Variable phase and frequency pulse-width modulation technique
Phased array systems rely on the production of an exact carrier frequency to function. Reconstructing digital signals by specified amplitude and phase is accomplished explicitly by inducing frequency shifts away from a base frequency implied by phase changes. Shifting the carrier frequency of a digitally controlled phased array while preserving the timing of the individual phase pulses enables more efficient driving of the phased array system when the phase of the drive signals change dynamically in time.
1. A system comprising:
an output of an electrical signal that is controlled by an interpolation of a polynomial spline evaluated over uniformly spaced samples in real-time;
wherein, within the interpolation, a term of an nth power of an independent variable at an mth timestep within an interval is inferred using identities for an mth n-dimensional triangular number and all mth triangular numbers in lower numbers of dimensions whose values correspond to those producible using recursively defined counters;
wherein a rearrangement rearranges identities into a polynomial that is progressively evaluated using an integer combinations of recursive counters given by the rearrangement;
wherein at least one of the recursive counters is incremented on each timestep by a sample value that is to be interpolated;
wherein the interpolation results in an up-sampling of a function described by values of the uniformly-spaced samples, and increasing a number of the uniformly spaced samples of the function by a power-of-two.
2. The system of claim 1 , wherein the polynomial spline has a plurality of intervals, and wherein at least one of the plurality of intervals is a base frequency period.
3. The system of claim 1 , wherein the polynomial spline has a plurality of intervals, and wherein at least one of the plurality of intervals is a power-of-two count of base frequency periods.
4. The system of claim 1 , wherein the polynomial spline has a plurality of intervals, and wherein at least one of the plurality of intervals is a power-of-two fraction of a base frequency period.
5. The system of claim 1 , wherein the electrical signal has a phase angle, and wherein the phase angle is modified by a rolling counter incremented by a value driven by an external environmental monitoring sensor.
6. The system of claim 1 , wherein the electrical signal has a phase angle, and wherein the phase angle is modified by a rolling counter decremented by a value driven by an external environmental monitoring sensor.
7. A system comprising:
an electrical signal, wherein an instantaneous phase angle of the electrical signal is substantially calculated by a real-time progressive evaluation of a polynomial spline;
wherein, within the evaluation, a term of an nth power of an independent variable at an mth timestep within an interval is inferred using identities for an mth n-dimensional triangular number and all mth triangular numbers in lower numbers of dimensions whose values correspond to those producible using recursively defined counters;
wherein a rearrangement rearranges identities into a polynomial that is progressively evaluated using an integer combinations of recursive counters given by the rearrangement;
wherein at least one of the recursive counters is incremented on each timestep by a sample value that is to be interpolated;
wherein the evaluation results in an up-sampling of a function described by values of uniformly-spaced samples, and increasing a number of the uniformly spaced samples of the function by a power-of-two.
8. The system of claim 7 , wherein the polynomial spline has a plurality of intervals, and wherein at least one of the plurality of intervals is a base frequency period.
9. The system of claim 7 , wherein the polynomial spline has a plurality of intervals, and wherein at least one of the plurality of intervals is a power-of-two count of base frequency periods.
10. The system of claim 7 , wherein the polynomial spline has a plurality of intervals, and wherein at least one of the plurality of intervals is a power-of-two fraction of a base frequency period.
11. The system of claim 7 , wherein the electrical signal has a phase angle, and wherein the phase angle is modified by a rolling counter incremented by a value driven by an external environmental monitoring sensor.
12. The system of claim 7 , wherein the electrical signal has a phase angle, and wherein the phase angle is modified by a rolling counter decremented by a value driven by an external environmental monitoring sensor.