Transceivers with orthonormal precoders - Signal Processing and Optimization for Transceiver Systems - page 481

q ( n )

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

x ( n )

α I

−1

U f

α I

H G 1

power control unitary

precoder F

equalizer G

Figure 15.2 . Inserting the multiplier α in the unitary precoder for power control.

We can eliminate α using Eq. (15.12) and write

M− 1

E mmse = Mσ q σ s
p 0

1
σ h,k

(15 . 13)

k =0

For the case where the precoder is not restricted to be orthonormal (Sec. 12.4),
the ZF-MMSE system has MSE

M− 1 2

E mmse = σ q σ s
p 0

1
σ h,k

. (15 . 14)

k =0

Summarizing, for the ZF-MMSE transceiver the minimized MSE is given by

⎧
⎨

M− 1

Mσ q σ s
p 0

1
σ h,k

(orthonormal precoder)

k =0

E mmse = (15 . 15)

⎩ M− 1 2

σ q σ s
p 0

1
σ h,k

(unrestricted precoder).

k =0

Defining the vector

z = 1
σ h, 0 σ h,M− 1 T

1
σ h, 1

1

...

(15 . 16)

we see that Eq. (15.15) can be written as

⎧
⎨

z 2 A (orthonormal precoder)

E mmse = (15 . 17)

⎩

M z 1

A

(unrestricted precoder),

where A is a constant and z p denotes the p -norm of the vector z ,thatis,

z p =

p 1 /p

M− 1

| z k | . (15 . 18)

k =0