Duality Theorem

(7.21) which can be shown to be equal to (7.14).

7.2.2 Zero-Forcing in the Uplink If zero-forcing constraints

G^H_kH_UL,jT_j =0, ∀k,∀j6=k ⇔ G^H_kH_UL,jCx_jH_UL,j^H G_k =0, ∀k,∀j6=k (7.22) are imposed, we can use a receive filterG^H_k =G^H_k,ZF of sizeM ×D_k, whose rows are an orthonormal basis of the null space ofP

j6=kHUL,jCxjH_UL,j^H , whereDkis the dimensionality of this null space. We obtain

r_ZF,k =







µlog₂det

I_Dk+_µ¹G^H_kH_UL,kCx_kH_UL,k^H G_k

ifD_k>0

0 otherwise (7.23)

withC_xk = T_kT_k^H, where we have usedG^H_kG_k = I_Dk and the fact that the uplink noise covariance matrix is a scaled identity. The resulting reduced rate region isRUL,TIN−ZF(Q)⊆ RUL,TIN(Q). If a smaller number of streamss_k< D_kis transmitted, we can again combine a zero-forcing filter with a second filter stageG˜^H_k of sizes_k×D_k, so thatG^H_k = ˜G^H_kG^H_k,ZFhas sizes_k×M.

7.2.3 Duality Theorem

It was shown in [51], [126, Sec. 3.3] that the complex MIMO BC and MIMO MAC share the same achievable rate region under a sum power constraint if linear transceivers and proper complex signals are employed in both systems. Since the same derivation can also be applied to real-valued systems, we have the following duality lemma for the RoP MIMO BC-TIN.

Lemma 7.2.2. For any given channel matricesH_k^Hand noise covariance matricesCη_k, the rate region of the RoP MIMO BC-TIN(7.3)and the rate region of the dual RoP MIMO MAC-TIN (7.16)under a sum power constraint are the same, i.e.,RTIN(Q) =RUL,TIN(Q).

Sketch of Proof. A proof for complex systems with proper signals was given in [51, Th. IV.1]

and can be transferred to real-valued systems. We summarize the main steps in the following.

To study the uplink-to-downlink conversion, letG^H_k =G^H_k,MMSE be the uplink MMSE receive filters (7.19), and use a modal matrixW_kofG^H_kH_UL,kT_kto decorrelate the streams by

7.2. Uplink-Downlink Duality 95

means of the filters from (7.20). Note that we can assumee^T_sT_k⁰T_k^0He_s>0for all streamssof all usersk, which can be ensured by removing zero columns fromT_k⁰ (and the corresponding rows fromG^0H_k ) [51], i.e., we assumeT_k⁰ to be a full-rank tall (or square) matrix.

From (7.21), the rate of thesth stream of userkis then given byr_k,s^UL =µlog₂(1 +γ_k,s^UL) with the signal-to-interference-and-noise ratio (SINR)

γ_k,s^UL= (e^T_sG^0H_k H_UL,kT_k⁰e_s)²

e^T_sG^0H_k X_kG⁰_ke_s (7.24) in the uplink. Assuming stream-wise decoding in the downlink using the filters

B_k=G⁰_kA_k R^H_k =A⁻¹_k T_k^0H 1

µC_ηk −¹₂

(7.25) where A_k = diag(α_k,s), α_k,s ∈ R, the intra-user interference in the denominator of (7.6) cancels out, and we obtain the per-stream rater^DL_k,s =µlog₂(1 +γ_k,s^DL)with the SINR We have used (7.2), (7.13), and (7.25) to obtain this reformulation.

To achieveγ^UL_k,s =γ_k,s^DL(and thusr_k,s^UL=r_k,s^DL), we need e^T_s

A_kG^0H_k X_kG⁰_kA_k−T_k^0HC_z⁰

kT_k⁰

e_s= 0 (7.28)

for all streamssof all usersk. As shown in detail in the proof of [51, Th. IV.1], this condition can be rewritten as a system of linear equations in the variablesα²_k,s. Based on results on nonnegative matrices from [157], it can then be shown that there exists a unique set of positive scalarsα²_k,ssolving this system of equations. Moreover, it can be shown that plugging in this solution into (7.25) leads to downlink transmit filters that require the same sum transmit power as the original uplink transmit filters.

The proof is completed by repeating the same reasoning for the downlink-to-uplink transformation, where the transformation rule is given by

T_k= 1

µC_ηk ¹₂

R⁰_kA¯_k G^H_k = ¯A⁻¹_k B_k^0H (7.29) with R⁰_k and B_k⁰ from (7.8). The scaling factors contained in the diagonal matrix A¯_k are obtained from solving

As a basis for the further derivations in this chapter, we need to analyze the relation between maximum-entropy transmission in the downlink and in the dual uplink. The result is stated in the following theorem.

Theorem 7.2.1. Let(Pk =PUk)_∀kandPBbe power shaping spaces fulfilling the compatibility assumption (Definition 3.1.3). If the noise vectorsη_kare maximum-entropy signals with respect toPk, respectively, a rate vector is achievable with sum powerQin the RoP MIMO BC-TIN with maximum-entropy per-user input signals (with respect to the power shaping spacePB) if and only if it is achievable with the same sum powerQin the dual RoP MIMO MAC-TIN with maximum-entropy transmit signals (with respect to the power shaping spaces(Pk)_∀k).

Proof. We need to show that the transformation derived in the proof of Lemma 7.2.2 leads to maximum-entropy transmission in the downlink if maximum-entropy transmission is used in the uplink and vice versa. For later use, note thatC⁻

1

η_k2 ∈ Pkby Corollaries 2.4.1 and 2.4.2, andH_UL,kis compatible with(PB,Pk)(see Proposition 2.7.2).

For the uplink-to-downlink transformation, we have to show that all B_k from (7.25) correspond to maximum-entropy transmission with respect to PB if all T_k correspond to maximum-entropy transmission with respect to Pk. To this end, we need to study the transformations between several involved power shaping spaces (amongst others due to the transformation withW_k) and the properties of the matrixA_kobtained from solving (7.28).

Consider the singular value decompositionT_k = U_kΣ_kV_k^H with a tall matrix U_k and square matricesΣ_k andV_k. If maximum-entropy transmission is employed by userk, we have U_kΣ_k²U_k^H = T_kT_k^H ∈ Pk. Then, P¯k = {P¯ |P¯ = V_kU_k^HP U_kV_k^H, P ∈ Pk} is a power shaping space due to Proposition 2.6.1, andT_kis compatible with(Pk,P¯k). The latter can be seen by noting thatΣ_k ∈P˜k ={P˜ |P˜ =U_k^HP U_k, P ∈ Pk}(Corollary 2.6.2 and Corollary 2.4.1) and thatP¯is obtained fromP˜by applying a transformation with the unitary matrixV_k. Moreover, if maximum-entropy transmission is employed by all users,X⁻¹∈ PB

by Corollary 2.4.2, andG^H_kH_UL,kT_k=T_k^HH_UL,k^H X⁻¹H_UL,kT_k∈P¯k.

Decorrelating the streams by means of (7.20) leads to another power shaping space P_k⁰ = {P⁰ |P⁰ = W_k^HP W¯ _k, P¯ ∈ P¯k}. In case of maximum-entropy transmission with T_k⁰T_k^0H =T_kT_k^H∈ Pk, the matrixT_k⁰ is compatible with(Pk,Pk⁰). Moreover, if maximum-entropy transmission is employed by all users, G^0H_k = T_k^0HH_UL,k^H X⁻¹ is compatible with (P_k⁰,PB) (see Proposition 2.7.2). For later use, note that we can chooseW_k such that the projectionproj_P⁰

ktoPk⁰ commutes with the projectionproj_diagto the space of diagonal matrices (see Corollary 2.6.4).

Let Ξ_k = T_k^0HH_UL,k^H X_k⁻¹H_UL,kT_k⁰. Applying the matrix inversion lemma (7.10) to X⁻¹ = (X_k+H_UL,kT_k⁰T_k^0HH_UL,k^H )⁻¹, we have

X⁻¹ =X_k⁻¹−X_k⁻¹H_UL,kT_k⁰ I_M+T_k^0HH_UL,k^H X_k⁻¹H_UL,kT_k⁰−1

T_k^0HH_UL,k^H X_k⁻¹ (7.32) and we can show thatΞ_kis diagonal (e.g., [51]) since

Ξ_k−Ξ_k(I_M +Ξ_k)⁻¹Ξ_k=G^0H_k H_UL,kT_k⁰ (7.33) is diagonal. Moreover, we have that

Φ_k=G^0H_k X_kG⁰_k =T_k^0HH_UL,k^H X⁻¹X_kX⁻¹H_UL,kT_k⁰

=Ξ_k−2Ξ_k(I_M +Ξ_k)⁻¹Ξ_k+Ξ_k(I_M +Ξ_k)⁻¹Ξ_k(I_M +Ξ_k)⁻¹Ξ_k (7.34)

7.2. Uplink-Downlink Duality 97

is a nonnegative diagonal matrix. From the first line of (7.34), we can see thatΦ_k∈ P_k⁰ if all users employ maximum-entropy transmission sinceG^0H_k is compatible with(P_k⁰,PB)in that case. We can rewrite (7.28) as

k andproj_diagcommute, the unique solution for the set of scalarsα²_k,smust solve the two systems of equations The homogeneous system of equations (7.38) is solved by settingN_k⁰ = 0for allk. From this, we see that the solution to (7.36) fulfillsA²_k = P_k⁰ ∈ P⁰ for all usersk. As a result, B_kB_k^H=G⁰_kA²_kG^0H_k ∈ PB, i.e., the solution corresponds to maximum-entropy transmission in the downlink.

The proof is completed by repeating the same reasoning for the downlink-to-uplink transformation.

Due to this Theorem, we can prove all further statements in this chapter via the dual uplink whenever this is more convenient. If we do so, no further justification for switching to the uplink is given.