Transceivers with orthonormal precoders - Signal Processing and Optimization for Transceiver Systems - page 490

Since the eigenvalues of the inverse matrix are 1 /λ k , the trace in the preceding
equation becomes

P− 1

1
λ k

τ = J − P )+

k =0

M− 1

1
1+ μ k σ s
σ q

= J − P )+ +( P − M )

k =0

M− 1

1
1+ σ s
σ q

= +( J − M ) .

μ k

k =0

The MSE in Eq. (15.46) therefore becomes

M− 1

σ s
1+ σ s
σ q

E mse = (15 . 48)

μ k

k =0

As in earlier sections let the channel SVD be written as follows:

V h
P×P

H = U h Σ h . (15 . 49)

J×J J×P

As usual let the matrix ( Σ h ) M denote the diagonal matrix of the dominant
channel singular values σ h, 0 ≥

σ h, 1 ≥ ... ≥ σ h,M− 1 . We are now ready to prove

the main result:

Theorem 15.3. MMSE transceiver with orthonormal precoder. Consider the
transceiver with orthonormal precoder

♠

⎡ ⎤

I M

⎣ ⎦

F = V h (15 . 50)

0

and equalizer

0 ] U h , G =[ Σ g (15 . 51)

where Σ g is the M × M diagonal matrix defined as

− 1

I + σ q
σ s

Σ g =( Σ h ) − M ( Σ h ) − M . (15 . 52)

This transceiver has the MMSE property. The minimized mean square error is
given by

M− 1

σ q
σ h,k + σ q

E mse = (15 . 53)

k =0

σ s