Optimization of the Structure of the Periphery of a Perception System
Abstract Generated abstract
The paper formulates an optimization problem for the peripheral structure of a perception system, seeking to minimize signal transformation error at fixed system complexity. It models the measuring block as parallel links governed by an operator parameter, relates link complexity to dynamic range and reliability, and derives the expected error under independent failures together with noise related measurement error. Using estimates drawn from the human auditory system, the analysis suggests that the optimal value of the structural parameter lies between 1 and 2, supporting the proposed operator model as a plausible description of transformations at the auditory periphery.
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OPTIMIZATION OF THE STRUCTURE OF THE PERIPHERY OF THE PERCEPTION SYSTEM
V. P. BONDARENKO, V. M. RAZIN
The optimality condition for the structure of the periphery of the perception system can be written in the form [1]:
\[ \min \delta,\quad \text{for } S=\text{const}, \]
where \(\delta\) is the error of signal transformation at this level,
\(S\) is the complexity of the given level of the system.
The operator \(L_2\), which describes the transformation of the signal at this level, is characterized by the parameter \(\alpha\) [2], which makes it possible to reduce the given problem to the search for the optimal value of \(\alpha\). By the construction of the operator \(L_2\), the measuring block is divided into \(l\) parallel-operating links. The energy threshold of any measuring device is equal to [3]
\[ C=\gamma^2Pt, \]
where \(\gamma^2\) is the square of the error,
\(P\) is the power consumed from the object of measurement,
\(t\) is the measurement time.
The quantity \(\tau=\gamma^2 t\) is usually constant [3]. Assuming that, in the process of measurement, the signal energy is used completely, one may write [3]
\[ D_i=\ln \frac{E_i}{C_i}=-\ln \tau = D_1, \tag{1} \]
where \(D_i\) is the required dynamic range of the \(i\)-th measuring link,
\(E_i\) is the energy of the \(i\)-th component of the input signal [2].
On the basis of (1), one may conclude [1, 3] that
\[ S_i=S_j=S_1, \]
where \(S_i\) is the complexity of the \(i\)-th measuring link.
The total complexity of the block is equal to [1, 2]
\[ S=\sum_{i=1}^{l} S_i = lS_1, \tag{2} \]
where
\[ l=\frac{B}{\ln \alpha},\qquad B=\ln \omega_{\mathrm{в}}-\ln \omega_{\mathrm{н}}. \]
Suppose that the distribution laws of the moments of failure of the measuring elements are identical and equal to
\[ a(t)=\frac{1}{T_0}\exp\left[-\frac{t}{T_0}\right], \]
where \(T_0\) is the mean duration of failure-free operation of an element, which is inversely proportional to its complexity,
\[ T_0=\frac{K_1}{S_1}=\frac{BK_1}{S\ln \alpha}, \tag{3} \]
where \(K_1\) is the proportionality coefficient.
Let us estimate the efficiency of the measuring block over a time interval \(t_1 \ll T_0\), which substantially simplifies the solution without disturbing the general picture [4]. Then the probability of failure of an element during the time \([0,t_1]\), taking (3) into account, is equal to
\[ g_1=1-P_0^{\ln \alpha}=1-P_1, \tag{4} \]
where \(P_1\) is the probability of failure-free operation of an element during time \(t_1\),
\[ P_0=\exp\left[-\frac{t_1S}{K_1B}\right]. \]
For such a system, under independence of failures, the mean error will be equal to [5]
\[ \delta_1=H_0\delta_0+\sum_{i=1}^{l}\delta_iH_i+\sum_{\substack{j,i=1\\ j\ne i}}^{l}\delta_{i,j}H_{i,j}+\ldots+H_l\delta_l, \tag{5} \]
where \(H_0, H_{i,j,\ldots}\) are the probabilities of each of the possible states of the measuring block, and \(\delta_{i,j}\) is the error arising from the simultaneous failure of the \(i\)-th and \(j\)-th measuring elements.
Taking (4) into account, one may write
\[ H_n=H_{i,j,\ldots,k}=g_1^nP_1^{\,l-n}. \]
Consider the expression
\[ \xi_n=\sum_{\substack{i,j,\ldots,k=1\\ i\ne j\ne \ldots \ne k}}^{l}\delta_{i,j,\ldots,k}. \]
The error \(\delta_{i,j,\ldots,k}\) is equal to [2]
\[ \delta_{i,j,\ldots,k}=\int_{0}^{T}(\varphi_i+\ldots+\varphi_k)^2\,dt. \]
The total number of such terms in the sum \(\xi_n\) is equal to
\[ m=C_l^n. \]
Taking this into account, the expression for \(\xi_n\) can be transformed to the form
\[ \xi_n=m\left[ \frac{n}{l}\sum_{i=1}^{l}\langle \varphi_i^2\rangle + \frac{n(n-1)}{l(l-1)} \sum_{i=1}^{l}\sum_{\substack{j=1\\ j\ne i}}^{l} \langle \varphi_i,\varphi_j\rangle \right], \]
where
\[ \langle \varphi_i,\varphi_j\rangle=\int_{0}^{T}\varphi_i\varphi_j\,dt. \]
Substituting the obtained expressions for \(\xi_n\) and \(H_n\) into formula (5), we have
\[ \delta_1=\sum_{n=0}^{l} p_n\left[ \frac{n}{l}\sum_{i=1}^{l}\langle \varphi_i^2\rangle+ \frac{n(n-1)}{l(l-1)} \sum_{i=1}^{l}\sum_{\substack{j=1\\ j\ne i}}^{l} \langle \varphi_i,\varphi_j\rangle \right], \]
where \(p_n=C_l^n g_1^n p_1^{\,l-n}\) is the probability of simultaneous failure of any \(n\) measuring links.
Transforming the expression for \(\delta_1\), we obtain
\[ \delta_1=g_1p_1\sum_{i=1}^{l}\langle \varphi_i^2\rangle+ g_1^2\sum_{i=1}^{l}\sum_{j=1}^{l}\langle \varphi_i,\varphi_j\rangle . \tag{6} \]
Let us replace, in expression (6), the components of the sum by their upper bounds:
\[ \sum_{i=1}^{l}\sum_{j=1}^{l}\langle \varphi_i,\varphi_j\rangle=E, \]
where \(E\) is the upper bound of the signal energy [2].
\[ \langle \varphi_i^2\rangle=\int_{-\infty}^{\infty}Y_i(j\omega)Y_i^*(j\omega)\,d\omega, \]
where \(Y_i(j\omega)\) is the transfer coefficient of the \(i\)-th filter [2], and \(Y_i^*(j\omega)\) is the complex-conjugate function to \(Y_i(j\omega)\).
After substituting the value of \(Y(j\omega)\), integrating, and transforming, expression (6) is written in the form
\[ \xi_1=\frac{\delta_1}{E} =p_0^{\ln \alpha}\left(1-p_0^{\ln \alpha}\right)A+ \left(1-p_0^{\ln \alpha}\right)^2, \]
where \(\xi_1\) is the relative error.
In addition to the error \(\delta_1\), there will also be an error \(\delta_2\) associated with the finite accuracy of measurement by one link. Suppose that the error \(\delta_2\) is determined mainly by the intrinsic noise of the measuring link and the filter; then
\[ \delta_2=lE_{\text{n}}, \]
where \(E_{\text{n}}\) is the noise energy of one channel.
Then the total relative error will be equal to
\[ \xi=\xi_1+l\frac{E_{\text{n}}}{E} =\xi_1+\frac{BE_{\text{n}}}{\ln \alpha\,E}. \]
For the analysis of \(\xi\), let us estimate \(\frac{E_{\text{n}}}{E}\) and \(B\) using data for the human auditory system [6]. If \(E_{\text{n}}\) is assumed to correspond to the threshold of audibility, then the ratio is
\[ \frac{E_{\text{n}}}{E}\approx 10^{-7}, \]
and
\[ B=\ln \omega_{\text{в}}-\ln \omega_{\text{н}}\approx 10. \]
Thus, approximately one may write
\[ \xi=\xi_1+\frac{10^{-6}}{\ln \alpha}. \]
Figure 1 presents the dependences of $\xi_1$ on $\alpha$ for different values of $P_0$. The second term will make a substantial change in the value of $\xi$ only for $\alpha$ sufficiently close to unity, driving the value of $\xi$ to infinity as $\alpha \to 1$. Consequently, it may be concluded that the value of $\alpha$ corresponding to the optimal structure of the periphery of the perception system lies within the limits
\[ 1 < \alpha < 2. \]
Fig. 1
Thus, the operator $L_2$ proves to be completely determined and corresponds sufficiently well to the signal transformations at the periphery of the human auditory organ.
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