Systems and methods for communicating by modulating data on zeros in the presence of channel impairments
Communication systems and methods in accordance with various embodiments of the invention utilize modulation on zeros. Carrier frequency offsets (CFO) can result in an unknown rotation of all zeros of a received signal's z-transform. Therefore, a binary MOCZ scheme (BMOCZ) can be utilized in which the modulated binary data is encoded using a cycling register code (e.g. CPC or ACPC), enabling receivers to determine cyclic shifts in the BMOCZ symbol resulting from a CFO. Receivers in accordance with several embodiments of the invention include decoders capable of decoding information bits from received discrete-time baseband signals by: estimating a timing offset for the received signal; determining a plurality of zeros of a z-transform of the received symbol; identifying zeros from the plurality of zeros that encode received bits by correcting fractional rotations resulting from the CFO; and decoding information bits based upon the received bits using a cycling register code.
1. A communication system, comprising:
a transmitter, comprising:
an encoder configured to receive a plurality of information bits and output a plurality of encoded hits in accordance with a cycling register code (CRC);
a modulator configured to modulate the plurality of encoded bits to obtain a discrete-time baseband signal, where the plurality of encoded bits are encoded in the zeros of the z-transform of the discrete-time baseband signal; and
a signal generator configured to generate a continuous-time transmitted signal based upon the discrete-time baseband signal;
a receiver, comprising:
a demodulator configured to down convert and sample a received continuous-time signal at a given sampling rate to obtain a received discrete-time baseband signal, where the received discrete-time baseband signal includes at least one of a timing offset (TO) and a carrier frequency offset (CFO);
a decoder configured to decode a plurality of bits of information from the received discrete-time baseband signal by:
estimating a TO for the received discrete-time baseband signal to identify a received symbol;
determining a plurality of zeros of a z-transform of the received symbol;
identifying zeros from the plurality of zeros that encode a plurality of received bits; and
decoding a plurality of information bits based upon the plurality of received hits using the CRC.
2. The communication system of claim 1 , wherein the receiver receives the continuous-time transmitted signal over a multipath channel.
3. The communication system of claim 1 , wherein the modulator is configured to modulate the plurality of encoded bits so that the z-transform of the discrete-time baseband signal comprises a zero for each of a plurality of encoded bits.
4. The communication system of claim 1 , wherein the modulator is configured to modulate the plurality of encoded bits so that each zero in the z-transform of the discrete-time baseband signal is limited to being one of a set of conjugate-reciprocal pairs of zeros.
5. The communication system of claim 4 , wherein:
each conjugate reciprocal pair of zeros in the set of conjugate-reciprocal pairs of zeros comprises:
an outer zero having a first radius that is greater than one; and
an inner zero having a radius that is the reciprocal of the first radius;
where the inner and outer zero have phases that are the same phase;
the radii of the outer zeros in each pair of zeros in the set of conjugate-reciprocal pairs of zeros are the same; and
the phases of the outer zeros in each pair of zeros in the set of conjugate-reciprocal pairs of zeros are evenly spaced over one complete revolution.
6. The communication system of claim 1 , wherein the cycling register code is a cyclically permutable code (CPC).
7. The communication system of claim 6 , wherein the CPC is extracted from a Bose Chaudhuri Hocquenghem (BCH) code.
8. The communication system of claim 6 , wherein the CPC is extracted form a primitive BCH code.
9. The communication system of claim 6 , wherein the CPC has a code length that is a Mersenne prime.
10. The communication system of claim 9 , wherein the CPC has a code length selected from the group consisting of 3, 7, 31, and 127.
11. The communication system of claim 1 , wherein the CRC is generated by an inner code and an outer code which are combined in a non-linear fashion.
12. The communication system of claim 11 , wherein the outer code is a cycling register code having a lower code rate than the inner code.
13. The communication system of claim 11 , wherein the outer code is a cyclically permutable code (CPC).
14. The communication system of claim 11 , wherein the CRC is an affine CPC (ACPC) code.
15. The communication system of claim 14 , wherein the ACPC is characterized by being attainable using a cyclic inner code having codewords of an inner codeword length, which is affine translated by a given binary word of the inner codeword length, and then further encoded by a cyclic outer code.
16. The communication system of claim 1 , wherein the decoder is configured to estimate the TO by measuring energy over an expected symbol length with a sliding window in the sampled signal.
17. The communication system of claim 16 , wherein the decoder is configured to measure energy over an expected symbol length by convolving samples with a universal Huffman sequence of the expected symbol length comprising two impulses at the beginning and the end of the expected symbol length.
18. The communication system of claim 1 , wherein the decoder is configured to estimate the TO by identifying a set of three energy peaks that yield a maximum energy sum over an expected symbol length.
19. The communication system of claim 1 , wherein:
the demodulator is configured to oversample the received discrete-time signal by zero-padding; and
the decoder is configured to identify zeros from the plurality of zeros that encode a plurality of received hits by identifying a fractional rotation resulting from the CFO.
20. The communication system of claim 19 , wherein the decoder is configured to determine a most likely set of zeros for the z-transform of the discrete-time baseband signal used to generate the transmitted signal based upon the received symbol.