Interference alignment for channel-adaptive waveform modulation
Embodiments provide an apparatus and method for interference alignment for channel-adaptive waveform modulation. The method includes obtaining at least a part of a first matrix and a part of a second matrix for the impulse response function of a communication channel. The method further includes designing a set of one or more linearly independent waveforms based on at least the obtained parts of the first and second matrices such that a first subspace spanned by the linearly independent waveforms when multiplied by the obtained part of the first matrix at least partially overlaps a second subspace spanned by the linearly independent waveforms when multiplied by the obtained part of the second matrix.
1. A method for transmitting a sequence of data blocks, comprising:
obtaining at least a part of a first matrix and a part of a second matrix for an impulse response function of a communication channel, the part of the first matrix relating to channel-induced interference between a current data block and a previously transmitted first data block, the part of the second matrix relating to channel-induced interference between the current data block and a previously transmitted second data block, the second data block being transmitted before the first data block;
designing a set of one or more linearly independent waveforms based on at least the obtained part of the first matrix and the obtained part of the second matrix for the impulse response function such that a first subspace spanned by the linearly independent waveforms when multiplied by the obtained part of the first matrix at least partially overlaps a second subspace spanned by the linearly independent waveforms when multiplied by the obtained part of the second matrix; and
transmitting a sequence of the data blocks over the channel from a transmitter, each data block of the sequence being a weighted linear superposition of the one or more waveforms of the designed set.
2. The method of claim 1 , wherein the designing step designs the set of linearly independent waveforms such that the first subspace and the second subspace occupy a same linear space.
3. The method of claim 1 , wherein the designed set includes a subset of eigenvectors of a product of (1) an inverse of the first matrix and (2) the second matrix.
4. The method of claim 3 , wherein the subset comprises right eigenvectors of the product.
5. The method of claim 1 , wherein the designing step further includes:
obtaining eigenvectors and corresponding eigenvalues based on an eigenvector decomposition of a product of (1) an inverse of the first matrix and (2) the second matrix; and
selecting a subset of the obtained eigenvectors.
6. The method of claim 5 , wherein the designing step further includes:
configuring a second set of waveforms based on the selected subset, wherein the configured second set of waveforms is an orthogonal complement of the product of the selected subset and either of the first or the second matrices.
7. The method of claim 1 , wherein the first data block immediately precedes the current data block and the second data block immediately precedes the first data block.
8. The method of claim 1 , further comprising:
transmitting a set of pilot signals over the communication channel that is between the transmitter and a receiver, the part of the first matrix and the part of the second matrix for the impulse response function being obtained responsive to measurements of said pilot signals.
9. The method of claim 1 , further comprising:
for each one of the data blocks of the sequence, modulating the waveforms of the designed set to have amplitudes responsive of a received input data symbol and linearly superimposing the modulated waveforms to produce the each one of the data blocks.
10. The method of claim 1 , wherein the designed set has a number of the waveforms equal to one half of a symbol interval when delay spread is twice the symbol interval.
11. An apparatus for communicating data, comprising:
a transmitter including
an array of modulators, each modulator being configured to modulate an amplitude of a corresponding one of linearly independent waveforms over a sequence of sampling intervals in response to receipt of each of a sequence of input data symbols; and
an adder configured to form a sequence of data blocks, each data block being a linear superposition of modulated transmitter waveforms produced by the modulators responsive to receipt of one of the input data symbols, the adder configured to transmit the data blocks via a communication channel; and
wherein the transmitter configures the modulated waveforms in a manner responsive to a part of a first matrix and a part of a second matrix for an impulse response function of the communication channel, the part of the first matrix relating to channel-induced interference between a current data block and a previously transmitted first data block, the part of the second matrix relating to channel-induced interference between the current data block and a previously transmitted second data block, the second data block being transmitted before the first data block,
wherein the transmitter configures the modulated waveforms such that a first subspace spanned by the modulated waveforms when multiplied by the part of the first matrix at least partially overlaps a second subspace spanned by the modulated waveforms when multiplied by the part of the second matrix.
12. The apparatus of claim 11 , wherein the transmitter configures the modulated waveforms such that the first subspace and the second subspace occupy a same linear space.
13. The apparatus of claim 11 , where the modulated waveforms include a subset of eigenvectors of a product of (1) an inverse of the first matrix and (2) the second matrix.
14. The apparatus of claim 13 , wherein the subset comprises right eigenvectors of the product.
15. The apparatus of claim 11 , wherein the transmitter configures the modulated waveforms by obtaining eigenvectors and corresponding eigenvalues based on an eigenvector decomposition of a product of (1) an inverse of the first matrix and (2) the second matrix and selects a subset of the obtained eigenvectors.
16. The apparatus of claim 15 , wherein the transmitter configures the modulated waveforms by constructing a second set of waveforms based on the selected subset, wherein the constructed second set of waveforms is an orthogonal complement of the product of the selected subset and either of the first or second matrices.
17. The apparatus of claim 11 , further comprising:
a receiver having an array of demodulators, the demodulators projecting the data blocks onto conjugate waveforms to produce estimates of a linear combination of components of the input data symbol carried by the data blocks being demodulated.
18. The apparatus of claim 17 , wherein the transmitter transmits a set of pilot signals over the communication channel that is between the transmitter and the receiver, the part of the first matrix and the part of the second matrix for the impulse response function being obtained responsive to measurements of said pilot signals.
19. The apparatus of claim 11 , wherein each modulator modulates the amplitude of the corresponding one of linearly independent waveforms to have amplitudes responsive of a received input data symbol and linearly superimposing the modulated waveforms to produce each one of the data blocks.
20. The apparatus of claim 11 , wherein a number of the modulated waveforms is equal to one half of a symbol interval when delay spread is twice the symbol interval.