A periodic sequence of rectangular pulses. Electrical and temporal parameters of rectangular pulses
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 (MPFVI), it is possible to determine the spectrum of the probing signal (MPFRI) relatively simply and quickly. When a probing signal is reflected from a moving target, the frequencies of the harmonic spectrum of the carrier vibration 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.
· Maximum value 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), the ASF “compresses” 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 signal, and accordingly the spectrum will become continuous.
Classification of signals and their parameters.
Electrical signals are electrical processes used to transmit or store information.
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. Here are a few typical examples: a harmonic signal with a known amplitude A and period T(Fig. 1.1 A); sequence of rectangular pulses with a known repetition period T, duration t and amplitude A(Fig. 1.1 b); sequence of pulses of arbitrary shape with known duration t and amplitude A and period T(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 of electrical quantities). Random signals also include noise.
Deterministic signals, in turn, are divided into periodic ones, for which the condition is satisfied S(t)=S(t+kT), Where T– period, k- any integer, and under S(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: S(t)¹ S(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 S(t)=S(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 S(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 as follows 
(1.4)
When using a complex form of notation, the expression for the signal S(t) takes the form 
. Here are the coefficients 
, 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 and n equal to
, and the amplitude of the sin component b n equal to 
.
Amplitude n th harmonics 
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
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 that a single pulse has.
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 apply:
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 T(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 f= 1/T 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(f) |
| d( t) | |
| d( f) | |
| cos 2 p f 0 t | /2 |
| sin 2 p f 0 t | /2 |
| d( t - t 0) | |
| d( f - f 0) | |
| , a>0 | |
 |  |
 | |
| exp(- at)u(t), a>0 | |
| rect( t/ T) | T sinc fT |
| W sinc Wt | rect( f / W) |
 |
sinc x =
Table 2.2 Properties of the Fourier transform f)
