Transceivers with decision feedback equalizers - Signal Processing and Optimization for Transceiver Systems - page 600

the precoder matrix F and the DFE matrix B appropriately. The minimized
MSE under the zero-forcing constraint and the power constraint is therefore

2 /M

M− 1

E mmse = M 2 σ s σ q
p 0

1
σ h,k

. (19 . 26)

k =0

Compare this with the minimum MSE for the case where there is no DFE,
which was derived in Chap. 12 under the zero-forcing constraint and the power
constraint:

2 .

M− 1

E no df e = σ s σ q
p 0

1
σ h,k

(19 . 27)

k =0

Thus

1
M

2

M− 1

1
σ h,k

E no df e
E mmse

k =0

= (19 . 28)

2 /M

M− 1

1
σ h,k

k =0

The benefit of going from a linear transceiver to a transceiver with DFE is
therefore measured by the square of the ratio of the arithmetic mean to geometric
mean of the reciprocal of the M dominant channel singular values. The ratio
(19.28) will be referred to as the gain due to DFE in the zero forcing case:
G DF E,zf = AM 1

2

σ h,k

. (19 . 29)

GM 1
σ h,k

For the case of scalar channels with time-domain DFE, a similar formula has
been observed in the pioneering work of Price [1972]. See p. 729 of the review
by Lucky [1973].
It is interesting to recall here a result from the theory of optimal bit allocation
(Chap. 14): the “coding gain” due to bit allocation in a zero-forced system was
shown to be an AM/GM ratio (Sec. 14.7). In that AM/GM ratio, the numbers
1 /σ h,k appeared instead of 1 /σ h,k .

19.3.3 Proof of Eq. (19.25)

We will find it convenient to express the P × M precoder F in terms of its SVD:

Σ f
0

V f
M×M

F = U f

P×P

(19 . 30)

P×M

Here U f and V f are unitary matrices, and Σ f is diagonal with nonnegative
diagonal elements.

Since F has rank M as explained after Eq. (19.13), the