This page accompanies the paper
"MIMO BIORTHOGONAL PARTNERS AND APPLICATIONS"
by Bojan Vrcelj and P.P. Vaidyanathan

submitted to IEEE Trans. Signal Processing


The equivalent MIMO (3-by-3) channel Fsq(z) was taken to be:

F00(z) = - 0.2101 + 0.0067z-1 + 0.2199z-2 - 0.1177z-3 - 0.4573z-4
F01(z) = 0.0357 - 0.3351z-1 + 0.2309z-2 - 0.1434z-3
F02(z) = - 0.2653 - 3.2019z-1 - 0.4438z-2 - 0.1378z-3 + 0.0520z-4
F10(z) = 0.1863 - 3.2573z-1 - 0.3591z-2
F11(z) = 0.4892 + 0.1416z-1 - 0.1555z-2 - 0.0992z-3 - 0.1915z-4 - 0.0514z-5
F12(z) = - 0.2849 - 0.4620z-1 - 0.0211z-2 + 0.1480z-3
F20(z) = 0.4235 - 0.2924z-1 + 0.1358z-2
F21(z) = - 0.1649 - 2.6988z-1 + 0.6550z-2 - 0.2351z-3
F22(z) = - 0.0458 + 0.3100z-1 - 0.7134z-2 + 0.0277z-3 - 0.4219z-4

As explained in the paper, the SSE version of this channel, F2(z) is obtained from Fsq(z) by decimation. Similarly, Frec(z) is obtained from Fsq(z) by keeping the first two rows of the 3-by-3 matrix Fsq(z).


According to the algorithm implicit from the simple Bezout identity, one possible FIR FSE of Fsq(z) (up to 4 decimal places) is given by:

Hsq00(z) = 0.0449 - 0.0008z-2 + 0.0060z-4 - 0.0036z-6 - 0.0005z-8 - 0.0003z-10 - 0.0001z-12
Hsq01(z) = - 0.3152 - 0.0011z-2 + 0.0002z-4 + 0.0001z-6 - 0.0002z-8
Hsq02(z) = - 0.0175 - 0.0756z-1 - 0.0089z-2 + 0.0219z-3 + 0.0113z-4 + 0.0021z-5 - 0.0004z-6 + 0.0017z-7 + 0.0007z-8 + 0.0005z-9 + 0.0001z-10
Hsq10(z) = - 0.0361 + 0.0242z-2 + 0.0220z-4 + 0.0115z-6 + 0.0038z-8 - 0.0001z-10
Hsq11(z) = 0.0193 - 0.0096z-2 - 0.0058z-4 - 0.0014z-6 - 0.0004z-8 - 0.0001z-10
Hsq12(z) = - 0.3590 - 0.0987z-1 + 0.0079z-2 - 0.0474-3 - 0.0145z-4 - 0.0328z-5 - 0.0090z-6 - 0.0003z-7 - 0.0002z-8 - 0.0001z-10
Hsq20(z) = - 0.3274 - 0.0840z-2 - 0.0638z-4 + 0.0011z-6
Hsq21(z) = 0.0635 + 0.0212z-2 + 0.0034z-4 + 0.0021z-6
Hsq22(z) = 0.0132 + 0.5031z-1 + 0.1467z-2 + 0.0132z-3 + 0.0010z-4 + 0.0003z-5 + 0.0028z-6 - 0.0005z-8

Next, one possible FIR FSE of Frec(z) (up to 4 decimal places) is given by:

Hrec00(z) = 0.0012 + 0.5045z-1 - 0.0321z-2 - 0.4023z-3 + 0.0826z-4 - 0.2868z-5 + 0.0059z-6 + 0.1380z-7 - 0.0335z-8 + 0.0594z-9 - 0.0015z-10 + 0.0007z-11 + 0.0013z-12 + 0.0001z-13 - 0.0004z-14
Hrec01(z) = - 0.3360 + 0.0613z-1 + 0.0366z-2 - 0.2897z-3 - 0.0364z-4 + 0.1726z-5 + 0.0113z-6 + 0.0883z-7 + 0.0353z-8 - 0.0069z-9 - 0.0131z-10 + 0.0056z-11 - 0.0087z-12 + 0.0028z-13 - 0.0004z-14 + 0.0001z-15
Hrec10(z) = - 0.8867 + 10.9643z-1 - 0.9136z-2 - 1.6972z-3 + 0.9150z-4 - 3.7057z-5 + 0.4294z-6 - 0.0353z-7 - 0.1334z-8 - 0.0035z-9 + 0.0266z-10 + 0.0001z-12
Hrec11(z) = - 0.6618 + 0.8284z-1 + 0.4488z-2 - 5.9379z-3 - 0.7197z-4 + 0.0966z-5 - 0.0560z-6 - 0.0620z-7 + 0.4998z-8 - 0.1622z-9 + 0.0270z-10 - 0.0074z-11 + 0.0003z-12 - 0.0001z-13
Hrec20(z) = - 0.2804 - 0.1969z-1 + 0.0468z-2 - 0.1003z-3 + 0.0077z-4 - 0.0005z-5 - 0.0026z-6 + 0.0007z-8
Hrec21(z) = 0.0019 - 0.1783z-1 + 0.0223z-2 - 0.0126z-3 + 0.0201z-4 - 0.0085z-5 + 0.0144z-6 - 0.0046z-7 + 0.0007z-8 - 0.0002z-9


Now, here are the zeroth and the third order estimators for the square (Asq(z)) and the rectangular (Arec(z)) case:
Asq,000(z) = - 0.0515; Asq,001(z) = 0.0059; Asq,002(z) = 0.0094;
Asq,010(z) = 0.1115; Asq,011(z) = 0.0209; Asq,012(z) = 0.0568;
Asq,020(z) = - 0.0002; Asq,021(z) = - 0.0153; Asq,022(z) = - 0.0301;

Arec,000(z) = - 0.0933;
Arec,010(z) = - 1.6680;
Arec,020(z) = 0.0130;
(Recall that in the 2-by-3 case, estimator Arec(z) is a 3-by-1 vector.)

Asq,300(z) = 1.0717 - 0.0597z-1 - 0.0080z-2 - 0.0010z-3;
Asq,301(z) = - 0.0198 - 0.0029z-1 - 0.0001z-2;
Asq,302(z) = 0.0198 - 0.0037z-1 - 0.0011z-2;
Asq,310(z) = 0.8191 - 0.1412z-1 + 0.0351z-2 - 0.0001z-3;
Asq,311(z) = 0.0047 + 0.0005z-1 + 0.0001z-2 - 0.0001z-3;
Asq,312(z) = 0.0633 - 0.0033z-1 + 0.0012z-2 - 0.0001z-3;
Asq,320(z) = 0.2774 - 0.0999z-1 - 0.0013z-2;
Asq,321(z) = - 0.0217 + 0.0016z-1 + 0.0002z-2;
Asq,322(z) = - 0.0275 - 0.0007z-1 - 0.0001z-2 + 0.0001z-3;

Arec,300(z) = - 0.0836 + 0.0410z-1 + 0.0273z-2 - 0.0001z-3;
Arec,310(z) = - 1.6828 - 0.1342z-1 - 0.0157z-2 - 0.0301z-3;
Arec,320(z) = 0.0150 + 0.0124z-1 + 0.0045z-2 + 0.0031z-3;

The other estimators are obtained using a similar procedure (see the paper).
The above estimators for the square and rectangular case are used to obtain more sophisticated solutions for FSEs, that would suppress the noise power at the output. For more details, see the paper, Sec. 4.1.

Go to the home pages of
Department of Electrical Engineering,
California Institute of Technology.

Maintained by Bojan Vrcelj
Last updated January, 2001