GTD-Based Transceivers

3.2 MIMO Transceivers with Decision Feedback and Bit Loading

3.2.3 GTD-Based Transceivers

optimizingF,G,and the bit allocation under the zero-forcing constraint. ♦ Thus, when bit loading is allowed, DFE with linear precoding has the same performance as linear transceivers! However, the DFE system with linear precoding actually provides more choices of possible configurations that achieve theP_min in (3.18). This interesting observation will be elabo- rated further in the following subsections.

where

G0= [Q^†]_M×J. (3.22)

SincePandQhave orthonormal columns, the columns ofFare orthonormal, and so are the rows ofG0.Finally, the feedback matrixBis determined by the zero forcing conditionB =GHF−I. To simplify this, observe first that

GHF = (diag([R]_M×M))⁻¹G₀QRP^†F

= (diag([R]_M×M))⁻¹ IM0

R

Ñ I_M

0 é

= (diag([R]_M×M))⁻¹[R]_M×M.

Here we have used the facts that

G0Q= (IM 0) and P^†F= Ñ

IM

0 é

which follow from the choices of (3.20) and (3.22), and the column orthonormality ofPand Q.

Thus the expression for the feedback matrix becomes

B=GHF−I= (diag([R]M×M))⁻¹[R]M×M −I. (3.23)

This is strictly upper triangular sinceRis upper triangular. Fig. 3.2 shows the structure of the GTD transceiver just described.

n White noise

x y ⁺ ^s

s

M N J M

x y

F P

⁺

R

K

Q G

₀

K J

diag([R]_MxM)^Ͳ1

s

Ͳ

+

s~

M

B

M

H H

Figure 3.2: The proposed form of optimal solution for the DFE transceiver.

With the above choice of transceiver matrices the error variance (3.5) in thekth substream becomes

σ_e²_k = σ²_n

[R]²_kk. (3.24)

Substituting into Eq. (3.8) the transmitted power needed to satisfy the specified QoS and bit rate constraints can be expressed as

P_trans=

M

X

k=1

d_k2^bk[F^†F]_kkσ²_e

k =

M

X

k=1

d_k2^bk [R]²_kkσ_n². Sinceσ²_ndk=ck(from (3.11)), this simplifies to

P_trans=

M

X

k=1

ck2^bk

[R]²_kk. (3.25)

We now show that the system in Fig. 3.2 withF,G, andBchosen as described achieves optimality for problem (3.12), provided the bit allocation is chosen appropriately:

Theorem 3.2.3 With the bit allocation chosen as

bk = log₂ Ñ

c

M2^b 1

QM k=1σ_h,k²

!_M¹é

−log₂(ck) + log₂([R]²_kk), (3.26)

for1≤k≤M, the system in Fig. 3.2 withFas in (3.20),Gas in (3.21), andBas in (3.23), achieves the minimized power for the specified{Pe(k)}and bit rate constraint. ♦

Proof: Observe first that (3.26) satisfies the total bit constraint because

M

X

k=1

bk = log₂ c M

^M 2^{M b} QM

k=1σ_h,k²

!

−log₂

M

Y

k=1

ck+ log₂(

M

Y

k=1

[R]²_kk)

= M b−log₂ 1 QM

k=1σ_h,k²

+ log₂Y^M

k=1

[R]²_kk

= M b,

using (3.19) andc=M(QM

k=1ck)^M1.Next, (3.26) implies ck2^bk

[R]²_kk = c

M2^b 1 QM

k=1σ_h,k² _M¹

. (3.27)

Substituting into (3.25) we get

Ptrans=

M

X

k=1

ck2^bk

[R]²_kk =M× c

M2^b 1 QM

k=1σ²_{h,k M}¹

. (3.28)

Since this is the minimum achievable powerPmin(see discussion leading to Eq. (3.18)), the proof

is complete.

The extra flexibility in designing the transceivers, offered by this GTD-based DFE system, must be carefully understood. Recall that the bit loading formula for the linear transceiver to achieve the minimum transmitted power is [48]

bk=D−log₂ck+ log₂(σ_h,k² ), (3.29)

whereσ_h,kare fixed numbers given to us by the channel. The values computed from (3.29) are not guaranteed to be integers, or even nonnegative. For the GTD-based DFE system, the bit loading scheme (3.26) can be written as

b_k =D−log₂c_k+ log₂([R]²_kk). (3.30)

The freedom of the GTD-based system allows us to reshape the value of[R]kkas long as the multi- plicative majorization property (2.3) is satisfied. This flexibility may be used, for example, to ensure that the bit loading scheme in (3.30) is realizable. So, even though the linear transceiver with bit allocation (3.29) can achieve the same minimum power (3.28) as any optimal DFE transceiver, the bit allocation formula in the GTD-based DFE opens up more freedom.

We now make an interesting observation about the powersPkin the optimal system. Substitut- ing (3.24) into (3.7) and using the definition ofckin (3.11) we find

P_k= 2^bkck

[R]²_kk. (3.31)

Substituting from (3.30) it then follows thatPk = 2^Dfor allk.Thus in the optimal system which has orthonormal columns for the precoderF, the powersP_kare identical for allk.SinceP_trans=P

kP_k from (3.31) and(3.25), we therefore haveP_k =P_trans/Mfor allk.

In Chapter 2 we mentioned many examples of the GTD, such as SVD, Schur decomposition, GMD, and so on. Some of these have already appeared in the literature in different contexts. Each of

these serves as a specific realization of the optimal DFE transceiver acheiving minimum transmitted power, provided the bits are allocated as in Eq. (3.26). Each realization has a different choice ofr_k (= [R]_kk) satisfying the majorization condition (2.3), and in all cases, we restrict the precoderFto be the orthonormal choice (3.20).Gis chosen as in (3.21), andBas in (3.23). We now elaborate on these different realizations arising from different GTD forms ofH=QRP^†.

1. SVD Transceiver - the Linear Transceiver. The singular value decomposition (SVD) of the chan- nel matrix can be written as H = UΣV^†, whereUand Vare unitary andΣis a diagonal matrix. Since R = Σis diagonal, the feedback matrixB = 0from (3.23), and the system reduces to a linear transceiver as in Fig. 3.3. This optimal solution for linear transceivers was proposed in [48].

n White noise n White noise

^

J M

y

V

⁺

ɇ

N

G U

J J

F

N M

s ^ s

H

Figure 3.3: The SVD system, which represents a linear transceiver.

2. GMD Transceiver. The geometric mean decomposition (GMD) was introduced in [36]. The GMD of the channelHhas the formH=QRP^†, whereQandPhave orthonormal columns, andRis an upper triangular matrix. Furthermorethe firstMdiagonal elements ofRare identical, and equal to the geometric mean of theM dominant channel singular values. For the case where the specified error probabilitiesP_e(k)(hencec_k) are identical for allk, it follows from (3.26) that there is no need for bit allocation, that is,b_k =bfor allk.Unlike other special cases of the GTD such as the SVD, the question ofbkbecoming unrealizable (i.e., taking noninteger or negative values) therefore does not arise.

3. QR Transceiver - ZF-VBLAST System. The QR decomposition of the channel matrix can be written as H = QR, whereQhas orthonormal columns, andRis upper triangular. This yields a special case of the GTD transceiver, where the precoder isF=

Ñ IM

0 é

, and can be implemented at no cost. See Fig. 3.4. This system leads to the ZF-VBLAST system, widely used in MIMO wireless communication [135].

n

White noise

^s y +

I

_M

R Q G

( )

s

Ͳ

J M

0

R

N

Q G

K J

M

()

s~

M

B

M

H

Figure 3.4: The QR transceiver, which has the lazy precoder. This is identical to the ZF-VBLAST system.

The optimal transceiver design usually assumes thatHis known at the transmitter side. This assumption is not generally true. The more practical scheme would be the so calledlimited feedback scheme, in which the receiver uses a low rate feedback channel to tell the transmitter to use one of the precoders in a pre-determined codebook of precoders [56].

The QR based transceiver with bit loading is very suitable in limited feedback systems be- cause the precoder matrix is identity, and only the bit loading vector[b1 . . . bM]needs to be known.² The receiver can compute{bk}from (3.15), quantize it to the bit loading vector near- est to the vectors in a predetermined codebook, and feed back the index of that vector to the transmitter. The design of this codebook is an interesting problem, but is beyond the scope of this thesis. Intuitively, this scheme will perform better than limited feedback schemes using the Grassmann codebook [91, 56], since the Grassmann codebook aims to cover the Grassmann manifold [92] while the bit loading codebook only tries to coverM-vectors with integer valued entries. This intuition is supported by Monte Carlo simulations in Sec. 3.2.6.

2In the scheme described in [14], the power allocationPkalso should be fed back, but in the GTD based optimal system Pk=Ptrans/Mfor allkas shown at the end of Sec. 3.2.3.

It is reassuring to know that since all GTD-based systems are optimal when the bit loading formula is realizable, this QR based special case has no loss of optimality even though it offers a simple precoder and a simple way to perform limited feedback.

4. Bidiagonal Transceiver. It is well-known [25] that anyJ×P matrixHcan be factored asH= QRP^†, whereQandPhave orthonormal columns, andRhas the bidiagonal form

R=





























d1 f1 0 · · · 0 0 d2 f2 · · · 0 ... . .. ... . .. ... 0 · · · dP−1 fP−1

0 · · · 0 d_P 0



























 .

With the channel represented in this bi-diagonal form, the feedback matrix given in (3.23) becomes

B=























0 f1 0 · · · 0 0 0 f2 · · · 0 ... . .. ... ... ... 0 · · · 0 fM−1

0 · · · 0 0





















 .

Therefore the implementation of the DFE will be very simple since we need only to feed backoneprevious decision for detecting the current symbol. Also, the computation of the bidiagonal decomposition is inexpensive [25]. To the best of our knowledge, this kind of system has not previously been reported in transceiver literature.

Summarizing, any of the above four GTD-based systems achieves optimality. However, each one of them has some special features, which might be useful in different situations. Also, it is possible that other GTD-based systems exist with potential benefits in specific situations.