Features of the Technical Implementation of the Autocorrelation Principle

Yu. K. Petrov

(Presented by the scientific seminar of the Department of Mathematical and Computing Instruments and Devices)

The autocorrelation principle is widely used for detecting hidden periodicity in mathematical investigations.

The autocorrelation principle can be used to analyze the feedback signal from an automatic-control object. Suppose it is known that one of the controlled parameters \(q\) of the system produces a response of the system that has period \(T_0\) and appears in the complex feedback signal \(U(t)\), which, along with the periodicity \(T_0\), contains several other components. In this case, to control the parameter \(q\), it is necessary to estimate the weight of the periodicity in the complex signal \(U(t)\). Using the autocorrelation principle, we define the integral

\[ I=\int_{0}^{nT_0} U_0(t)\,U_0(t+T_0)\,dt . \tag{1} \]

The value of the integral \(I\) can serve as the control signal for the parameter \(q\). Automatic computation of relation [1] is easily implemented with standard analog means of computing technology (Fig. 1).

Since such complex computing devices as a constant-delay unit and a multiplication unit are used to compute the correlation function, the need arises to design and manufacture simpler correlators.

Fig. 1. \(\alpha\)—constant-delay unit; \(\beta\)—multiplication unit; \(\gamma\)—integrating unit

of delay and multiplication unit, there arises the need to design and manufacture simpler correlators.

The autocorrelation problem is especially simplified when the feedback signal has a pulsed character. In this case the characteristics of a periodic sequence are only the period and the phase shift between two periodic pulse sequences with the same period. A pulse autocorrelator is shown in Fig. 2, where the delay line is some electrotechnical device. Depending on the magnitude of \(T_0\), this may be

Fig. 2

a long line, a magnetostrictive delay line, or a device for recording and reading from magnetic tape, etc. The delay line delays all pulses of the original sequence by a period slightly smaller than \(T_0\). The pulses from the output of the delay line control the pulse-potential coincidence circuit \(H\). Pulses from the coincidence circuit enter the input of the summing device \(\Sigma\). The threshold element \(\Sigma\) operates in a pulsed mode in such a way that after each time interval \(nT_0\) the information in it is erased and it is reset to its initial state. If the number of pulses exceeds some threshold value \(m<n\), then a signal \(I\) arises, which can be used to control the parameter \(q\). The threshold element is needed to improve the noise immunity of the correlator, and the numerical value \(n\) depends on the level of interference in the original signal \(U(t)\).

The device in Fig. 2 is considerably simpler than the device in Fig. 1. However, both devices prove inapplicable in certain specific situations. Suppose that the character of the controlling action must change when, in the feedback signal \(U(t)\), there is present, instead of one sequence \(T_0\), two sequences with period \(T_0\) that are shifted in phase. In this case it is necessary to have a correlator that distinguishes two periodic sequences which have the same period and differ only in phase. Such a problem cannot be solved by means of the circuits in Figs. 1 and 2.

Let us pose the question: why do the circuits in Figs. 1 and 2 not distinguish phase shifts of pulse sequences? The reason is that the described circuits use delay devices whose operation does not depend on the phase shifts of the original sequence: they delay all pulses of the original sequence independently of phase shifts. We shall call a delay device of this kind open. The input of this device is always open, regardless of whether it is delaying some other pulse of the original sequence or not.

In order to understand how the circuit in Fig. 2 should be modified so that it becomes phase-sensitive, let us consider the operating principle of a certain phase-sensitive correlator. In the circuit of Fig. 2, let us replace the open delay device by a closed one: that is, one such that during the active state, in which a pulse is being delayed, its input is closed. Such a device may be an ordinary trigger relay. Take \(N\) trigger relays with relaxation period \(T_0\) and assemble \(N\) circuits of Fig. 2. For this we shall need the corresponding number of coincidence circuits \(H\) and threshold elements \(\Sigma\). Connect all the circuits in parallel, and place at the input of the device a pulse distributor, which

which will direct the first pulse and all subsequent pulses to the input of the first delay, the second pulse and all subsequent pulses to the input of the second delay, etc. In such a circuit a signal will arise at the outputs of those circuits whose flip-flop relays are triggered by pulses belonging to periodic sequences with period \(T_0\). If the original periodic sequence contains several periodicities \(f_0\) with period \(T_0\), shifted in phase, then the signal \(I\) will arise at several outputs.

A correlator constructed in this way separates periodic sequences by phase. However, the number \(N\) must exceed the maximum number of interference pulses arising in the original sequence between two pulses of the periodic sequence; \(f_0\) in the general case may be very large. Therefore the technical implementation of such a circuit proves impossible. Nevertheless, the ideas developed in the preceding paragraph indicate a way to modify the circuit of Fig. 2 in such a direction that it becomes phase-sensitive. The open delay device in the circuit of Fig. 2 should be made closed, and the start of operation of the delay line should be phased to the moment of arrival of one of the pulses belonging to the sequence \(f_0\).

The corresponding device, which can easily be implemented in practice, is shown in Fig. 3. The circuit of Fig. 2 is supplemented by a control device \(UU\) and a gate device \(V\). The gate device is normally open. The first pulse from the coincidence circuit starts the control device, which erases all information in the delay line and changes the operating mode of the gate device. Now the gate device opens only for a short interval of time, synchronously with circuit \(I\), at the moment when pulses of the sequence \(f_0\) arrive at the input of the delay.

Fig. 3

If it is necessary to use as the controlling signal some correlation \(I\) from a second periodic sequence, it is necessary to have two copies of the devices of Fig. 3. Both devices are connected in series, with the original sequence \(U(t)\), from which one of the sequences \(f_0\) has been excluded, being fed to the input of the subsequent device. The operation of excluding the sequence \(f_0\) can be carried out with the aid of the inhibit circuit \(Z\), as shown in Fig. 3.

In the described instrument only such delay devices can be used in which information can be erased by an external signal. Such devices may include all kinds of shift registers based on triggers, ferrite cores, magnetic films, tunnel diodes, etc.