Systems and methods for detecting cellular service access requests from terrestrial user equipment at a satellite

1 . A satellite computer system for a satellite, the satellite including at least one antenna configured to send and receive signals to terrestrial user equipment (UE) on a radio access network (RAN) via a plurality of beams, each of the beams having a beam footprint that defines a cell of the RAN, the RAN defining a physical random access channel (PRACH) slot and an access signal format used by the UE to request an uplink allocation from a base station platform implemented on the satellite during the PRACH slot, the access signal format (1) based on a Zadoff-Chu (ZC) sequence having a uth root, and (2) having at least one copy of a preamble having a preamble length, the satellite computer system comprising at least one processor in communication with a memory, the memory storing computer-readable instructions that are executable to cause the at least one processor to perform steps including:

receiving, via the at least one antenna, a signal in the PRACH slot, wherein an orbital motion of the satellite causes one or more access signals, transmitted in the access signal format by the one or more UE and embedded in the received signal, to include a timing offset variation and a frequency offset variation across one or more locations of the one or more UE;

sampling the received signal to obtain sample values, wherein the sample values represent complex values;

detecting, based on the sample values, a coarse time offset of the one or more access signals within the PRACH slot;

detecting, based on the sample values, information identifying the uth root of each of the one or more access signals;

generating, for each of the one or more access signals based on the identified uth root, potential pairs of a number of periods frequency offset L n and a time offset T n ;

selecting, for each of the one or more access signals based on the detected coarse time offset, one of the potential pairs (L n , T n ) as corresponding to an actual frequency offset and an actual time offset; and

sending, to the one or more UE based on the selected pair for the corresponding access signal, an access response formatted according to the RAN.

2 . The satellite computer system according to claim 1 , wherein the step of generating the potential pairs (L n , T n ) includes correlating a package of the sample values with a reference ZC sequence based on the corresponding identified uth root.

3 . The satellite computer system according to claim 2 , wherein the step of correlating includes performing a frequency-domain correlation between the reference ZC sequence and the package of sample values.

4 . The satellite computer system according to claim 3 , wherein the step of generating the potential pairs (L n , T n ) further includes:

performing an inverse Fourier transform on a result of the frequency-domain correlation to obtain a time-domain result; and

identifying a reference time offset T 0 of each of the one or more access signals as corresponding a peak value of the time-domain result, wherein the reference time offset T 0 is associated with a zero number of periods frequency offset L 0 .

5 . The satellite computer system according to claim 4 , wherein the step of generating the potential pairs (L n , T n ) further includes determining the time offset T n for each of the potential pairs based on T 0 −L n ·u −1 , wherein u −1 is a modular multiplicative inverse of the identified uth root associated with the corresponding access signal, and wherein each L n is a preselected potential value for the number of periods frequency offset.

6 . The satellite computer system according to claim 1 , wherein the step of detecting the information identifying the uth root includes:

processing a package of the sample values into root-detection frequency bins;

comparing values of the root-detection frequency bins to a root-detection threshold; and

determining the information identifying the uth root based on a bin index of a corresponding one of the root-detection frequency bins meeting the root-detection threshold.

7 . The satellite computer system according to claim 6 , wherein the step of processing the package of the sample values includes:

generating the package of sample values to include a number of samples greater than the length of the ZC sequence;

transforming the package of sample values to a first set of frequency-domain bins corresponding in number to the number of samples; and

selecting a subset of the first set of frequency-domain bins, the subset corresponding in number to the length of the ZC sequence.

8 . The satellite computer system according to claim 7 , wherein the step of processing the package of the sample values further includes applying adjacent-sample conjugate multiplication to values of the subset of the first set of frequency-domain bins.

9 . The satellite computer system according to claim 8 , wherein the step of processing the package of the sample values further includes transforming results of the adjacent-sample conjugate multiplication into the values of the root-detection frequency bins.

10 . The satellite computer system according to claim 9 , wherein the step of transforming the results of the adjacent-sample conjugate multiplication into the values of the root-detection frequency bins includes applying a Nzc-point Fourier transform to the results of the adjacent-sample conjugate multiplication, wherein Nzc corresponds to the length of the ZC sequence.

11 . The satellite computer system according to claim 6 , wherein the step of determining the information identifying the uth root based on the bin index of the corresponding one of the root-detection frequency bins meeting the root-detection threshold includes, for each root-detection frequency bin meeting the root-detection threshold:

assigning the bin index of the root-detection frequency bin as a modular multiplicative inverse of the uth root, wherein the information identifying the uth root is the assigned modular multiplicative inverse.

12 . The satellite computer system according to claim 6 , wherein the at least one antenna includes a horizontally polarized (H-POL) antenna and a vertically polarized (V-POL) antenna, wherein the step of sampling the received signal includes separately sampling the received signals from the H-POL and V-POL antennas such that the sample values include separate sets of H-POL sample values and V-POL sample values, wherein the step of processing the package of the sample values into the root-detection frequency bins includes:

processing a package of the H-POL sample values into H-POL root-detection frequency bins;

processing a package of the V-POL sample values into V-POL root-detection frequency bins; and

combining results of the H-POL root-detection frequency bins and the V-POL root-detection frequency bins.

13 . The satellite computer system according to claim 1 , wherein the step of detecting the coarse time offset includes performing preamble-length-shifted conjugate multiplication of the sample values against corresponding sample values obtained by delaying the received signal by the preamble length.

14 . The satellite computer system according to claim 13 , wherein the access signal format further has a cyclic prefix (CP) attached to a front of the at least one copy of the preamble, wherein the CP has a CP length and is copied from an end of the preamble sequence, and wherein the step of detecting the coarse time offset further includes:

summing, for each sample window of a plurality of sample windows having the CP length, vector values generated by the preamble-length-shifted conjugate multiplication in the sample window to obtain a plurality of conjugate-multiplication vector sums, wherein a magnitude of one or more vector sums of the plurality of conjugate-multiplication vector sums meets a magnitude threshold; and

determining the coarse time offset of the one or more access signals based on an offset of one or more of the sample windows corresponding to the one or more vector sums.

15 . The satellite computer system according to claim 13 , wherein the actual frequency offset for each of the one or more access signals corresponds to a sum of the selected number of periods frequency offset L n and a fractional portion F, and wherein the steps further include estimating the fractional portion F based on a phase of an aggregate vector value obtained by summing vector values generated by the preamble-length-shifted conjugate multiplication.

16 . The satellite computer system according to claim 15 , wherein the at least one antenna includes a horizontally polarized (H-POL) antenna and a vertically polarized (V-POL) antenna, wherein the step of sampling the received signal includes separately sampling the received signals from the H-POL and V-POL antennas such that the sample values include separate sets of H-POL sample values and V-POL sample values, wherein the step of performing the preamble-length-shifted conjugate multiplication includes performing the preamble-length-shifted conjugate multiplication separately for the H-POL sample values and the V-POL sample values, and wherein the steps further include obtaining the aggregate vector value by combining results of the preamble-length-shifted conjugate multiplication for the H-POL sample values and the preamble-length-shifted conjugate multiplication for the V-POL sample values.

17 . A method implemented by a satellite computer system on a satellite, the satellite including at least one antenna configured to send and receive signals to terrestrial user equipment (UE) on a radio access network (RAN) via a plurality of beams, each of the beams having a beam footprint that defines a cell of the RAN, the RAN defining a physical random access channel (PRACH) slot and an access signal format used by the UE to request an uplink allocation from a base station platform implemented on the satellite during the PRACH slot, the access signal format (1) based on a Zadoff-Chu (ZC) sequence having a uth root, and (2) having at least one copy of a preamble having a preamble length, the method comprising steps performed by the satellite computer system including:

receiving, via the at least one antenna, a signal in the PRACH slot, wherein an orbital motion of the satellite causes one or more access signals, transmitted in the access signal format by the one or more UE and embedded in the received signal, to include a timing offset variation and a frequency offset variation across one or more locations of the one or more UE;

sampling the received signal to obtain sample values, wherein the sample values represent complex values;

detecting, based on the sample values, a coarse time offset of the one or more access signals within the PRACH slot;

detecting, based on the sample values, information identifying the the uth root of each of the one or more access signals;

generating, for each of the one or more access signals based on the identified uth root, potential pairs of a number of periods frequency offset L n and a time offset T n ;

selecting, for each of the one or more access signals based on the detected coarse time offset, one of the potential pairs (L n , T n ) as corresponding to an actual frequency offset and an actual time offset; and

sending, to the one or more UE based on the selected pair for the corresponding access signal, an access response formatted according to the RAN.

18 . The method according to claim 17 , wherein the step of detecting the coarse time offset includes performing preamble-length-shifted conjugate multiplication of the sample values against corresponding sample values obtained by delaying the received signal by the preamble length.

19 . The method according to claim 18 , wherein the access signal format further has a cyclic prefix (CP) attached to a front of the at least one copy of the preamble, wherein the CP has a CP length and is copied from an end of the preamble sequence, and wherein the step of detecting the coarse time offset further includes:

summing, for each sample window of a plurality of sample windows having the CP length, vector values generated by the preamble-length-shifted conjugate multiplication in the sample window to obtain a plurality of conjugate-multiplication vector sums, wherein a magnitude of one or more vector sums of the plurality of conjugate-multiplication vector sums meets a magnitude threshold; and

determining the coarse time offset of the one or more access signals based on an offset of one or more of the sample windows corresponding to the one or more vector sums.

20 . The method according to claim 18 , wherein the actual frequency offset for each of the one or more access signals corresponds to a sum of the selected number of periods frequency offset L n and a fractional portion F, and wherein the steps further include estimating the fractional portion F based on a phase of an aggregate vector value obtained by summing vector values generated by the preamble-length-shifted conjugate multiplication.