Twomode limited feedback blockfading adaptive transmission with ...

WIRELESS COMMUNICATIONS AND MOBILE COMPUTING Wirel. Commun. Mob. Comput. 2012; 12:352–366 Published online 12 May 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/wcm.968

RESEARCH ARTICLE

Two-mode limited feedback block-fading adaptive transmission with minimum guaranteed rate in MIMO channel Milan Knize and Jan Sykora* Czech Technical University in Prague, Faculty of Electrical Engineering, Technicka 2, Prague 6, Czech Republic

ABSTRACT This paper develops an adaptive scheme relying on a part-time availability of the channel state information and supplying the constant minimum rate achievable for an arbitrary fading realization. The two-mode transmission is investigated with the instantaneous perfect channel state information being known at the transmitter in the limited portion of time. We derived a general procedure for a suitable probabilistic description of the channel under such a scheme. The first mode is without any feedback and it uses a simple uniform symbol energy assignment. The second one is the emergency mode with the perfect instantaneous channel state information. It is implemented by the channel inversion (or eigenmode sub-space channel inversion) guaranteeing the minimum required rate regardless of the channel observation. The parameters of the transmission are the total available transmitted symbol energy, the portion of time the perfect feedback channel is used, the minimum required rate, and the average symbol energy transmitted in the first and the second mode. For these cases, we have found very essential trade-off among the system parameters and also we have derived a general procedure how to evaluate long-term performance. Copyright © 2010 John Wiley & Sons, Ltd. KEYWORDS adaptive; channel capacity; MIMO *Correspondence Jan Sykora, Czech Technical University in Prague. E-mail: [email protected]

1. INTRODUCTION The partial CSIT (Channel State Information at Transmitter) adaptation is widely discussed in many papers. The influence on achievable capacity, optimal coding and symbol energy allocation scheme is investigated for different models of the CSIT distortion. A common model for this partial knowledge of the feedback channel state information encompasses the delay in the feedback channel and imperfect channel prediction [1,2], channel estimation error [3,4], limited bit representation of the CSIT (quantized feedback) [5], mean or covariance feedback [6,7], etc. Most of these contributions have been so far focused solely on the quality of feedback information delivered to the transmitter.

1.1. Paper contribution This paper does not deal with quality of channel feedback information in the similar sense as the previously mentioned contributions. We want to address different issue regarding 352

feedback information knowledge at the transmitter which is the limited time availability of channel feedback information at the transmitter. This concern is actually complementary to the question of channel information quality. Therefore our approach can be further applied on top of all mentioned models of limited quality of CSIT as complement constraint feature of the real system. With our two mode scheme we can substantially reduce the total average amount of information which needs to be fed-back to the transmitter. The reduction is not driven by partial nature of feedback information but by the substantial relative reduction of time we need to convey this information to the transmitter. In numerical section we will show that this reduction is very substantial. Our partial knowledge of CSIT targets the time restriction of the feedback channel utilization. The robust system with limited feedback channel usage is developed and the performance deterioration is fully qualified. We believe such CSIT impairment has not been investigated rigorously so far. The channel and coding conditions are similar as used e.g., in Reference [8]. But in our approach, the Copyright © 2010 John Wiley & Sons, Ltd.

M. Knize and J. Sykora

assumption of the perfect CSIT available all the time is omitted (see Reference [9]). We investigate the impact of the time-limited CSIT availability on the overall system performance. Moreover, strictly zero transmission outage, fixed supported data rate, and maximum on-line processing simplicity and limited off-line optimization complexity is required. Our approach significantly reduces the overall on-line processing complexity compared to optimal space– time water-filling which requires on-line optimization procedures to be run once the feedback information is used at the transmitter. This simplicity is achieved by the choice of uniform power allocation for mode without channel state information and simple channel inversion policy for the mode using channel information at the transmitter. Such a technique moreover a priori guarantees the fixed dimensionality of the coding book achieving the capacity performance of the system because a fixed number of eigen-modes are used during the whole transmission in adaptive mode regardless of the particular channel fading observation. Throughout our investigation we assume MIMO (Multiple Input Multiple Output) channel in both potential options—symmetric (having the same number of transmitting and receiving antennas) and asymmetric (with different number of transmitting and receiving antennas).

Two-mode limited feedback block-fading adaptive transmission

1.2.1. Time-limited usage of feedback channel. The limited feedback channel usage is a reasonable requirement since in the significant portion of time the feedback channel is free to be used e.g., for the relevant data transmission of other users instead of the channel state information only. This might be an important point especially for duplex-like transmission between two transceivers and in complex networks with large number of messages passing among nodes. Obviously, the motivation is to save scarce resources such as time or frequency bandwidth which are necessary to be at the disposal for the perfect channel state transmission from the receiver to the transmitter. So the motivation is very similar to papers addressing various kinds of partial or imperfect feedback channel tracking. To limit amount of information we have to track back to ensure the capacity or given transmission parameters fulfillment. For example, limited feedback like quantization could reduce the required channel feedback capacity to 3–5 bits per feedback channel usage. With our approach we can additionally save channel feedback resources through switching the feedback off for a substantial part of the transmission. Keep in mind, the two-mode feedback transmission involves both substances, conveying the perfect channel state information under mode 2 as well as single bit information transmission regarding the switching between two modes.

1.2. Motivation, system parameters, and constraints In this subsection we introduce our motivation in more details and also clarify what are the desired features of designed two-mode transmission system. Motivation and required performance characteristics are then precisely translated into key desired constraints we have to reflect in our overall rigorous investigation. It comprehensively describes what are the key constraints we would like to fulfill in our two-mode transmission scheme design. Our motivation is to develop very simple and robust transmission scheme with limited feedback channel usage and minimal overall coding, on-line and off-line processing complexity. Let us structure the whole motivation and desired system features into four specific constraints which are separately discussed in four following subsections. The first is about time-limited availability of channel information at the transmitter, the second about the constraints on coding and channel model illustrating delay-limited nature of the system and desired simplicity of coding procedure achieving system capacity. The third is about minimum single block rate condition which in consequences defines zero-outage per block probability for arbitrary fading realization requirement. The last subsection is about our requirement on overall limited on-line processing and offline optimization complexity. All these requirements will lead to the explanation carried out in Subsection 2.4 clarifying our choice of eigen-mode channel inversion as the right applicable adaptive scheme for emergency mode 2 relying on feedback channel information at the transmitter.

1.2.2. Channel model and coding approach. The block-fading channel model with the long term (ergodic) observation is adopted. The symbol energy constraint is considered as the maximum long term available symbol energy. Each block is assumed to support reliable communication (large number of symbols per block) so that we are able to evaluate the capacity development per one block as well as the long term (non-random scalar) capacity value. Transmission outage is considered as the event when the capacity per single block is below the required rate (see References [10,11]). This is the crucial definition for our entire investigation similar to the service outage definition in Reference [8]. Our approach tries to consider both temporal qualities, the long-term performance (ergodic capacity), as well as the block-term related values such as the probability of the first and second mode and transmission outage performance. The second constraint we impose on our coding approach is the fixed dimensionality of the codebook achieving the capacity of the channel. A codebook with fixed dimensionality of codewords is much simpler to construct and as such leads to the overall simplicity and practical implementation of real codes which might be applied in the real implementation. That again disqualifies some optimal adaptive scheme from the discussion about potential candidates for transmission policy utilizing channel state information at the transmitter. Details are discussed in Subsections 2.3 and 2.4.

Wirel. Commun. Mob. Comput. 2012; 12:352–366 © 2010 John Wiley & Sons, Ltd. DOI: 10.1002/wcm

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Two-mode limited feedback block-fading adaptive transmission

1.2.3. Minimum single block rate condition. We want to develop a transmission scheme supporting a given transmission rate for all single blocks of overall transmission . This is equivalent to the solution yielding strictly zero transmission outage probability. We are interested in the mutual relation between the portion of time when the perfect CSIT is necessarily required and the minimum supported rate. The next task is to reveal the capacity of such scheme with the minimum required rate and even the mutual trade-off among the ergodic capacity, minimum rate and frequency of the feedback channel usage. Section 7 is about selected optimization procedures where given sub-sets of the parameters are assumed to be exogenously given as system required performance. Section 7 characterizes the generic applicability of our approach in particular system design. The solution relies on the combination of two different communication modes. The scheme with no symbol energy control (mode 1) when no CSIT is available is combined with the channel inversion (mode 2) with the perfect CSIT. The channel inversion is understood as emergency mode and it is used for bad channel state occurrences to guarantee the achievable rate even for very deep and harmful fades. Such scheme is shown to offer the desired minimum rate per block with reasonable frequency of feedback channel usage.

1.2.4. System complexity constraints. We want to develop a scheme with the minimum requirements imposed on on-line processing complexity. Both preferred transmission modes (with and without channel state information available at the transmitter) should be applicable without any need to perform online optimization procedures per each realization of the channel. Also we prefer such adaptation rule which leads to reasonably simple off-line optimization tasks. Off-line optimization has to be run before the actual transmission to get the system parameters which are necessary to setup the system to meet desired on-line transmission performance properly. Both these requirements clearly lead to our subsequent choice of uniform energy assignment for mode without channel state known at the transmitter and channel inversion as the simplest adaptation rule meeting our previously mentioned requirements. Details of our choice are described in Subsections 2.3 and 2.4.

2. SYSTEM MODEL In this section we describe the MIMO channel model used throughout the whole investigation and channel decomposition into set of eigen-modes. Based on the required system constraints from Section 1.2, we also describe the rationale of our choice of transmission policies which are applied in mode 1 and mode 2. Special attention is given to explaining why, given our motivation and system constraints, the channel inversion is the right policy for emergency mode 2. 354

M. Knize and J. Sykora

2.1. MIMO channel model The common MIMO (multiple-input multiple-output) flat fading channel model with NT transmitting and NR receiving antennas can be described by the input–output relation y = Hx + w

(1)

Vector y is the received column vector with dimensionality NR × 1 whereas transmitted vector is denoted as x and has the dimensionality NT × 1. The channel matrix H is of the dimension NR × NT with IID (independent identically distributed) entries with complex circularly symmetric Gaussian distribution and unity variance (IID Rayleigh fading). The noise vector w has the covariance matrix Rw = σ 2 I. All vectors are considered as columns. The average transmitted energy constraint is expressed by the ¯ The inequality E[xH x] ≤ E¯ or equivalently tr(E[xxH ]) ≤ E. notation of symbol energy is given as follows. The symbol E¯ denotes the total constrained average energy which is at the disposal for one multidimensional (over spatial dimension or equivalently over eigenmode space) symbol. The symbol E (resp. E(H)) is the energy which is allocated for given channel realization based on the considered energy allocation algorithm and finally the symbol Ei (or Ei (H)) is the energy allocated in ith eigenmode of the full eigenmode space. We have to emphasize once again that we understand Equation (1) as the model acquired from the decomposition of continuous time input and output into the constellation space orthonormal basis and thus we use the transmit energy per symbol constraint. Otherwise, when the sufficient statistic would be formed from the correctly obtained samples of continuous time input and output quantities than we can write the above mentioned condition for average transmitted power constraint.

2.2. MIMO channel decomposition The convenient SVD (singular value decomposition) of the channel matrix yields y = UDVH x + w and enables us to perform the adaptation in eigenmodes. The eigenmode space is of the dimension Ne = min[NT , NR ]. Further we will use the parameter Nn = max[NT , NR ]. Matrices U ∈ CNR ×NR , V ∈ CNT ×NT are unitary and D ∈ RNR ×NT is nonnegative and diagonal with entries given as non-negative square roots of eigenvalues of matrix HHH . The ordered eigenvalues are denoted explicitly with index as λi and it holds λ1 ≥ λ2 ≥ . . . ≥ λNe . When unordered full space of eigenmodes is assumed the symbol λ is used to denote the general eigenmode gain with the common shared marginal distribution. Accordingly, λ˜ will denote the general subchannel gain of unordered subspace of eigenmodes. The eigenmode equivalent variables are y˜ = UH y, x˜ = VH x, w ˜ = UH w and the equivalent input–output equation is

Wirel. Commun. Mob. Comput. 2012; 12:352–366 © 2010 John Wiley & Sons, Ltd. DOI: 10.1002/wcm

M. Knize and J. Sykora

y˜ = Dx˜ + w. ˜ The energy adaptation along the vector x˜ is performed.

2.3. Mode 1—uniform symbol energy allocation A common uniform symbol energy allocation rule is applied in mode 1 with no channel state information known at the transmitter. Symbol energy is uniformly distributed over all transmitting antennas. Ergodic capacity of such scheme is very simple to evaluate as the energy allocation rule is not the function of the channel state. As such the capacity of the channel with such energy allocation policy is just given by simple expectation of MIMO channel capacity term with uniform allocation over the distribution of MIMO channel fading. From shorter term observation perspectives it is obvious that such fixed energy allocation suffers from very poor delay-limited capacity performance since once the very harm channel fading occurs in given observation window the requirement to support some nonzero minimum rate R0 in such a case is not fulfilled. In such occurrences we need to change the uniform symbol energy allocation used in mode 1 to so called emergency mode 2 using channel information at the transmitter in very simple way to support minimum rate R0 even under such bad conditions.

2.4. Mode 2—emergency mode—channel inversion versus waterfilling 2.4.1. Space-time waterfilling. Assuming perfect knowledge of the channel state information at the transmitter it is well known that space– time water-filling is a capacity achieving adaptation policy under an assumption of long term energy constraint and ergodic channel observation. But such optimal space– time water-filling (cf. to References [8,12]) could not be used as the emergency mode regarding its essential nature increasing the symbol energy for good channel states and decreasing the symbol energy for bad states. For the worst states an outage might even occur. Outage occurrence is strictly contradictory to our general zero transmission outage requirement for given minimum required single block rate R0 described in Subsection 1.2.3. It unfortunately disqualifies the optimal space–time water-filling scheme from our further investigation as potential candidate for emergency transmission mode.

2.4.2. Spatial water-filling. The second natural choice for emergency mode candidate could be spatial water-filling. Such policy assigns different levels of energy over eigenmode space for given channel matrix realization. The strongest eigenmodes get mode symbol energy and the weaker ones could be even excluded from the transmission for given channel state realization.

Two-mode limited feedback block-fading adaptive transmission

With given fixed amount of energy available for single block (i.e., not assuming the possibility of space–time water-filling) water-filling over spatial domain is the optimal policy in terms of achievable delay-limited capacity (capacity per given block). So from the perspective of the criterion introduced in Subsection 1.2.3 it would definitely make sense to potentially consider spatial water-filling as the adaptation emergency mode. The spatial water-filling suffers from one crucial disadvantage which is in contradiction to our elementary motivation summarized in Subsections 1.2.2 and 1.2.4. Under spatial water-filling the capacity achieving codewords have different dimensionality for different channel states. That is given by the fact that water-filling over spatial domain suffers from variable dimensionality of eigen-mode space over which the code is applied (various number of eigen-mode sub-channels dependent on the water-filling optimization procedure). That leads to more complicated optimal code design That is in contradiction with our constraint of fixed dimensionality of codebook under mode 2 per block for all fading realizations described in Subsection 1.2.2. Moreover, on-line application of spatial water-filling algorithm is complicated by the need to run optimized allocation procedure for each fading realization distributing a given amount of energy over eigenmodes according to water-filling rule. Such procedure is not elementary especially for higher dimensionality of eigenmode space. Also the off-line optimization procedure evaluating parameter like average transmitted energy under mode 2 EM2 , probability of mode 2 etc. which has to be evaluated to deliver key system setting parameters prior to transmission offline is very difficult. That makes spacial water-filling not aligned with our minimum on-line and off-line optimization complexity requirements described in Subsection 1.2.4. These are the major reasons why spatial water-filling was not proposed for adaptive emergency mode.

2.4.3. Eigenmode channel inversion. Eigenmode channel inversion is well aligned with all constraints we imposed on the system design and performance described in Section 1.2. Previous investigation of space–time and spatial only water-filling has led us clearly to the choice of eigenmode channel inversion supporting R0 per each block. Such approach relies only on the very limited computing resources (both on-line and off-line) to operate the transmission system. Also as it is well known that the capacity advantage of space–time and spatial only water-filling over the channel inversion is very limited. Another reason why we proposed the channel inversion as the right emergency transmission scheme is that such allocation scheme relies on fixed dimensionality of channel coding. For symmetric MIMO channels (the same number of transmitting and receiving antennas) the general total channel inversion is not applicable because in Reference [10], we have found that the capacity of such scheme

Wirel. Commun. Mob. Comput. 2012; 12:352–366 © 2010 John Wiley & Sons, Ltd. DOI: 10.1002/wcm

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Two-mode limited feedback block-fading adaptive transmission

M. Knize and J. Sykora

Figure 1. The zero-outage capacity per single block development and symbol energy allocation rule.

is zero; similar to the capacity of single-input-singleoutput (SISO) Rayleigh channel [13]. The approach called truncated channel inversion where the transmission of information can be switched off for deep fades cannot be used either. See References [11] and [14] for deep analysis of general channel inversion framework in both SISO and MIMO channels and the trade-off between achievable ergodic capacity, outage probability, and impact of channel correlation on the inversion performance. In future research it would be very interesting to investigate these trade-offs for multiple access channel. So far we do not know about rigorous analysis of eigenmode channel inversion capacity region considering multiple access SISO and MIMO channels. We have found the solution supporting our minimum single block rate requirement in MIMO Rayleigh channel in Reference [10]. To achieve non-zero capacity with channel inversion it was proved to be sufficient to exclude the weakest eigenmode of decomposed MIMO Rayleigh channel from the transmission. Such a solution enables us to guarantee a given minimum transmission rate for arbitrary fading occurrence at the mild expense of lower total capacity.

2.5. Overall two-mode transmission scheme Based on previously described mode 1 and mode 2 the overall two-mode symbol energy transmission is illustrated in the Figure 1. The average transmitted energy under mode 1 is denoted as EM1 and under mode 2 as EM2 . The first mode with constant symbol energy allocation is preferred in the cases when the zero-outage capacity per single transmission block is higher than required rate per block denoted as R0 . In this mode, we save the resource-consuming feedback channel for any other utilization. Whenever the zero-outage channel capacity per 356

single transmission block without symbol energy allocation rule (uniform assignment) would be below the required minimum rate, the second emergency mode is applied. With such overall symbol energy controlling, the ergodic capacity is apparently lower bounded by minimum per block required rate. The first example (on the top) in the Figure 1 is parametrized by given value of EM1 and R0 , the next middle level illustrates the case of higher EM1 and constant R0 . We can see that with the higher EM1 the probability of the transmission under the mode 1 is also higher and the symbol energy allocation is the uniform one more often. Similarly the bottom example in Figure 1 shows the influence of higher minimum rate requirement R0 on the symbol energy allocation rule. Next sections investigate both modes from eigenmode space perspective.

3. MODE 1—NO FEEDBACK CHANNEL 3.1. Achievable rate under mode 1 The eigenvalue vector of HHH for the single block-fading MIMO realization H is denoted as λNe = [λ1 , . . . , λNe ]T . Transmission in mode 1 corresponds to the case when the required rate is bellow the instant capacity with no CSIT. The probability of mode 1 transmission parametrized by the minimum rate per block is evaluated as Pr[R0 ≤ CM1 (H, EM1 )]. It stands for the time share when no feedback is used. The random value of the channel capacity achievable without CSIT is given by equal symbol energy allocation over the space dimension† as Reference [15] CM1 (H, EM1 ) = log2 det(I + EM1 /(σ 2 NT )HHH )

† Equivalently

(2)

over the eigenmode space.

Wirel. Commun. Mob. Comput. 2012; 12:352–366 © 2010 John Wiley & Sons, Ltd. DOI: 10.1002/wcm

M. Knize and J. Sykora

Two-mode limited feedback block-fading adaptive transmission

where the symbol energy EM1 is given as the average symbol energy transmitted under the mode 1. In order to discuss the eigenmode symbol energy allocation, the following form of the capacity term is favorable

CM1 (H, EM1 ) = log2

Ne 


1+

i=1

EM1 λi σ 2 NT

 (3)

The portion of time the system is transmitting in mode 1 is given by the probability that the minimum required rate (guaranteed regardless instantaneous fading realization) R0 is lower or equal to CM1 , i.e.,

 Pr[M1 ] = Pr 2

R0

≤

Ne 
i=1

EM1 1 + 2 λi σ NT

 (4)

Let the subset of eigenvalue vectors, fulfilling the minimum rate per block requirement with no CSIT, be further denoted as

 M1 =

λNe :

Ne 
i=1

EM1 1 + 2 λi σ NT



 ≥2

R0

(5)

Obviously, the set M1 is parametrized by the value of EM1 and R0 . When the vector λNe belongs to the set M1 , then we can transmit with the average symbol energy EM1 and the minimum guaranteed rate R0 is supported for a certainty.

+

problem is that for the higher number of transmitting and receiving antennas the function fb (.) is very difficult to handle and so the integration with such upper bound stays analytically unsolvable. This is the only major issue that prevents us from analytical evaluation of integration involved and we need to resolve all evaluations strictly with numerical integrations. So the problem is not primarily given by the structure of the integrands given by probability density functions but by the complexity of the integration bounds. Figure 2 illustrates the shape of the regions M1 , M2 (M2 is defined later in Equation (10)) for the case of NT = NR = 2. The function λNe = fb (λNe −1 ) can be implicitly specified as follows

fb (λNe −1 ) =

N −1

3.3. Eigenmode PDFs under mode 1 Using the joint distribution of the eigenvalue vector we can write the probability of transmission in mode 1 as‡ Pr[M1 ] =

The boundary of the set M1 can be defined by the function λNe = fb (λNe −1 ), λ1 , . . . , λNe ≥ 0, which can be Ne derived from the implicit boundary definition i=1 (1 + EM1 λi /(σ 2 NT )) = 2R0 . Thus, we can easily replace the f (λ ) integration  (.)dλNe by RNe −1 0 b Ne −1 (.)dλNe . The



where (x)+ = x, for x ≥ 0 and (x)+ = 0, for x < 0. This formula can be then rewritten into many other forms suitable for its features discussion (symmetry, convexity, etc. . .).



3.2. The boundary of M1

M1

Figure 2. Two-mode regions
M1 ,
M2 , for the case NT = NR = 2.

M1

p(λNe )dλNe =

Rn+

pM1 (λNe )dλNe (7)

Where p(λNe ) is the unordered joint probability density function of the vector of eigenvalues λNe and pM1 (λNe ) denotes the modified joint distribution of λNe under the mode 1, i.e.,

pM1 (λNe ) =

p(λNe );

λNe ∈ M1

0;

e λN e ∈ R N + \ M1

(8)

Notice that no rescaling factor is used in this definition (8), so that obviously, pM1 (λNe ) carries both information about the probability density function (PDF) of the vector λNe in the support set M1 as well as the information about the portion of time we transmit in mode 1. This is the reason why it is called just modified PDF. As the consequence, the integration over the whole support gives the lower or equal value to 1. Equivalently, Equation (8) obeys the general formula based on conditioned PDF

+

pM1 (λNe ) = p(λNe | M1 )Pr[M1 ]

e (1 + EM1 λi /(NT σ 2 )) NT σ 2 2R0 − i=1 Ne −1 EM1 (1 + EM1 λi /(NT σ 2 ))

(9)

i=1

(6)

‡ Rn

+

Wirel. Commun. Mob. Comput. 2012; 12:352–366 © 2010 John Wiley & Sons, Ltd. DOI: 10.1002/wcm

denotes the n-dimensional space of real positive numbers.

357

Two-mode limited feedback block-fading adaptive transmission

4. MODE 2—PERFECT FEEDBACK CHANNEL—N T = N R

4.2. Eigenmode PDFs under mode 2

We assume common Rayleigh MIMO channel where the eigenvalues obtained via SVD obey the Wishart distribution (see References [16,17]). For the symmetric case, the subspace channel inversion has to be applied due to the unfavorable symmetric (NT = NR ) channel feature that the total channel inversion is not feasible over the full eigenmode space. In our paper [10], we provide the comprehensive approach to subspace channel inversion. It was proved, it is sufficient to exclude only the weakest eigenmode from the transmission to get the fully invertible eigenmode subspace. The capacity loss for such technique is reasonably low.

4.1. Achievable rate under mode 2 Equivalent to Equation (5), we define

 e M2 = RN + \ M1 =

λNe :

Ne 
i=1

EM1 1 + 2 λi σ NT



 < 2 R0 (10)

Under the perfect feedback channel case, we want to ensure the minimum rate to be achievable, i.e., the channel inversion based on the instantaneous fading realization is the second mode adaptation policy. To ensure the zero probability of outage we have to apply the approach of eigenmode subspace total inversion [10]. The minimum rate should fulfill



Ne −1

CM2 =

log2 (1 + K) = (Ne − 1) log2 (1 + K) ≡ R0

i=1

(11) This is the instantaneous rate, respective the capacity. Just recall that no expectation is needed even for the ergodic capacity under the total inversion. From that we can write K = 2R0 /(Ne −1) − 1 and finally the average transmitted symbol energy in the second mode as

EM2 Pr[M2 ] = (Ne − 1) R+

Ne − 1 1 − Pr[M1 ]

R+

(12)

2R0 /(Ne −1) − 1 M2 ˜ ˜ psub (λ)d λ (13) λ˜

There was used the modified subspace marginal distribution M2 ˜ p sub (λ) in formulas (12, 13). In the following subsection, we are going to reveal how these PDFs can be derived. 358

M2 ˜ The symbol p sub (λ) denotes the PDF of the unordered eigenvalue of the subspace, where the weakest eigenmode is excluded from the transmission. It can be obtained in two steps as follows. First, the joint PDF of subspace of unordered eigenmode is given by§

M2 p sub (λNe −1 ) = Ne



min[λNe −1 ]

pM2 (λNe )dλNe

(14)

0

where λNe denotes the weakest eigenvalue and λNe −1 denotes the vector of remaining Ne − 1 unordered eigenvalues. Notice the reason of the multiplication by Ne in Equation (14). Throughout our investigation we assume the unordered joint PDF p(λNe ). The ordered joint PDF is then defined over the support λ1 ≥ λ2 ≥ . . . ≥ λNe ≥ 0 as po (λNe ) = Ne !p(λNe ). Then, the joint unordered distribution of the subspace, when the weakest eigenmode is removed, is obtained generally as psub (λNe −1 ) = 1/(Ne − min[λNe −1 ] 1)! 0 po (λNe )dλNe and so equivalently Equation (14) holds. Based on the Subsection 3.2, the term (14) can be equivalently expressed into M2 p sub (λNe −1 ) = Ne



min[λNe −1 ;fb (λNe −1 )]

p(λNe )dλNe (15) 0

The integration bounds correspond to the subset definition I = {λNe : λNe ≤ min[λNe −1 ; fb (λNe −1 )]}

(16)

which is depicted for the special case NT = NR = 2 in Figure 2. Based on the definition (8), it is worth mentioning M2 that the PDF p sub (λNe −1 ) depends on particular value of R0 and EM1 through the set M1 , resp. M2 . The marginal PDF of unordered eigenvalues with the weakest eigenvalue removed is given by M2 ˜ p sub (λ) =

N −2 R+e

M2 ˜ p sub (λ, λ2 , . . . , λNe −1 )dλ2 . . . dλNe −1

(17) K M2 ˜ ˜ p (λ)d λ λ˜ sub

From that it follows EM2 =

M. Knize and J. Sykora

We have used the notation relying on the definition (8). The symbol λ˜ stands for the eigenvalue of reduced unordered eigenmode space (i.e., eigenmode space without the weakest eigenmode and assuming the rest eigenmodes are kept unordered). Equivalently, we can use the notation with the original joint PDF p(λNe ) and two mode transmission might be reflected by proper particular integration bounds. Although we find the second way to be more illustrative

§ The

symbol min[x] stands for the lowest element of the vector x.

Wirel. Commun. Mob. Comput. 2012; 12:352–366 © 2010 John Wiley & Sons, Ltd. DOI: 10.1002/wcm

M. Knize and J. Sykora

Two-mode limited feedback block-fading adaptive transmission

one, it is very difficult to develop a formalism to handle the generic case. The necessary average symbol energy transmitted in mode 1 is determined by the minimum rate constraint R0 and the average symbol energy transmitted in the first mode EM1 . Finally, the total symbol energy constraint is given by the equation E¯ = EM1 Pr[M1 ] + EM2 (1 − Pr[M1 ])

5.2. Eigenmode PDFs under mode 2 pM2 (λ) can be found by pM2 (λ) = The PDF  N −1

R+e

p

M2

(λ, λ2 , . . . , λNe )dλ2 . . . dλNe .

To

conclude,

the long term total symbol energy limit is





E¯ = EM1 M1

p(λNe )dλNe + Ne

R+

2R0 /Ne − 1 M2 p (λ)dλ λ

(18)

(22)

Using previous terms we can write

E¯ = EM1 M1

6. THE ERGODIC CHANNEL CAPACITY

p(λNe )dλNe



+ (Ne − 1) R+

2R0 /(Ne −1) − 1 M2 ˜ ˜ psub (λ)d λ (19) λ˜

From Equation (19), we can at least numerically find the proper value of EM1 since it is the only unknown variable in Equation (19) assuming given E¯ and R0 . The closed form expression is not simply available for such general case since the set M1 is parametrized by EM1 itself so the bound of the integration depends on this value.

The overall ergodic capacity is generally C = E[CM1 | M1 ]Pr[M1 ] + E[CM2 | M2 ]Pr[M2 ] (23) Using the results on the capacity under the mode 2 we can rewrite the last formula into the form C = E[CM1 | M1 ]Pr[M1 ] + R0 (1 − Pr[M1 ]). Using the joint PDFs, we got



5. MODE 2—PERFECT FEEDBACK CHANNEL—N T
= N R From the joint PDF p(λNe ), it can be proved that in such nonsymmetric MIMO case the full eigenmode set can be easily inverted regardless the channel matrix realization, [10]. As the consequence, some of equations from Subsection 4 slightly change as follows.



Ne 

C=

M1 i=1



+ R0

1+

log2

1− M1

p(λNe )dλNe

p(λNe )dλNe

(24)

The expected capacity of the mode 1 transmission is given by the expected capacity of the single eigenmode times, the number of such eigenmodes, so that we can write



C = Ne

5.1. Achievable rate under mode 2

R+

log2



1+



For the full eigenmode space the Equation (11) is redefined to Ne 

 





CM2 =

EM1 λi σ 2 NT

+ R0

EM1 λ pM1 (λ)dλ σ 2 NT

pM2 (λNe )dλNe

N

R+e

(25)

where log2 (1 + K) = Ne log2 (1 + K) ≡ R0



(20) pM1 (λ) =

i=1

N −1

R+e

Equivalently, the constant value K = 2R0 /Ne − 1. The average transmitted symbol energy under mode 2 is

pM1 (λNe )dλNe −1

(26)

is the marginal PDF of the eigenvalue from the unordered full set of eigenvalues.

(2R0 /Ne − 1)/λpM2 (λ)dλ

EM2 = Ne /(1 − Pr[M1 ) R+

(21) The PDF pM2 (λ) is the marginal distribution of the eigenvalue from the unordered full set under the mode 2.

7. VARIOUS APPLICATION CRITERIA In this section, the various optimization criteria are unveiled. Each criterion uncovers a special application requirement. The closed form expressions are unfortunately cumbersome

Wirel. Commun. Mob. Comput. 2012; 12:352–366 © 2010 John Wiley & Sons, Ltd. DOI: 10.1002/wcm

359

Two-mode limited feedback block-fading adaptive transmission

(see Section 3.2) and usually do not exist at all. That is the reason that the optimization procedure is rather generically stated. The particular numerical results are shown in the next section.

7.1. Given E¯ and R0

= Rn+

(1) We start with the Equation (19) that sets mutual relations among total average symbol energy E¯ average symbol energy transmitted in mode 1 EM1 and the rate R0 . From this equations (assuming given E¯ and R0 ) we can find the value EM1|E,R ¯ 0

pM1 (λNe )dλNe

Ne − 1 1 − Pr[M1 | EM1|E,R ¯ 0 , R0 ]



× R+

M1|E,R ¯



C = Ne R+

log2



+ R0

N

EM1|E,R ¯ 0 λ pM1 (λ)dλ σ 2 NT

pM2 (λNe )dλNe

2

+ (Ne − 1)

R0 /(Ne −1)

λ˜

R+

−1

M2 ˜ ˜ p sub (λ)d λ

¯ R0 →EM1|E,R E, ¯ 0 → M1 , M2 →

(27) We know that the set M1|E,R ¯ 0 is conditioned by the value of EM1|E,R ¯ 0 itself because of its definition given in Equation (5). Let us recall that set M1|E,R ¯ 0 in Equation (27) has to be evaluated as Ne 
i=1

EM1|E,R ¯ 1 + 2 0 λi σ NT



 ≥2

R0

.

Based on that it is obvious that EM1|E,R ¯ 0 cannot be simply obtained in a closed form from Equation (27). The only way how to compute this is to use some numerical package with strong numerical integration capabilities. (2) Once we have found EM1|E,R ¯ 0 we can evaluate the sets M1 , resp. M2 , from definitions (5), resp. (10). These two sets depend solely on the value EM1|E,R ¯ 0 and R0 . (3) Then, based on Equations (7) and (4), we can easily find also

¯ M2|E,R → Pr[M1 | E¯ M1|E,R ¯ 0 , R0 ] → E ¯ 0 → C

We are going to find the minimum symbol energy to guarantee the minimum rate R0 to be supported regardless of the instantaneous fading realization. From the equation for total symbol energy (19) follows that this value E¯ depends only on two parameters R0 and EM1 . This is given by the fact that the integration interval M1 again depends only on R0 and EM1 (see Equation (5)) and the marginal PDF of unordered eigenvalues with the weakest eigenvalue removed (17) is obtained from Equation (15). The integration bound given by boundary function fb() depends again only on R0 and EM1 . So once the value R0 is exogenous the total symbol energy is parametrized only by the value of EM1 , i.e., by the symbol energy transmitted under mode 1. So we want to find EM1 ∈R+

Pr[M1 | EM1|E,R ¯ 0 , R0 ]



= Pr 2

≤

Ne 
i=1

EM1|E,R ¯ 1 + 2 0 λi σ NT



(32)

The symbol energy transmitted in the first mode is then given as ¯ M1 , R0 ) EM1 min|R0 = arg min E(E EM1 ∈R+

 The symbol XM|z,y stands for the value of X under mode M conditioned by the knowledge of values y, z.

360

(31)

7.2. Minimum total symbol energy with required R0

¯ M1 , R0 ) E¯ min|R0 = min E(E

R0

(30)

In a nutshell the overall evaluation process is shortly summarized as follows:

0

λNe :



1+

p(λNe )dλNe





2R0 /(Ne −1) − 1 M2 ˜ ˜ psub (λ)d λ (29) λ˜

(5) The final step is to get the capacity as given in Equation (25).

R+e

E¯ = EM1

(28)

(4) The symbol energy EM2|E,R ¯ 0 can be then calculated from the integration (13) as follows: EM2 =

A very common situation is considered. We assume we want to design such a system which operates with its given total symbol energy and strictly supports minimum achievable rate R0 for arbitrary fading. The procedure to find other key performance indicators follows.

M1|E,R ¯ 0 =

M. Knize and J. Sykora

(33)

Now, we know both R0 and EM1 min|R0 so we can find the M1 and M2 . Moreover, we can also evaluate Pr[M1 |

Wirel. Commun. Mob. Comput. 2012; 12:352–366 © 2010 John Wiley & Sons, Ltd. DOI: 10.1002/wcm

Two-mode limited feedback block-fading adaptive transmission

EM1 min|R0 , R0 ], Pr[M1 | EM1 min|R0 , R0 ], and finally EM2 min|R0 from Equation (12). We can also evaluate the capacity which definitely has to be greater or equal to the required minimum rate R0 . 7.3. Required R0 and Pr[M1 ] This scheme sets the given R0 together with the given Pr[M1 ], i.e., the proportion of time with no feedback. The steps to find all key performance indicators are as follows: (1) The function (4) is monotonic in the variable EM1 so we can simply find EM1|R0 ,Pr[M1 ] from integration (7). (2) Since we know R0 and EM1|R0 ,Pr[M1 ] , we also know the multidimensional sets M1 and M2 from definitions (5), resp. (10) M1 =

λNe :

Ne 
i=1



10

3x3 5 below this line C=R0 the minimal single block rate condition is not fulfilled

2x2 2

3

4

5

6

7

8

9

10

R0 [bits/channel usage]

Figure 3. The achievable capacity when given total power E¯ and minimum rate R0 are required, NT = NR = 2 and NT = NR = 3.



EM1|R0 ,Pr[M1 ] 1+ λi ≥ 2R0 σ 2 NT

e M2 = RN + \ M1

E=10dB E=12.5dB E=15dB E=17.5dB

0

;

(34)

(3) We can evaluate the symbol energy EM2|R0 ,Pr[M1 from Equation (13), the total symbol energy from Equation (19) and the capacity (25). To sum up the whole evaluation procedure we can briefly write ¯ C R0 , Pr[M1 ] → E¯ M1|R0 ,Pr[M1 ] → M1 , M2 → EM2 , E, (35)

8. NUMERICAL RESULTS This section shows the numerical results of particular schemes from the Section 7 for particular cases NT = NR = 2, NT = 3, NR = 2 and NT = NR = 3. As we will show, some interesting observations follow from such results. The cases of MIMO 2 × 2, 3 × 2 and 3 × 3 channels were adopted due to its very different features. In the 2 × 2 case, two eigenmodes are used in mode 1 which is given from Ne = min[NT, NR ] = 2 but only single mode (reduced space) in mode 2. This is given by the fact that for symmetric MIMO channel defined by NT = NR , the invertible eigenmode space dimensionality is given by Ne − 1 (so for 2 × 2 case Ne − 1 = 1). This is equivalent to the number of invertible sub-channels (eigenmodes). For details be referred to Reference [10]. In MIMO 3 × 2, we are free to use both two sub-channels under both modes. In the symmetric 3 × 3 case we can again use only Ne − 1 = 2 eigenmode for channel inversion. In Section 3.2 we have shown why the evaluation of performance of key system parameters is not

straightforwardly feasible. Thus, all numerical results in this section were obtained through very demanding numerical integrations. To find numerical performance of system with higher number of antennas (e.g., system 4 × 4) stays still task with extreme numerical complexity. 8.1. Given E¯ and R0 In this subsection, the scheme from Subsection 7.1 is evaluated. In Figure 3, we can see the achievable capacity curve development regarding increasing R0 parametrized by the value of total available symbol energy E¯ for 2 × 2 ¯ and 3 × 3 MIMO system. It is interesting that for given E, the capacity is constant for given interval of R0 till the very sharp decline to zero (especially for 2 × 2 case we can see very sharp drop of capacity due to the fact we have only one eigenmode at disposal for channel inversion). In the zero-capacity zone our two-mode processing is no longer applicable. In Figure 4, there is obvious that the probability 0

10

−1

10

−2

10 Pr[Λ2] [−]



15

C [bits/channel usage]

M. Knize and J. Sykora

−3

10

−4

10

E=10dB E=12.5dB E=15dB E=17.5dB

−5

10

−6

10

2

3

4 5 R0 [bits/channel usage]

6

7

Figure 4. The probability of mode 2 for given total power E¯ and given R0 is required, NT = 2, NR = 2.

Wirel. Commun. Mob. Comput. 2012; 12:352–366 © 2010 John Wiley & Sons, Ltd. DOI: 10.1002/wcm

361

Two-mode limited feedback block-fading adaptive transmission

10

for higher energy R0=5 reliably supported

Capacity [bits/channel usage]

9

Waterfilling capacity

8

for higher energy R0=6 reliably supported

7 6 5 Two−mode capacity 4 for R0=2,3,4

Two−mode scheme capacity lower than required R0=6

3 Two−mode scheme capacity lower than required R0=5

2 1 0 10

11

R0=2 R0=3 R0=4 R0=5 R0=6 R0=7 optimal waterfilling

12 13 14 15 16 total average symbol energy E [dB]

17

18

Figure 5. The achievable capacity when given total power E¯ and required R0 are assumed, NT = NR = 2.

of mode 2 increases significantly for higher R0 for each of ¯ but for quite large interval of R0 the particular value of E, the second mode (emergency mode) is used really only very rarely. This nicely shows how reasonable our motivation is. We want to reduce the necessary availability of channel feedback information at the transmitter and through this further reduce the total average amount of information which has to be fed-back. For example, for E¯ = 10 dB and when we require the minimum supported rate R0 = 4 bits/channel usage, the system transmits in mode 2 only in 10% of the total transmission time. This is very substantial reduction of time we need to use feedback channel. So by our scheme we have relatively reduced the average amount of information which need to be fed-back to transmitter under perfect channel feedback 10 times. This behavior is driven by the fact that for given E¯ and reasonably low rate R0 uniform power allocation is sufficient to support this rate in the majority of time. Once we require higher R0 which is closer to the ergodic capacity of the system, we need to rely on channel inversion more often. The overall capacity performance for 2 × 2 and 3 × 3 systems is depicted in Figure 5 for 2 × 2 and in Figure 6 for 3 × 3 MIMO system. We can see the threshold in terms of total symbol energy E¯ after which (for given R0 ) the capacity

M. Knize and J. Sykora

is very close to the optimal water-filling capacity. That is given by the fact that for reasonable low R0 and reasonable high total symbol energy optimal water-filling allocation tends to go to uniform allocation which we use in mode 1. The capacity behavior from Figures 5 and 6 perfectly corresponds to behavior of capacity in Figure 3. From all these figures, it follows that some maximum R0 is supported for given total symbol energy (see Figure 3) or equivalently the capacity is lower than required R0 for not sufficiently high total available symbol energy E¯ (see examples depicted in Figures 5 and 6). Where capacity is lower that R0 the system is not able to operate regarding our desired system features.

8.2. Minimum total symbol energy with required R0 This subsection shows the numerical performance regarding the motivation of Subsection 7.2. The gap between required rate R0 and achievable overall system ergodic capacity C for NT = NR = 2 is shown in Figure 7. Clearly, for higher R0 the minimum required symbol energy is also higher. It is interesting observation that the positive gap between C and R0 for higher required R0 increases as well. In the asymmetric case with NT = 3, NR = 2, we observe an opposite capacity gap development compared to Figure 7 (see these results in Figure 8). This performance is caused by the position of local minimum of the ¯ M1 , R0 ) regarding variable EM1 for given R0 . function E(E For symmetric case NT = NR , we have found that for higher desired R0 for the minimum points it holds E¯ → EM1 so that the whole available energy is spent in mode 1 (uniform assignment). That perfectly corresponds to the case described above where we highlighted the fact that for higher energy and reasonably low R0 , uniform energy allocation we use in mode 1 without feedback

10 For higher R0 there is higher probability of mode 1, i.e. inversion and feedback channel have to be used less often, since we use more often the uniform power allocation we achieve higher average capacity than R0 which is exactly required to by supported under inverion mode

9

Capacity|R0 [bits/channel usage]

14

for higher energy R0=10 Waterfilling capacity for higher energy R0=9 reliably supported reliably supported

12 10 8 6 4 2 10

Two−mode capacity for R0=5,6,7,8 11

Two−mode scheme capacity lower than required R0=9

Two−mode scheme capacity lower than required R0=10 R0=5 R0=6 R0=7 R0=8 R0=9 R0=10 optimal waterfilling

12 13 14 15 16 total average symbol energy E [dB]

17

18

Figure 6. The achievable capacity when given total power E¯ and required R0 are assumed, NT = NR = 3.

362

C,R0 [bits/channel usage

8 16

7 For lower R0 we have to use the inversion more often.

6 This is given by the shape of pdf on only one eigenmode that can be used for inversion.

5 Once we use in the majority the mode which inverts channel to support only R0, the capacity

4 of the entire systems goes to R0. 3 2 1 −5

Capacity,Nt=2,Nr=2 R0 0

5 10 min[E|R0] [dB]

15

20

Figure 7. The capacity achievable with minimum total power E¯ required to guarantee R0 , comparison with optimal water-filling scheme achieving the ergodic capacity with perfect channel feedback, NT = NR = 2.

Wirel. Commun. Mob. Comput. 2012; 12:352–366 © 2010 John Wiley & Sons, Ltd. DOI: 10.1002/wcm

Two-mode limited feedback block-fading adaptive transmission

7.5 7

For lower R0 there is higher probability of mode 1, i.e. unifrom power allocation with no feedback, since we use uniform power allocation more often there is positive gap between capacity and required R0

C,R0 [bits/channel usage]

6.5 6 5.5 5

For higher R0 there is higher probability of mode 2, i.e. inversion and feedback channel have to be used more often, since we invert the channel more to support the exact value of R0 the gap between capacity and required R0 diminishes.

4.5 4 3.5 3

Capacity,Nt=3,Nr=2 R0 6

7

8

9

10 11 min[E|R0] [dB]

12

13

14

Figure 8. The capacity achievable with minimum total power E¯ required to guarantee R0 , NT = 3, NR = 2.

becomes very close to the optimal water-filling allocation. That is caused by the observation that water-filling which utilizes perfect channel state information tends to be uniform allocation for higher signal to noise ratio (i.e., transmitted symbol energies). In other words, channel inversion (emergency mode) is not applied so often for higher R0 as long as theminimum total symbol energy E¯ is desired. This might be very contradictory with the opinion one could naturally presume without our investigation. Surprisingly, for asymmetric case, e.g., NT = 3, NR = 2, ¯ M1 , R0 ) does not exist with respect the local minimum of E(E to variable EM1 so that we can shape the particular curve of Pr[M1 ] with selection of the value EM1 from available ¯ The explanation of such difference between interval (0, E). 2 × 2 and 3 × 2 case follows. The actual behavior consists of two effects. The first one is given by the general fact that optimal water-filling energy assignment tends to converge to uniform energy allocation for higher signal to noise ratio, i.e., higher transmitted symbol energies. So as we use uniform energy policy in mode 1 without channel state information known to the transmitter with assumed higher total symbol energy we can expect that mode 1 will be used more often (compared to mode 2 with inversion) as uniform policy tends to be the optimal one. All this holds as long as there is no further requirement on minimum required single block supported rate R0 . Once such a requirement is added into our consideration we can also observe the second effect caused by mutual relation between required R0 and minimum total ¯ M1 , R0 ) with respect to EM1 . This is also strongly energy E(E influenced by the particular fashion of marginal probability density functions for given dimensionality of eigenmode space which is very different, for example, symmetric (NT = NR ) and asymmetric (NT = NR ) channels. The basic relation is that once we require higher R0 we could expect that for given available total symbol energy the emergency mode 2 has to be applied more often. That perfectly corresponds with results presented in previous Subsection 8.1. These two effects are mutually interrelated and as we

observed they lead to different results of probability of mode 1 and mode 2 for symmetric versus asymmetric MIMO channel. First let us focus on 2 × 2 scheme. For lower R0 we have to use the inversion mode more often. The inversion mode inverts the channel to support exactly the value of R0 . From that it is obvious that the capacity of the entire system goes to R0 for lower R0 . For higher R0 the probability of mode 1 is higher, i.e., inversion and feedback channel have to be used less often. It is more efficient to invest energy into the uniform allocation (no-feedback) than into the inversion. Since we use the uniform power allocation more often we achieve higher average capacity than R0 which is the rate minimum required to be supported under inversion mode. So obviously the driver of such capacity gap behavior lies in the time share being in mode 1 versus mode 2. This is depicted in Figure 9. There we can explicitly see that for lower R0 the minimum total symbol energy was found when we transmit only under channel inversion. For higher R0 , the time share of uniform energy assignment without feedback substantially increase and as under this mode we can achieve higher that required R0 even though the total average capacity of our two-mode scheme becomes higher than just R0 . Such behavior corresponds perfectly to the first effect described in previous paragraph. For higher R0 , the higher minimum required average total symbol energy E¯ is required; the uniform energy allocation performance is very close to optimal water-filling performance so the mode 1 is used more often. Also the fact that for lower R0 we transmit only under inversion is given by dimensionality of eigenmode space we have available (only two eigenmodes for uniform assignment and one eigenmode eligible for inversion) and by features of probability density functions describing these eigenmodes. For uniform assignment we have two eigenmodes but if they are unordered both are not invertible and so the inversion cannot be applied [10]. For inversion we have to use only the strongest eigenmode

0

10

−1

10 Pr(M1) [−]

M. Knize and J. Sykora

−2

10

Probability of mode 1 (without feedback) for 2x2 Probability of mode 1 (without feedback) for 3x2 −3

10

3

3.5

4

4.5 5 5.5 R0 [bits/channel usage]

6

6.5

7

Figure 9. The probability of transmission under mode 1, i.e., without any feedback and with uniform power allocation when minimum E¯ required to guarantee R0 is transmitted as average total power, NT = 2, NR = 2 and NT = 3, NR = 2.

Wirel. Commun. Mob. Comput. 2012; 12:352–366 © 2010 John Wiley & Sons, Ltd. DOI: 10.1002/wcm

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Two-mode limited feedback block-fading adaptive transmission

364

30 20 10

E [dB]

0

−10 −20

min[E|R0] E M1

EM2

−30

E

*Pr[Λ

M1

]

M1

EM2*Pr[ΛM2]

−40 1

2

3

4 5 6 R0 [bits/channel usage]

7

8

¯ when Figure 10. The average power assignment (EM1 , EM2 , E) minimum E¯ required to guarantee R0 is transmitted as average total power, NT = 2, NR = 2.

probability Pr[M1 ] for 2 × 2 case (Figure 9) compared to the previous case (Figure 4). In the latter case, the total average symbol energy is given and we do not search for the minimum of this energy over the value of EM1 . Be aware of the fact that Figure 4 is about Pr[M2 ] whereas Figure 9 shows Pr[M1 ]=1-Pr[M2 ].

8.3. Required R0 and Pr[M1 ] We move on to the situation described in Subsection 7.3. The capacity achievable with different Pr[M1 ] and R0 is illustrated in Figure 11. It is natural that with higher Pr[M1 ] we achieve higher capacity for the same R0 but price we have to pay is the increment of the total symbol energy. The total symbol energy E¯ is then divided into EM1 and EM2 in such a way that for higher Pr[M1 ], the value EM2 is ¯ significantly lower and EM1 is almost equal to E. 12

Pr[Λ

]=0.7

M1

Capacity [bits/channel usage]

(after ordering according to eigenmode channel gains). Moreover based on this inefficiency of eigenmodes PDFs the uniform power assignment is not able to support R0 in minimum total symbol energy point. That is the reason why we have to rely on inversion in the majority of time for lower required R0 . The situation of 3 × 2 case is rather different. See the completely opposite development of capacity gap in Figure 8 and compare it to Figure 7. For lower R0 the probability of mode 1 is higher, i.e., uniform power allocation with no feedback is used more often (see Figure 9). Since we use uniform power allocation more often we can observe a positive gap between provided capacity and required R0 . For higher R0 the probability of mode 2 is higher, i.e., inversion and feedback channel have to be used more often. Since we invert the channel more to support the exact value of R0 the gap between capacity and required R0 diminishes. Although the dimensionality of eigenmode space is the same (two eigenmodes) both these eigenmodes can be inverted, even if they are not ordered, see Reference [10]. So if we are required to transmit with minimum total symbol energy for lower required R0 we can rely on uniform power assignment more often and only for higher values of R0 we have to rely on inversion (see Figure 9). This capacity behavior is given by the fact that the first effect influencing the resultant performance (given by the fact that for higher energies the uniform energy allocation goes to uniform and the respective capacities mutually converge as well) is neglected by the position of ¯ M1 , R0 ) in respect to variable EM1 minimum of function E(E for given R0 . As we already mentioned this second effect and its strength is conditioned by particular development of marginal probability functions of eigenmodes for given channel configuration. Since for asymmetric channels we have the same dimensionality of eigenmode space for both modes it means that the natural advantage of mode 1 having more eigenmode modes (i.e., signal diversity degrees) than mode 2 (channel inversion) in symmetric channels is no longer valid. As a result we can observe that for lower required rates R0 the uniform allocation is able to support it whereas for higher R0 the minimum required energy lies in the point where the channel inversion has to be used more often. ¯ average Next, in Figure 10, the total symbol energy E, symbol energy transmitted in mode 1 EM1 and in mode 2 EM2 are shown for NT = NR = 2. For higher R0 the value of EM1 increases substantially; however, the average symbol energy transmitted under the channel inversion is even higher than the total symbol energy. Do not be confused by this fact since we did not weigh the symbol energy by the corresponding probability of being under the given mode Pr[M1 ] and Pr[M2 ]. Taking this fact into account, we observe very unexpected results. The curve corresponding to EM2 Pr[M2 ] is decreasing for R0 higher than approximately 5 bits/channel use. The explanation of this is similar to the explanation of different capacity behavior for 2 × 2 and 3 × 2 described in previous paragraph. Again, this is due to the inverse trend of the

M. Knize and J. Sykora

11

Pr[ΛM1]=0.9

10

Pr[Λ

]=0.99

M1

R

0

9 8 7 6 5 4 3 2

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

R0 [bits/channel usage]

Figure 11. The achievable capacity when given Pr[
M1 ] and R0 are to be guaranteed, NT = NR = 2.

Wirel. Commun. Mob. Comput. 2012; 12:352–366 © 2010 John Wiley & Sons, Ltd. DOI: 10.1002/wcm

M. Knize and J. Sykora

Two-mode limited feedback block-fading adaptive transmission

REFERENCES

N =3,N =2 T

R

11 N =N =2 T

R

10

20

9

15

8

10

7

E

25

6

5

5

0 1 7

0.8

4

6 5

0.6

4

3

3 Pr[Λ

]

M1

0.4

2

R0

Figure 12. The achievable trade-off between capacity (given by color scale), Pr[
M1 ] and R0 and total power E¯ when Pr[
M1 ] and R0 are guaranteed, NT = NR = 2 and NT = 3, NR = 2 (gray faced).

Figure 12 shows a very interesting trade-off among the required rate R0 , the probability of mode 1 Pr[M1 ], the ¯ and achievable capacity. necessary total symbol energy E, We see that for higher rate, not only the capacity is higher but also the total symbol energy is higher. It is also clear that for NT = 3, NR = 2 we can achieve similar capacity ¯ with lower symbol energy E.

9. CONCLUSIONS We have developed the novel two mode transmission adaptive scheme with the minimum guaranteed rate requirement and a limited portion of time when the perfect and instantaneous channel state information is available at the transmitter. The constant symbol energy assignment is exploited in the mode 1 and the full(sub)space eigenmode channel inversion in the mode 2. We have derived generic method how to handle rigorously the probabilistic description of the eigenmode space regarding the two mode transmission. The trade-off among system parameters (capacity, total available symbol energy, average symbol energy, and frequency of the first and the second mode) is generally provided. Three selected application criteria are applied to clarify practical applicability of our investigation in the design of robust scheme satisfying exogenous desired parameters. Non-trivial trade-offs were found yielding counter-intuitive results showing ultimate performance we can get.

ACKNOWLEDGEMENTS This work was supported by Grant Agency of the Czech Republic, grant 102/09/1624 and the Ministry of Education, Youth and Sports of the Czech Republic, prog. MSM6840770014, and grant OC188.

1. Zhou S, Giannakis GB. How accurate channel prediction needs to be for transmit-beamforming with adaptive modulation over Rayleigh MIMO channels? IEEE Transactions on Communications 2004; 3: 1285–1294. 2. Falahti S, Svensson A, Sternad M, Ekman T. Adaptive modulation systems for predicted wireless channels. IEEE Transactions on Communications 2004; 52: 307– 316. 3. Yoo T, Goldsmith A. Capacity and power allocation for fading MIMO channels with channel estimation error. IEEE Transactions on Information Theory 2006; 52: 2203–2214. 4. Baccarelli E, Biagi M, Polizzoni C. On the information throughput and optimized power allocation for MIMO wireless systems with imperfect channel estimation. IEEE Transactions on Signal Processing 2005; 53: 2335– 2347. 5. Bhashyam S, Sabharwal A, Aazhang B. Feedback gain in multiple antenna systems. IEEE Transactions on Communications 2002; 50: 785–798. 6. Zhou S, Giannakis GB. Adaptive modulation for multiantenna transmissions with channel mean feedback. IEEE Transactions on Communications 2004; 3: 1626– 1636. 7. Visotsky E, Madhow U. Space-time transmit precoding with imperfect feedback. IEEE Transactions on Information Theory 2001; 47: 2632–2639. 8. Luo J. Service outage based adaptive transmission in fading channels. Ph.D. Thesis, The State University of New Jersey, May 2004. 9. Sykora J, Knize M. Two-mode limited feedback blockfading transmission with minimum guaranteed rate in MIMO channel. In COST 273, TD-05-114, Lisbon, Portugal, November 2005; 1–9. 10. Knize M, Sykora J. Subspace inversion symbol energy adaptation in MIMO Rayleigh channel with zero outage probability. In Proceedings of Vehicular Technology Conference , Los Angeles, USA, September 2004 Fall. 11. Knize M, Sykora J. General framework and advanced information theoretical results on eigenmode MIMO channel inversion. Radioengineering 2005; 14: 16–27. 12. Caire G, Shamai S. On the capacity of some channels with channel state information. IEEE Transactions on Information Theory 1999; 45: 2007–2019. 13. Goldsmith AJ, Varaiya PP. Capacity of fading channels with channel side information. IEEE Transactions on Information Theory 1997; 43: 1986–1992. 14. Knize M. Adaptation and diversity coding for MIMO communication systems and networks. Ph.D. Thesis, Czech Technical University in Prague, December 2006.

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15. Biglieri E, Calderbank R, Constantinides A, Goldsmith A, Paulraj A, Poor HV. MIMO Wireless Communications. Cambridge University Press: Cambridge, UK, 2007. 16. Telatar IE. Capacity of multi-antenna Gaussian channels. Technical Reports BL0112170-950615-07TM, AT&T Bell Labs, 1995. 17. Edelman A. Eigenvalues and condition numbers of random matrices. Ph.D. Thesis, Massachusetts Institute of Technology, 1989.

AUTHORS’ BIOGRAPHIES Milan Knize received the master degree in Electrical Engineering from the Czech Technical University in Prague with highest honours in 2003. His master thesis on adaptive MIMO communication systems was awarded by Werner von Siemens Excellence award. He completed his Ph. D. degree at the Czech Technical University in Prague in 2006. His research is focused on information theory applications in MIMO, adaptive, and multiuser systems. He also obtained master degree in Finance and Macroeconomics from the University of Economics in Prague in 2006.

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Jan Sykora received the M.Sc. and Ph.D. degrees in Electrical Engineering from Czech Technical University in Prague, Czech Republic, in 1987 and 1993, respectively. Since 1991, he has been with the Faculty of Electrical Engineering, Czech Technical University in Prague, where he is now a Professor of Radio Engineering. His research includes work on wireless communication and information theory, cooperative and distributed modulation, coding and signal processing, MIMO systems, nonlinear space–time modulation and coding, and iterative processing. He has served on various IEEE conferences as a Technical Program and Organizing Committee member and chair. He has led a number of industrial and research projects.

Wirel. Commun. Mob. Comput. 2012; 12:352–366 © 2010 John Wiley & Sons, Ltd. DOI: 10.1002/wcm