On the Separation of an Additive Mixture of Periodic Pulse Sequences
Abstract Generated abstract
The paper proposes a method for separating an additive mixture of periodic pulse sequences, motivated by limitations of correlation-based radar signal separation such as long processing time, sensitivity to short sequences, and poor resolution for close frequencies. The method converts the one-dimensional pulse train into a two-dimensional representation by dividing the time axis into equal intervals and stacking them, so that pulses belonging to the same periodic component form straight-line segments whose angle, length, and position determine the period, number of pulses, and initial phase. The paper derives conditions for choosing the stacking interval from the known minimum and maximum possible periods, gives a practical angular range for reliable operation, and illustrates the approach on a mixture of six periodic components with interference pulses. It argues that repeated application over different period ranges can recover components with arbitrary periods.
Full Text
On the Problem of Separating an Additive Mixture of Periodic Pulse Sequences
B. N. Epifantsev
(Presented by the scientific seminar of the Scientific Research Institute of Electrical Engineering)
As is known, the correlation method is taken as the basis for separating signals received simultaneously from several radar stations. Its application is associated with carrying out a number of successive procedures: obtaining the autocorrelation function of the input signal; determining the delay interval (the period of the maximum frequency) at which the autocorrelation function has a maximum; extracting the pulses of the sequence thus found, with their subsequent subtraction from the mixture; and carrying out analogous operations on the transformed signal until all components have been completely separated. The obvious shortcomings of this method are the following.
- A long separation time, which increases when random pulse interference is present in the total signal.
- The separation time depends on the number of periodic components in the mixture.
- The probability of correct extraction depends on the number of pulses of the component (the operating time of the transmitting station), and when their number is small it reaches a practically unacceptable value.
- When close or identical frequencies are present in the mixture, “target skipping” occurs.
- Hardware implementations of the method are cumbersome and unreliable.
In another known method, characterized by converting the time interval between pulses into an amplitude, with subsequent amplitude selection, the indicated shortcomings are inherent to an even greater degree.
It is possible to propose yet another variant for solving the stated problem, based on the artificial conversion of a one-dimensional signal into a two-dimensional one. The essence of this variant is set forth below.
Let us turn to Fig. 1, в, which presents an image of two segments of straight lines, denoted by the indices I and II, in the \(XOY\) plane. With line-by-line scanning of this image with period \(T p_{px}\) and step \(\delta_y\), from left to right, we obtain the sum of two periodic pulse sequences, each of which completely characterizes a straight line in the plane.
Indeed, when the scanning element, with an aperture commensurate with the width of the line, passes in the \(H_3K_3\) plane (Fig. 1, в), a change in density will be recorded at point 1, which will repeat at point 2 after a time interval \(T_\alpha\).
Fig. 1. Illustration of the method for separating periodic pulse sequences.
\[ T_{\alpha}=T_{1K_3}+T_{K_3H_4}+T_{H_4 2}, \]
where \(T_{1K_3}\) is the time for the scanning element to pass from point 1 to point \(K_3\); \(T_{K_3H_4}\) is the time of the return stroke of the beam \(T_{o x}\); \(T_{H_4 2}\) is the time for the scanning element to pass from the beginning of the scan in the plane \(H_4K_4\) to point 2.
It is also obvious that
\[ T_{\alpha}=T_{p x}+T_{o x}-\Delta t . \tag{1} \]
The third change in density will be repeated at point 3 after a time determined by formula (1), etc.
Thus, there is a periodic change in density; the period is a function of the line-scan time and of the angle of inclination of the line \(\alpha\). For \(\alpha=90^\circ\) (the case in which the straight line II is located),
\[ T_{\alpha}=T_{p x}+T_{o x}. \]
For \(\alpha>90^\circ\)
\[ T_{\alpha}=T_{p x}+T_{o x}+\Delta t . \tag{2} \]
And in general form
\[ T_{\alpha}=T_{p x}+T_{o x}-\frac{\delta_y}{\operatorname{tg}\alpha}. \tag{3} \]
The simple considerations presented make it possible to draw the first generalization.
The sum of segments of periodic pulse sequences with unknown periods and random initial phases is equivalent to an image of a set of differently oriented straight-line segments in the plane \(XOy\); moreover, the period of any term of this sum uniquely determines the angle of inclination of the corresponding straight-line segment, the number of pulses determines its length, and the initial phase determines the location of this line. Consequently, in solving the problem of separating an additive mixture of periodic pulse sequences and determining their parameters, one should mark the positions of the pulses on the axis of the argument, divide this axis into equal intervals, place each subsequent interval beneath the preceding one, and determine the sets of marks lying on straight lines. Then one should measure the parameters of the straight lines.
As for the magnitude of the interval into which the argument is divided, it can be found from the following proposition.
The problem under consideration assumes that the maximum and minimum possible periods in the mixture are known. Then from (1) and (2) it follows that
\[ \begin{aligned} T_{\min}&=t_p-\Delta t,\\ T_{\max}&=t_p+\Delta t, \end{aligned} \tag{4} \]
where
\[ t_p=T_{p x}+T_{o x}. \]
Solving (4) with respect to \(t_p\), we obtain
\[ t_p=\frac{T_{\max}+T_{\min}}{2}. \tag{5} \]
This is the required value of the scan period, for which, over the range of periods \(T_{\min}<T_{\alpha}<T_{\max}\), the straight-line segment will be formed by the set of marks lying on each neighboring interval (but not skipping one). This condition excludes the possibility of drawing false straight-line segments and makes it possible to find the range of period values \(T_{\max}\) and \(T_{\min}\) for which the proposed method is valid under a single application.
Indeed, for
\[ k\pi < z \leq \frac{2k+1}{2}\pi \qquad (k=0,1,2,\ldots), \]
\[ 0<T_\alpha \leq t_p, \]
for
\[ \frac{(2k+1)\pi}{2}<\alpha<k\pi \qquad (k=1,2,3,\ldots), \]
\[ t_p \leq T_\alpha < 2t_p, \]
i.e.,
\[ 0<T_\alpha<2t_p. \tag{6} \]
In practice condition (6) should be made more stringent, namely, the consideration should be restricted to sets of marks with angles \(45^\circ \leq z \leq 135^\circ\), i.e., taking into account (3) and (4),
\[ (t_p-\delta_y)\leq T_\alpha \leq (t_p+\delta_y). \tag{7} \]
Let us illustrate the application of the method by an example. In Fig. 1a there is shown an additive mixture, normalized in amplitude and duration, of six segments of periodic pulse sequences with three interference pulses, beginning from an arbitrarily chosen point \(H_0\). The axis of the argument is divided into intervals \(t_p\), which are then placed one under another with the marking of the locations of the pulses (Fig. 1b). It is not difficult, looking at such a picture, to say that the original mixture contains 6 components and three interference pulses. Moreover, three components have a period equal to \(t_p\), two have a period smaller than \(t_p\), and one has a period larger than \(t_p\). With the aid of a protractor, for example, one can determine the values of all the periods with sufficient accuracy. If the angle can be measured with an accuracy of up to half a degree, then the described method makes it possible, taking (7) into account, to separate 180 components differing in period, i.e., it has a resolving power of slightly more than half a percent. Measuring the length and position of the line segments, we obtain exhaustive information about the sequence in Fig. 1a.
With repeated application of the method (separation of periodicities in the range \((t_{p1}-\delta_y)\leq T_\alpha \leq (t_{p1}+\delta_y)\), their subtraction from the original sequence, subsequent separation of components with periods \((t_{p2}-\delta_y)\leq T_\alpha \leq (t_{p2}+\delta_y)\), etc.), which is characteristic of prototypes, one can obtain exhaustive information about the components of a mixture with any possible periods.