Use of an Analog Computer to Determine the Spectral Density of a Stationary Random Process
Abstract Generated abstract
This paper proposes a method for rapidly estimating the spectral density of a stationary random process using only an analog computer. The approach treats the computer model as a tunable conservative second-order element, squares and integrates its output, and relates the resulting variance growth to the input spectral density at the element’s natural frequency. By varying this frequency, the spectral density can be estimated over a range of interest, particularly for infralow-frequency signals modeled on analog computers. Experiments on an MNB-1 analog computer reportedly confirmed the method, giving usable estimates up to 5 Hz with an accuracy of about 20 to 30 percent.
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USE OF AN ANALOG COMPUTER TO DETERMINE THE SPECTRAL DENSITY OF A STATIONARY RANDOM PROCESS
A. M. MALYSHENKO, E. I. GROMAKOV
(Presented by the scientific seminar of the Department of Automation and Telemechanics)
In modeling the motion of dynamic systems under random actions on analog computers (ACs), there arises a need for a rapid (though not sufficiently accurate) determination of the statistical characteristics of the coordinates of interest. In this case, a preferable measurement variant may be one which, apart from the analog computer, requires no other apparatus for this purpose. The method described below satisfies this requirement and makes it possible to determine the statistical characteristics of an infralow-frequency process over a fairly wide range of frequencies.
The statistical characteristics of a stationary random process are determined by the distribution law, the mathematical expectation and the correlation function, or by the spectral density uniquely associated with it [1]. Determining the mathematical expectation presents practically no serious difficulties. At the same time, the technique for determining correlation functions and spectral densities is rather complex and labor-intensive. Special apparatus intended for these purposes is rather expensive and has so far been available only in limited quantities [2].
The operation of a large number of instruments for determining spectral density (spectral analyzers) is based on linear filtering of the signal under study by means of frequency-selective resonator systems and subsequent integration of the square of the resonator output signal [2]. The method described below for determining the spectral density of a random process on an analog computer is based on the same principle.
In order to justify the proposed method, let us describe the dynamics of a conservative element (a second-order oscillatory element without damping) under random disturbances. The dynamics of such an element is described by the equation
\[ \frac{d^{2}y(t)}{dt^{2}}+\nu^{2}y(t)=x(t), \tag{1} \]
where \(x\), \(y\) are, respectively, the input and output signals;
\(\nu\) is the natural frequency of oscillations.
If we assume that \(x(t)\) is a stationary random process with zero mathematical expectation, i.e.
\[ m_x(t)\equiv 0, \]
then the variance of the output signal \(\sigma_y^2\) is determined, according to [3], by the formula
\[ \sigma_y^2=\frac{1}{\nu^2}\left[\pi S_x(\nu)\cdot t-a_0+a_1\cos 2\nu t-b_1\sin 2\nu t\right], \tag{2} \]
where
\[ a_0=\int_0^\infty\left(\tau\cos\nu\tau+\frac{1}{\nu}\sin\nu\tau\right)R_x(\tau)\,d\tau; \]
\[ a_1=\frac{1}{\nu}\int_0^\infty \sin\nu\tau\cdot R_x(\tau)\,d\tau; \]
\[ b_1=\frac{1}{\nu}\int_0^\infty \cos\nu\tau\cdot R_x(\tau)\,d\tau, \]
\(R_x(\tau)\) is the autocorrelation function of the input signal \(x(t)\).
After the lapse of a time interval during which \(R_x(\tau)\) becomes a small quantity, it may be considered, with a sufficient degree of accuracy, that
\[ \sigma_y^2\simeq \frac{\pi}{\nu^2}S_x(\nu)\cdot t, \tag{3} \]
i.e., the variance of the output signal becomes proportional to the time and to the value of the spectral density of the input signal at the natural-oscillation frequency of the conservative element.
It follows from (3) that
\[ S_x(\nu)\simeq \frac{\nu^2}{\pi\cdot t}\sigma_y^2. \tag{4} \]
By virtue of the stationarity of the input signal and the linearity of the conservative element,
\[ \sigma_y^2=\lim_{T\to\infty}\frac{1}{T}\int_0^T y^2(t)\cdot dt, \tag{5} \]
therefore the spectral density of interest to us is
\[ S_x(\nu)\simeq \frac{\nu^2}{\pi T^2}\int_0^T y^2(t)\cdot dt. \tag{6} \]
To increase the accuracy of determining the spectral density by formula (6), it is necessary to increase the measurement interval \(T\).
According to (6), in order to determine the spectral density of a random signal it is necessary to perform the following operations: pass this signal through a conservative element, square the output signal of the element, and then integrate with respect to time. The necessary elements and their functional connection are shown in Fig. 1.
By changing the natural-oscillation frequency of the element and each time determining, from the output signal of the integrator, the value of the spectral density of the input signal at the natural-oscillation frequency of the conservative element, one can determine the desired spectral density \(S_x(\omega)\) over the entire frequency range of interest to us.
Implementation of the indicated method on an analog computer leads to the setup diagram presented in Fig. 2.
Here, on the first, second, and third operational amplifiers a conservative element with tunable natural frequency is implemented. The fourth amplifier serves for the necessary scaling. The division and multiplication unit (DMU) operates in multiplication mode and performs the function of a squarer. The fifth amplifier is used in integration mode.
Fig. 1.
To determine \(S_x(\nu)\) using the model shown in Fig. 2, it is necessary, after a sufficiently long interval of time \(T\) has elapsed (within \(1\text{--}2\) min) after starting the model, to stop the solution and measure the time \(T\) and the output signal of the model using the analog computer’s measuring instrument or the electronic indicator included with it. After this, to determine \(S_x(\nu)\), it is sufficient to reduce the model output signal by a factor of \(T^2\).
Fig. 2.
The indicated procedure is not limited only to the implementation, presented here, of a conservative element on the operational amplifiers of an analog computer. However, in determining the statistical characteristics of infralow-frequency stationary signals reproduced on an analog computer, the indicated method has quite significant advantages.
The possibility of determining the spectral density by the described method was confirmed experimentally by the authors on an MNB-1 analog computer. The output signal of the model had a clearly expressed parabolic dependence on time and made it possible to compute \(S_x(\omega)\) within the operating frequency range of the MNB-1 (up to \(5\) Hz) with an accuracy on the order of \(20\text{--}30\%\).
REFERENCES
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