Combined Real Representation of Proper Signals and CN Signals

3.4 Description of Complex Multicarrier MIMO Systems

3.4.3 Combined Real Representation of Proper Signals and CN Signals

In order to be able to study proper signals and CN signals in the combined real representation (3.37), we define the following three spaces.

Definition 3.4.1. For a terminalT ∈ T withmT ∈ Nantennas in a complex multicarrier MIMO system withCcarriers, we define

• the space of symmetricBSC2matricesP`T=SBSC^Cm₂ ^T,

3.4. Description of Complex Multicarrier MIMO Systems 53

• the space of symmetric matrices with block-diagonal submatrices PT^CN=

( P =

A B B^T C

A,C ∈S^M,B ∈R^M×M )

withM= (mT)_∀c,

• the space of symmetricBSC2matrices with block-diagonal submatricesP`T^CN= `PT∩ PT^CN.

Proposition 3.4.1. The spacesP`T,P_T^CN, andP`_T^CNfrom Definition 3.4.1 are power shaping spaces.

Proof. The spaceP`T is a power shaping space due to Theorem 2.9.1, andP_T^CNis a power shaping space due to Proposition 2.8.1. As a consequence,P`T^CNis a power shaping space due to Proposition 2.4.3.

The following proposition states that proper signals (Definition 2.9.2), CN signals (Defini-tion 3.3.2), and signals that are both proper and CN can be considered as maximum-entropy signals with respect to these three power shaping spaces, respectively. The signalzTis used as a placeholder for any signal at nodeT, i.e., the transmit signalxT, the received signalyT, or the noiseηT. Recall that we call a multicarrier signal proper if the per-carrier signals are jointly proper (see Section 1.4.1), i.e., if the combined complex vector is proper.

Proposition 3.4.2. For a signalzT = stack(z_T^(c))and its combined real covariance matrix C_z_ˇ_T ∈ S^2Cm^T, we have the following equivalences using the power shaping spaces from Definition 3.4.1.

• Czˇ_T ∈P`T⇔zTis proper (but possibly CC)⇔z_T⁽¹⁾, . . . ,z_T^(C)are jointly proper,

• Czˇ_T ∈ P_T^CN ⇔zT is CN (but possibly improper)⇔z_T⁽¹⁾, . . . ,z_T^(C) are uncorrelated across carriers,

• C_z_ˇ_T ∈ P`T^CN ⇔zT is proper and CN⇔ z_T⁽¹⁾, . . . ,z_T^(C) are proper and uncorrelated across carriers.

Proof. If and only ifzTis proper, the pseudocovariance matrix˜CzTis zero, so that the second summand in (2.34) vanishes, i.e., Cz_ˇ_T is aBSC2 matrix. If and only ifz_T⁽¹⁾, . . . ,z_T^(C) are uncorrelated across carriers, the covariance matrixC_z_T and the pseudocovariance matrix˜CzT

are block-diagonal. We then obtain from (2.34) thatCzˇ_Tconsists of block-diagonal submatrices.

If and only ifzTis proper and CN,CzTis block-diagonal and˜CzT =0, i.e.,C_z_ˇ_TisBSC2with block-diagonal submatrices.

To see how the power shaping spaces defined above can be used in mathematical derivations, let us consider the following example.

Example 3.4.1. If a signalzTis proper, the combined real representationzˇ_Tis a maximum-entropy signal with respect toPT = `PT, so that we haveCzˇ_T=Pzˇ_T ∈ PT. Impropriety ofzT

can be described by adding a component from the orthogonal complement, i.e.,N_z_ˇ_T ∈ NT, so thatC_z_ˇ_T =P_z_ˇ_T+N_z_ˇ_T ∈ P/ T. Thus, we can check ifzTis proper by verifying whether the projectionproj_N_T(Czˇ_T)is zero, which is equivalent to checking whetherproj_P_T(Czˇ_T) =Czˇ_T.

If all signals in a system are proper, we may decide to study the system in the combined complex representation, which then complies to the definition of a RoP MIMO system. The following example shows that the framework of power shaping spaces can be applied in this case as well.

Example 3.4.2. If a proper signalzT = stack(z_T^(c)) is CN, the covariance matrix CzT is block-diagonal, i.e., the combined complex representationzTis a maximum-entropy signal with respect to the power shaping spacePT=H^(m^T⁾^∀c (the space of block-diagonal matrices with appropriate block sizes, see Example 2.8.5). A CC signal with correlations across carriers can be described by adding a component from the orthogonal complement, i.e.,N_z_T ∈ NT, so that CzT =P_z_T+N_z_T ∈ P/ T.

3.4.4 Compatibility with Power Shaping Spaces

For our further analysis, it is important to study how the compatibility assumption given in Definition 3.1.3 translates to the combined real representation (3.37) of a complex multicarrier MIMO system.

The combined real channel matrixH`_DSis aBSC2matrix since the corresponding complex channel is assumed to be linear, i.e., there is no conjugate linear component in (2.38). Due to the assumption of orthogonal carriers, it holds in addition thatH`_DSconsists of block-diagonal submatrices with dimensionsMD× MS, whereMD= (mD)_∀c, andMS = (mS)_∀c. This leads to the following Proposition.

Proposition 3.4.3. For the following choices ofPSandPD, the channel matrixH`_DSin(3.37) is compatible with(PD,PS):

• PS= `PSandPD= `PD,

• PS=PS^CNandPD=PD^CN,

• PS= `P_S^CNandPD= `P_D^CN.

Proof. The first bullet follows from Proposition 2.9.1, the second from Proposition 2.8.2, and the last from Proposition 2.7.4.

Therefore, the following statement holds no matter which of the three possible choices in Proposition 3.4.3 we make: the signal portionH`_DSxˇ_Sin the combined real received signal at nodeDis a maximum-entropy signal with respect toPDif the respective transmit signalxˇSis a maximum-entropy signal with respect toPS.

The fact that the constraints act only on the complex per-carrier covariance matrices C_x(1)

, . . . ,C_x(C) S

S∈S (but not on the pseudocovariance matrices or on cross-covariance matrices between per-carrier signals) translates to the combined real representation as follows.

Proposition 3.4.4. Let PS for each transmitter S ∈ S be chosen according to one of the possibilities in Definition 3.4.1. Then, the combined real representation(Cxˇ_S)_S∈S ∈ Qof the constraint(3.38)is compatible withN|S|

i=1PSi (in the sense of Definition 3.1.2).

3.4. Description of Complex Multicarrier MIMO Systems 55

Proof. The matrices C_x(c) S

, c = 1, . . . , C are the diagonal blocks of the covariance matrix CxS ofxS in (3.35). Thus, when calculating the covariance matrix ofxˇ_S in (3.37), we see from (2.34) that the elements of the combined real covariance matrixCx_ˇ_S that correspond toC_x^(c)

, c = 1, . . . , C are the diagonal blocks of the submatrices in theBSC2 component.

Thus, adding arbitrary off-diagonal blocks to the submatrices or adding an arbitraryBHSC2

component does not affect the feasibility ofCx_ˇ_Sin terms of the considered constraint.

If all signals in a system are proper and we decide to switch to the combined complex representation (3.35), we can exploit that the block-diagonal channel matrixHDSin (3.35) is compatible with(PD,PS)ifPS =H^(m^S⁾^∀candPD=H^(m^D⁾^∀c. The signal portionHDSxSin the received signal at nodeDis a maximum-entropy signal with respect toPDif the respective transmit signalxS is a maximum-entropy signal with respect toPS, i.e., a CN signal. When choosingPSin this manner at all transmitting nodesS∈ S, it is easy to verify that the combined complex version(CxS)_S∈S ∈ Qof the constraint (3.38) is compatible withN|S|

i=1PSi. 3.4.5 Mutual Information Expressions

When studying MIMO systems, we often have to evaluate mutual information expressions of the form given in (3.2). When applied to the combined real representation (3.37) of a complex multicarrier MIMO system, this expression reads as

I ( ˇx_S⁰; ˇy_D) = 1 withS⁰ ∈ SD. If all transmit signals are maximum-entropy signals with respect to one of the power shaping spaces defined above, this expression can be simplified as follows.

Proposition 3.4.5. The mutual information betweenxˇ_S⁰, S⁰ ∈ SDandyˇ_Din(3.44)equals

Proof. IfC_x_ˇ_S ∈ P`S ∀S ∈ SDandC_η_ˇ_D ∈ P`D, all matrices in (3.44) areBSC2. Therefore, we can use the equivalences from Section 2.9.5 to get (3.45). IfCxˇ_S ∈ P_S^CNfor allS ∈ SD

and Cη_ˇ_D ∈ PD^CN, all matrices in (3.44) consist of block-diagonal submatrices. Thus, we can use a permutation matrix (as in the proof of Proposition 2.8.1) to bring the matrices whose determinants are to be calculated to block-diagonal form without changing the values of the determinants. We then obtain (3.46) by exploiting the fact that the determinant of a block-diagonal matrix equals the product of the individual determinants of the blocks. This fact is also used to obtain (3.47).

Chapter 4

Point-to-Point Transmission: the Gaussian MIMO Channel

In this chapter, we consider a single-user MIMO transmission as in [29], which we refer to as a (Gaussian) MIMO channel. For the complex multicarrier MIMO channel, the optimal transmit strategy under the assumption of proper and/or CN noise can be concluded from [29]. It turns out that the transmit signal should be proper and/or CN as well. We rederive and generalize these results using the framework of power shaping spaces in order to see the functioning of this framework in a rather simple scenario before turning our attention to the more complicated multiuser systems, and in order to obtain some fundamental insights that can be reused at several points later on. It turns out that the abovementioned existing results can be obtained as simple special cases of a more general theorem based on the notion of maximum-entropy signals.

In addition, we study optimal transmit strategies for cases with reduced-entropy noise, such as improper noise or CC noise. By adapting the transmit strategy to the properties of the noise, performance gains can be obtained in these cases. However, even if the transmitter is oblivious of the noise properties and sticks to a maximum-entropy transmit signal, the achievable rate in case of reduced-entropy noise is still higher than for maximum-entropy noise. This remarkable aspect is shown at the end of this chapter, where we prove that maximum-entropy noise is the worst-case noise in the Gaussian MIMO channel.

All results in this chapter are first derived in a general manner for a RoP MIMO system without specifying which particular kind of reduced-entropy signals we consider. Then, they are specialized to the complex multicarrier MIMO channel, where we are interested in particular in improper signaling and CC transmission.

4.1 System Model and Channel Capacity

As we have only one transmitter Sand one receiverD, we can introduce the abbreviations y =y_D,H =H_DS,x=x_S,η=η_D, andM =M_D. The general model for RoP MIMO systems in (3.1) then simplifies to the description of the RoP MIMO channel

y=Hx+η. (4.1)

The case of the complex MIMO channel with proper noise was studied in detail in [29], and the derivations given therein analogously hold in the real-valued MIMO channel.

A rateris achievable in the RoP MIMO channel if it does not exceed the mutual information (2.9) between the transmitted signalxand the received signaly[29], i.e., we can achieve

r= I (x;y) = h (y)−h (y|x) = h (y)−h (η). (4.2) Sinceh (η)does not depend on the transmit strategy,rcan be maximized by maximizingh (y).

For any fixedC_x, which leads to a fixed

C_y =HC_xH^H+C_η (4.3)

h (y)is maximal ifyfollows a Gaussian distribution (see [59,62] for a real-valued system and [29,63] for a complex system). Under the assumption of Gaussian noise that is independent of x, we can choosexto be Gaussian in order to achieve this. The constraints on the transmit strategy affect onlyC_x, and the above reasoning applies for anyC_xincluding the optimal one.

Thus, a Gaussianxmaximizes (4.2).

Remark 4.1.1. For complex signals, [29,63] show that the entropy-maximizingyis not only Gaussian, but also proper. In [29], it was concluded from this that proper transmit signals are optimal in a complex MIMO channel with proper noise. When we study the complex multicarrier MIMO channel later on, we obtain the same result as a simple special case of a more general theorem, which we prove based on the framework of power shaping spaces. As long as we consider the RoP MIMO channel, we are not interested in the question of (im)proper signals since we consider either a setting where all signals are proper by assumption (see Definition 3.1.1) or a real-valued setting.

As the optimalxis Gaussian, we can use the mutual information expression for Gaussian signals given in (3.2), and we obtain

r = I (x;y) =µlog₂ det C_η+HC_xH^H

detC_η . (4.4)

The aim is then to find the optimal transmit covariance matrix

Cmaxx0 r s.t. C_x ∈ Q (4.5)

whereQis defined as in (3.3). The solution to (4.5) is the channel capacity of the RoP MIMO channel.

4.2 Optimality of Maximum-Entropy Transmission

Our aim is now to study how knowledge about noise properties can enable us to draw direct conclusions about the optimal transmit signal without even carrying out the optimization (4.5).

In particular, we assume that the noise is a maximum-entropy signal with respect to some power shaping space. According to the following theorem, the optimal transmit signal then is a maximum-entropy signal, too.

4.3. Optimal Transmit Covariance Matrix under a Sum Power Constraint 59

Theorem 4.2.1. LetPSandPDbe power shaping spaces fulfilling the compatibility assumption (Definition 3.1.3). If the noise is a maximum-entropy signal with respect toPD, the capacity of the RoP MIMO channel(4.1)is achieved by a maximum-entropy transmit signal with respect to PS.

Proof. LetPx= proj_P_S(Cx)andNx= proj_N_S(Cx), so thatCx=Px+Nx. Sincexand ηare independent, andH is compatible with(PD,PS), we haveCy=Py+Nywith

P_y =HP_xH^H+P_η∈ PD N_y =HN_xH^H+N_η ∈ ND (4.6) whereN_η = proj_N_D(C_η) =0andP_η = proj_P_D(C_η) =C_ηby assumption. We can assume Px0due to Proposition 2.5.1. For any givenPx, the power shaping matrixPyis fixed, and the entropy ofyis maximized byNy=0(Theorem 2.5.2), which can be achieved by choosing N_x=0. To see that this choice is feasible, note thatC_x =P_x+N_x 0holds forN_x =0 and that the compatibility assumption makes the covariance constraint equivalent toPx∈ Q, which does not depend onNx. This reasoning holds for anyPx0including the optimal one.

Thus, we can always chooseN_x =0, i.e.,C_x ∈ PS.

4.3 Optimal Transmit Covariance Matrix under a Sum Power Constraint

While the statement of Theorem 4.2.1 could be shown regardless of the choice ofQas long as the compatibility assumption is fulfilled, we need to know the constraint set if we want to carry out the covariance optimization (4.5). In this section, we consider the optimization for the special case of a sum power constraint (3.4). For more complicated constraints, where an analytic solution may not be easy to obtain, the optimization could be solved by a convex programming solver since the objective function is concave inC_x and the constraint set is convex by assumption (see Section 3.1.1).

In [29], the optimal transmit covariance matrix Cx under a sum power constraint was derived for the complex MIMO channel with proper noise, and the derivation can be transferred to the real-valued MIMO channel. We can express the achievable rate (4.4) as

r =µlog₂det I_M_S+H^HC_η⁻¹HCx

(4.7)

where we have used that

det(I_m+AB^H) = det(I_n+B^HA) (4.8) forA,B ∈C^m×n(e.g., [29]).

Based on the eigenvalue decomposition

Udiag(λ_i)U^H=H^HC_η⁻¹H (4.9) the optimal transmit covariance matrix can be expressed as [29]

Cx=Udiag(pi)U^H (4.10)

with the power allocation

p_i = max

ω− 1 λi

, 0

. (4.11)

In this so-called waterfilling solution (see, e.g., [62]), the water levelωmust be chosen such that the sum power constraint (3.4) is fulfilled with equality. The fact thatCxmust have the modal matrixU can be shown using the Hadamard inequality [68, Sec. 7.8].

Unlike in the previous section, we have not made any assumptions on the noise covariance matrix Cη. Therefore, this solution is generally valid no matter whether we are facing maximum-entropy noise or reduced-entropy noise.

4.3.1 Maximum-Entropy Noise

As a plausibility check, we can plug in the assumption of maximum-entropy noise into the solution forC_x from above. Due to Theorem 4.2.1, the resultingC_xmust then correspond to maximum-entropy transmission. This can be verified as follows.

If Cη ∈ PD, we have C_η⁻¹ ∈ PD by Corollary 2.4.2 andP = H^HC_η⁻¹H ∈ PS by Proposition 2.7.1. For any given value ofω, (4.10) describes a mappingP 7→ C_x with a structure as in Theorem 2.4.1 withf defined in (4.11). Thus,Cx∈ PS.

Note that we have used other properties of power shaping spaces than in the proof of Theorem 4.2.1, i.e., we have found an alternative way of showing the statement of Theorem 4.2.1 for the special case of a sum power constraint.

4.3.2 Reduced-Entropy Noise

If the noise covariance matrix does not lie in a power shaping space fulfilling the compatibility assumption, the conclusion that the optimal transmit signal should be a maximum-entropy signal is not possible. In the following, we see that the optimal transmit signal is then a reduced-entropy signal in general.

As in the proof of Theorem 4.2.1, the aim is still to maximize the entropy ofy, i.e., it would still be desirable to chooseC_x such thatN_y=0. However, due to the assumption of reduced-entropy noise, we haveNη 6=0in (4.6), which changes the situation significantly.

To makeNyvanish, we would have to chooseNx such thatHNxH^H=−Nη, but this is not always possible. On the one hand, the range space ofH might be a strict subspace of C^M (orR^M for a real-valued system). In this case, it can happen that the equation does not have a solution. On the other hand, when choosingNx 6= 0, the positive-semidefiniteness constraintC_x=P_x+N_x 0introduces a coupling betweenP_xandN_x, so that focusing solely on achievingNy=0might no longer be the best strategy. It can even be impossible to find a feasible combination ofP_xandN_xthat makesN_yvanish, even though the covariance constraintC_x ∈ Qdoes not act on N_x directly. To get a better intuition about this, let us consider the following minimal example.

Example 4.3.1. LetPS = PD = {P |P = αI₂, α ∈ R}, and consider the RoP MIMO channel with channel matrixH =I₂and reduced-entropy noise with covariance matrix

Cη=

4.4. Worst-Case Noise 61

N_x =−N_η, which would require P_x I₂ in order to fulfill the positive-semidefiniteness constraintPx+Nx0(sincePxis a scaled identity and must fulfillPx −Nx). However, this is not possible due to the power constraint. The best that can be done in terms of compensating the reduced-entropy noise is to chooseN_x =−¹₂N_ηandP_x= ¹₂I₂. It can be verified using(4.10)that this best-effort approach is indeed the optimal transmit strategy in the considered scenario. If the power constraint is relaxed totr[Cx]≤2, the optimal solution is N_x=−N_ηandP_x =I₂, and we obtainN_y =0.

If the system is larger or if we consider more general channel matrices and more complicated power shaping spaces, the optimal solution can of course not be identified so easily, but the qualitative result will be the same. The optimal transmit covariance (4.10) will in general have a nonzero entropy reduction componentN_xthat is chosen in a way that it compensates the entropy reduction componentNηof the noise covariance matrix at least partially.

4.4 Worst-Case Noise

It is well known that Gaussian noise is the worst-case noise in a single-antenna point-to-point channel with additive noise (see, e.g., [103], [62, Prob. 9.21]). In [66, 67], this result was extended to more general wireless networks, and it was remarked in [67] that the results can be easily transferred to MIMO systems. The intuition behind these results is that the worst possible disturbance (with fixed second-order properties) is the one which has the highest entropy, i.e., the Gaussian distribution.

It therefore is to be expected that the worst possible noise among all Gaussian noise distributions, i.e., the one with the worst-case choice of the second-order properties, has to be a maximum-entropy signal in the sense of Definition 2.5.1. Even though this seems to be intuitively clear, we have to be aware that (4.2) is a difference of two differential entropies that both get smaller if the entropy of the noise is reduced. For a formal proof, we have to study which of the two terms is affected more strongly.

We still consider the RoP MIMO channel (4.1), i.e., we assume the noise to be proper in case of a complex MIMO channel. Later on, we will present a result stating that proper noise is the worst-case when compared to improper noise (see Corollary 4.5.5), but at the current point, the noise is proper Gaussian by assumption, and we are only interested in the worst-case noise covariance matrix. For the real-valued MIMO channel, the real-valued Gaussian distribution is known to be the worst-case noise distribution (see above), and we ask for the worst-case noise covariance matrix as well.

We are interested in two types of worst-case noise optimization. In the first one, we assume the transmit covariance matrix to be fixed and ask for the noise covariance matrix that minimizes the achievable rate, i.e., we need to solve

Cminη0 r s.t. C_η∈ Qnoise (4.13) withrfrom (4.4) for a fixedC_x ∈ Q. As a second question, we ask for the worst-case noise in the case that the transmit strategy is adapted to the noise properties in an optimal manner, i.e.,

Cminη0 max

Cx0 r s.t. C_x ∈ Q and C_η∈ Qnoise. (4.14)

In both formulations,Qnoiseis a convex set of Hermitian matrices such that we obtain a compact constraint set with nonempty interior if we in addition demand thatCη0.

Unlike previous publications, such as [86, 104–106], we are not interested in solving worst-case noise optimizations for particular choices of the constraint sets and the channel matrices. Instead, we aim at deriving structural properties of the worst-case noise in terms of the concept of power shaping spaces. The result is stated in the following worst-case noise theorem, whose proof uses similar arguments as [28], but in a more general framework.

Theorem 4.4.1. LetPSandPDbe power shaping spaces fulfilling the compatibility assumption (Definition 3.1.3), and let the constraintC_η ∈ Qnoisebe compatible withPD(in the sense of Definition 3.1.2). Ifxis an arbitrary maximum-entropy signal with respect toPS,(4.13)is minimized by maximum-entropy noise with respect toPD.

Proof. LetP_ε =HC_xH^H+εI_M. Using the compatibility assumption, we haveP_ε ∈ PD

sinceCx ∈ PSand sinceI_M ∈ PD. As determinant and logarithm are continuous functions, we can calculate the achievable rate fromr(C_η) = lim_ε→0 r_ε(C_η)with

rε(Cη) =µlog₂det (Pε+Cη) detCη

=µlog₂det (IM +Cηε)

Im Dokument Reduced-Entropy Signals in MIMO Communication Systems (Seite 52-0)