Determination of Design Parameters of Analog and Optical Correlators

V. P. Ivanchenkov, A. M. Kuvshinov

(Presented by the seminar of the LVTA NIIYaF)

Recently, in the computation of correlation functions, optical correlators have been finding wide application. The advantages of optical correlators in comparison with other variants of analog and discrete-type correlators are high speed of operation, the two-dimensional nature of the analysis performed, and the simplicity of implementing multichannel information processing.

In [1], the methodological errors of multiplicative and interference correlators with time integration were studied in detail. Estimation of the error already at the stage of choosing a specific circuit and designing a correlator makes it possible to calculate some of its most important design parameters. The consideration of such a possibility for optical analog correlators is the subject of the present article. The main design parameters of optical correlators may include: the integration interval \(D\), the drawing speed of the optical transparencies \(V\), and the dynamic range of the device.

A sufficiently general scheme of an analog-type optical correlator is shown in Fig. 1. We shall assume that the task of the optical correlator in this case consists in obtaining functions of the form:

\[ R_{XY}(\Delta x,\Delta y)=a\int_{0}^{D_y}\int_{0}^{D_x} f_1(x,y)\, f_2(x+\Delta x, y+\Delta y)\,dxdy, \tag{1} \]

where \(f_1(x,y)\) and \(f_2(x+\Delta x, y+\Delta y)\) are realizations of stationary random functions describing the transparency of two transparencies; \(\Delta x\), \(\Delta y\) are linear displacements along the axes \(XX\) and \(YY\) of one realization relative to the other;

\(D_x, D_y\) are the spatial integration intervals;

\(a\) is a constant taking into account scale-related instrumental coefficients.

In the case of using the coordinate \(YY\) to carry out multichannel processing, expression (1) takes the form:

\[ R_X(\Delta x)=a_1\int_{0}^{D_x} f_1(x)f_2(x+\Delta x)\,dx. \tag{2} \]

Most often, signal realizations are specified in the form of recordings on photographic films, whose transparency varies according to \(f_1(x)\) and \(f_2(x)\).

A fundamentally unavoidable correlation error is the error arising from the finiteness of the averaging interval \(D\) in expression (1). In the case of integration with respect to time, expressions for calculating these errors are given in [1]. For noncoherent optical correlators, in the calculation it is necessary to take into account the effect of an additional error arising from the presence of a constant component when bipolar signals are recorded on an optical transparency. The expressions for the normalized variance of the correlation function \(\sigma_{\rho XY}^{2}\) of noncentered random processes having time \(t\) as their argument can be written as

\[ \sigma_{\rho XY}^{2}\leq \frac{2\rho_h(0)}{H^2(0)} \left[ \left(\frac{m_x}{\sigma_x}\right)^2\theta_Y+ \left(\frac{m_Y}{\sigma_Y}\right)^2\theta_x+ 2\frac{m_xm_Y}{\sigma_x\sigma_Y}\theta_{XY}+ \sqrt{\theta_{X_{\mathrm{kv}}}\theta_{Y_{\mathrm{kv}}}+\theta_{XY_{\mathrm{kv}}}} \right], \tag{3} \]

where \(\sigma_{\rho XY}^{2}\) is the normalized variance of the correlator output signal;

\[ \rho_h \]
is the autocorrelation function of the integrator;

\[ H(\omega) \]
is the transfer function of the integrator;

\[ \theta_X,\ \theta_Y \]
are the absolute correlation intervals;

\[ \theta_{X_{\mathrm{kv}}},\ \theta_{Y_{\mathrm{kv}}} \]
are the quadratic correlation intervals;

\[ \theta_{XY_{\mathrm{kv}}} \]
is the quadratic interval of mutual correlation;

\[ m_X \text{ and } m_Y \]
are the mathematical expectations;

Fig. 1. Diagram of an optical correlator
1 — radiation source, 2, 6 — condenser, 3 — diaphragm, 4, 5 — optical transparency, 7 — radiation receiver

\[ \sigma_X \text{ and } \sigma_Y \]
are the root-mean-square deviations.

For an ideal time integrator [1]:

\[ \frac{\rho_h(0)}{H^2(0)}=\frac{1}{T}, \tag{4} \]

where \(T\) is the time interval of integration.

When correlation functions are computed by means of an optical correlator, a good approximation close to ideal integration can be obtained. Then expression (3), for the case of computing the autocorrelation function, can be written in the form

\[ \sigma_{\rho X}\leq 2\sqrt{\frac{Q}{D}\left[2\left(\frac{m_X}{\sigma_X}\right)^2\theta_X+\theta_{X_{\mathrm{kv}}}\right]}, \tag{5} \]

where \(D=QT\);

\(Q\) is the speed of recording the signal on the optical transparency.

For a specified accuracy of computing the correlation function and speed of recording the signal on the optical transparency \(Q\), the value \(D\) can be determined from expression (5):

\[ D \simeq \frac{4Q}{\sigma_{pX}^{2}}\left[2\left(\frac{m_X}{\sigma_X}\right)^2\theta_X+\theta_{X_{\mathrm{kv}}}\right]. \tag{6} \]

The quantities \(\theta_X\) and \(\theta_{X_{\mathrm{kv}}}\) in expression (6) are calculated by the formulas:

\[ \theta_X=\frac{1}{2}\int_{-\infty}^{+\infty}\left|\rho_X(\Delta x)\right|dx; \]

\[ \theta_{X_{\mathrm{kv}}}=\frac{1}{2}\int_{-\infty}^{+\infty}\rho_X^2(\Delta x)dx, \tag{7} \]

where \(\rho_X(\Delta x)\) is the normalized correlation function.

In analysis, the correlation intervals \(\theta_X\) and \(\theta_{X_{\mathrm{kv}}}\) can be easily calculated from plots of the correlation functions.

With uniform relative displacement of two transparencies with the functions \(f_1(x)\) and \(f_2(x)\) recorded on them, the lower bound of the signal spectrum at the correlator output is determined as

\[ f_{\min}D=V, \tag{8} \]

where \(V\) is the speed of relative displacement of the transparencies. Given the value \(f_{\min}\), which determines the frequency resolution of signals in correlation analysis, and having calculated \(D\) by formula (6), one can find the value of the speed \(V\) from (8).

The maximum possible modulation frequency of the light flux (the maximum frequency of the input signal) that the correlator can process without substantial distortions is determined by the time constant of the photodetector:

\[ f_{\max}=\frac{1}{\tau_n}. \tag{9} \]

The required dynamic range of the correlator is determined by the maximum nonuniformity of the spectrum of the signal being analyzed. At the same time, the maximum nonuniformity of the signal spectrum limits a certain noise level that appears when recording the signal on the optical transparency. The noise is characterized by the variance \(\langle n^2\rangle\). In the case where photographic film is used as the optical transparency during recording, the noise may be considered uniformly distributed over all frequencies. Then the nonuniformity of the spectrum is determined by the ratio of the maximum response of the correlator to the signal to the response to the noise [2]:

\[ B=10\lg\frac{A}{\langle n^2\rangle/M}, \tag{10} \]

where \(A\) is the maximum amplitude of the signal at the correlator output;

\(M\) is the relative width of the signal spectrum.

The dynamic range of the correlator must be no less than that determined by formula (10); otherwise the result of measuring the correlation function will be distorted.

The dynamic range of the correlator is determined by the ratio of the maximum signal to the level of its own noise. It should be noted that the actual dynamic range of the correlator will have different values for different frequency components of the signal. In this case, however, it should be taken into account that when the pulling speed of the transparencies \(V\) is changed, the quantities \(f_{\max}\) and \(f_{m}\) of the input signal are shifted, but their ratio \(M\), and consequently the magnitude of the required dynamic range of the correlator, remains constant.

Thus, using expressions (6), (8), and (10), it is possible, in the calculation, to estimate the averaging interval, the pulling speed of the transparencies, and the required dynamic range of the correlator.

REFERENCES

1. G. A. Ball. Hardware Correlation Analysis of Random Processes. Moscow, “Energiya,” 1968.

2. V. A. Zverev, E. F. Orlov. Optical Analyzers. Moscow, “Sovetskoe radio,” 1971.