Differential MIMO transceiver

1. A non-transitory computer-readable medium having stored thereon computer-readable instructions that when executed by a computing device cause the computing device to:

spatially filter a first signal using a spatial filter matrix to define a first filtered signal, wherein the first signal is a result of a first transmitted signal transmitted by a first plurality of antennas and received by a second plurality of antennas connected to the receiver;

compute a conjugate of the first filtered signal to define a conjugate first signal;

spatially filter a second signal using the spatial filter matrix to define a second filtered signal, wherein the second signal is a result of a second transmitted signal transmitted by the first plurality of antennas and received by the second plurality of antennas connected to the receiver, wherein the second signal is received after the first signal;

compute a Hadamard product of the first filtered signal and the conjugate first signal to define a differential measurement signal; and

detect a plurality of information symbols from the differential measurement signal.

2. The non-transitory computer-readable medium of claim 1 , wherein the spatial filter matrix is computed from an estimated channel matrix.

3. The non-transitory computer-readable medium of claim 2 , wherein the estimated channel matrix is computed by transmitting a plurality of pairs of training signals before the first signal is received.

4. The non-transitory computer-readable medium of claim 3 , wherein the computer-readable instructions further cause the computing device to:

for each pair of training signals of the plurality of pairs of training signals,

compute a conjugate of a first received signal of the pair of training signals to define a conjugate first received training signal; and

compute a Kronecker product of the conjugate first received training signal and a second received signal of the pair of training signals to define a differential measurement signal for the pair of training signals;

compute a first estimated channel matrix from the defined differential measurement signal computed for each pair of training signals of the plurality of pairs of training signals; and

compute a second estimated channel matrix from the first estimated channel matrix;

wherein the spatial filter matrix is computed from the second estimated channel matrix.

5. The non-transitory computer-readable medium of claim 4 , wherein each pair of the plurality of pairs of training signals is selected so that a single column of the first estimated channel matrix is estimated from the defined differential measurement signal for the associated pair.

6. The non-transitory computer-readable medium of claim 4 , wherein the spatial filter matrix is computed using H o H (ρH o H o H +σ 2 I n ) −1 , where H o is the second estimated channel matrix, H o H is a hermitian matrix computed from H o , ρ is an estimated value of a signal to noise ratio of the plurality of pairs of training signals, σ 2 is an estimated noise power matrix computed from the plurality of pairs of training signals, and I n is an identity matrix.

7. The non-transitory computer-readable medium of claim 4 , wherein computing the first estimated channel matrix comprises solving z=H d x, where H d is the first estimated channel matrix, z=vec (rr T H ), x=vec(ss T H ), where r is the second received signal of the pair of training signals, r T H is a hermitian transpose of the first received signal of the pair of training signals, where s is a second transmitted signal of the pair of training signals, and s T H is a hermitian transpose of a first transmitted signal of the pair of training signals.

8. The non-transitory computer-readable medium of claim 7 , wherein computing the second estimated channel matrix comprises:

define a column matrix index as one;

define a row matrix index as one;

define a second column matrix index as one;

define a number of antennas as a number of the second plurality of antennas;

(a) select n column elements from H d (r i , c 1 ), where n is the number of antennas, c 1 is the column matrix index, and r i =r, . . . , r+n−1, where r is the row matrix index;

(b) select a column value from H d (c 1 , c 1 );

(c) normalize each of the selected n column elements using a square root of the selected column value;

(d) store the normalized n column elements in H o (r j , c 2 ), where H o is the second estimated channel matrix, c 2 is the second column matrix index, and r j =1, . . . , n;

(e) increment the row matrix index using r=r+n;

(f) increment the column matrix index using c 1 =c 1 +n+1;

(g) increment the second column matrix index using c 2 =c 2 +1; and

(h) repeat (a) to (g) until the second column matrix index is greater than the number of antennas.

9. A receiver comprising:

a processor; and

a non-transitory computer-readable medium operably coupled to the processor, the computer-readable medium having computer-readable instructions stored thereon that, when executed by the processor, cause the receiver to

spatially filter a first signal using a spatial filter matrix to define a first filtered signal, wherein the first signal is a result of a first transmitted signal transmitted by a first plurality of antennas and received by a second plurality of antennas connected to the receiver;

compute a conjugate of the first filtered signal to define a conjugate first signal;

spatially filter a second signal using the spatial filter matrix to define a second filtered signal, wherein the second signal is a result of a second transmitted signal transmitted by the first plurality of antennas and received by the second plurality of antennas connected to the receiver, wherein the second signal is received after the first signal;

compute a Hadamard product of the first filtered signal and the conjugate first signal to define a differential measurement signal; and

detect a plurality of information symbols from the differential measurement signal.

10. The receiver of claim 9 , wherein the spatial filter matrix is computed from an estimated channel matrix.

11. The receiver of claim 10 , wherein the estimated channel matrix is computed by transmitting a plurality of pairs of training signals before the first signal is received.

12. The receiver of claim 11 , wherein the computer-readable instructions further cause the receiver to:

for each pair of training signals of the plurality of pairs of training signals,

compute a conjugate of a first received signal of the pair of training signals to define a conjugate first received training signal; and

compute a Kronecker product of the conjugate first received training signal and a second received signal of the pair of training signals to define a differential measurement signal for the pair of training signals;

compute a first estimated channel matrix from the defined differential measurement signal computed for each pair of training signals of the plurality of pairs of training signals; and

compute a second estimated channel matrix from the first estimated channel matrix;

wherein the spatial filter matrix is computed from the second estimated channel matrix.

13. The receiver of claim 12 , wherein each pair of the plurality of pairs of training signals is selected so that a single column of the first estimated channel matrix is estimated from the defined differential measurement signal for the associated pair.

14. The receiver of claim 12 , wherein the spatial filter matrix is computed using H o H (ρH o H o H +σ 2 I n) −1 , where H o is the second estimated channel matrix, H o H is a hermitian matrix computed from H o , ρ is an estimated value of a signal to noise ratio of the plurality of pairs of training signals, σ 2 is an estimated noise power matrix computed from the plurality of pairs of training signals, and I n is an identity matrix.

15. The receiver of claim 12 , wherein computing the first estimated channel matrix comprises solving z=H d x, where H d is the first estimated channel matrix, z=vec(rr T H ), x=vec(ss T H ), where r is the second received signal of the pair of training signals, r T H is a hermitian transpose of the first received signal of the pair of training signals, where s is a second transmitted signal of the pair of training signals, and s T H is a hermitian transpose of a first transmitted signal of the pair of training signals.

16. The receiver of claim 15 , wherein computing the second estimated channel matrix comprises:

define a column matrix index as one;

define a row matrix index as one;

define a second column matrix index as one;

define a number of antennas as a number of the second plurality of antennas;

(a) select n column elements from H d (r i , c 1 ), where n is the number of antennas, c 1 is the column matrix index, and r i =r, . . . , r+n−1, where r is the row matrix index;

(b) select a column value from H d (c 1 , c 1 );

(c) normalize each of the selected n column elements using a square root of the selected column value;

(d) store the normalized n column elements in H o (r j , c 2 ), where H o is the second estimated channel matrix, c 2 is the second column matrix index, and r j =1, . . . , n;

(e) increment the row matrix index using r=r+n;

(f) increment the column matrix index using c 1 =c 1 +n+1;

(g) increment the second column matrix index using c 2 =c 2 +1; and

(h) repeat (a) to (g) until the second column matrix index is greater than the number of antennas.

17. A method of detecting a plurality of information symbols, the method comprising:

spatially filtering, by a receiver, a first signal using a spatial filter matrix to define a first filtered signal, wherein the first signal is a result of a first transmitted signal transmitted by a first plurality of antennas and received by a second plurality of antennas connected to the receiver;

computing, by the receiver, a conjugate of the first filtered signal to define a conjugate first signal;

spatially filtering, by the receiver, a second signal using the spatial filter matrix to define a second filtered signal, wherein the second signal is a result of a second transmitted signal transmitted by the first plurality of antennas and received by the second plurality of antennas connected to the receiver, wherein the second signal is received after the first signal;

computing, by the receiver, a Hadamard product of the first filtered signal and the conjugate first signal to define a differential measurement signal; and

detecting, by the receiver, a plurality of information symbols from the differential measurement signal.

18. The method of claim 17 , wherein the spatial filter matrix is computed from an estimated channel matrix.

19. The method of claim 18 , wherein the estimated channel matrix is computed by transmitting a plurality of pairs of training signals before the first signal is received.

20. The method of claim 19 , further comprising:

for each pair of training signals of the plurality of pairs of training signals,

computing, by the receiver, a conjugate of a first received signal of the pair of training signals to define a conjugate first received training signal; and

computing, by the receiver, a Kronecker product of the conjugate first received training signal and a second received signal of the pair of training signals to define a differential measurement signal for the pair of training signals;

computing, by the receiver, a first estimated channel matrix from the defined differential measurement signal computed for each pair of training signals of the plurality of pairs of training signals; and

computing, by the receiver, a second estimated channel matrix from the first estimated channel matrix;

wherein the spatial filter matrix is computed from the second estimated channel matrix.