Transceivers with decision feedback equalizers - Signal Processing and Optimization for Transceiver Systems - page 655

The details of this proof are given below. Substituting from Eq. (19.196), Eq.
(19.191) can be rewritten as

1
σ h,k

1 /M

K− 1

σ s ( σ h,K ) K/M

E pure =

k =0

σ s 1

−K/M 1

σ h,k

1 /M

K− 1

=

σ h,K

k =0

σ s σ h,K 1

( M−K ) /M 1

σ h,k

1 /M

K− 1

=

σ h,K

k =0

1
σ h,k

1 /M

M− 1

σ s σ h,K

= (using Eq. (19.196))

k =0

so that, from Eq. (19.193), we get

σ h,k 1 /M

M− 1

σ s
E br

1
σ h,K

= − 1 . (19 . 197)

k =0

Next, from Eq. (19.190) we have

σ h,k 1 /M

M− 1

σ s
E ZF

p 0
Mσ q

= . (19 . 198)

k =0

Using the expression for the power p 0 given in the equation following Eq. (19.96),
we get

K− 1

Kσ s
σ q λ −

p σ q 1

σ h,k

=

k =0

K− 1

K
σ h,K −

1
σ h,k

= (from Eq. (19.196))

k =0

K− 1

M
σ h,K −

1
σ h,k −

M − K
σ h,K

=

k =0

M− 1

M
σ h,K −

1
σ h,k

= (again from Eq. (19.196))

k =0

Substituting this into Eq. (19.198) we get

= 1 σ h,k 1 /M

M− 1 M− 1

σ s
E ZF

1
M

1
σ h,k σ h,K −

k =0 k =0

σ h,k 1 /M

1
M

σ h,k 1 /M

M− 1 M− 1 M− 1

1
σ h,K

1
σ h,k

= − .

k =0 k =0 k =0