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 at , where is the pulse duration and V(t)=0 at , then
It is convenient to write this formula in the form:
(2.1.)
). 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.
Where m– an integer (exceptm
A periodic sequence of rectangular video pulses is a modulating function for the formation of a periodic sequence of rectangular radio pulses (PPRP), which are probing signals for detecting and measuring the coordinates of moving targets. Therefore, using the spectrum of the modulating function (PPVI), it is possible to determine the spectrum of the probing signal (PPVI) relatively simply and quickly. When a probing signal is reflected from a moving target, the frequencies of the harmonic spectrum of the carrier wave change (Doppler effect). As a result, it is possible to identify a useful signal reflected from a moving target against the background of interfering (interference) vibrations reflected from stationary objects (local objects) or slow-moving objects (meteorological formations, flocks of birds, etc.).
PPPVI (Fig. 1.42) is a set of single rectangular video pulses following each other at equal intervals of time. Analytical expression of the signal.
where is the pulse amplitude; – pulse duration; – pulse repetition period; – pulse repetition rate, ; – duty cycle.
To calculate the spectral composition of a periodic sequence of pulses, the Fourier series is used. With known spectra of single pulses forming a periodic sequence, we can use the relationship between the spectral density of the 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 = 0; ± 1; ± 2; ...
The amplitude-frequency spectrum (Fig. 1.43) will be represented by a set of components:
in this case, positive values correspond to zero initial phases, and negative values correspond 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:
· The PPPVI spectrum is discrete, consisting of individual harmonics with frequency .
· The ASF envelope changes according to the law.
· The maximum value of the envelope at is equal to the value of the constant component.
· The initial phases of harmonics within the odd lobes are equal to 0, within the even lobes .
· The number of harmonics within each lobe is equal to .
Signal spectrum width at 90% of signal energy
· Signal base, so the signal is simple.
If you change the duration of the pulses or their repetition frequency F(period), then the parameters of the spectrum and its ASF will change.
Figure 1.43 shows an example of a change in the signal and its ASF when the pulse duration is doubled.
Periodic sequences of rectangular video pulses and their ASF parameters, T,. And , T, are shown in Figure 1.44.
From the analysis of the given graphs it follows:
1. For PPPVI with pulse duration:
· Duty ratio q=4, therefore, 3 harmonics are concentrated within each lobe;
· Frequency of the k-th harmonic;
· Signal spectrum width at 90% energy level;
The constant component is equal to
2. For PPPVI with pulse duration:
· Duty ratio q= 2, therefore, within each lobe there is 1 harmonic;
· The frequency of the k-th harmonic remains unchanged;
· The signal spectrum width at the level of 90% of its energy decreased by 2 times;
· The constant component increased by 2 times.
Thus, we can conclude that with increasing pulse duration, the ASF is “compressed” along the ordinate axis (the width of the signal spectrum decreases), while the amplitudes of the spectral components increase. The harmonic frequencies do not change.
In Figure 1.44. An example of a change in the signal and its ASF with an increase in the repetition period by 4 times (a decrease in the repetition rate by 4 times) is presented.
c) the signal spectrum width at the level of 90% of its energy has not changed;
d) the constant component decreased by 4 times.
Thus, we can conclude that with an increase in the repetition period (a decrease in the repetition frequency), “compression” occurs in the ASF along the frequency axis (the amplitudes of the harmonics decrease with an increase in their number within each lobe). The signal spectrum width 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 infinitesimal values. In this case, the signal will turn into a single one, and accordingly the spectrum will become continuous.
Let us consider a periodic sequence of rectangular pulses with a period T, pulse duration t u and a 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, and only the coefficients need to be calculated.
constant component
(2.28)
The constant component is the average value over the period, i.e. is the area of the pulse divided by the entire period, i.e. 
, i.e. the same thing that happened with a strict formal calculation (2.28).
Let us remember that the frequency of the first harmonic is ¦ 1 = , where T is the period of the rectangular signal. Distance between harmonics D¦=¦ 1. If the harmonic number n turns out to be such that the argument of the sine is , then the amplitude of this harmonic goes to zero for the first time. This condition is satisfied when . 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. Since the sine goes to zero for all values of the argument that are multiples of p, 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 (2.22) and (2.23) it follows that the spectrum of rectangular pulses with a duty cycle of 2 consists only of odd harmonics. Because the S=2, then N=2, 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. With duty cycle S=3, zero amplitudes will be at 3, 6, 9, 12, ... harmonics.
With increasing duty cycle, the “first zero” shifts to the region of harmonics with higher numbers and, consequently, the rate of decrease in harmonic amplitudes decreases. Simple calculation of the amplitude of the first harmonic at U m=100V for duty cycle S=2, U m 1=63.7V, at S=5, U m 1=37.4V and at S=10, U m 1=19.7V, i.e. As the duty cycle increases, the amplitude of the first harmonic decreases sharply. If we find the amplitude ratio, for example, of the 5th harmonic U m 5 to the amplitude of the first harmonic U m 1, then for S=2, U m 5/U m 1=0.2, and for S=10, U m 5 / U m 1 = 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.
Literature: [L.1], p. 40
As an example, we give the Fourier series expansion of a periodic sequence of rectangular pulses with amplitude, duration and repetition period, symmetrical about zero, i.e.
, (2.10)
Here 
Expanding such a signal into 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) will be written as follows
, (2.13)
In Fig. 2.3 shows a sequence of rectangular pulses. The spectrum of the sequence, as well as any other periodic signal, is discrete (line) in nature.
The spectrum envelope (Fig. 2.3, b) is proportional 
. The distance along the frequency axis between two adjacent spectrum components is , and between two zero values (the width of the spectrum lobe) is . 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 (if the duty cycle is an integer value). As the period increases, the fundamental frequency 
decreases, the spectral components in the diagram come closer together, the amplitudes of the harmonics also decrease. In this case, the shape of the envelope is preserved.
When solving practical problems of spectral analysis, cyclic frequencies are used instead of angular frequencies 
, measured in Hertz. Obviously, the distance between adjacent harmonics on the diagram will be , and the width of one spectrum lobe will be . These values are presented in parentheses in the chart.
In practical radio engineering, in most cases, instead of the 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 
.
As for the phase spectrum (Fig. 2.3, d), it is believed that the initial phases of the harmonic components change abruptly by the amount -π when the sign of the envelope changes sinc kπ/q. The initial phases of the harmonics of the first lobe are assumed to be zero. Then the initial phases of the harmonics of the second lobe will be φ = -π , third petal φ = -2π etc.
Let's consider another Fourier series representation of the signal. To do this, we use Euler’s formula
.
In accordance with this formula, the kth component (2.9) of the signal expansion into a 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) will take the following form
, (2.16)
, (2.17)
It is easy to verify that expansion (2.16) is carried out in terms of the basis functions 
, which are also orthogonal on the interval 
, i.e.
Expression (2.16) is complex form Fourier series, which extends to negative frequencies. Quantities and 
, where denotes the complex conjugate of a quantity, are called complex amplitudes spectrum Because is a complex quantity, it follows from (2.15) that
And 
.
Then the totality constitutes the amplitude spectrum, and the totality constitutes the phase spectrum of the signal.
In Fig. Figure 2.4 shows a spectral diagram of the spectrum of the sequence of rectangular pulses discussed above, represented by a complex Fourier series
The spectrum also has a line character, but unlike the previously considered spectra, it is determined both in the region of positive and in the region of negative frequencies. Since is an even function of the argument, the spectral diagram is symmetrical about zero.
Based on (2.15), we can establish a correspondence between the coefficients and expansion (2.3). Because
And 
,
then as a result we get
. (2.18)
Expressions (2.5) and (2.18) allow you to find the values in practical calculations.
Let us give a geometric interpretation of the complex form of the Fourier series. Let us select the kth component of the signal spectrum. In complex form, the kth component is described by the formula
where and are determined by expressions (2.15).
In the complex plane, each of the terms in (2.19) is represented as vectors of length 
, rotated at 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 whose length is . But this vector corresponds to the harmonic component
As for the projections of vectors onto the imaginary axis, these projections have equal lengths, but opposite directions and add up to zero. This means that signals presented in complex form (2.16) are actually real signals. In other words, the complex form of the Fourier series is mathematical an abstraction that is very convenient for solving a number of problems of spectral analysis. Therefore, sometimes the spectrum defined by the trigonometric Fourier series is called physical spectrum, and the complex form of the Fourier series is mathematical spectrum.
And in conclusion, we will consider the issue of energy and power distribution in the spectrum of a periodic signal. To do this, we use Parseval’s equality (1.42). When the signal is expanded into a trigonometric Fourier series, expression (1.42) takes the form
.
DC energy
,
and the energy of the kth harmonic
.
Then the signal energy
. (2.20)
Because average signal power
,
then taking into account (2.18)
. (2.21)
When the signal is expanded into a complex Fourier series, expression (1.42) takes the form
,
Where 
- energy of the kth harmonic.
The 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 as much as the energy or power of the corresponding spectral component of the physical spectrum. This is due to the fact that the physical spectrum is distributed equally between the mathematical spectrum.
| -τ and /2 |
| τ and /2 |
| T |
| t |
| U 0 |
| S(t) |
Task No. 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 have the form of a non-periodic, and clock signals - a periodic sequence of pulses.
To correctly assess the possibility of transmitting such pulses via communication channels, we will determine their spectral composition. A periodic signal in the form of pulses of any shape can be expanded into a Fourier series according to (7).
Signals of various shapes are used for transmission over overhead and cable communication lines. The choice of one form or another depends on the nature of the messages being transmitted, the frequency spectrum of the signals, and the frequency and time parameters of the signals. Signals close in shape to rectangular pulses are widely used in the technology of transmitting discrete messages.
Let's calculate the spectrum, i.e. a set of constant amplitudes and
harmonic components of periodic rectangular pulses (Figure 4,a) with duration and period. Since the signal is an even function of time, then in expression (3) all even harmonic components vanish ( 
=0), and the odd components take the following values:
(10)
The constant component is equal to
(11)
For a 1:1 signal (telegraph points) Figure 4a:
,
.
(12)
Modules of the amplitudes of the spectral components of a sequence of rectangular pulses with a period 
are shown in Fig. 4, b. The abscissa axis shows the main pulse repetition frequency 
() and frequencies of odd harmonic components 
,
etc. The spectrum envelope changes according to the law.
As the period increases compared to 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 find that the constant component is equal to
In the frequency band from zero to frequency there are five harmonic components (Figure 4, d), while there is only one tide.
With a further increase in the pulse repetition period, the number of harmonic components becomes larger and larger. In the extreme 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 Spectrum of a single pulse
A single video pulse is specified (Figure 5):
Figure 5
The Fourier series method allows for a deep and fruitful generalization, which makes it possible to obtain the spectral characteristics of non-periodic signals. To do this, let us mentally supplement a single pulse with the same pulses, periodically following after a certain time interval, and obtain the previously studied periodic sequence:
Let's imagine a single pulse as a 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)
Let's denote
.
(17)
The quantity is the spectral characteristic (function) of a single pulse (direct Fourier transform). It depends only on the temporal description of the pulse and in general is complex:
, (18) where 
; (19)
; (20)
,
Where 
- module of the spectral function (amplitude-frequency response of the pulse);
- phase angle, phase-frequency characteristic of the pulse.
Let us find for a single pulse using 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, they speak of a continuous (solid) spectrum possessed by a single pulse.
The total pulse energy (the energy released at the active resistance Ohm) is equal to
(22)
Changing the order of integration, we obtain
.
The internal integral is the spectral function of momentum taken with the argument -, i.e. is a complex conjugate quantity: 
Hence
Squared modulus (the product of two conjugate complex numbers is equal to the squared modulus).
In this case, it is conventionally said that the pulse spectrum is two-sided, i.e. located in the frequency band from to.
The given relationship (23), which establishes the connection between the pulse energy (at a resistance of 1 Ohm) and the modulus of its spectral function, is known as Parseval’s equality.
It states that the energy contained in a pulse is equal to the sum of the energies of all components of its spectrum. Parseval's equality characterizes an important property of signals. If some selective system transmits only part of the signal spectrum, weakening its other components, this means that part of the signal energy is lost.
Since the square of the modulus is an even function of the integration variable, then by doubling the value of the integral, one can introduce integration in the range from 0 to:
.
(24)
In this case, they say 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 by frequency, and its value at frequency is equal to the pulse energy per frequency band equal to 1 Hz. Consequently, the pulse energy is the result of integrating the signal’s energy spectrum over the entire frequency range. In other words, the energy is equal to the area enclosed between the curve depicting the signal’s energy spectrum and the abscissa axis.
To estimate the energy distribution over the spectrum, use the relative integral energy distribution function (energy characteristic)
,
(25)
Where 
- pulse energy in a given frequency band from 0 to, which characterizes the fraction of pulse energy concentrated in the frequency range from 0 to.
For single pulses of various shapes, the following laws hold true:
