CN115664546A - Dynamic complex battlefield electromagnetic environment construction method - Google Patents
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Abstract
The invention relates to the technical field of electromagnetism correlation, in particular to a method for constructing an electromagnetic environment of a dynamic complex battlefield, which comprises the following steps: performing signal simulation on the communication signal, the radar signal and the interference signal; carrying out random simulation on the amplitude of the simulation signal, and carrying out random simulation on the occurrence and duration of the signal after the random simulation; signals with random amplitudes and random occurrence duration are superposed from few to many according to signal types, four electromagnetic environment signals with different complexities, namely simple, mild, moderate and severe, are constructed, dynamic changes of simulation electromagnetic signals can be better adapted, and the practicability is high.
Description
Technical Field
The invention relates to the technical field of electromagnetism correlation, in particular to a method for constructing a dynamic complex battlefield electromagnetic environment.
Background
The definition of a complex electromagnetic environment is: the sum of electromagnetic activities and phenomena that affect combat in a certain battlefield space. The battlefield electromagnetic environment is objectively existed on the basis of the development of electromagnetic space and the application and reaction activities of electromagnetism on battlefield, and the development of the battlefield electromagnetic environment depends on the wide application of the electromagnetic activities in various fields. With the wide penetration of electromagnetic application technology into various military and civil fields, especially the countermeasure activity of military information weapon equipment in the electromagnetic field aggravates the complexity of the electromagnetic environment in the battlefield, and the influence of the electromagnetic environment on the performance of weapon equipment and the influence on the operation action are beginning to arouse the high attention of people.
The analysis of the composition of the electromagnetic environment of the battlefield is the main content of qualitative research on the electromagnetic environment, is the basis for knowing the electromagnetic environment of the battlefield, and can essentially grasp the formation of the electromagnetic environment of the battlefield and various influencing factors. As can be known from the definition of a battlefield electromagnetic environment, under certain external conditions, electromagnetic radiation generated by electromagnetic radiation sources with different purposes in a certain battlefield space constitutes the battlefield electromagnetic environment.
In the current research of the construction of the electromagnetic environment of the dynamic complex battlefield, one main problem is that the electromagnetic signals of the real battlefield have very obvious dynamic change characteristics, thereby greatly increasing the simulation difficulty.
Disclosure of Invention
The invention aims to provide a dynamic complex battlefield electromagnetic environment construction method to solve the problems in the background technology.
In order to achieve the purpose, the invention provides the following technical scheme: a dynamic complex battlefield electromagnetic environment construction method comprises the following steps: carrying out signal simulation on the communication signal, the radar signal and the interference signal;
carrying out random simulation on the amplitude of the simulation signal, and carrying out random simulation on the occurrence and duration of the signal after the random simulation; and superposing the signals with random amplitudes and random occurrence duration according to the types of the signals from few to many to construct four electromagnetic environment signals with simple, mild, moderate and severe different complexities.
The method for constructing the electromagnetic environment of the dynamic complex battlefield comprises the following steps: the communication signal simulation comprises digital modulation, and the digital modulation method comprises the following steps: modulating different parameters of a carrier wave by using a digital signal, wherein the formula is as follows:
S(t)＝Acos(ωt+θ)；
wherein the parameters of S (t) include: amplitude a, frequency ω, initial phase θ.
The method for constructing the electromagnetic environment of the dynamic complex battlefield comprises the following steps: the radar signals comprise single carrier frequency conventional pulse signals and chirp pulse compression signals.
The method for constructing the dynamic complex battlefield electromagnetic environment comprises the following steps: the single carrier frequency conventional pulse signal is expressed as:
where rect (T) represents a rectangular function, T r Is the pulse repetition period, f c Representing the frequency.
The method for constructing the dynamic complex battlefield electromagnetic environment comprises the following steps: the chirp compression signal is represented as:
wherein rect (T) represents a rectangular function, T r Is the pulse repetition period, f c Representing the frequency and K the chirp rate.
The method for constructing the dynamic complex battlefield electromagnetic environment comprises the following steps: the interference signals comprise radio frequency noise interference signals, noise amplitude modulation interference signals, noise frequency modulation interference signals and noise phase modulation interference signals.
The method for constructing the dynamic complex battlefield electromagnetic environment comprises the following steps: the mathematical expression of the radio frequency noise interference signal is represented as:
J(t)＝U n (t)cos[2πf j t+φ(t)]；
the complex representation is:
J(t)＝U n (t)·exp[2πf j t+φ(t)]；
wherein the envelope function U n (t) obey rayleigh distribution; the phase function phi (t) obeys 0,2 pi]Are uniformly distributed with U n (t) are independent of each other; center frequency of f j Constant and greater than the spectral width of J (t)。
The method for constructing the dynamic complex battlefield electromagnetic environment comprises the following steps: the mathematical expression of the noise amplitude modulation interference signal is as follows:
the complex form is:
wherein, U 0 、f j Is a constant, respectively carrier frequency amplitude and frequency; modulating noise U n (t) is zero mean and variance
The generalized stationary random process of (2);
is [0,2 π ]]Are uniformly distributed with U n (t) mutually independent random variables.
The method for constructing the electromagnetic environment of the dynamic complex battlefield comprises the following steps: the mathematical expression of the noise frequency modulation interference signal is as follows:
the complex form is:
wherein, U j 、f j Is constant, respectively carrier frequency amplitude and frequency; the modulation noise u (t) is a random process with zero mean value and generalized stability;
is [0,2 π ]]Therein are allRandom variables which are uniformly distributed and are independent of u (t); k FM Is constant and is called chirp rate.
The method for constructing the dynamic complex battlefield electromagnetic environment comprises the following steps: the mathematical expression of the noise phase modulation interference is as follows:
the complex form is:
wherein, U j 、f j Is a constant, respectively carrier frequency amplitude and frequency, the modulation noise u (t) is a random process with zero mean value and generalized stability,
is [0,2 π ]]Uniformly distributed random variable in and independent of u (t), K FM Is a constant; when u (t) is gaussian noise, the power spectrum of the noise-modulated signal is approximated as:
in the formula,. DELTA.F n Is the modulation noise bandwidth; d = K PM σ n Called effective phase shift;
is the variance of the modulation noise u (t).
Compared with the prior art, the invention has the beneficial effects that: the simulation system is novel in design, simulates communication signals, radar signals and electromagnetic interference signals, simultaneously carries out random simulation on simulation signal amplitudes, carries out random simulation on signal occurrence and duration after random simulation, constructs four simple, mild, moderate and severe electromagnetic environment signals with different complexities, can better adapt to dynamic changes of simulated electromagnetic signals, and is strong in practicability.
Drawings
Fig. 1 is a diagram of 50MHz frequency 2ASK waveforms and spectra.
Fig. 2 is a graph of a 100MHz frequency 2ASK waveform and spectrum.
Fig. 3 is a diagram of 50MHz frequency 4ASK waveforms and spectra.
Fig. 4 is a diagram of a 100MHz frequency 4ASK waveform and spectrum.
Fig. 5 is a diagram of a 50MHz frequency 2PSK waveform and spectrum.
FIG. 6 is a diagram of a 100MHz frequency 2PSK waveform and spectrum.
Fig. 7 is a diagram of 50MHz frequency 4PSK waveforms and spectra.
FIG. 8 is a diagram of a 100MHz 4PSK waveform and spectrum.
FIG. 9 is a diagram of a 50MHz frequency 2FSK waveform and spectrum.
FIG. 10 is a diagram of a 100MHz frequency 2FSK waveform and spectrum.
FIG. 11 is a diagram of a 50MHz frequency 4FSK waveform and spectrum.
FIG. 12 is a diagram of a 100MHz 4FSK waveform and spectrum.
Fig. 13 is a diagram of 50MHz frequency 16QAM waveforms and spectrum.
FIG. 14 is a graph of a 100MHz 16QAM waveform and spectrum.
FIG. 15 shows the frequency f c Is 220MHz and the pulse period T r The waveform and the frequency spectrum of the single-carrier frequency rectangular pulse signal are 0.05 us.
FIG. 16 shows the frequency f c Is 440MHz and the pulse period T r The waveform and the frequency spectrum of the single-carrier frequency rectangular pulse signal are 0.1 us.
FIG. 17 shows the frequency f c Is 220MHz and a pulse period T r The chirp compressed signal waveform and spectrogram is 0.05 us.
FIG. 18 shows the frequency f c Is 440MHz and the pulse period T r The chirp compresses the signal waveform and spectrum plot for 0.1 us.
FIG. 19 shows the frequency f c Is 220MHz and the pulse period T r The waveform and spectrogram of the second-time sweep nonlinear frequency modulation signal of 0.05 us.
FIG. 20 shows the frequency f c Is 440MHz and the pulse period T r The waveform and the spectrogram of the quadratic sweep frequency nonlinear frequency modulation signal are 0.1 us.
FIG. 21 shows the frequency f c Is 220MHz and a pulse period T r The log swept non-chirp signal waveform and spectrogram is 0.05 us.
FIG. 22 shows the frequency f c Is 440MHz and a pulse period T r The waveform and spectrogram of the logarithmic sweep frequency non-linear frequency modulation signal is 0.1 us.
FIG. 23 shows the frequency f c Is 220MHz and the pulse period T r The waveform and spectrogram of the phase modulation radar signal is 0.05 us.
FIG. 24 shows a pulse period T with a frequency of 440MHz r The waveform and spectrogram of the phase modulation radar signal is 0.1 us.
FIG. 25 shows the frequency f c Is 220MHz and the pulse period T r The waveform and spectrogram of the radar signal are modulated at the frequency of 0.05 us.
FIG. 26 shows the frequency f c Is 440MHz and a pulse period T r The waveform and spectrogram of the frequency modulation radar signal is 0.1 us.
FIG. 27 is a graph of the waveform and spectrum of an RF noise interference signal.
FIG. 28 is a diagram of the waveform and spectrum of a noise-amplitude-modulated interference signal.
Fig. 29 is a diagram of the waveform and spectrum of a noise fm interference signal.
Fig. 30 is a diagram of the waveform and spectrum of a noise-modulated interference signal.
Fig. 31 is a graph of the waveform and spectrum of 2ASK after random amplitudes.
Fig. 32 is a diagram of signal waveform and spectrum for 2PSK after random amplitude.
Fig. 33 is a signal waveform and spectrum diagram of CW after amplitude randomization.
FIG. 34 is a graph of the signal waveform and spectrum of NRF after amplitude randomization.
FIG. 35 is a graph of the waveform and spectrum of a 2ASK signal after random amplitude and random occurrence and random duration.
FIG. 36 is a graph of the waveform and spectrum of a 2PSK signal after random amplitude and random occurrence and duration.
Fig. 37 is a graph of the signal waveform and spectrum of CW after random amplitude and occurrence and random duration.
Fig. 38 is a graph of the waveform and spectrum of an LFM signal after random amplitude and random occurrence and random duration.
FIG. 39 is a time domain and frequency domain waveform diagram of a simple complex electromagnetic environment signal.
Fig. 40 is a time domain and frequency domain waveform diagram of a mildly complex electromagnetic environment signal.
FIG. 41 is a time domain and frequency domain waveform diagram of a moderately complex electromagnetic environment signal.
FIG. 42 is a time domain and frequency domain waveform diagram of a heavily complex electromagnetic environment signal.
FIG. 43 is a flow chart of a method for constructing a dynamic complex battlefield electromagnetic environment.
Detailed Description
Various exemplary embodiments, features and aspects of the present application will be described in detail below with reference to the accompanying drawings. In the drawings, like reference numbers indicate functionally identical or similar elements. While the various aspects of the embodiments are presented in drawings, the drawings are not necessarily drawn to scale unless specifically indicated.
The word "exemplary" is used exclusively herein to mean "serving as an example, embodiment, or illustration. Any embodiment described herein as "exemplary" is not necessarily to be construed as preferred or advantageous over other embodiments.
In addition, numerous specific details are set forth in the following specific examples in order to provide a better understanding of the present application. It will be understood by those skilled in the art that the present application may be practiced without some of these specific details. In some instances, methods, means, elements well known to those skilled in the art have not been described in detail so as not to obscure the present application.
In the embodiment of the present invention, please refer to fig. 43, a method for constructing a dynamic complex battlefield electromagnetic environment includes the following steps:
step one, carrying out signal simulation on communication signals, radar signals and interference signals.
The communication signal simulation comprises digital modulation, and the digital modulation method comprises the following steps: modulating different parameters of a carrier wave by using a digital signal, wherein the formula is as follows:
S(t)＝Acos(ωt+θ)；
wherein the parameters of S (t) include: amplitude A, frequency omega and initial phase theta; the modulation is such that a, ω, or the like varies with the digital baseband signal. Wherein, the ASK modulation mode uses two different amplitudes of the carrier wave to represent 0 and 1; the FSK modulation mode is that two different frequencies of a carrier wave represent 0 and 1; and the PSK modulation scheme represents 0 and 1 by the change in the initial phase of the carrier.
It should be noted that ASK represents amplitude shift keying, PSK represents phase shift keying, FSK represents frequency shift keying, and the modulation method mainly changes the amplitude, phase or frequency of the cosine wave to transmit information, and the method corresponds to three digital modulation modes, ASK, PSK and FSK respectively.
Referring to fig. 1-14, waveforms and frequency spectrums of 2ASK, 4ASK, 2PSK, 4PSK, 2FSK, 4FSK, and 16QAM at 50MHz and 100MHz frequencies are shown, respectively.
In order to construct electromagnetic environment signals with different complexity degrees, the invention simulates communication signals with 20 frequencies and 17 modulation modes, and the specific parameters are as shown in the following table 1;
TABLE 1 communication Signal simulation parameters
The radar signal is typically a narrow-band signal, i.e. most of the energy of the radar signal is concentrated at the carrier frequency f c Within a certain frequency band Δ f in the vicinity, and Δ f < fc, so that the real signal emitted by the radar can be expressed as:
in the formula, a (t) and
respectively representing the amplitude and phase of the modulation function. For a narrow-band signal, a (t) depicts the envelope of the cosine function oscillating rapidly in equation (2-2), which is conventionally referred to as a (t); for ease of description and analysis, the pair of equations (2-2) is generally represented in complex signal form:
in the formula
Referred to as the complex envelope of the radar signal.
For the most commonly used pulse system radar, the general form of the complex envelope in equation (2-3) can be expressed as:
where rect (t) represents a rectangular function, f Δk Is the frequency variation of the carrier frequency of the k-th pulse, T r For the pulse repetition period, a (t) is the modulation function of a single pulse, N P Indicating the number of pulses, T p Is the pulse width.
The invention mainly considers radar emission signals of two forms of single carrier frequency conventional pulse signals and linear frequency modulation pulse compression signals.
Wherein, for the radar with single carrier frequency conventional pulse system, f in the formula (2-4) Δk For 0,a (t) =1, the radar transmission signal by the formula (2-3) is expressed as:
where rect (T) represents a rectangular function, T r Is the pulse repetition period, f c Referring to FIG. 15 and FIG. 16, the pulse waveforms of the single carrier rectangular pulse signal are shown, respectively, at a frequency f c 220MHz, pulse period of 0.05us and frequency of 440MHz, pulse period of 0.1 us.
For a chirp-compressed signal, a (t) in equation (2-4) can be expressed as:
b is linear bandwidth, T p As above, the pulse width is shown.
Let K = B/τ, then there is a (t) = exp (j π Kt) 2 ) And K is called chirp rate. The chirp compression signal can be expressed as:
wherein rect (T) represents a rectangular function, T r Is the pulse repetition period, f c Referring to fig. 17 and 18, the waveforms and spectra of the chirp compression signals are respectively a chirp compression signal with a frequency of 220MHz and a pulse period of 0.05us, and a chirp compression signal with a frequency of 440MHz and a pulse period of 0.1 us.
In order to construct electromagnetic environment signals with different complexity degrees, the invention simulates radar signals with 20 frequencies, 8 pulse widths (pulse repetition frequencies) and 6 modulation modes, the specific parameters are shown in Table 2,
TABLE 2 Radar Signal simulation parameter settings
The interference signals comprise radio frequency noise interference signals, noise amplitude modulation interference signals, noise frequency modulation interference signals and noise phase modulation interference signals.
The entropy of the radio frequency noise interference is maximum, the covering effect is best, and therefore the radio frequency noise interference method has a good interference effect. However, the noise power level generated by the microwave device is very low, it is difficult to obtain large interference power, and the interference effect is improved by fully amplifying the interference power, so that the power utilization rate is low, in order to improve the interference efficiency, the noise needs to be modulated, and the mathematical expression of the radio frequency noise interference signal is represented as:
J(t)＝U n (t)cos[2πf j t+φ(t)]；
the complex representation is:
J(t)＝U n (t)·exp[2πf j t+φ(t)]；
wherein the envelope function U n (t) obey rayleigh distribution; the phase function phi (t) obeys 0,2 pi]Are uniformly distributed in the same direction as U n (t) are independent of each other; center frequency of f j Constant and larger than the spectral width of J (t).
Please refer to fig. 27, which is f c And (4) interfering the simulation result of the time domain waveform and the power spectrum by the radio frequency noise of =200 MHz.
The noise amplitude modulation interference signal is a random signal formed by carrying out amplitude modulation on a carrier by using band-limited white Gaussian noise, and the mathematical expression of the noise amplitude modulation interference signal is as follows:
the complex form is:
wherein, U 0 、f j Is constant, respectively carrier frequency amplitude and frequency; modulating noise U n (t) is zero mean and variance
The generalized stationary random process of (2);
is [0,2 π ]]Are uniformly distributed with U n (t) random variables independent of each other.
Please refer to fig. 28, which is a time domain waveform and a frequency domain spectrum diagram of a carrier frequency of 200MHz, a modulation noise bandwidth of 40MHz, a modulation noise power of 1W, a modulation factor, and a noise amplitude modulation interference.
The noise frequency modulation interference signal is a random signal obtained by frequency modulating a carrier wave by noise, and the mathematical expression of the noise frequency modulation interference signal is as follows:
the complex form is:
wherein, U j 、f j Is a constant, respectively carrier frequency amplitude and frequency; the modulation noise u (t) is a random process with zero mean value and generalized stability;
is [0,2 π ]]Random variables which are uniformly distributed and are independent from u (t); k FM Is constant and is called chirp rate.
Please refer to fig. 29, which shows the frequency modulation index m fe And =20, the carrier frequency is 200MHz, and the modulation noise bandwidth is 40 MHz.
The mathematical expression of the noise phase modulation interference is as follows:
the complex form is:
wherein, U j 、f j Is a constant, respectively carrier frequency amplitude and frequency, the modulation noise u (t) is a random process with zero mean value and generalized stability,
is [0,2 π ]]Uniformly distributed and independent of u (t) random variable, K FM Is a constant;
when u (t) is gaussian noise, the power spectrum of the noise-modulated signal is approximated as:
in the formula,. DELTA.F n To modulate the noise bandwidth; d = K PM σ n Called effective phase shift;
is the variance of the modulation noise u (t).
Carrying out random simulation on the amplitude of the simulation signal, and carrying out random simulation on the occurrence and duration of the signal after the random simulation;
and superposing the signals with random amplitudes and random occurrence duration according to the types of the signals from few to many to construct four electromagnetic environment signals with simple, mild, moderate and severe different complexities.
Referring to fig. 30, a time domain waveform and a power spectrum of a noise phase-modulated signal with a carrier frequency of 200MHz and a modulation noise bandwidth of 40MHz and an effective phase shift are shown.
In order to construct electromagnetic environment signals with different complexity degrees, 20 frequencies and 4 types of interference signals are simulated, and specific parameters are shown in table 3.
Table 3 interfering signal simulation parameter settings
And step two, performing random simulation on the amplitude of the simulation signal to achieve the purpose of constructing a complex electromagnetic environment signal with stronger dynamic property and complexity, wherein 2ASK,2PSK, CW and NRF signals with random amplitudes are shown in FIGS. 31, 32, 33 and 34, performing random simulation on the occurrence and duration of the signal after the random simulation, and performing 2ASK,2PSK, CW and LFM signals with random amplitudes and random occurrences and duration, as shown in FIGS. 35, 36, 37 and 38.
And step three, superposing the signals with random amplitudes and random occurrence duration according to the signal types from few to many to construct four electromagnetic environment signals with simple, mild, moderate and severe degrees and different complexities.
The number of simple electromagnetic environment signals is 50-240 radar signals, 20-85 communication signals and 5-20 interference signals, and the number of final signal types is 75-345. The number of the signals of the mild complex electromagnetic environment is 241-480 radar signals, 85-170 communication signals, 20-40 interference signals and 346-690 final signal types. The moderate complex electromagnetic environment signal selects 481-720 radar signals, 170-255 communication signals, 40-60 interference signals and 691-1035 final signal types. The number of the severe complex electromagnetic environment signals is 721-960 radar signals, 255-340 communication signals, 60-80 interference signals and 1036-1380 final signal types.
The types of signals constituting the electromagnetic environment signal are shown in table 4,
TABLE 4 four complex electromagnetic environment signal composition
In conclusion, the simulation method and the simulation system have the advantages that the communication signals, the radar signals and the electromagnetic interference signals are simulated, the amplitude values of the simulated signals are simulated randomly, the occurrence and the duration of the signals are simulated randomly after the random simulation, the simple, light, medium and heavy electromagnetic environment signals with different complexities are constructed, the dynamic change of the simulated electromagnetic signals can be better adapted, and the practicability is high.
It will be evident to those skilled in the art that the invention is not limited to the details of the foregoing illustrative embodiments, and that the present invention may be embodied in other specific forms without departing from the spirit or essential attributes thereof. The present embodiments are therefore to be considered in all respects as illustrative and not restrictive, the scope of the invention being indicated by the appended claims rather than by the foregoing description, and all changes which come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein. Any reference sign in a claim should not be construed as limiting the claim concerned.
Furthermore, it should be understood that although the present description refers to embodiments, not every embodiment may contain only a single embodiment, and such description is for clarity only, and those skilled in the art should integrate the description, and the embodiments may be combined as appropriate to form other embodiments understood by those skilled in the art.
Claims (10)
1. A dynamic complex battlefield electromagnetic environment construction method is characterized by comprising the following steps:
carrying out signal simulation on the communication signal, the radar signal and the interference signal;
carrying out random simulation on the amplitude of the simulation signal, and carrying out random simulation on the occurrence and duration of the signal after the random simulation;
and superposing the signals with random amplitudes and random occurrence duration according to the types of the signals from few to many to construct four electromagnetic environment signals with simple, mild, moderate and severe degrees and different complexities.
2. The method according to claim 1, wherein the communication signal simulation comprises digital modulation, and the digital modulation method comprises: the different parameters of the carrier are modulated by digital signals, the formula is as follows:
S(t)＝Acos(ωt+θ)；
wherein the parameters of S (t) include: amplitude a, frequency ω, initial phase θ.
3. The method of claim 1, wherein the radar signal comprises a single carrier frequency conventional pulse signal and a chirp compressed signal.
4. The method as claimed in claim 3, wherein the single carrier frequency normal pulse signal is represented as:
where rect (T) represents a rectangular function, T r For the pulse repetition period, f c Representing the frequency.
5. The method of claim 3, wherein the chirp-compressed signal is expressed as:
where rect (T) represents a rectangular function, T r Is the pulse repetition period, f c Representing frequency and K the chirp rate.
6. The method as claimed in claim 1, wherein the interference signal includes a radio frequency noise interference signal, a noise amplitude modulation interference signal, a noise frequency modulation interference signal, and a noise phase modulation interference signal.
7. The method of claim 6, wherein the mathematical expression of the RF noise interference signal is expressed as:
J(t)＝U n (t)cos[2πf j t+φ(t)]；
the complex representation is:
J(t)＝U n (t)·exp[2πf j t+φ(t)]；
wherein the envelope function U n (t) obey rayleigh distribution; the phase function phi (t) obeys 0,2 pi]Are uniformly distributed in the same direction as U n (t) are independent of each other; center frequency of f j Constant and larger than the spectral width of J (t).
8. The method as claimed in claim 6, wherein the mathematical expression of the noise amplitude modulation interference signal is:
the complex form is:
wherein, U 0 、f j Is constant, respectively carrier frequency amplitude and frequency; modulating noise U n (t) is zero mean and variance
The generalized stationary random process;
is [0,2 π ]]Are uniformly distributed and are equal to U n (t) random variables independent of each other.
9. The method as claimed in claim 6, wherein the mathematical expression of the noise FM interference signal is:
the complex form is:
wherein, U j 、f j Is constant, respectively carrier frequency amplitude and frequency; the modulation noise u (t) is a random process with zero mean value and generalized stability;
is [0,2 π ]]Random variables which are uniformly distributed and are independent from u (t); k FM Is constant and is called chirp rate.
10. The method of claim 6, wherein the mathematical expression for the noise modulation and interference modulation is as follows:
the complex form is:
wherein, U j 、f j Is a constant, respectively carrier frequency amplitude and frequency, the modulation noise u (t) is a random process with zero mean value and generalized stability,
is [0,2 π ]]Random with uniform distribution in and independent of u (t)Variable, K FM Is a constant;
when u (t) is gaussian noise, the power spectrum of the noise-modulated signal is approximated as:
in the formula,. DELTA.F n Is the modulation noise bandwidth; d = K PM σ n Called effective phase shift;
is the variance of the modulation noise u (t).
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