An FCM pulse is a rectangular radio pulse with internal phase-code keying (carrier oscillation) of high-frequency filling.
Manipulation is the same as modulation when parameters change abruptly.
An FCM pulse is a set of adjacent rectangular radio pulses with the same duration T, the same amplitude and the same filling frequency.
The initial HF filling phase of these pulses can take only two values: either 0 or π. The alternation of these values from pulse to pulse obeys a specific code.
The choice of code is made based on the condition of obtaining the best ACF of the signal.
Let's consider an example of a FCM pulse with a volume of n elementary signals, where phase manipulation is carried out by the Barker code.
The width of the spectrum of the FCM pulse is determined by the duration of the elementary pulse T and
FCM is a complex signal. Its base is determined by the number of pulses n (n>>1).
Let us synthesize a linear filter matched with the FCM pulse according to the required impulse response.
The impulse response is a mirror image of the input signal.
Conditional image g of (t):
As we can see, the impulse response of the synthesized optimal filter is also a FCM pulse, the code of which is a mirror image of the signal code, therefore the response of our filter to a δ-pulse will be n adjacent rectangular radio pulses of the same duration, amplitude and frequency.
The initial phase of the RF fill pulses varies from pulse to pulse in accordance with the mirror code.
The test showed that our filter is optimal for this signal.
Let us find the response of the resulting optimal filter to a given FCM pulse. It is known that the response of the optimal filter follows the shape of the ACF of the FCM pulse
Conventional image of an FCM pulse
Conditional image of the adder response (signal at the output of the adder).
The output of the adder also produces seven rectangular radio pulses, spaced from each other by the interval T and. The duration of these pulses is the same and equal to T i.
Their filling frequency is the same. The initial filling phase of the central pulse is 0, and for all others π. The amplitude of the central pulse is seven times greater than the amplitude of all other pulses.
Conclusion: the signal at the output of the optimal filter, matched to the PCM pulse, represents n adjacent triangular radio pulses of the same duration 2T and, with the same filling frequency and the same initial phase, and the amplitude of the central pulse (main lobe) is seven times higher than for other pulses (side lobes).
It turns out that in the optimal filter, phase-code keying is transformed into amplitude keying.
As you can see, one FCM pulse turned into seven triangular pulses: one central and six side ones.
It is impossible to completely eliminate side lobes; there are no such codes. The Barker code is the best of all codes in terms of the ratio of the amplitude of the side lobe to the central lobe.
Unfortunately, the Barker code length cannot be greater than 13.
To obtain a large signal base, maximum length sequence (M-sequence) codes are widely used.
If we count the duration of the output signal of the optimal filter at a level of 0.5 from the maximum, then it turns out that this duration is equal to T and = T s /n (n-base), therefore the optimal filter compresses the input signal in time by a number of times equal to the base.
The effect of compression of a complex signal in an optimal filter allows it to be increased by a number of times equal to the base of the signal, time resolution of signals.
Time resolution means the ability to separately observe two signals shifted relative to each other for some time.
At the input of the optimal filter, signals can be observed separately if they are shifted relative to each other by more than T s.
After an optimal filter, signals can be observed separately if they are shifted relative to each other by more than T and.
Advantage of complex signals:
1) With optimal filtering, a gain in signal-to-noise ratio equal to the base is obtained. This means that the communication system can operate with low signal-to-noise ratios at the input. This gives:
You can receive a signal from afar (from space);
You can carry out secret communication.
2) Using complex signals, for example FCM, it is possible to implement code division of communication channels.
3) Thanks to complex signals, it is possible to solve the age-old problems of communication and location; for example, it is known that to increase the communication range it is necessary to increase the energy of the transmitted signal. When working with a rectangular radio pulse, the energy is determined by the amplitude of the pulse and the duration of the signal. The amplitude of the transmitted pulse cannot be increased indefinitely; therefore, the pulse duration is increased. However, increasing the signal duration degrades the time resolution of the signal.
The use of complex signals makes it possible to separate these quantities: the energy depends on the duration of the signal T s, and the resolution of the signal depends on the value of the signal base n = T s / T u.
Section 6.
UDC 621.396.96:621.391.26
A method for increasing the efficiency of radar for detecting people behind optically opaque obstacles
O. V. Sytnik I. A. Vyazmitinov, E. I. Miroshnichenko, Yu. A. Kopylov
Institute of Radiophysics and Electronics named after. A. Ya. Usikova NAS of Ukraine
The possibilities of reducing the level of side lobes of the autocorrelation function of FCM probing signals and the problems of their practical implementation in equipment are considered. An optimal phase-amplitude intrapulse modulation has been proposed, which makes it possible to reduce the side lobes and at the same time increase the repetition rate of probing messages. The factors influencing the characteristics of such signals are studied and a criterion for their feasibility in equipment is proposed.
Introduction.
Signal processing algorithms in a radar with a quasi-continuous probing signal designed to detect objects hidden behind optically opaque obstacles are usually built on the principle of optimal correlation processing or matched filtering [ – ].
Probing signals for such radars are selected based on the requirement to ensure the necessary resolution and noise immunity. In this case, they try to make the signal uncertainty function pencil-shaped in the corresponding plane with a minimum level of side lobes. For this, various complex types of modulation are used [, ,]. The most common of them are: frequency-modulated signals; multi-frequency signals; phase-shift keyed signals; signals with code phase modulation; discrete frequency signals or signals with code frequency modulation; composite signals with code frequency modulation and a number of signals that are a combination of several types of modulation. The narrower the main peak of the signal uncertainty function and the lower the level of its side lobes, the correspondingly higher the resolution and noise immunity of the radar. The term “noise immunity” in this work means the radar’s resistance to interference caused by reflections of the probing signal from objects that are not targets and located outside the analyzed strobe (frequency, time). Such signals are called long-baseline signals or ultra-wideband (UWB) signals in the literature.
One of the varieties of UWB signals are phase-keyed signals, which represent a coded sequence of radio pulses, the initial phases of which change according to a given law. Code sequences of maximum length or M-sequences have very important properties for radar:
· M-sequences are periodic with period , where is the number of elementary pulses in the sequence; − duration of an elementary pulse;
· The level of the side lobes of the uncertainty function for a periodic sequence is − , and for a single sequence of pulses − ;
· Pulses in one period of the sequence, differing in phases, frequencies, durations, are distributed with equal probability, which gives grounds to consider these signals to be pseudorandom;
· Formation M-sequences are carried out quite simply on shift registers, and the number of bits of the register is determined by the length of one period of the sequence - from the relation.
The purpose of this work is to study the possibilities of reducing the level of side lobes of the uncertainty function of signals modulated M-sequences.
Formulation of the problem.
Figure 1 shows a fragment of a modulating function formed by a periodic sequence (here there are two periods M-sequences with
).
Section along the time axis of the uncertainty function of a radio signal modulated by such M-sequence is shown in Fig. 2. The side lobe level, as predicted by theory, is 1/7 or minus 8.5 dB.
Let us consider the possibility of minimizing the side lobes of the uncertainty function of the FCM signal. Let us denote by the symbol M-sequence, the duration of one period is equal to . In discrete time, provided that , the algorithm for calculating the elements of the sequence can be written in the following form:
(1)
The radio signal emitted by the locator is the product of the carrier harmonic signal
, (2)
Where
− vector of parameters for the modulating function (1) -
. (3)
The signal power is distributed between the side lobes of the uncertainty function -
(4)
and the main petal -
, (5)
where the symbol *− denotes the operation of complex conjugation, and the limits of integration in the time and frequency domains are determined by the corresponding type of signal modulation.
Attitude
(6)
can be considered as the objective function of a parametric optimization problem.
Algorithm for solving the problem.
The solution to the optimization problem (6) is to estimate the parameter -
, (7)
where is the domain of definition of the vector.
The traditional way to calculate estimate (7) is to solve the system of equations -
. (8)
The analytical solution (8) turns out to be quite labor-intensive, so we will use a numerical minimization procedure based on Newton’s method
, (9)
where is the quantity that determines the step length of the procedure for searching for the extremum of the objective function.
One way to calculate stride length is to calculate:
. (10)
In the simplest case, when the vector is composed of one parameter, for example or , the probing signal is generated relatively simply. In particular, when optimizing the objective function by parameter, the signal is generated in accordance with the relation
. (11)
In Fig. Figure 3 shows a fragment of the module of the autocorrelation function of the signal (11) at , which corresponds to a PCM radio signal without intrapulse phase modulation.
The side lobe level of this function corresponds to the theoretical limit equal to , where . In Fig. Figure 4 shows a fragment of the module of the autocorrelation function of the signal (11) with the parameter obtained by optimizing the function (). The side lobe level is minus 150 dB. The same result is obtained with amplitude modulation M-sequences. In Fig. Figure 5 shows the appearance of such a signal at the optimal value.
Rice. 5. Fragment of an amplitude-modulated FCM signal
The probing signal is generated in accordance with the algorithm
. (12)
Simultaneous amplitude-phase modulation leads to a decrease in the side lobe by another order of magnitude. It is not possible to reach the zero level of the side lobe due to the inevitable computational errors of the recurrent procedure for minimizing the objective function (), which do not allow one to find the true value of the parameter , but only its certain vicinity - . In Fig. Figure 6 shows the dependence of the values of the optimal phase modulation coefficients on the parameter , which determines the length of the sequence.
Rice. 6. Dependence of optimal phase shift on length M- sequences
From Fig. 6 it can be seen that as the sequence length increases, the value of the optimal phase shift asymptotically tends to zero and at we can assume that the optimal signal with intrapulse phase modulation is practically no different from a conventional PCM signal. Research shows that as the length of the modulating PSP period increases, the relative sensitivity to signal distortion will decrease.
An analytical criterion for choosing the limit sequence length can be the following relation
, (13)
where is a number that determines the possibility of technical implementation of a signal with intrapulse modulation in equipment.
Assessing the feasibility of complicating the signal.
The inevitable complication of the signal with a decrease in the side lobes of the autocorrelation function significantly tightens the requirements for generation devices and signal transmission and reception paths. Thus, if there is an error in setting the phase multiplier to one thousandth of a radian, the side lobe level increases from minus 150 dB to minus 36 dB. With amplitude modulation, the error relative to the optimal value of the coefficient A one thousandth leads to an increase in the side lobe from minus 150 dB to minus 43 dB. If the errors in setting the parameters are 0.1 from the optimal ones, which can be implemented in the equipment, then the side lobe of the uncertainty function will increase to minus 15 dB, which is 7 - 7.5 dB better than in the absence of additional phase and amplitude modulation.
On the other hand, the side lobe of the uncertainty function can be reduced without complicating the signal by increasing . So at the side lobe level will be approximately minus 15 dB. It should be noted that ordinary (i.e., without additional AM-FM modulation) PCM signals are sensitive to errors that arise during their formation. Therefore the length M-sequences in real radar devices are also impractical to increase indefinitely.
Let us consider the influence of errors that occur in equipment during the formation, transmission, reception and processing of FCM radio signals on their properties.
Assessment of the influence of errors in the formation of a FCM signal on its properties.
The entire set of factors influencing the characteristics of the signal can be divided into two groups: fluctuation and deterministic.
Fluctuation factors include: phase-frequency instabilities of reference oscillators; noises of various kinds; signals leaking from the transmitter directly to the receiver input and, after correlation processing with the reference signal, forming noise-like processes, and other factors.
Deterministic factors include: insufficient broadband of the forming circuits; asymmetry of the modulating function; incoherence of the modulating function and the carrier oscillation; difference in the shape of the reference and probing signals, etc.
More generally, the analytical expression for a signal modulated by a pseudorandom M- sequence, represent it in the form
, (14)
Where ; - constant amplitude; or p- signal phase; N=2k-1; k-integer; -duration of the elementary pulse forming the sequence.
Its two-dimensional correlation function is written as:
(15)
at
, , and its normalized spectrum is shown in Fig. 7. Here, for clarity, a fragment of the frequency axis is shown, where the main components of the signal spectrum are concentrated. A characteristic feature of such a signal, as can be seen from Fig. 7, is the reduced level of the unmodulated carrier oscillation, which in the ideal case tends to zero.
Fig.7. Normalized signal spectrum
The wide spectrum band and the absence of periodic unmodulated oscillations makes it possible to implement algorithms for detecting and identifying objects in location systems like , with the useful signal weakened in obstacles by 40-50 dB and levels of correlated interference exceeding the signal by 50-70 dB.
Rice. 8. Spectral density of the distorted signal
In the case when signal distortions are specified by deterministic functions in the coordinates Doppler shift - delay, it is more convenient to take into account their influence on the parameters of the autocorrelation function of the signal, for example, in the form of the following error functions.
Thus, for a phase-keyed pseudo-random signal with N=15, the dependence of the level of the residual side lobe of the autocorrelation function on the bandwidth of the forming circuits and the radio path is shown in Fig. 9.
Fig.9. Dependence of the ACF side lobe level on the bandwidth
transmission of the forming path for k=4
Here, the ordinate axis shows the value that determines the maximum achievable level of the side lobe of the autocorrelation function - a signal modulated by a pseudorandom M- sequence, and along the abscissa axis - expressed as a percentage, the ratio of the bandwidth of the forming circuit to the maximum value of the frequency of the effective spectrum of the signal. The dots on the graph show the ACF side lobe level values obtained from numerical simulation of hardware effects. As can be seen from Fig. 9, in the absence of frequency distortions in the radio paths, the level of the side lobe of the ACF signal modulated by the phase of the periodic PSP with a period N, is – 1/ N. This corresponds to the known theoretical limit. When the spectrum of the modulated signal is limited, the side lobe level increases and at 50% limitation reaches the level, which corresponds to a non-periodic autocorrelation function. Further limitation of the radio signal spectrum leads to almost complete collapse of the ACF and, as a result, to the inability to use the signal for practical purposes.
Distortions of the spectrum of the signal emitted by the locator and the reference oscillations arriving at the correlator, due to the asymmetry between positive and negative levels and durations of modulating oscillations, lead to a significant increase in interference in the area of the side lobes of the ACF and deterioration of the spatial resolution and detection characteristics of the locator. The dependence of the side lobe level on the asymmetry coefficient is shown in Fig. 10
The asymmetry coefficient was determined as
, (16)
where is the duration of the undistorted elementary pulse forming M- subsequence; the indices “+” and “−” mean the duration of the positive and negative elementary pulse with asymmetric distortions.
Fig. 10. Dependence of the ACF side lobe level on the magnitude of asymmetric signal distortions for k=4.
Conclusion.
The choice of signal and the degree of complexity of its modulating function is determined primarily by the nature of the tasks for which the radar is intended. The use of a fairly complex FCM signal with intrapulse modulation requires the creation of precision equipment, which will inevitably lead to a significant increase in the price of the design, but at the same time will make it possible to create universal units that can be used both in radars for rescuers and in radars for detecting fast-flying aircraft. goals. This possibility arises because the characteristics of a complex signal with a short sequence length, i.e. high sending repetition rate, allow you to have the necessary resolution and noise immunity with the ability to measure Doppler frequencies in a wider range. In addition, the construction of radar systems with continuous radiation and pseudo-random phase modulation of the carrier wave requires a detailed analysis and consideration of all factors that cause signal distortion in both the transmitting and receiving paths of the locator. Taking into account distorting factors comes down to solving engineering problems to ensure sufficient broadband, stability of electrical parameters and stability of the characteristics of the forming paths. In this case, the radar probe signals must be coherent with the modulating and auxiliary signals. Otherwise, technical solutions are needed that would minimize the difference distortions between the radiated and reference oscillations. One of the possible ways to implement such technical solutions is to introduce symmetrical amplitude limitations of signals in the output stages of the transmitter and at the input of the receiver correlator. In this case, although part of the signal energy is lost, it is possible to form an ACF of the modulated signal with acceptable parameters. Such technical solutions are acceptable in portable radars, where the cost and dimensions of the system play a decisive role.
The most promising at present, from the authors’ point of view, should be considered the construction of devices for generating and processing radio signals of complex structure for radar equipment, based on high-speed signal processors operating at clock frequencies of several gigahertz. The structural diagram of the radar with this approach becomes extremely simple. These are a linear power amplifier, a low noise linear receiver amplifier, and a processor with peripheral devices. This scheme allows not only to almost completely realize the properties of signals inherent in their fine structure, but also to create technologically easy-to-set up radar systems, the information processing of which is based on optimal algorithms.
Literature
1. Frank U.A., Kratzer D.L., Sullivan J.L. The Twopound Radar // RCA Eng.- 1967. No. 2; P.52-54.
2. Doppler radar for reconnaissance on the ground. Ser. Tech. intelligence means services cap. state // VINITI. – 1997. – No. 10. – P. 46-47.
3. Nordwall Bruce D.Ultra-wideband radar detects buried mines // Aviat. Week and Space Technol- 1997. No. 13.-P. 63-64.
4. Sytnik O.V., Vyazmitinov I.A., Myroshnychenko Y.I. The Features of Radar Developments for People Detection Under Obstructions // Telecommunications and Radio Engineering.¾ 2004. ¾. Estimation of Implementation Errors Effect on Characteristics of Pseudorandom Radar Signal // Telecommunications and Radio Engineering.¾ 2003. ¾ Vol.60, No. 1&2. ¾ P. 132–140.
9. Handbook of Radar / Ed. M. Skolnik. Per. from English Ed. K.N. Trofimova. , M.: Sov. radio, 1978, Vol.3. 528s.
Wide-shelf signals also include signals with intra-pulse linear modulation frequency (chirp). It can be presented in the form
where φ(t) is the total phase.
The frequency inside the pulse changes according to the following law
,
where Δf is the frequency deviation.
The total phase at time t is obtained by integrating the frequency:
Thus, the total phase of the signal changes according to a quadratic law. Taking into account the full chirp phase, the signal can be written in the following form
Signal base 
. The appearance of the chirp signal is shown in Fig. 4.179.
Optimal processing of a chirp signal requires the presence of a matched filter with a characteristic mirrored with respect to the signal. Among analog filters, this is a dispersive delay line, whose delay time depends on frequency.
A simplified diagram of a matched filter for a chirp signal is shown in Fig. 4.180.
We find the spectrum of the signal at the output of the matched filter using the formula
where K(jω) is the transfer function of the matched filter;
S(jω) – spectrum of external chirp signal.
The appearance of the spectrum S(jω) is shown in Fig. 4.181
where is the moment when the maximum output signal appears;
K is a constant.
Letting the modulus of the spectral density equal to a constant value, we obtain
where B is the amplitude of the spectral components.
In accordance with Parseval's theorem
We will find the signal at the output of the matched filter in the time domain using the Fourier transform of the spectral plane
Integrating over positive frequencies and isolating the active part, we get
Thus, the output pulse became K times narrower than the input pulse, and its amplitude increased by a factor.
The appearance of the pulse is shown in Fig. 4.172
The width of the main lobe at the zeros is 2/Δf, and at the level 0.64-1/Δf. The compression ratio at this level will be equal to
The chirp signal uncertainty diagram is shown in Fig. 4.183.
With the occupied frequency band, chirp is the best signal for time resolution.
The signal compression mechanism in the optimal filter can be explained as follows. The optimal filter delays the spectral components for a time:
(4.104)
where is the average frequency;
Frequency deviation;
Pulse duration;
Time to reach the maximum of the compressed pulse.
The dependence of the delay time on frequency (4.104) is shown in Fig. 4.184. The delay time is a linearly decreasing function of frequency. The dependence of the delay time on frequency is called dispersion.
At time t, the instantaneous frequency of the signal at the filter input is equal to . The oscillation of this frequency arrives at the output of the filter with a delay of , i.e. in the moment . Let's define this moment:
Consequently, all spectral components of the signal (regardless of their frequency) are delayed in the filter for such a time that they arrive at its output simultaneously at time . As a result of arithmetic addition, a peak signal surge is formed. (Fig. 4.185)
The shape of the compressed radio pulse in the absence of frequency mismatch is determined by the amplitude-frequency spectrum of the input signal. The phase-frequency spectrum, in this case, is compensated by the phase-frequency response of the filter and does not affect the shape of the input signal. Compensation of the phase-frequency spectrum of the signal is the main reason
time compression, leading to a coordinated superposition of harmonic components.
FCM signal processing
A phase-code-manipulated signal is a pulse signal divided into parallel pulses, each of which has its own initial phase (Fig. 4.186)
For such a signal the relation holds
where N is the number of partial pulses in the signal;
Δf – signal spectrum width.
Phase codes are usually binary, but can be more complex. The FCM signal can be represented as a train of coherent pulses. For such a packet, the optimal detector is shown in Fig. 4.187
The features of the scheme are as follows:
· Delay between adjacent line taps, delays must be equal to the duration of the partial pulse τ 1 ;
· Some taps of the delay line must include phase shifters that provide common-mode summation of signals.
The block diagram of the optimal PCM signal detector is shown in Fig. 4.188
The diagram shows: PV – phase shifters; SF – matched filter. Figures 4.189 and 4.190 show circuits of the optimal detector and voltage diagrams for a signal consisting of three partial pulses.
One of the main parameters characterizing a radar system is the discernibility coefficient, which is defined as the ratio of the minimum signal power at the receiver input P min to the noise power
Detection performance depends on signal energy
FCM radio pulses are characterized by an abrupt change in phase within the pulse according to a certain law, for example (Fig. 1.66):
– three-element signal code
– law of phase change
or seven-element signal (Fig. 1.67):
Thus, we can draw conclusions:
· ASF of signals with chirp is continuous.
· The ASF envelope is determined by the shape of the signal envelope.
· The maximum ASF value is determined by the signal energy, which in turn is directly proportional to the amplitude and duration of the signal.
The spectrum width is 
where the frequency deviation and does not depend on the signal duration.
· Signal base (bandwidth factor) can be n>>1. Therefore, chirp signals are called broadband.
FCM radio pulses with a duration are a set of elementary radio pulses following each other without intervals, the duration of each of them is the same and equal to . The amplitudes and frequencies of the elementary pulses are the same, but the initial phases may differ by (or some other value). The law (code) of alternation of initial phases is determined by the purpose of the signal. For FCM radio pulses used in radar, appropriate codes have been developed, for example:
1, +1, -1 - three-element codes
- two variants of four-element code
1 +1 +1, -1, -1, +1, -2 - seven-element code
The spectral density of coded pulses is determined using the additivity property of Fourier transforms, in the form of the sum of the spectral densities of elementary radio pulses.
Currently remain relevant in radar, the task is resolution, and in information transmission systems, the task is to distinguish signals.
To solve these problems, one can use FCM signals encoded by ensembles of orthogonal functions, which, as is known, have zero cross-correlation.
To resolve signals in radar, you can use a burst signal, each pulse of which is encoded by one of the rows of an orthogonal matrix, for example, the Vilenkin-Chrestenson or Walsh-Hadamard matrix. These signals have good correlation characteristics, which allows them to be used for the above-mentioned tasks. To distinguish between signals in data transmission systems, you can use the same signal with a duty cycle equal to one.
The Vilenkin-Chrestenson matrix can be used to form a polyphase ( p-phase) FCM signal, and the Walsh-Hadamard matrix, as a special case of the Vilenkin-Chrestenson matrix for the number of phases equal to two, to form a biphasic signal.
Polyphase signals are known to have high noise immunity, structural secrecy and a relatively low level of side lobes of the autocorrelation function. However, to process such signals, it is necessary to spend a greater number of algebraic addition and multiplication operations due to the presence of real and imaginary parts of the signal samples, which leads to an increase in processing time.
Discrimination and resolution challenges can be exacerbated by the a priori unknown Doppler shift of the carrier frequency due to the relative motion of the source and subscriber or radar and target, which also complicates real-time signal processing due to the presence of additional Doppler processing channels.
To process the above-mentioned signals having a Doppler frequency addition, it is proposed to use a device that consists of an input register, a discrete conversion processor, a cross-connection unit and a set of identical ACF signal generation units, which are sequentially connected shift registers.
If we take the orthogonal Vilenkin-Chrestenson matrix as a basis matrix for processing a polyphase burst signal, then the discrete transformation will turn into a discrete Vilenkin-Chrestenson-Fourier transform.
Because Since the Vilenkin-Chrestenson matrix can be factorized using the Goode algorithm, the discrete Vilenkin-Chrestenson-Fourier transform can be reduced to the fast Vilenkin-Chrestenson-Fourier transform.
If we take the orthogonal Walsh-Hadamard matrix as a basis matrix - a special case of the Vilenkin-Chrestenson matrix for processing a biphasic burst signal, then the discrete transformation will turn into a discrete Walsh-Fourier transform, which by factorization can be reduced to the fast Walsh-Fourier transform.
