Patent GB-1095151-A (GB)
Title: Improvements relating to electrical signal interpolating devices
Inventor: UDALL ANTHONY JOHN SHAWCROSS
IPC: G06G7/30
Filed: 1964-02-07
Abstract: 1,095,151. Interpolating device. ELECTRIC & MUSICAL INDUSTRIES Ltd. Feb. 8, 1965 [Feb. 7, 1964], No. 5248/64. Heading G4G. An electric interpolating device responsive to a series of discrete input electric signal values to produce a smoothly varying output electric signal value performing interpolation of a higher order than linear, includes means for correcting drift of the output signal value and of its first derivative. Fig. 4 shows a quadratic interpolator wherein a tape advancing mechanism 3 responsive to clock pulses at 4 moves a punched tape 1 past a reading head 2 to derive successive discrete sample signals applied to digital/analogue converter 5, whose resultant output multiplied by weighting factor 2 is applied to adder 11 feeding analogue store 7 with a signal value which is changed at the incidence of each clock pulse. A Blumlein integrator 8 operates on the output of the analogue store producing a signal on which a further Blumlein integrator operates to produce the required output at 9, which is fed back to the adder together with the output of the integrator 8; each signal being multiplied by a weighting factor of -2; the respective weightings arising since the rate of change of slope of a parabola is uniform, and the slope at mid-section is that of a chord joining the section end points. The interpolated output signal represents a parabolic curve joining a current sample signal to the next succeeding sample signal with the same initial slope as the terminal slope of the preceding interpolated section, and the analogue store 7 receives an input representing the required rate of change of slope of the next succeeding parabolic section. The parabolic interpolation is thus defined by the initial value of a section, its final value, and its initial slope. In a modification (Fig. 5) further compensation is effected by introducing a further definition of the final slope of a section whereby the interpolation curve is sectionally cubic, for which the following equations are given: where Cr is the constant third derivative, ##ris the section initial second derivative, fr is the section initial value, fr+  1  is the section final value, f r  is the section initial slope, fr +  1  is the section final slope, # is the time interval between samples. The section initial slope and value are equivalent to those finalizing the preceding section, the section final value is derived from the tape record, and the section final slope is calculated from the chord joining the two adjacent sampling points, so that: The tap is read by two heads 2, 13 to derive signals representing f r  +  1  and f r  + 2 which are converted to analogue signals at 5, 14. Signal f r  +  1  is respectively applied to adders 15, 16 with multiplying weighting factors - 12, + 6; and signal f r  +  2  is applied to the adders with respective weighting factors + 3, - 1. The output of 15 energizes analogue store 7 followed by integrators 8, 10, 12 in cascade, and the output of 16 establishes the section initial condition of integrator 8, which is reset together with the analogue store by the leading edge of the clock pulse from 4. The output of integrator 10 is fed back to adders 15, 16 with weighting factors + 6, - 4 and the output of integrator 12 at terminal 9 representing value f r  is fed back to adders 15, 16 with weighting factors + 3, - 5 whereby output signals are produced representing ##r and Cr. The equations 4 and 5 above are thus simulated, and the signals to which the analogue store and integrator 8 are reset at the beginning of each interpolation section satisfy the required outputs from integrators 10, 12 at the end of each section, so that there is a comparison between demanded and achieved final values and final slopes, and cumulative value error and slope error are corrected. In modifications (Fig. 6, not shown), interpolation is effected according to two parabolic curves respectively extending over adjacent halves of a section defined by two adjacent sampling points and utilizing six parameters; of which two maintain second order midsection continuity while four satisfy the boundary conditions of the section; and (Fig. 7, not shown) by interlacing alternate sample points and processed data points to give second order continuity. In a further modification (Fig. 8, not shown) applicable to the above devices, residual drift arising over individual interpolation sections is removed by comparing sample values for adjacent sections to detect section drift, accumulating drift errors, and adding into the interpolator loop. The analogue stores may be of the kind described in Specification 761,853. The Provisional Specification describes several further modifications and includes a mathematical analysis.