Transceivers with decision feedback equalizers - Signal Processing and Optimization for Transceiver Systems - page 654

Resubstituting C = HF this reduces to Eq. (19.72) indeed.

19.D Bias-removed MMSE is better than ZF MMSE

In this section we will use the simple notations E pure to denote the

average minimized mean square error per scalar symbol for the DFE system
with and without the zero-forcing constraint. Dividing the expressions in Eq.
(19.124) by M (to get per-symbol values) we obtain

E ZF and

1 /M

M− 1

E ZF = Mσ s σ q
p 0

1
σ h,k

(19 . 190)

k =0

and

E pure = σ s λσ q

K/M 1

σ h,k

1 /M

K− 1

. (19 . 191)

σ s

k =0

Substituting for λ from Eq. (19.97), we have

K/M K− 1

1
σ h,k

1 /M

K

E pure = σ s

. (19 . 192)

+ K− 1
k =0

p σ q

1
σ h,k

k =0

Since the optimal DFE system equalizes the mean square errors of the M in-
dividual components in the block, these expressions represent the errors in the
individual symbols. For the case where there is no zero forcing, we have to per-
form bias removal, as explained in Sec. 16.3. Denoting the mean square error
per symbol after bias removal by

E br , we have
σ s
E br

σ s
E pure −

= 1 (19 . 193)

(review Sec. 16.3.2). Our goal now is to prove that

E br ≤E ZF . (19 . 194)

Even though it is obvious that E pure ≤E ZF , it takes more effort to verify Eq.

(19.194). For a given K , the error E pure (hence E br ) does not depend on σ h,k with

k E ZF does depend on all the components σ h,k , it is necessary

(and su cient) to prove

≥ K. But since

E br ≤E ZF for the extreme case where σ h,K ,...,σ h,M− 1
are as large as possible (i.e., E ZF as small as possible). Now recall from Eq.

(19.96) that K is such that (1 /λ ) ( σ q /σ s σ h,k ) − ≤ 0for k ≥ K. That is,

λσ q
σ s

σ h,k ≤

, K ≤ k ≤ M − 1 . (19 . 195)

So, it is necessary and su cient to prove E br ≤E ZF for the case where

σ h,K = σ h,K +1 = ... = σ h,M− 1 = λσ q

(19 . 196)

σ s