Literature: [L.1], pp. 50-51
[L.2], p. 65-66
[L.3], p. 24-25
To solve practical problems of radio engineering, it is extremely important to know the duration and width of the signal spectrum, as well as the relationship between them. Knowing the duration of the signal allows you to solve problems effective use the time available for message transmission, and knowledge of the spectrum width - the effective use of the radio frequency band.
The solution of these problems requires a strict definition of the concepts of "effective duration" and "effective width of the spectrum." In practice, there are a large number of approaches to determining the duration. In the case when the signal is limited in time (finish signal), as is the case, for example, for a rectangular pulse, the determination of the duration is not difficult. The situation is different when, theoretically, the signal has an infinite duration, for example, an exponential pulse
In this case, the effective duration can be taken as the time interval during which the value of the signal . In another method, the time interval during which . The same can be said about the definition of the effective width of the spectrum.
Although in the future, some of these methods will be used in the analysis of radio signals and circuits, it should be noted that the choice of method depends significantly on the signal shape and spectrum structure. So for an exponential pulse, the first of these methods is more preferable, and for a bell-shaped signal, the second method.
More universal is the approach using energy criteria. With this approach, as the effective duration and effective width of the spectrum, the time interval and the frequency range are considered, respectively, within which the overwhelming part of the signal energy is concentrated
, (2.52)
, (2.53)
where is a coefficient showing what part of the energy is concentrated in intervals or . Typically, the value is chosen within 
.
Let us apply criteria (2.52) and (2.53) to determine the duration and width of the spectrum of rectangular and exponential pulses. For a rectangular pulse, all the energy is concentrated in the time interval or , so its duration is . As for the effective width of the spectrum, it was found that more than 90% of the pulse energy is concentrated within the first lobe of the spectrum. If we consider the one-sided (physical) spectrum of the pulse, then the width of the first lobe of the spectrum is in circular frequencies or in cyclic frequencies. It follows that the effective width of the spectrum of a rectangular pulse is equal to
Let's move on to the definition of exponential momentum. The total pulse energy is
.
Using (2.52), we get
.
By calculating the integral on the left side of the equation and solving it, we can come to the following result
.
We find the spectrum of the exponential momentum using the Fourier transform
,
whence it follows
.
Substituting this expression into (2.53) and solving the equation, we obtain
.
Let us find the product of the effective duration and the effective width of the spectrum. For a rectangular pulse, this product is
,
or for cyclic frequencies
.
For exponential momentum
Thus, the product of the effective duration and the effective width of the spectrum of a single signal is a constant value that depends only on the shape of the signal and the value of the coefficient . This means that as the duration of the signal decreases, its spectrum expands and vice versa. This fact has already been noted when considering property (2.46) of the Fourier transform. In practice, this means that it is impossible to form short signal, which has a narrow spectrum, which is a manifestation of the physical uncertainty principle.
It was noted in the work that with an increase in the number of zeros, the spectrum of the complex envelope of the PM signal shifts to higher frequencies. This refers to the shift of that part of the spectrum in which the main part of the signal energy is concentrated, since in principle the spectrum of the PM signal is not identically equal to zero (except for the set of points with zero measure) on the entire frequency axis. To determine
spectrum shift, you can use the concept of effective spectrum width, for example, ), which is determined by the relation
In the case of PM signals, the integral in the numerator diverges and definition (11.8) is meaningless. But given that the main part of the PM signal energy is concentrated between the first zeros, then the infinite limits of the integral in the numerator can be replaced. Turning to the variable and considering the even function, and the integral in the denominator (11.8) is equal to, we determine the effective spectrum width of the complex envelope of the PM signal with blocks as follows :
Substituting (11.6) into (11.9), we obtain
i.e., with such a definition, it is proportional to the integral of the periodic function (11.7) over the period. After integration, we find
Therefore, the more blocks a PM signal has, the more . In table. 11.1 shows the values for several PM signals that differ significantly from each other in their structure.
In the first line of the table Figure 11.1 shows the data for a rectangular pulse with a duration of only one block. The larger the smaller. This example corresponds to a PM signal that has the smallest number of blocks. In
Table 11.1 (see scan)
the second line of the table. 11.1 shows the data for the PM signal with the largest number of blocks. This PM signal (meander) represents a sequence of alternating pulses. For a meander, what is the maximum value of . The third line shows the data for the optimal PM signal, for which For such a signal is two times less than the maximum. Thus, the effective spectrum width of optimal PM signals lies approximately in the middle between the values corresponding to the two extreme values for a rectangular pulse and a meander. The last line shows the values of the effective spectrum width of an ideal (hypothetical) signal consisting of pulses, the energy spectrum of which coincides with the energy spectrum of a single pulse of duration
Signal Spectrum Width 1. The value characterizing the part of the signal spectrum containing the spectral components, the total of which is a given part of the total signal power
Used in document:
Appendix No. 1 to GOST 24375-80
Telecommunication dictionary. 2013 .
See what the "Signal Spectrum Width" is in other dictionaries:
signal spectrum width- A value that characterizes part of the signal spectrum containing spectral components, the total power of which is a given part of the total signal power. [GOST 24375 80] Topics television, radio broadcasting, video General terms ... ...
Signal Spectrum Width- 2. Width of the signal spectrum A value that characterizes the part of the signal spectrum containing spectral components, the total power of which is a given part of the total signal power Source: GOST 24375 80: Radio communication. Terms and ... ...
spectrum width (optical channel signal)- 44 spectrum width (optical channel signal): The frequency band or wavelength range in which the main part of the average power of the optical radiation of the optical channel signal is transmitted Source: OST 45.190 2001: Fiber transmission systems ... ... Dictionary-reference book of terms of normative and technical documentation
width of the spectrum of the output signal of the module (unit) microwave- spectrum width Δfbread Frequency interval of the microwave output module (block) spectrum, in which a given part of the oscillation power is concentrated. [GOST 23221 78] Subjects components of communication technology Generalizing terms microwave modules, microwave units Synonyms width ... Technical Translator's Handbook
spectrum width- The frequency band in which the main energy of the emitted signal is concentrated and the frequency components with maximum values are located. Spectrum width is usually measured in terms of 0.5 (ZdB) of maximum power, or in terms of 0… Technical Translator's Handbook
The width of the spectrum of the output signal of the module (unit) microwave- 20. The width of the spectrum of the output signal of the module (unit) microwave Δfwide
It is already clear from the previous paragraphs that the shorter the duration of the signal, the wider its spectrum. To establish quantitative relationships between specified parameters signal, it is necessary to agree on the definition of the concepts of signal duration and the width of its spectrum. In practice, various definitions are used, the choice of which depends on the purpose of the signal, its shape, and also on the structure of the spectrum. In some cases, the choice is arbitrary. So, the width of the spectrum of a rectangular pulse is determined either as the base of the main lobe (for example, in paragraph 1 of § 2.10), or at a level from the maximum value of the spectral density. The duration of the bell-shaped pulse (see § 2.10, paragraph 3) and the width of its spectrum are sometimes determined at the level of 0.606 from the maximum value, respectively, or. The energy criterion is often used, understanding the spectrum width as a frequency band containing a given fraction of the total signal energy.
For practice, it is also important to estimate the length of the "tails" of the spectrum outside the frequency band containing the bulk of the signal energy.
1. DEFINITION OF THE PRODUCT BAND X DURATION
To identify the limiting relationships that relate the duration of the signal and the width of the spectrum, the method of moments has become widespread in the modern theory of signals.
By analogy with the concept of the moment of inertia in mechanics, the effective signal duration can be determined by the expression
where the midpoint of the pulse is determined from the condition
It means that the function is square-integrable (signal with finite energy).
Similarly, the effective width of the spectrum is determined by the expression
Since the modulus of the spectrum does not depend on the shift in time, we can put Finally, the signal can be normalized in such a way that its energy E is equal to unity and, therefore,
Under these conditions, the expressions for and take the form
and hence the product duration x band
It must be borne in mind that are the standard deviations from and , respectively. Therefore, the total duration of the signal should be equated and the total width of the spectrum (including the region of negative frequencies) to the value .
The product depends on the waveform, but it cannot be less than 1/2. It turns out that the smallest possible value corresponds to a bell-shaped impulse.
The method of moments is not applicable to all signals. It can be seen from the expressions for that the function with increasing t must decrease faster than , and the function must decrease faster than, since otherwise the corresponding integrals tend to infinity (diverge).
In particular, this applies to the spectrum of a strictly rectangular pulse, when
In this case, the expression for does not make sense, and the estimate of the effective width of the spectrum of a rectangular pulse has to be based on other criteria.
Let us consider some simple signals like video pulses, i.e., signals whose spectrum is concentrated in the region low frequencies, and use Parseval's equality to determine the energy contained in the band from to some cutoff frequency :
Relating then to the total pulse energy E, we determine the coefficient
characterizing the concentration of energy in a given band.
We will take a rectangular pulse as the initial signal, then consider a triangular and bell-shaped (Gaussian) pulse. The latter is especially indicative, since it provides the maximum possible concentration of the energy of the spectrum in a given band.
For a rectangular pulse in accordance with (2.68)
Calculating the integral, we get
where is the integral sine.
Passing to the argument, we write
For a triangular pulse whose spectral density is given by formula (2.73) and whose total energy
Rice. 2.23. Fraction of signal energy in the band (a) and pulse deformation at spectrum truncation (b)
For a Gaussian pulse, in accordance with (2.77), we obtain
where is the total energy of the Gaussian pulse, and the function
Given that the duration of the Gaussian pulse is defined in paragraph 3 of § 2.10 and is equal to , the argument of the function can be written in the form Functions for three pulses are shown in Fig. 2.23, a.
So, the value of the product required for a given maximum for a rectangular pulse (at ) and minimum for a Gaussian. In particular, the level corresponds to values equal to 1.8; 0.94 and 0.48.
The choice of the spectrum boundary according to the energy criterion is not always acceptable in some practical problems. So, if during the processing of an impulse it is required to keep its shape close enough to a rectangular one, then it should be much greater than unity. To illustrate this important point, in Fig. 2.23b shows the initial pulse (dashed line) and its deformation when the spectrum is truncated at the levels.
In any case, for a given signal shape, compressing it in time in order, for example, to increase the accuracy of determining the moment of its occurrence is inevitably accompanied by an expansion of the spectrum, which forces the bandwidth of the measuring device to be expanded.
Similarly, compression of the pulse spectrum in order to increase the accuracy of frequency measurements is inevitably accompanied by signal stretching in time, which requires an extension of the observation (measurement) time. The impossibility of simultaneously concentrating a signal in a narrow frequency band and in a short time interval is one of the manifestations of the uncertainty principle known in physics.
The question of the value of the product duration X band is relevant in connection with the problem of electromagnetic compatibility that arises with mutual interference of radio stations. From this point of view, the most desirable pulse shape is close to bell-shaped.
2. RATE OF REDUCTION OF THE SPECTRUM OUTSIDE THE MAIN BAND
To reveal the relationship between the behavior in the region of relatively high frequencies and the structure of the signal s(t), we use the properties of such test signals as a single pulse and a single jump.
A single impulse is the only function that has a non-decreasing spectral density on the entire frequency axis -
Therefore, it can be argued that the signal whose spectrum outside the main band does not decrease with increasing , contains a delta function (in real conditions sufficiently powerful short impulse).
Further, the only function of time that has a spectral density of the form is a unit jump and . Consequently, the decrease in the tail of the signal spectrum according to the law indicates the presence of jumps in the function, i.e. discontinuities. But at the discontinuity points, the derivative of the function turns into a delta function (with a constant coefficient equal to the jump value). Therefore, the decrease in the spectrum proportionally indicates the presence of a delta function in the composition of the derivative. This reasoning can be continued for derivatives of the signal of higher orders.
Let us illustrate the above with examples of three signals shown in Fig. 2.24: with a break, with a break and a “smooth” signal (without breaks and breaks).
In the first example (Fig. 2.24, a), the derivative is determined by the expression
and the spectral density of the function in accordance with Table. 2.1
To determine the spectral density of the signal , which is an integral of , we can proceed from the expression
In this case, the operation is legal because [see (2.60)].
At spectral density . As can be seen from fig. 2.24, a, this is explained by the presence of a function in the first derivative of the signal s(t).
