Transceivers with decision feedback equalizers - Signal Processing and Optimization for Transceiver Systems - page 648

Let σ be any number such that

σ 0 >σ>σ 1 . (19 . 161)

We will show how to find real unitary matrices U 1 and U 2 such that

σ 0

U 2 = σ

.

0 ×

U 1 (19 . 162)

0 σ 1 0 σ 0 σ 1 /σ

Thus, the product of the diagonal elements is unchanged by the premultiplication
with U 1 and postmultiplication with U 2 . Since U 1 and U 2 are real and unitary,
they can both be regarded as planar rotation operator s [Ho rn and Johnson, 1985].
Note that the case where σ is the geometric mean √ σ 0 σ 1 is a special case which
still satisfies Eq. (19.161); the diagonal elements on the right hand side of Eq.
(19.162) become identical in this case.

Construction of U 1 and U 2 . Choose
U 2 = c

, cσ 0 ,

s
sc

− 1

σ

sσ 1

U 1 = (19 . 163)

− sσ 1 cσ 0

where c =cos θ and s =sin θ . Wewillshowhowtochoosecos θ such that
all the claims are satisfied. We have

σ 0 c 2 σ 0 .

1
σ

+ s 2 σ 1 sc ( σ 1 − σ 0

0 )

U 1 U 2 =

0 σ 1 0 σ 0 σ 1

So we have to choose cos θ such that

c 2 σ 0 + s 2 σ 1 = σ 2 ,

(19 . 164)

that is,

c 2 = σ 2 −

σ 1

σ 1
The right-hand side of the above expression is in the range 0 <x< 1as
required (because of Eqs. (19.160) and (19.161)). With c chosen to satisfy
Eq. (19.164) we have

σ 0 −

σ 0 σ ( σ 1 − σ 0 ) /σ

0

0

U 1 U 2 =

0 σ 1 σ 0 σ 1 /σ

as claimed. Note that U 2 is unitary, and with c chosen to satisfy Eq. (19.164)
U 1 is unitary as well.