Second-order Sections Research Articles

Canonic realizations of second-order digital filters due to finite precision arithmetic

Extending a procedure first used by Jackson, it is demonstrated that because of quantization after multiplications, practical realizations of second-order digital-filter sections are limited to 24 basic structures with the same transfer function.

Selectivity improvement in a useful second-order active RC section†

Abstract A negative feedback circuit using one operational amplifier has its Q factor enhanced through the introduction of an additional voltage amplifier with positive finite gain in the feedback loop. The new network has two useful outputs. If a twin,T is connected in cascade with the network input terminals the overall circuit is suitable for the realization of biquadratics or third-order low-pass elliptic functions. The network sensitivity is very low.

Theory of Resonant Raman Scattering in Crystals: A Generalized Bare-Exciton Approach

The bare-exciton theory of light scattering in crystals is generalized to include dispersive effects, and to incorporate damping in a first-principles calculation. The theory is applied to exciton-mediated Raman scattering; expressions for the first- and second-order Raman cross sections due to a single discrete exciton level are derived, and the resonance behavior investigated. For small couplings and intermediate dampings the predictions of the present generalized theory are very similar to available bare-exciton and undamped-polariton predictions, while differing, however, for large couplings. The cross sections are resonant when the incoming or outgoing photon frequencies lie near an exciton frequency, and peak sharply for small photon-exciton couplings and dampings. Actual calculations are presented for various choices of the parameters, among them parameters appropriate to CdS. A comparison of the present theory with other theories and with experiment is carried out.

Roundoff-noise analysis for fixed-point digital filters realized in cascade or parallel form

The roundoff-noise outputs from two transpose configurations, each for the cascade and parallel forms of a digital filter, are analyzed for the case of uncorrelated roundoff noise and fixed dynamic range. Corresponding transpose configurations are compared on the basis of the variance, or total average power, and the peak spectral density of the output roundoff noise. In addition to providing general computational techniques to be employed in choosing an appropriate configuration for the digital filter, these results also indicate useful "rules of thumb" relating to this choice of configuration. Included are indications of good (although not necessarily optimum) sequential orderings and pole-zero pairings for the second-order sections comprising the cascade form. Computational results are presented which indicate that the analysis is quite accurate and useful.

FEN filter design using tantalum and silicon integrated circuits

An economical approach to integrated active RC filter design is described. Complex filter networks are broken down into a series of cascadable second-order filter sections consisting of tantalum thin-film RC networks and semiconductor integrated operational amplifiers. Two building blocks are available for any desired frequency within a decade and for any desired filter function (e.g., low-pass, high-pass, band-pass, band-reject, all-pass, etc.). One building block is for low Q realizations and contains one amplifier; the other is for high Q realizations and contains two. The considerable versatility of this approach is obtained by 1) a network synthesis approach based on decomposing a given second-order function into a low Q asymptotic approximation of this function in cascade with an active frequency emphasizing network and 2) by the characteristics of tantalum and silicon integrated circuits.

Q factor, Q stability and gain in active filters

This letter sets out a simple theory relating the maximum Q factor of a second-order active-filter section to the stability of that Q factor and the available gain in the active elements used. The theory is shown to apply to both the ‘negative-feedback’ or ‘Rauch’-type filters and ‘positive-feedback’ or ‘Sallen–Key’-type filters and by implication to negative-impedance convertors (n.i.c.s) and the whole family of circuits based on minimum-phase feedback networks.

Theory of Collision Broadening in the Sudden Approximation

A method is developed for calculating the cross section for line broadening in the limiting case in which important collisions may be considered to occur very rapidly with respect to the periods corresponding to the transitions in question. This method is essentially an extension of the familiar sudden approximation; reference is made throughout to the particular example of optical double resonance only for definiteness and because there are available experimental results for comparison with the theory. Assuming the impact model and considering the case when the energy levels in question are well resolved with respect to their width, an exact expression is found for resonant self-broadening in the usual weak rf field limit of interest in double resonance. For the case of very small oscillator strengths for the optical transition (intercombination lines), the resonant broadening becomes sufficiently small so that second-order, intermediate-state processes dominate, and approximations must be made to effect sums over intermediate states. Here the broadening is of the same sort as foreign gas broadening, and in many cases of interest, the approximations can be made in a sufficiently realistic manner to obtain useful expressions for the broadening. It is shown that the resonant cross section and this second-order cross section are not additive, but that the one which is larger dominates the total cross sections very strongly. Finally, we evaluate explicitly the self-broadening and foreign gas broadening (by the noble gases) of double resonance lines in the group II metals and the self-broadening in the optical spectrum of helium. In all cases our theoretical predictions are in good agreement with measurements which have been made.

Automatic Control by System Variable Adaptation†

ABSTRACT The performance of an automatic control system can be improved by the addition of subsidiary feed-back loops whose characteristics are controlled by signal polarities. This paper describes an analogue computer study that was made of the properties of a system of this type. The system consists of a second-order linear feed-forward section and a feed-back section which contains two subsidiary loops. Signal polarities are used to control the amounts of rate and proportional feed-back that are transmitted by these subsidiary loops. The sensitivity of the system to changes in the damping and natural frequency of the linear feed-forward section was evaluated using sinusoidal and transient inputs. The results show that the addition of these subsidiary loops makes the system very insensitive to changes in the damping. The sensitivity of the system to changes in the natural frequency is not reduced.

Automatic control through signal polarities

This paper describes an analogue computer study that was made of an automatic control system in which signal polarities control the amount of proportional feedback that is used. The system consists of a conventional second-order linear feedforward section, and a subsidiary feedback loop which produces the desired control action. The sensitivity of the system to changes in the damping and bandwidth of the linear feedforward section was determined using sinusoidal and transient inputs. It is shown that the system is insensitive to damping variations provided that the damping is less than about critical. It is, however, quite sensitive to variations in the bandwidth.