration t u , i.e. 
. 
Therefore, the “first zero” is numerically equal to the duty cycle of the pulse N S and denote it with the letter N, emphasizing the special properties of this harmonic:=2
. "first zero"=2
Since the sine goes to zero for all values of the argument that are multiples of , the amplitudes of all harmonics with numbers that are multiples of the number of the “first zero” also go to zero. That is
at , Where k– any integer. So, for example, from (22) and (23) it follows that the spectrum of rectangular pulses with a duty cycle of 2 consists only of odd harmonics. Because the and denote it with the letter N, emphasizing the special properties of this harmonic:=2, , Where k 1 , then and denote it with the letter N, emphasizing the special properties of this harmonic:=5, , Where k 1 , i.e. the amplitude of the second harmonic goes to zero for the first time - this is the “first zero”. But then the amplitudes of all other harmonics with numbers divisible by 2, i.e. all even ones must also go to zero. and denote it with the letter N, emphasizing the special properties of this harmonic:=10, , Where k 1 =19.7V, i.e. As the duty cycle increases, the amplitude of the first harmonic decreases sharply. , Where k 5 If we find the amplitude ratio, for example, of the 5th harmonic , Where k 1 to the amplitude of the first harmonic and denote it with the letter N, emphasizing the special properties of this harmonic:=2, , Where k 5 /, Where k 1 , then for and denote it with the letter N, emphasizing the special properties of this harmonic:=10, , Where k 5 / , Where k 1 = =0.2, and for
0.9, i.e. the rate of attenuation of higher harmonics decreases with increasing duty cycle.
Thus, with increasing duty cycle, the spectrum of a sequence of rectangular pulses becomes more uniform.
2.5. Spectra with decreasing pulse duration and signal period. and denote it with the letter N, emphasizing the special properties of this harmonic:= Adjust duty cycle/ T The harmonic frequencies are n/T, where t T The harmonic frequencies are n/T, where you can either change the pulse duration Adjust duty cycle at T The harmonic frequencies are n/T, where=const, or by changing the period T at
Adjust duty cycle =const.T The harmonic frequencies are n/T, where Let us consider the signal spectra in this case.=const, =var. 1 =1/ Adjust duty cycle= First harmonic frequency =var.= =var. 1 = f "first zero"= Adjust duty cycle/ T The harmonic frequencies are n/T, where const and T The harmonic frequencies are n/T, where const. T The harmonic frequencies are n/T, where 0 "first zero" First zero =var.= =var. 1 and as the pulse shortens
T The harmonic frequencies are n/T, where shifts to the region of harmonics with large numbers. AtAdjust duty cycle , the spectrum is discrete and, infinitely wide and with infinitesimal harmonic amplitudes. =const,=var. =var. 1 We will increase the period =var. T =var.= =var. 1 , then the frequency of the first harmonic and the distance between spectral lines will decrease. Because=1/T =const,, then the spectral lines will shift to an area more =var. 1 = =var. low frequencies
and the “density” of the spectrum will increase. If , then the signal from periodic becomes non-periodic (single pulse).
In this case
, (30)
0, i.e. the spectrum turns from discrete to continuous, consisting of an infinitely large number of spectral lines located at infinitesimal distances from each other. 
(31)
This leads to the following rule: periodic signals generate discrete (line) spectra, and non-periodic signals generate continuous (continuous) spectra.(When passing from a discrete spectrum to a continuous spectrum, the Fourier series is replaced by the Fourier integral. This replacement is carried out most simply if we use the representation of the Fourier series in complex form (16) and (17). The Fourier integral for a continuous spectrum is written ) Where Function F j called spectral function or spectral density, which depends on frequency. Formulas (30) and (31) are collectively called one-way Fourier transform, which is a special case of more When passing from a discrete spectrum to a continuous spectrum, the Fourier series is replaced by the Fourier integral. This replacement is carried out most simply if we use the representation of the Fourier series in complex form (16) and (17). The Fourier integral for a continuous spectrum is written .
general transformation Laplace and is obtained by replacing the complex variable in the Laplace transform R =const, on periodic signals generate discrete (line) spectra, and non-periodic signals generate continuous (continuous) spectra.(When passing from a discrete spectrum to a continuous spectrum, the Fourier series is replaced by the Fourier integral. This replacement is carried out most simply if we use the representation of the Fourier series in complex form (16) and (17). The Fourier integral for a continuous spectrum is written
)
The spectral function can be represented as an envelope of the coefficients of the Fourier series, i.e. as a limit 
line spectrum 
-periodic function at, i.e.
(
)
– dependence of the amplitude of spectral components on frequency, and phase spectrum , i.e.
.
the law of changes in the phase of the spectral components of a signal depending on frequency. It can be shown that the amplitude spectrum is always an even function, and the phase spectrum is always an odd function Spectral function for many non-periodic signals (single pulses periodic signals generate discrete (line) spectra, and non-periodic signals generate continuous (continuous) spectra.(various shapes)
) is most easily and simply found using tables of originals and images in the Laplace transform, which are given in educational and reference literature. After finding the image according to Laplace =var.(T)
p
(32)
for a given non-periodic function =var.(T)
, the spectral function is found 
So, according to (30), the non-periodic function =var.(T)
appears to be a collection of an infinitely large number of harmonics with infinitely small amplitudes
over the entire frequency range from - to +, i.e. performance
in the form of a Fourier integral implies the summation of undamped harmonic oscillations of an infinite continuous spectrum of frequencies.
description of the laboratory setup
The work is performed on the “Signal Synthesizer” block, the functional diagram of which is shown in Fig. 16.
.
The block contains generators G1-G6 of the first six harmonics of the signal. The frequency of the first harmonic is 10 kHz. The harmonic signal from the output of the nth generator through the phase shifter Ф n and attenuator A n is supplied to the adder. Phase shifters set the initial phases of n harmonics, and attenuators set their amplitudes A n.
In general, the sum of six harmonics of the signal is obtained at the output of the adder
From the output of the adder, the signal is fed to the Y input of the oscilloscope. For its external synchronization, a special pulse signal is used, supplied from the “Sync” socket.
Signals can be divided into two large classes: deterministic and random. Deterministic signals are those whose instantaneous values at any time can be predicted with a probability equal to one and which are specified in the form of some specific function of time. Let us give some typical examples: a harmonic signal with a known amplitude A and period Adjust duty cycle(Fig. 1.1 A); sequence of rectangular pulses with a known repetition period Adjust duty cycle, duration t and amplitude A(Fig. 1.1 b); sequence of pulses of arbitrary shape with known duration t and amplitude A and period Adjust duty cycle(Fig. 1.1 V). Deterministic signals do not contain any information.
Random signals are chaotic functions of time, the values of which are unknown in advance and cannot be predicted with a probability equal to one (single pulse with duration t and amplitude A(Fig. 1.1 G) speech, music in expression electrical quantities). Random signals also include noise.
Deterministic signals, in turn, are divided into periodic ones, for which the condition is satisfied and denote it with the letter N, emphasizing the special properties of this harmonic:(T)=and denote it with the letter N, emphasizing the special properties of this harmonic:(t+kT), Where Adjust duty cycle– period, N- any integer, and under and denote it with the letter N, emphasizing the special properties of this harmonic:(T) refers to current, voltage or charge changing over time (Fig. 1.1 a B C).
Obviously, any deterministic signal for which the condition is satisfied is non-periodic: and denote it with the letter N, emphasizing the special properties of this harmonic:(T)¹ and denote it with the letter N, emphasizing the special properties of this harmonic:(t+kT).
The simplest periodic signal is a harmonic signal of the form 
.
Any complex periodic signal can be decomposed into harmonic components. Below, such a decomposition will be carried out for several specific types of signals.
A high-frequency harmonic signal in which information is embedded through modulation is called a radio signal (Fig. 1.1 d).
Periodic signals.
Any complex periodic signal and denote it with the letter N, emphasizing the special properties of this harmonic:(T)=and denote it with the letter N, emphasizing the special properties of this harmonic:(t+kT) (Fig. 1.2), specified on the range of values T from –¥ to +¥, can be represented as a sum of elementary harmonic signals. This representation is carried out in the form of a Fourier series, if only the given periodic function satisfies the Dirichlet conditions:
1. On any finite time interval the function and denote it with the letter N, emphasizing the special properties of this harmonic:(T) must be continuous or have a finite number of discontinuities of the first kind.
2. Within one period, the function must have a finite number of maxima and minima.
Typically, all real radio signals satisfy these conditions. IN trigonometric form The Fourier series has the form (1.1)
where the constant component is equal to 
(1.2)
and the coefficients a n, And b n for cosine and sinusoidal terms, the expansions are determined by the expressions 
(1.3)
Amplitude (modulus) and phase (argument) nth harmonics are expressed through coefficients a n, And b n in the following way 
(1.4)
When using a complex form of notation, the expression for the signal S(t) takes the form 
. Here are the odds 
, called complex amplitudes, are equal 
and are related to the quantities a n and b n by the formulas: for n>0, and for n<0. С учётом обозначений 
.
The spectrum of a periodic function consists of individual lines corresponding to discrete frequencies 0, w, 2w, 3w ..., i.e., it has a line or discrete character (Fig. 1.3). The use of Fourier series in combination with the principle of superposition is a powerful means of analyzing the influence of linear systems on the passage of various types of periodic signals through them.
When expanding a periodic function into a Fourier series, you should take into account the symmetry of the function itself, since this allows you to simplify the calculations. Depending on the type of symmetry, the functions represented by the Fourier series can:
1. 
Do not have a constant component if the area of the figure for the positive half-cycle is equal to the area of the figure for the negative half-cycle.
2. Do not have even harmonics and a constant component if the function values are repeated after half a period with the opposite sign.
Spectral composition of a sequence of rectangular pulses at different periods of their duty cycle.
A periodic sequence of rectangular pulses is shown in Fig. 1.4. The constant component of the Fourier series is determined from the expression and for this case it is equal to 
.
Amplitude of the cos component a n equal to
, and the amplitude of the sin component b n equal to 
.
Amplitude The harmonic frequencies are n/T, where th harmonics 
In this expression
 |
sinc function as shown in Fig. 2.6, reaches a maximum (unity) at y = 0 and tends to zero at at® ±¥, oscillating with a gradually decreasing amplitude. It passes through zero at points at= ±1, ±2, …. In Fig. 2.7, A as a function of the ratio p/t 0 shows the amplitude spectrum of the pulse sequence | with n|, and in Fig. 2.7, b the phase spectrum q is shown n. It should be noted that the positive and negative frequencies of a two-way spectrum are a useful way of expressing the spectrum mathematically; It is obvious that in real conditions only positive frequencies can be reproduced.
Attitude
An ideal periodic pulse train includes all harmonics that are multiples of the natural frequency. In communications systems, it is often assumed that a significant portion of the power or energy of a narrowband signal occurs at frequencies from zero to the first zero of the amplitude spectrum (Fig. 2.7, A). Thus, as a measure bandwidth pulse sequence, the value 1/ is often used Adjust duty cycle(Where T - pulse duration). Note that the bandwidth is inversely proportional to the pulse duration; The shorter the pulses, the wider the band associated with them. Note also that the distance between the spectral lines D =var.= 1/=const, 0 is inversely proportional to the pulse period; As the period increases, the lines are located closer to each other.
Table 2.1. Fourier images
| x(T) | X(=var.) |
| d( T) | |
| d( =var.) | |
| cos 2 p =var. 0 T | /2 |
| sin 2 p =var. 0 T | /2 |
| d( T - T 0) | |
| d( =var. - =var. 0) | |
| , a>0 | |
 |  |
 | |
| exp(- at)u(T), a>0 | |
| rect( T/ Adjust duty cycle) | Adjust duty cycle sinc fT |
| W sinc Wt | rect( =var. / W) |
 |
sinc x =
Table 2.2 Properties of the Fourier transform f)
