Transceiver optimization: summary and tables - Signal Processing and Optimization for Transceiver Systems - page 820

Table I.7. Minimum error probability for circulant channel (Chap. 17 )

The MMSE transceiver in Table I.6 can be converted to a transceiver with min-
imum symbol error probability by inserting additional DFT and IDFT matrices
as shown in the figure below (Sec. 17.4). The MSE (with or without zero forc-
ing) and the channel input power for this system are the same as in Table I.6. So
the system below has minimum MSE as well as minimum error probability. This
assumes that the SNRs are large enough to validate the convexity assumptions
of Sec. 16.2.2. Since the product W † Λ a W is circulant for any diagonal Λ a ,the
optimal precoder is circulant, and so is the optimal equalizer!

q ( n )

s ( n ) x ( n ) x ( n ) s ( n )

W
M

W
M

W
M

W
M

Λ a

Λ b

H

normalized DFT diagonal normalized IDFT circulant channel normalized DFT diagonal normalized IDFT

circulant precoder circulant equalizer

For PAM and QAM constellations, the minimized average symbol error proba-
bility per component of s ( n )isgivenby

⎨ A 1

E ave

c Q (ZF-MMSE case)

P e,min = (I . 31)

A 1

⎩

E ave − σ s

c Q (pure-MMSE case),

where E ave istheaverageMSE(perscalarcomponent s k ( n )). The constants

c and A depend on the constellation, as summarized in Eqs. (I.22) and (I.23),
where b is the number of bits used for the constellation.