CHAPTER 5

Power Allocation algorithms for MIMO Systems

with Imperfect Channel State Information

In this Chapter, the focus is the throughput analysis, outage evaluation and op-

timized power allocation for a Multiple-Input Multiple-Output (MIMO) pilot-based

wireless system equipped with a noisy feedback-path for communicating back to

the transmitter the imperfect MIMO channel estimates available at the receiver.

Both cases of Gaussian and Isotropic-Unitary (IU) input signals are analyzed and

conditions for the (asymptotical) achievement of the system capacity are provided.

Specifically, main contributions of this Chapter may be so summarized.

First, the development of closed-form analytical expressions for the computation

of the ergodic information throughput conveyed by the considered MIMO system

is presented and then it is specified the operating conditions guaranteeing the

achievement of the corresponding ergodic capacity.

Second, the Chapter follows with the presentation of an iterative algorithm for

the optimized power allocation over transmit antennas that explicitly accounts

for both noise introduced by the feedback channel and imperfect MIMO forward

channel estimates.

Third, asymptotically exact closed-form upper and lower bounds on the informa-

tion outage probability of the system in delay-limited applications is developed.

Finally, the developed analytical tool is employed to gain practical insight about

136

actual operating conditions where the complexity overhead demanded by the im-

plementation of the feedback link is really justified by ensuing throughput enhance-

ment of the overall system.

5.1 Recent Related Works

For achieving the improved bandwidth-vs.-power tradeoff demanded by emerg-

ing 4th Generation WLANs [1], it may be of interest to explore also the ultimate

throughputs conveyed by pilot-based MIMO systems that rely on short training

sequences for acquiring rough (i.e., imperfect) estimates of the actual channel co-

efficients. Some interesting contributions along this line of research have been

recently presented in [13,14,15,33], where a lower bound on the capacity of pilot-

aided systems is developed and then it is exploited for optimization purposes [14]

(see also previous Chapter).

Lastly, additional results about conveyed throughput, performance analysis and

optimized precoding of Multi-Antenna systems equipped with non ideal feedback

channels have been presented in [8,9]. However, these contributions focus on sys-

tems with perfect channel estimates at the receiver and, in addition, the non-ideal

feedback links therein considered are deterministic (that is, noiseless).

Before proceeding, some remarks about adopted notations. Bold capital letters

denote matrices, underlined bold lower case symbols indicate vectors, while bold

characters overlined by arrow denote block-matrices and block-vectors. Apexes

∗, T , † mean conjugation, transposition and conjugate-transposition respectively,

and lower-case letters are used for scalar quantities. Furthermore, det[A] and

Tra[A] denote determinant and trace of the matrix A [a1 . . . am], while vect(A)

indicates the (block) vector built up by the ordered stacking of the columns of A.

137

Finally, Im is the (m×m) unit matrix, ‖A‖E indicates the Euclidean norm of A,

A⊗B is the Kroenecker product of matrix A by matrix B, 0m is the m-dimensional

zero-vector while lg denotes natural logarithm.

5.2 System Modeling

The considered (complex base-band equivalent) point-to-point link is com-

posed by t ≥ 1 transmitter and r ≥ 1 receiver antennas impaired by slow-variant

Rayleigh flat-fadings sketched in Fig.5.1. The flat fading assumption is met when

bandwidth B of the signals radiated by the t transmitter antennas is below the

coherence-band Bc of the employed forward channel of Fig.5.1. This is the operat-

ing condition of emerging 4th Generation WLANs [1] adopting Multi-Carrier (in

particular, OFDM) modulation formats [16].

Encoderand

Transmitterwith t

antennas

Receiverwith r

antennasand

Decoder

FEEDBACK-CHANNEL

MIMO FORWARD-CHANNEL H

1

2

t

1

2

r

SourceMessage

DetectedMessage

ˆH

Forward-channel estimatesavailable at Rx

Feedback-channel estimatesavailable at Tx

ˆ

ˆH

S D

h11

h21

hrt

M Mˆ

Figure 5.1: Sketch of the considered Multi-Antenna system with feedback channel.

H =ˆH for an ideal feedback link.

Path gain hji from the transmitter antenna i to the receiver one j may be

modeled as a complex zero mean unit-variance proper random variable (r.v.)

138

[2,3,4,5,6,7] and, for sufficiently spaced apart antennas, these gains hji ∈ C1, 1 ≤j ≤ r, 1 ≤ i ≤ t may be considered mutually independent. Accuracy of this as-

sumption fast improves when the antennas spacing becomes large compared to the

RF wavelength λ of radiated signals. Specifically, for indoor applications, spacing

of the order of λ suffices for meeting above independence assumption, while out-

door cellular links sited in urban and suburban environments may require antenna

spacings over 10λ at the base stations and 3λ at the mobile units [3,24]. However,

several measures and theoretical contributions support the conclusion that perfor-

mance loss induced by signal correlation among receiving antennas is not dramatic

even for correlation coefficients as high as 0.5-0.6 [3,10,24].

Furthermore, for low-mobility applications as those served by emerging 4th Gen-

eration WLANs [1,2,24], path gains hji may be assumed constant over T ≥ 1

signalling periods after which they change to new statistically independent values

held for another T signalling periods, and so on. This is equivalent to assume that

the coherence-time Tcoh of the MIMO forward channel of Fig.5.1 equates the length

T of the transmitted packets (e.g., Tcoh = T ). This assumption will be relaxed in

Paragraph 5.6, where information outages of no ergodic delay-limited MIMO sys-

tems will be considered. The resulting ”block-fading” model generally gives rise to

adequate representations of several TDMA, frequency-hopping and packet-based

interleaved systems of practical interest, where each transmitted frame is detected

independently [4,5,16,24,36,37]. More in particular, it is assumed that the coded

and modulated streams radiated by transmit antennas of Fig.5.1 are split into

packets constituted by T ≡ Ttr + Tpay slots, where the first Ttr ≥ 0 slots (e.g. the

packet header) are used for the transmission of known pilot sequences, while the

last Tpay = T − Ttr slots convey payload data.

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5.2.1 Training phase and MMSE estimation of the path gains of the

forward channel

Therefore, the Ttr × r (complex) samples gathered at the outputs of the r

receive antennas during the training phase may be organized into the (Ttr × r)

observed matrix Y [y1. . . y

r] given by [4,5,14]

Y =

√γTtr

tXH + W, (5.1)

where X [x1 . . . xt] is the (Ttr × t) matrix of the known transmitted pilots, H

[h1 . . .hr] is the (t × r) complex matrix of the above introduced forward channel

path gains hji and the (Ttr × r) noise matrix W is composed by independent

zero-mean unit-variance proper complex Gaussian samples. It is also assumed that

the pilot matrix X in (5.1) meets the following normalizing relationship

Tra[X†X] = t (5.1.a)

so that γ in (5.1) represents the (average) SNR per received sample at the out-

put of each receiving antenna during the training phase. This last is automatically

satisfied when X is a unitary matrix so that it meets the defining relationship

X†X = It. The observations Y in (5.1) are employed by the receiver to build up

the Minimum Mean Square Error (MMSE) matrix estimate H EH|Y of the

corresponding forward channel gains matrix H. This last topic has been debated

in depth, for example, in [12,13,15,26] and, more recently, in [14,27], so that, for

the sake of brevity, only some key results will be exploited in the following Para-

graphs.

First, in [12,26] it is proved that the MMSE estimates H and related error-

covariances constitute a sufficient statistic for the ML detection M of the trans-

mitted source message M of Fig.5.1. Hence, no information loss is experienced by

140

the considered receiver composed by an MMSE channel estimator cascaded to an

ML detector. Second, since W in (5.1) is Gaussian and X is known to the receiver,

the MMSE estimates H may be computed via a direct application of the Principle

of Orthogonality [26]. Therefore, a key problem is, indeed, how to choose the pilot

matrix X in order to minimize the total mean square error sqer(X) ‖H − H‖2E

experienced when X is employed for the MMSE estimates of H. To this regard, it

may be proved [12,14,15,26] that the optimal pilot matrix X must be unitary, so

that the resulting MMSE channel estimates H are given by the simple expression

[12,14,15,26]

H = σ2ε

√γTtr

tX

†Y, (X unitary matrix) (5.2)

where

σ2ε E‖hji − hji‖2 ≡E‖ε2

ji‖ = (1 + Ttrγ/t)−1, (5.2.a)

is the mean squared error affecting each estimate hji. Furthermore, from (5.2)

and the uncorrelation of H with the corresponding error matrix ε H − H

it follows that all the (t × r) elements of matrices H and ε are mutually-

independent equally-distributed zero-mean complex Gaussian proper r.v.s with

variances E‖hji‖2 ≡ 1 − σ2ε and E‖εji‖2 ≡ σ2

ε , respectively (see [12,14,15,16]

for the formal proofs of these properties). The behavior of estimator in terms of

estimation error variance can be understand by observing the plot of Fig.5.2 where

the impact of number of antennas and signal to noise ratio on estimation error

variance is shown.

Therefore, since limσ2ε→0 H = H and limσ2

ε→1 H = 0, thereinafter the considered

limit cases of σ2ε = 0 and σ2

ε = 1 are named as perfect Channel-State Information

(PCSI) and No CSI (NCSI), respectively. Similarly, the case of 0 < σ2ε < 1 is

141

020

4060

80100

0

2

4

6

8

1010

−3

10−2

10−1

100

γ Ttr

t

Figure 5.2: Estimation Error Variance as in eq.(5.2.a) of the channel estimator in

eq.(5.2).

named as Imperfect CSI (ICSI). Finally, eq.(5.1.a) points out that large values of

the product γTtr guarantee low σ2ε and then accurate forward channel estimates

H. However, since large γ induces power-penalty and large Ttr causes throughput-

loss, the careful setting of these parameters is demanded for achieving optimized

performance-vs.-throughput tradeoffs. To this regard, the interesting analysis re-

cently presented in [14] leads to the important conclusion that, under mild operat-

ing conditions, Ttr = t is the optimal setting for the length of the training sequence

(see [14] and references therein for additional details on this point).

142

5.2.2 Some System Considerations about the feedback channel

After computing the forward channel estimates H, the receiver communicates

them back to the transmitter via the feedback channel, so thatH [

h1 . . .

hr] is

the (t×r) matrix composed by the forward channel estimates hji, 1 ≤ j ≤ r, 1 ≤i ≤ t available at the transmitter of Fig.5.1.

As the modeling of the feedback channel, it is pointed out that Time-Division-

Duplex (TDD) systems planned for low-mobility applications may exploit pilot

tones on the downlink to estimate the uplink and viceversa, so that in this case the

assumption of noiseless and delay free feedback links may be considered reasonable

[24]. However, this assumption falls short when the Doppler-spread of the forward

channel is of the order of the transmission rate of the feedback one [8,9], and

also for Frequency-Division-Duplex (FDD) systems where dedicated noisy links

are employed to feedback the available channel estimates [24].

5.2.3 Payload Phase

On the basis of the received forward channel estimatesH and message M

to be communicated, the transmitter of Fig.5.1 suitably shapes the signal streams

φi(n) ∈ C1, Ttr + 1 ≤ n ≤ T, 1 ≤ i ≤ t, radiated by t transmit antennas.

The corresponding (sampled) signals yj(n) ∈ C1, Ttr + 1 ≤ n ≤ T, 1 ≤ j ≤ r,

received during the payload phase at the outputs of the r receive antennas may be

modelled as

yj(n) =

√γTpay

t

t∑i=1

hjiφi(n) + wj(n), Ttr + 1 ≤ n ≤ T, 1 ≤ j ≤ r, (5.3)

where wj(n) are mutually independent zero-mean unit-variance proper Gaussian

noise samples. Therefore, after assuming the (usual) constraint

143

t∑i=1

E‖φi(n)‖2 =t

Tpay

, Ttr +1 ≤ n ≤ T, (5.3.a)

on the average power radiated by t antennas during each slot-period, it is easily

recognized that γ in (5.3) represents the resulting average SNR at the output

of each receiving antenna, independently from t. Thus, similarly to (5.1), the

(Tpay × r) observed samples in (5.3) may be merged into the (Tpay × r) observed

matrix Y [y1...y

r] given by

Y =

√γTpay

tΦH + W, (5.4)

where the (Tpay×r) matrix W collects the noise samples in (5.3). The correspond-

ing (Tpay × t) matrix Φ [φ1...φ

t] gathers the transmitted signals φi(n) in (5.3)

and it must satisfy the following power constraint (see (5.3.a))

ETra[Φ†Φ] ≡ E‖Φ‖2E = t. (5.4.a)

Furthermore, from (5.3) it is simple to deduce that the (r × 1) column vector

y(n) [y1(n)...yr(n)]T collecting the outputs of the r receive antennas during the

n-th payload slot is linked to the (t × 1) column vector φ(n) [φ1(n)...φt(n)]T of

the signals radiated by transmit antennas via the relationship

y(n) =

√γTpay

tHT φ(n) + w(n), Ttr + 1 ≤ n ≤ T, (5.5)

where w(n) [w1(n)...wt(n)]T is the corresponding noise vector. Thus, the re-

sulting (t × t) correlation matrix Rφ Eφ(n)φ(n)† of the radiated signals is

constrained as (see (5.3.a))

Tra[Rφ] ≡ Eφ(n)†φ(n) = t/Tpay, Ttr + 1 ≤ n ≤ T . (5.5.a)

144

Finally, after stacking the Tpay observed vectors y(n) in (5.5) into the corre-

sponding (Tpayr×1) block vector y [yT (Ttr +1)...yT (T )]T , the Tpay relationships

in (5.5) may be collected as in

y =

√γTpay

t[ITpay ⊗ H]T φ + w, (5.6)

where w [wT (Ttr + 1)...wT (T )]T . Hence, the (Tpayt × 1) block vector φ

[φT (Ttr + 1)...φT (T )]T of the radiated signals in (5.6) must satisfy the following

relationship (see (5.3.a)):

Eφ†φ = t. (5.6.a)

It is important to stress that Eqs. (5.4), (5.4.a) describe the MIMO link defined

by (5.3), (5.3.a) on a per-frame basis while eqs. (5.6), (5.6.a) give rise to the cor-

responding description on a per-slot basis. Obviously, these models are equivalent

and will be both employed in the following Paragraphs. To this regard, it is impor-

tant to explicitly remark that for the block-fading channel model here considered

the above reported constraints (5.4.a),(5.5.a) and (5.6.a) are fully equivalent. In

fact, (5.5.a) obviously implies (5.6.a).

Moreover, since the channel-matrix H remains constant during the overall payload

phase, there is no reason to assign more power to some payload slots and less to

others. Then, constraint (5.6.a) automatically implies (5.5.a). These argumenta-

tions also lead to the equivalence of constraints (5.4.a) and (5.5.a) one.

5.3 Throughput Analysis and Optimized Power Allocation

for ideal feedback channels

The considered systems is equipped with ideal feedback channels so thatH ≡

H. Since the above considered block-fading model of Paragraph 5.2 guarantees

145

that the forward channel path gains H change to new independent realizations

every T signalling periods, the forward-channel is information stable and ergodic

[31] and then, in principle, the payload data can be reliably transmitted over

the forward channel at any rate below the corresponding (ergodic) capacity C.

Following quite standard approaches [25], [33, p.361], [22, Chaps.2,8], this last can

be directly expressed as

C = EC(H) ≡∫

C(H)p(H)dH, (nats/payload-slot) (5.7)

where

p(H) = (1/(π(1−σ2ε)))

rt exp(− 1

(1 − σ2ε)

Tra[H†H]

), (5.7.a)

is the joined pdf of the MMSE channel estimates in (5.2), while the random

variable

C(H) supφ:E φ

† φ≤t

(1/Tpay) I(y; φ|H) ≡ supΦ:‖Φ‖2

E≤t

(1/Tpay) I(Y;Φ|H) (5.8)

is the capacity of the forward channel (5.4), (5.6) conditioned on the realization H

of the channel estimates available at both the transmitter and receiver. Finally,

I(·; ·|·) in (5.8), is the mutual information conveyed by the forward channel (5.4),

(5.6) during the payload phase. Since C(H) in (5.8) depends on the available real-

ization of the forward channel estimate, it must be regarded as a random variable

whose expectation is carried out in (5.7). Furthermore, due to adopted normal-

ization by 1/Tpay, C and C(H) in (5.7), (5.8) measure the maximal information

throughput conveyable (in a reliable way) during each slot of the payload phase.

However, by noting that Tpay/T is the fraction of the overall packet devoted to

payload data, it suffices re-scaling eqs.(5.7), (5.8) by Tpay/T to obtain the (aver-

age) information throughput conveyed by an overall packet.

146

Although the pdfs of the input signals achieving the sup in (5.8) is known for

the limit cases of PCSI [2,10,30] and NCSI [4,5,6], till now it is unknown for the

general case of IPCSI [13,14,15,29]. As a consequence, at the present no analytical,

semi-analytical or numerical tools are available for the evaluation of the capacity

in (5.8) in the general case of IPCSI (e.g., for 0 ≤ σ2ε ≤ 1). Therefore, similarly to

[7,Sect.IV] where an analogous problem is encountered, it is mandatory to proceed

to evaluate the conditional information throughput I(·; ·|·) in (5.8), by considering

two specific pdfs for the input signals that result to be optimal in several cases of

practical interest.

Specifically, the first considered for the input signals is the (usual) Gaussian

one. This means that the Tpay components φ(n) ∈ Ct, Ttr + 1 ≤ n ≤ T of

the signal vector φ in (5.6) are assumed mutually independent equally-distributed

zero-mean proper complex Gaussian t-dimensional vectors with correlation matrix

Rφ satisfying the constraint (5.5.a). Hence, after indicating as

TG(H) supTra[Rφ]≤ t

Tpay

I(y; φ|H), (5.9)

the maximal information throughput conveyed by the forward channel of Fig.5.1

for the above considered input Gaussian pdf, in general it is known that TG(H) ≤C(H). However, the above inequality is satisfied as equality when the available

channel estimates H are perfect (that is, for σ2ε → 0 [2,10,30]) or when no CSI is

available but the length Tpay of the payload phase largely exceeds the number t of

transmit antennas (see [4, Sect.V] for more details on this important asymptotic

result).

The second considered pdf for the random (Tpay × t) matrix Φ [φ1...φ

t] of

the input signal in (5.4) is the Isotropic and Unitary (IU) one, thus meaning that

147

the matrix Φ meets the definitory conditions [4,5,7]

a) Φ†Φ = It, b) p(Φ) = p(UΦV), (5.10)

for any square deterministic unitary matrices U ∈ CTpay × CTpay and V ∈ CTpay ×CTpay [4,5,7]. Thus, after indicating as

TIU(H) sup‖Φ‖2

E≤t

1/Tpay I(Y;Φ|H), (5.11)

the conditional maximal information throughput supported by the forward chan-

nel of Fig.5.1 when the IU random matrix Φ satisfies power constraint (5.10), it is

pointed out that, in general, TIU(H) ≤ C(H). However, starting from some con-

jectures previously reported in [5], it has been recently proved in [15] that TIU(H)

in (5.11) approaches actual ergodic capacity C(H) in (5.8) for Tpay ≥ min (r, t)+ r

and large SNRs (see [5,15] for more details on this important asymptotic property).

Therefore, motived by the above stated capacity-achieving properties of TG(H)

and TIU(H), the following Paragraphs will deal with their closed-form analytical

evaluations.

5.3.1 Analytical Evaluation of TG(H)

Although the conditional mutual information I(y; φ|H) in (5.9) generally

resists closed-form computation, the considerations developed in the first part of

the Appendix A lead to the following result.

Proposition 1. Once assumed as assigned the correlation matrix Rφ of the

mutually independent equally-distributed Gaussian input signals φ(n) ∈ Ct, Ttr+

1 ≤ n ≤ T in (5.5), the resulting conditional mutual information conveyed by the

MIMO forward channel of Fig.5.1 equates

I(y; φ|H) = Tpay lg det[Ir +

γTpay

t(1 + γσ2ε)

HTRφH

∗]+

148

+rTpay lg1 + γσ2

ε(det

[It + σ2

εγt

T 2payRφ

])1/Tpay(5.12)

when at least one of the following operating conditions is met:

a) both Tpay and t are large; (5.12.a)

b) σ2ε vanishes; (5.12.b)

c) SNR γ vanishes. (5.12.c)

Before proceeding, it may be of interest to shortly point out the relative roles

played by two terms present at the right-hand-side (r.h.s) of (5.12). Specifically,

the first one explicitly accounts for the available channel estimates H, so that this

term is the dominating one for σ2ε → 0, while it vanishes for σ2

ε → 1. On the

contrary, the second term at the r.h.s. of (5.12) dictates the information through-

put sustained by the considered system in non coherent applications, and then it

dominates r.h.s. for σ2ε → 1 (e.g., for H = 0), while it vanishes for σ2

ε → 0 (e.g.,

for H = H). Finally, about the asymptotic nature of the condition (5.12.a), Ap-

pendix F points out that it arises from a combined application of the Law of Large

Numbers and Central Limit Theorem. However, it may be numerically ascertained

that the condition (5.12.a) is, by fact, virtually met even for values of t and Tpay

limited and as low as t = 5 and Tpay = 7t. Therefore, since in this framework the

length T = Tpay + Ttr of the transmitted packets equates the coherence time Tcoh

of the underlying MIMO forward channel, in practice, channel coherence times of

the order of Ttr +35 suffice to apply (5.12) even for non vanishing values σ2ε and γ.

Optimized Powers Allocation over Transmit Antennas

Therefore, according to (5.9), next step concerns the maximization of through-

put (5.12) under power constraint (5.5.a). Towards this end, the Singular Value

149

Decomposition (SVD) is introduced

H = UΛV†

(5.13)

of the available forward channel estimates H, where U and V are the unitary

matrices of the right and left eigenvectors of H, while

Λ diagλ1, ..., λs,0t−s, (5.13.a)

is the corresponding (t× r) diagonal matrix of resulting s min r, t magnitude-

ordered singular values λ1 ≥ λ2 ≥ ... ≥ λs. Thus, after indroducing the dummy

positions

αi γTpayλ2i

t(1 + γσ2ε)

, i ≤ i ≤ s, β σ2εγ

tT 2

pay (5.14)

in the Appendix F it is proved that a suitable application of the Khun-Tucker

conditions [22, eqs.(4.4.10), (4.4.11)] to the objective function (5.12) allows to

compute the optimal transmit powers P (i), 1 ≤ i ≤ t achieving constrained sup

in (5.9) as detailed by the following Proposition.

Proposition 2 (Optimized Power Allocation for Gaussian Input Signals)

Now it is assumed that at least one of the above operating conditions (5.12.a),

(5.12.b), (5.12.c) is met. Thus, for i = s + 1, ..., t, the optimized powers achieving

supremum in (5.9) vanish while for i = 1, ..., s they must satisfy the following two

relationships:

P (i) = 0, for λ2i ≤ (1 + γσ2

ε)(rσ2

ε +t

γTpayρ

), (5.15)

P (i) = 12β

β[ρ(1 − r

Tpay

)− 1

αi

]− 1+

+

√β

[ρ(1 − r

Tpay

)− 1

αi

]− 12 + 4β

(ρ − 1

αi− ρβr

αiTpay

),

for λ2i > (1 + γσ2

ε)(rσ2

ε + tγTpayρ

).

(5.16)

150

Furthermore, the nonnegative parameter ρ in (5.15),(5.16) is chosen so to satisfy

the following power constraint (see (5.5.a)):∑i∈I(ρ)

P (i) = t/Tpay, (5.17)

where

I(ρ)

i = 1, ..., s : λ2i > (1 + γσ2

ε)(rσ2

ε + t/γTpayρ)

, (5.17.a)

is the ρ-depending set of i -indexes meeting the inequality (5.16). Finally, the

corresponding optimized correlation matrix for the input signals equates

Rφ(opt.) = U∗diagP (1), ..., P (s),0t−sU

T, (5.18)

while the resulting maximized throughput TG(H) in (5.9) is directly computable

as

TG(H) = r lg (1 + γσ2ε) +

s∑i=1

lg

[1 + αiP

(i)

(1 + βP (i))r/Tpay

], (nats/payload slot).

(5.19)

For σ2ε → 0 and σ2

ε → 1 the presented relationships (5.15), (5.16) for the op-

timized power allocation assume interesting limit forms examined in the following

Remarks.

Remark 1 (Case of PCSI )

For σ2ε → 0 it can be easily viewed that the corresponding limit forms assumed by

(5.15), (5.16) may be collected in the following simple water-filling like expression

[10,27 and references therein]:

P (i) = max

0, ρ − t

γTpayλ2i

, 1 ≤ i ≤ s. (5.20)

Thus, it is concluded that the relationships (5.15), (5.16) generalize the standard

water-filling ones and they reduce to them when the available channel estimates

151

H are exact.

Remark 2 (Case of NCSI)

Passing now to consider the opposite limit case of NSCI, it may observed

that for σ2ε approaching unit the relationship (5.12) becomes

limσ2

ε→1I(y; φ|H) = rTpay lg (1 + γ) − r lg det

[It +

γ

tT 2

payRφ

].

Therefore, since the corresponding maximizing Rφ exhibits only one non zero

eigenvalue equating to t/Tpay, the resulting maximal throughput TG(H) in (5.9)

assumes the simple expression

limσ2

ε→1TG(H) = r lg

[ 1 + γ

(1 + γTpay)1/Tpay

], (nats/payload slot). (5.21)

Since this last holds for large t and Tpay (see (5.12.a)), it may be concluded that

the relationship (5.21) agrees with the conclusion originally reported in [4] about

the capacity-achieving property retained for large Tpay by the input Gaussian pdf

even in application scenarios with no CSI. Interestingly enough, this limit property

exhibited by (5.21) is no retained by the lower bound on TG(H) originally presented

in [29] for SISO systems and recently generalized in [13,14,15] to the case of MIMO

systems. Specifically, this lower bound assumes the form [13,14,15]

TG(H) ≥ supTra[Rφ]≤ t

Tpay

lg det[Ir +

γTpay

tH

TRΦH

∗](5.21.a)

where the maximizing Rφ is evaluated via the application of the (usual) water-

filling algorithm [12] to the channel estimates matrix H [13,16,15]. Therefore, since

for σ2ε → 1 (e.g., for H → 0) the lower bound (5.21.a) vanishes regardless of value

assumed by Tpay, it is no able to reflect the above mentioned capacity-achieving

asymptotic property retained by the input Gaussian pdf. Furthermore, a direct

comparison of relationships (5.21), (5.21.a) leads to the conclusion that the lower

152

bound (5.21.a) approaches r.h.s. of (5.19) only in the limit case of vanishing σ2ε .

An Algorithm for computing the optimized Power Allocation in (5.15), (5.16)

The first step for computing the optimized powers (5.15), (5.16) consists in

solving the (nonlinear) equation (5.17) with respect to the (a priori unknown) ρ

parameter. Although the computation of ρ via (5.17) resists closed-form formula,

nevertheless the cardinality |I(ρ)| of set (5.17.a) vanishes at ρ = 0 and, then, it

increases for growing values of ρ. In turns, this means that the corresponding

power summation present on left-hand-side (l.h.s.) of (5.17) vanishes at ρ = 0 and

then it increases for large values of ρ. Therefore, the solution of (5.17) can be found

via a numerical iterative procedure that starts with ρ = 0 and then at each cycle

increases the current value of ρ by an assigned step-size ∆. The procedure stops

when the corresponding summation in (5.17) equates the requested value t/Tpay.

The resulting algorithm for computing the optimized power allocation (5.15), (5.16)

is reported in Table 5.1, where the while-cycle therein present implements the

above mentioned iterative procedure for the computation of ρ. In Paragraph 5.5,

some numerical performance plots obtained via computer implementation of the

algorithm of Table I are reported.

5.3.2 Analytical Evaluation of TIU(H)

Passing now to consider TIU(H) in (5.11), in general, this last resists closed form

evaluation. However, an application of the Law of Large Numbers and Central

Limit Theorem leads to the conclusion summarized by the following Proposition.

Proposition 3 (Asymptotic evaluation of TIU(H))

For large number of t of transmit antennas, the conditional throughput TIU(H)

153

1. Compute SVD of H in (5.13);

2. Perform the magnitude-ordering of the singular values of H;

3. Set P (i) = 0, 1 ≤ i ≤ t;

4. Set ρ = 0 and I(ρ) = ∅;5. Set the step-size ∆;

6 While(∑

i∈I(ρ) P (i) < t/Tpay

)do

7. Update ρ = ρ + ∆;

8. Update I(ρ) via eq. (5.17.a);

9. Compute the powers P (i), i ∈ I(ρ) via eq.(5.16);

10. end;

11. Compute the optimized powers P (i), 1 ≤ i ≤ s via eqs. (5.15), (5.16);

12. Compute Rφ(opt.) via eq. (5.18);

13. Compute the conveyed maximal throughput TG via eq.(5.19).

Table 5.1: Algorithm for the optimized power allocation for Gaussian Input and

ideal feedback link.

in (5.11) equates

TIU(H) = r lg

[1 + σ2

ε

(8 + γσ5εTpay

t)t/Tpay

]+

+ lg det[Ir +

γ

t(1 + γσ2ε)

H†H], (nats/payload slot) (5.22)

where the t columns of the isotropic-unitary random input matrix Φ [φ1...φ

t]

achieving (5.22) must satisfy the additional constraint

Eφjφ†

l = 1

TpayITpayδjl, (5.22.a)

with δjl being the Kroenecker-delta.

The above result is proved in Appendix G and it merits some explicative com-

154

ments.

First, since orthogonality of the columns of the (Tpay × t) unitary input matrix Φ

requires Tpay ≥ t [4,2] and, in addition, the coherence time of the channel equates

the length T ≡ Tpay + Ttr of the transmitted packets, in principle large t demand

for large Tpay that, in turns, require large channel coherence times. However, from

simulations emerges that t = 5, 6 suffices for the applicability of (5.22), so that

channel coherence times limited and of order of about (6 + Ttr) signalling periods

may be considered still adequate for the application of Proposition 3.

Second, condition (5.22.a) requires that the signal streams radiated by different

transmit antennas are mutually uncorrelated and also that the sequence radiated

by each transmit antenna is wide-sense stationary, white and with variance equal to

0/Tpay. In turns, this means that for isotropic-unitary input matrix signals Φ the

optimal power allocation strategy simply consists to evenly split the total available

power t/Tpay in (5.3.a) over overall transmit antennas, regardless of actual channel

estimates H available at the output of the feedback link of Fig.5.1. Therefore,

from an application point of view it may be concluded that there is no reason for

equipping the system of Fig.5.1 with a feedback channel when the signal matrix

Φ transmitted over forward link is unitary and isotropic. However, the numerical

results of Paragraph 5.5 point out that T(H) in (5.19) largely outperforms TIU(H)

in (5.22) at small σ2ε and then utilization choice of unitary-isotropic input signals

may be paid via a sensible reduction of the conveyed throughput.

155

5.4 Optimized power allocation for Gaussian input Signals

and Noisy feedback channels

The above conclusion may fall short when the assumption of an ideal feed-

back channel is relaxed and the noisy forward channel estimatesH available at

the transmitter of Fig.5.1 differ form those H computed at the receiver. To this

regard, it is now consider for the feedback channel the (simple) model where

the transmitted channel estimates hji, 1 ≤ j ≤ r, 1 ≤ i ≤ t are corrupted

by zero-mean unit-variance uncorrelated proper complex Gaussian noise samples

vji, 1 ≤ j ≤ r, 1 ≤ i ≤ t. Therefore, the (t × r) matrix Ω collecting the corre-

sponding (t × r) noisy outputs ωji, 1 ≤ j ≤ r, 1 ≤ i ≤ t of the feedback link is

directly given by

Ω √

γf

1 − σ2ε

H + V, (5.23)

where V is the (t× r) matrix composed by the noise samples vji, while γf is the

SNR for the transmission over the feedback channel. After noting that the MMSE

estimatesH EH|Ω of the transmitted data H are sufficient for the optimal

shaping of the corresponding input signal vector φ in (5.6) (see [32] and references

therein), a direct application of the Orthogonality Principle leads to the simple

expression H EH|Ω ≡

√γf (1 − σ2

ε)

1 + γf

Ω, (5.24)

so that the elements εji hji − hji, 5 ≤ j ≤ r, 1 ≤ i ≤ t of the corresponding

error matrix ε H− H are still zero-mean uncorrelated equally-distributed proper

complex Gaussian r.v.s with variance

σ2f E‖hji −

hji‖2 = (1 − σ2ε)/(1 + γf ), 1 ≤ j ≤ r, 1 ≤ i ≤ t, (5.25)

depending on the employed SNR γf .

156

A Property of Statistical Equivalence

Equation (5.7.a) points out that these statistical properties are also retained

by the elements of matrix H available at the receiver. So, from a statistical point

of view, H andH may be considered equivalent, the only difference being that

the average square-magnitude of the elements of H is (1− σ2ε) (see sub-Paragraph

5.2.2), while for the element ofH it can be seen that

E‖hji‖2 = 1 − σ2ε(tot), (5.26)

where (see eq.(25))

σ2ε(tot) E‖hji − hji‖2 ≡ σ2

ε + σ2f = (1 + γfσ

2ε)/(1 + γf ), (5.27)

is the average total square error affecting each channel estimatehji available at

the transmitter. In other words, from the transmitter perspective, the system of

Fig.5.1 working at SNR γ during the training phase and equipped with a noisy

feedback link operating at SNR γf is fully equivalent to a system with noise-

less feedback link but operating in the training phase at a reduced SNR equal to

(see eqs.(5.24),(5.27)): t/Ttr[1+γf

1+γf (1+γTtr/t)−1 − 1]. Therefore, after performing the

(quite obvious) replacements detailed in Table 5.2, the iterative algorithm previ-

ously reported in Table I for the noiseless feedback link may be also utilized for

computing the corresponding optimized powers P (i), 1 ≤ i ≤ t when a noisy

feedback channel is employed. Afterwards, after indicating as T(f)G (

H) the result-

ing conditional maximal throughput conveyed by the forward channel of Fig.5.1 by

Gaussian input signals and equipped with the noisy feedback link of eq.(5.23), this

throughput may be directly computed by introducing into the relationship (5.19)

the replacements listed in Table 5.2. In the following Remarks some interesting

limit forms assumed by T(f)G (

H) are pointed out.

Remark 3 (Limit form for large γf )

157

Ideal Feedback link Noisy Feedback link

SVD on H SVD onH

σ2ε σ2

ε(tot)

λj, 1 ≤ i ≤ s,λj, 1 ≤ i ≤ s

Table 5.2: Ideal-vs.-noisy feedback link.

SinceH is a noisy version of H (see eq.(5.24)), the throughput T

(f)G (

H) con-

veyed in the presence of a noisy feedback link does not exceed the corresponding

TG(H) sustained with an ideal feedback path, so that, in general, follows

T(f)G (

H) ≤ TG(H). (5.28)

However, eq.(5.25) leads to the conclusion thatH approaches H for large γf , so

that it is possible to write

limγf→∞

T(f)G (

H) = TG(H). (5.29)

Remark 4 (Limit form for vanishing γf )

Since eq.(5.24) points out thatH vanishes for small γf , we it may be concluded

that for vanishing γf no information is available at the transmitter about the

singular values of H. Hence, by pursuing quite standard approaches based on the

Hadamard inequality, it may be proved [30] that in this case the diagonal input

correlation matrix Rφ = (1/Tpay)It maximizes throughput in (5.12). Therefore,

after assuming that at least one of conditions (5.12.a),(5.12.b),(5.12.c) is met, it

may be cocluded that for vanishing γf the maximal throughput conveyable by the

system of Fig.5.1 equates

C(0)(H) limγf→0

C(f)g (

H) = r lg

[1 + γσ2

ε

(1 + γσ2εTpay

t)r/Tpay

]+lg det

[It +

γ

t(1 + γσ2ε)

HH†]

,

(5.30)

158

where H collects the channel estimates available only at the receiver and σ2ε is

the corresponding average square error in (5.24). Therefore, C(0)(H) represents

the maximal conditional throughput conveyed by the system of Fig.5.1 when no

feedback link is available and the receiver is equipped with ICSI H. Remark 4 (Some Considerations about the model adopted for the feedback channel)

About the validity limits of the model (5.23), it may be observed that, in emerging

4th Generation WLANs delivering QoS-guaranteed broadband services to nomadic

(e.g., low mobile) users, the symbol rate sustained by the feedback path to convey

the estimates H of the forward channel is expected to be limited and, in any

case, large below the corresponding symbol-rate to be supported by the forward

path [24]. Hence, since very reliable power-control loops may be implemented to

compensate for the fading phenomena (possibly) affecting the feedback path [24],

it is reasonable to model the resulting power-controlled feedback link as an unfaded

AWGN channel [24]. The generalization of the results present in this Paragraph to

the case of faded paths with imperfect power-control does not look, straightforward

and, indeed, it is still an open question currently investigated by the authors. Before presenting some numerical results and comparisons supporting above

analytical results, it can be pointed out that also the recent contributions [9,10] deal

with the impairing effects induced by non-ideal feedback channels. However, these

contributions only focus on systems with perfect CSI H ≡ H at the receiver and, in

addition, they consider only deterministic (i.e., noiseless) non-ideal feedback links.

Therefore, the analysis carried out in [8,9] does not cover the operating scenarios

considered by Chapter.

159

5.5 Numerical Results and Comparisons about average through-

put

Although the pdf of the channel estimates H assumes the simple form in (5.8),

nevertheless the corresponding expectations

TG ETG(H), TIU ETIU(H), (5.31)

of the conditional throughput (5.19), (5.22) resist closed-form analytical compu-

tations even in the limit cases of σ2ε → 0 [17,16] and σ2

ε → 1 [6]. Therefore,

according to [10,6,16], it can be possible to proceed with evaluation of the ex-

pected throughput (5.34) via a Monte-Carlo approach based on sample-averages

of 10,000 independent realizations of TG(H) and TIU(H).

5.5.1 Case of Gaussian Input Signals with Ideal Feedback Link

The resulting values obtained for TG in (5.31) are drawn in Figs.5.2 and

5.3 for some application scenarios of practical interest [1,23]. Both figures allow

to appreciate the combined effect of σ2ε and γ on TG. Specifically, Fig.5.3 points

out that the loss induced on TG by channel estimation errors is limited up to few

nats at medium SNRs and for σ2ε below 0.71, while this loss fast increases beyond

15 nats when σ2ε exceeds 0.1 (compare the upper and lowest curves of Fig.5.3 at

SNR around 1dB). Furthermore, Fig.5.4 shows that the sensitivity of TG on σ2ε

is high for low γ values while it decreases for growing SNRs. Therefore, since

the reported plots lead to the conclusion that actual effectiveness of the power

allocation algorithm fast decreases for increasing values of σ2ε , in practice, the

introduction of an (ideal) feedback link in the communication system of Fig.5.1

may be justified only when the average throughput TG in (5.31) achieved via the

160

0 1 2 3 4 5 6 7 80

5

10

15

20

25

30

35

γ(dB)

CG

(na

ts/s

lot)

σε2=0.001

σε2=0.01

σε2=0.1

σε2=0

σε2=1

Figure 5.3: Channel Capacity for Gaussian input signals for different values of σ2ε

(Tpay = 80, t=12, r=6).

power allocation algorithm of Table I outperforms the corresponding one C(0)

EC(0)(H) attained by evenly splitting the available total power over transmit

antennas (see eq.(5.39) and the following text).

Therefore, values beyond unit for the resulting Power Allocation Gain (PAG)

PAG TG/C(0) ≡ ETG(H)/EC(0)(H), (5.32)

support for the utilization of an (ideal) feedback link, while PAG values approach-

ing unity indicate uselessness of any feedback link. A 3D plot of the PAG (5.32) is

reported in Fig.5.5 for a system equipped with 12 transmit and 6 receiver antennas.

This plot allows to appreciate the dependency of the PAG on both σ2ε and

161

0 0.05 0.1 0.15 0.2 0.255

10

15

20

25

30

35

σε2

CG

(na

ts/s

lot)

γ=10dBγ=3dBγ=0.5dB

Figure 5.4: Channel Capacity for Gaussian input signals and various values of γ

(Tpay = 80, t=12, r=6).

operating SNR γ, and, by fact, it point out that PAG values approach unity for σ2ε

exceeding 0.25. Finally, the joined effects of σ2ε and Tpay on TG may be understood

by the 3D plot of Fig.5.6. Interestingly enough, this plot stresses that TG fast

approaches its ultimate limit attained at σ2ε = 0 also in application scenarios

impaired by channel estimation errors provided that Tpay is sufficiently large and,

by fact, of the order of about 1000-1500 slots. Therefore, the plot in Fig.5.6 directly

supports the conjecture of [3] about asymptotic optimality of Gaussian pdf for

the input signals of MIMO systems in Fig.5.3 even in the presence of substantial

channel estimation errors.

162

0

0.08

0.16

0.24

0.32 0 10 20 30 40 0 10 20 30

1

1.5

2

2.5

3

3.5

4

σε2

γ (dB)

PA

G

Figure 5.5: Power Allocation Gain (PAG) for Tpay = 80, t=12, r=6.

5.5.2 Case of Isotropic and Unitary Input Signals

The corresponding average throughput TIU in (5.31) supported by the con-

sidered system when the input signal matrix Φ in (5.4) is isotropic and unitary is

plotted in Figs.5.7 and 5.8. After comparing these curves against those previously

reported in Figs.5.3 and 5.4 for the corresponding cases of Gaussian input signals

with ideal feedback link, it may be possible to arrive at the interesting conclusion

that, at medium SNRs, TG typically outperforms TIU when σ2ε < 0.25. On the

contrary, TIU dominates TG at σ2ε > 0.25, so that, in these operating conditions,

isotropically distributed input signals with no feedback link results to be the most

appealing architecture for the system of Fig.5.1.

163

Figure 5.6: Channel Capacity for Gaussian input signals and various values of σ2ε

and Tpay.

5.5.3 Case of Gaussian Input with Noisy Feedback Uink

The plots drawn in Fig.5.9 report the average throughput T(f)G ET(f)

G (H)

conveyed by the system of Fig.1 when the input signals are Gaussian and the

feedback channel is the noisy one described by eq.(5.23). An examination of these

plots supports the conclusions that, at medium-high γ and for γf <10dB, T(f)G

is (largely) dominated by TG, even for σ2ε as high as 0.01. The behaviors of the

resulting Noisy Feedback Loss (NFL)

NFL TG/T(f)G ≡ ETG(H)/ET(f)

G (H) (5.33)

164

0 1 2 3 4 5 6 7 80

2

4

6

8

10

12

14

γ(dB)

CIU

(na

ts/s

lot)

σε2=0.001

σε2=0.01

σε2=0.1

σε2=0

Figure 5.7: Channel Capacity for Isotropic Unitary input signals (Tpay = 80, t=12,

r=6) for different values of σ2ε .

plotted in Fig.5.10 confirm this conclusion and they point out that, by fact, γf

values below 6dB suffice to vanish the performance gains achieved via accurate

channel estimates H.

5.6 Outage Analysis and Related System Performance

Since the ergodic capacity in (5.7) dictates the ultimate maximum achievable

information rate averaged over all fading states of the underlying MIMO forward

channel, it represents, indeed, a meaningful system performance index for non-

real-time data services. However, the ergodic capacity may not be longer relevant

for real-time applications (as, for example, voice and video) in (very) slow fading

165

0 0.05 0.1 0.15 0.2 0.253

4

5

6

7

8

9

10

11

12

σε2

CIU

(na

ts/s

lot)

γ=10dBγ=3dBγ=0.5dB

Figure 5.8: Channel Capacity for Isotropic Unitary input signals for various values

of γ (Tpay = 80, t=12, r=6).

environments where the maximum allowed delay T is (largely) below the coherence

time Tcoh of the underlying MIMO forward channel [26,27]. In fact, when the

coherence time Tcoh of the forward channel of Fig.5.1 largely exceeds length T of

the transmitted packet, the forward channel cannot be longer considered ergodic

and, in this case, the information outage probability

Pout(η) P (C(H) ≤ η), (5.34)

becomes a meaningful performance index [2,10,16,25,27]. Unfortunately no ana-

lytical formulas are currently available for the computation of C(H) in (5.8) in the

general case of 0 < σ2ε < 1. However, the considerations reported in the first part

of Paragraph 5.3 lead to the conclusion that TG(H) in (5.9) approaches C(H) for

166

0 1 2 3 4 5 6 7 80

5

10

15

20

25

30

γ(dB)

CG

(na

ts/s

lot)

γf=∞ dB

γf=10dB

γf=3dB

Figure 5.9: Channel Capacity for Gaussian input signals and noisy feedback link

for different values of γf (Tpay = 80, t=12, r=6).

large Tpay or vanishing σ2ε , so that at least in these operating conditions follows

Pout(η) = P (TG(H) ≤ η), (large Tpay or small σ2ε). (5.35)

Furthermore, for Tpay ≥ min t, r+r and large γ, it is TIU(H) in (5.11) to converge

to C(H) [5,15], so that in this last case it follows

Pout(η) = P (TIU(H) ≤ η). (5.36)

Therefore, the following two sub-Paragraphs attempt to analytically evaluate the

probabilities defined in (5.15), (5.16), so to achieve some insight about system

outage at least in the above mentioned operating conditions.

167

0.3

0.2

0.1

0 0 10

2030

40

1

1.5

2

2.5

3

3.5

4

σε2

γf (dB)

NF

L

Figure 5.10: Noisy Feedback Loss (NFL) for Tpay = 80, t=12, r=6.

5.6.1 Case of Gaussian Input Signals

Unfortunately, due to the non linear nature of the relationship (5.19) linking

TG(H) to the optimized powers P (i), 1 ≤ i ≤ t output by the algorithm of

Table 5.1, computation of P (TG(H) ≤ η) resists closed-form evaluation, even in

the limit case of σ2ε → 0 [8,10 and reference therein]. However, in the Appendix

H it is proved that for an ideal feedback link the probability (5.35) may be upper-

bounded as in

P (TG(H) ≤ η) ≤

1 − maxτ∈1...t

[e−Th1(η,τ)

rτ−1∑j=0

1

j!

(Th1(η, τ)

2

)j]u−1(Th1(η, τ)),

(5.37)

168

and lower-bounded as

P (TG(H) ≤ η) ≥

1 − e−Th2(η)

rt−1∑j=0

1

j!

(Th2(η)

2

)j]u−1(Th2(η)), (5.38)

where u−1(·) is the Heaviside-function, while the thresholds Th1(·) and Th2(·) are

defined as below reported

Th1(η, τ)2τ(1+γσ2ε)

γ(1−σ2ε)

[eη

(1+

γσ2εTpay

τ

)rτ/Tpay

(1+γσ2ε)r − 1

], (5.37.a)

Th2(η) 2t(1+γσ2ε)

γ(1−σ2ε)

[eη/t (1+γσ2

εTpay)r/tTpay

(1+γσ2ε)r/t − 1

]. (5.38.a)

As tightness of presented bounds, it is important to observe that the thresholds

(5.37.a), (5.38.a) approach the following limit values

limγ→0

Th1(η, τ) = limγ→2

Th2(η) = +∞ ; limγ→+∞

Th1(η, τ) ≤ 0 and limγ+∞

Th2(η) ≤ 0;

(5.39)

limη→∞

Th1(η, τ) = limη∞

Th2(η) = +∞ ; limη→4

Th1(η, τ) ≤ 0 and limη→0

Th2(η) ≤ 0 (5.39.a)

so that the corresponding bounds (5.37) and (5.38) are guaranteed to be asymptot-

ically exact for large and small SNRs γ and, also, for large and small values of the

outage parameter η. The numerical plots of Fig.5.11 agree with those analytical

conclusions.

In addition, they also show that the relative gap between bounds and actual

outage P (TG(H) ≤ η) is of the order of about 1% over the overall range of con-

sidered outages. Interestingly enough, the arithmetical average of bounds (5.37),

(5.38) virtually overlaps actual outage P (CG(H) ≤ η), so that, in practice, this

arithmetical average may give rise to reliable evaluations of the corresponding ac-

tual outage probability. Finally, according to the property of statistical equivalence

stressed in the previous Paragraph 5.4, it suffices to replace σ2ε in (5.37.a),(5.38.a)

169

29.5 30 30.5 31 31.5 321

2

3

4

5

6

7

8

9

10x 10

−4

η (nats)

Gau

ssia

n O

utag

e P

roba

bilit

y

Actual Outage (35)Lower Bound (37)Upper Bound (38)Bounds Average

Figure 5.11: Comparison between Gaussian Outage, and corresponding Lower and

Upper Bounds of eqs. (5.37), (5.38) for Tpay = 80, t=12, r=6, σ2ε = 10−3, γ=10dB.

by σ2ε(tot) in (5.27) to account for the noise possibly introduced by the feedback

channel of Fig.5.1.

5.6.2 Case of Isotropic and Unitary Input Signals

Since also P (TIU(H) ≤ η) resists closed-form analytical computation even

in the limit case of σ2ε = 1 [5,6], in the Appendix I there is the proof that then

following relationships hold:

P (TIU(H) ≤ η) ≤

1 − e−Th3(η)

rt−1∑j=0

1

j!

(Th3(η)

0

)j]u−1(η), (5.40)

170

and

P (TIU(H) ≤ η) ≥

1 − e−Th4(η)

rt−1∑j=3

1

j!

(Th4(η)

2

)j]u−1(η), (5.41)

where the thresholds Th3(·) and Th4(·) are defined as

Th3(η) 2t(1+γσ2ε)

γ(1−σ2ε)

[ea(η) − 1

], Th4(η) 2t2(1+γσ2

ε)γ(1−σ2

ε)

[ea(η)/t − 1

], (5.41.a)

with the dummy-variable a(η) given by

a(η) η + r lg

[(1 + γσ2ε

tTpay

)t/Tpay

1 + γσ2ε

]. (5.42)

Since it can be easily recognized that Th3(·), Th4(·) satisfy the same limit condi-

tions previously reported in (5.39), (5.39.a) for Th1(·) and Th2(·), one may con-

clude that both bounds (5.40), (5.41) are asymptotically exact for large and small

γ and also for large and small values of the outage parameter η. Finally, for large

values of the number of transmit antennas both limits (5.40), (5.41) converge to

the same step-like behavior given by the following expression:

limt→∞

P (TIU(H) ≤ η) = u−1

(η − r lg

[1 + γ(

1 + γσ2

tTpay

)t/Tpay

]). (5.43)

The plots drawn in Fig.5.10 confirm these analytical behaviors. In addition,

the comparison of the curves in Figs.5.11 and 5.12 leads to the conclusion that in

application scenarios with σ2ε below 10−3 the information throughput guaranteed

by Gaussian input signals at outages around 9 × 10−4 are about 2.8 times beyond

the corresponding ones achieved by IU input streams.

5.7 Conclusion

From the main results presented in this Chapter two basic guidelines emerge

about the optimized design of the MIMO systems of Fig.5.1.

171

8 9 10 11 121

2

3

4

5

6

7

8

9

10x 10

−4

η (nats)

IU O

utag

e P

roba

bilit

y

Actual Outage (36)Lower Bound (41)Upper Bound (40)Bounds Average

Figure 5.12: Comparison between Isotropic Unitary Outage and corresponding

Lower and Upper Bounds of eqs. (40), (41) for Tpay = 80, t=12, r=6, σ2ε = 10−3,

γ=10dB.

First, the numerical results previously detailed in sub-Paragraphs 5.5.1, 5.5.2

point out that Gaussian input signals with optimized power allocation over trans-

mit antennas are to be preferred to IU input signals with evenly distributed powers

when the estimates H available at the receiver are sufficiently reliable (i.e. for val-

ues of σ2ε in (5.2.a) below about 0.25). Otherwise (that is for σ2

ε > 0.25), power

allocation does not give rise to substantial performance improvement and then IU

input signals are to be preferred (see Figs.5.2, 5.6).

Second, the numerical plots detailed in sub-Paragraph 5.5.3 lead to the con-

clusion that throughput loss induced by noisy feedback channels in systems with

172

Gaussian input signals is limited for SNR γf in (5.23) over 10dB, but it fast in-

creases for γf below 7-8dB. Thus, any actual effectiveness arising from the utiliza-

tion of feedback paths definitively vanishes for γf below 5-6dB (see Figs.5.9 and

5.10).

Overall, an interesting problem arising from the above conclusions concerns ac-

tual design of practical space-time codes that approach the performance of Gaus-

sian input signals at low σ2ε and, then, progressively shift to closely mimic behavior

of IU input signals for increasing values of σ2ε .

173

5.8 Appendix F - Derivation of the Throughput Formula

of Sub-Paragraph 5.3.1

To begin with, it is considered the mutual information I(y; φ|H) conditioned

on the available realization H of the channel estimates as

I(y; φ|H) = h(y|H) − h(y|φ, H), (F.1)

where h(·) denotes the differential entropy operator. Since from the assumed chan-

nel model in (5.6) it follows that the conditional r.v. y|φ, H is proper, Gaussian

and with covariance matrix given by

Cov(y|φ, H) =

ITpay +

σ2εTpay

t

⎡⎢⎢⎢⎣φT (Ttr + 1)

...

φT (T )

⎤⎥⎥⎥⎦[φ∗(Ttr + 1) . . .φ∗(T )

]⊗ Ir, (F.2)

it is possible to develop h(y|φ, H) as in (F.1) as below reported

h(y|φ, H)(a)= E φ

lg det

[(πe)rTpayCov(y|φ, H)

](b)= rTpay lg (πe) + rE φ

lg det

[It +

γσ2εT

2pay

t

1

Tpay

T∑n=Ttr+1

φ(n)φ(n)†]

, (F.3)

where (a) follows from the expression for the differential entropy of a proper

complex Gaussian r.v.s [30] while for deriving (b) the properties employed are

det[I + AB] = det[I + BA] and also det[A] = det[AT ]. Now, although the trans-

mitted signal vector φ is assumed proper and Gaussian, nevertheless the corre-

sponding expectation in (F.3) resists closed-form analytical evaluation [10,16,30],

and in any case its computation requires multiple nested numerical integrations

[10,16]. However, since the random vectors φ present in the summation (F.3)

are uncorrelated and equally-distributed (see Paragraph 5.3), the Law of Large

174

Numbers [18, eq.(8.96), p.265] guarantees that for large Tpay = T − Ttr the sample

average 1Tpay

∑Tn=Ttr+1 φ(n)φ(n)† in (F.3) converges (in the mean square sense) to

the corresponding expectation Rφ. Hence, for large Tpay eq.(F.3) assumes the

following limit form:

h(y|φ, H) ≡ rTpay lg (πe)+r lg det[It+

γσ2εT

2pay

tRφ

], (large Tpay). (F.4)

Passing now to consider the first differential entropy h(y|H) −Ey| ˆH

lg p(y|H)reported at the r.h.s. of (F.1), it must be stressed that the conditional pdf p(y|H)

therein present does not admit closed-form representation even in the limit case

of r = t = 1 [6], [7]. However, directly from the channel model in (5.6), it can be

deduced that the conditional random variable y|H is zero-mean and its covariance

equates

Cov(y|H) = (1+γσ2ε)

ITpay ⊗

[Ir +

γTpay

t(1 + γσ2ε)

HTRφH

∗]

. (F.5)

Therefore, argumentations based on the Central Limit Theorem assure that the

resulting bound

h(y|H

) lg

(πe)rTpay det

[Cov

(y|H

)], (F.6)

is satisfied as equality for large values of the number t of transmit antennas. Hence,

the insertion of (F.4),(F.5) into (F.1) directly leads to the relationship (5.12), valid

for large t and Tpay.

Finally, for vanishing values of the product γσ2ε , the relationships (F.3), (F.4)

converge to the same limit rTpay lg(πe) , regardless of Tpay. In addition, for γ → 0

the observation y converges (with Prob.1) to the Gaussian random variable w

(see eq.(5.6)), while for σ2ε → 0 the channel estimates H approach (in the mean-

square sense) actual ones H (see eq.(5.2.a)) so that r.v. y|H converges to the

corresponding Gaussian random variable y|H. These last considerations explicitly

175

prove the validity of (5.12) even under the operating conditions reported in (5.12.b),

(5.12.c).

F.1 Proof of the optimized Power Allocation of eqs. (5.15), (5.16)

After introducing into (5.12) the SVD in (5.13) for H, quite standard (but

tedious) algebraic manipulations allow to put the relationship (5.9) in the following

equivalent form:

TG

(H)

= r lg(1 + γσ2

ε

)+

+ supt∑

i=1P (i) t

Tpay

f (P (1), ..., P (s)) − r

Tpay

t∑i=s+1

lg

(1 +

γσ2εT

2pay

tP (i)

), (F.7)

where

f(P (1), ..., P (s)) s∑

i=1

fi (P(i)) ≡

s∑i=1

[lg (1 + αiP

(i)) − r

Tpay

lg (1 + βP (i))

], (F.8)

is an additive objective function depending only on the powers P (i), 1 ≤ i ≤ sto be allotted to the first s transmit antennas. Thus, since the sup in (F.7) may

be achieved only for P (i) = 0, s + 1 ≤ i ≤ t, (F.7) can be rewritten as in

TG

(H)

= r lg(1 + γσ2

ε

)+ sup

s∑i=1

P (i) tTpay

f (P (1), ..., P (s)) . (F.9)

Now, an examination of the sign of the second derivatives ∂2fi (P(i)) /∂P (i)2

of the logarithmic functions fi (P(i)) in (F.8) leads to the conclusions that the

resulting sum-function f (P (1), ..., P (s)) is guaranteed to be ∩-convex (at-least)

over the region D of Rs given by

D

(P (1), ..., P (s)) : P (i) ≥ max

0,

β−αi

√Tpay/r

βαi

(√Tpay/r−1

), i = 1, ..., s

. (F.10)

176

This last region approaches overall positive orthant Rs+ of Rs when σ2

ε vanishes or

when Tpay grows unbounded. Therefore, after assuming met at least one of the

above operating conditions the Khun-Tucker conditions [22, eqs. (4.4.10),(4.4.11)]

may be applied to search for the constrained maximum of the objective func-

tion in (F.8). From a practical point of view, simple-to-test sufficient conditions

guaranteeing ∩-convexity of the objective function (F.8) over overall positive or-

thant Rs+ for σ2

ε ≥ 0 for any finite value of Tpay ≥ r are the following ones:

λ2i ≥ (1 + σ2

ε)√

rTpay, i = 1, .., s. In fact, when these last are met, all maxs in

(F.10) vanish and then f(·) in (F.8) is ∩-convex over overall Rs+. In all the carried

out numerical tests ascertained for the satisfaction of the above conditions.

So doing, one arrives at the following relationships:

αi(1+αiP(i))−1− r

Tpay

β(1+βP (i))−1 ≤ 1

ρ, for all i such that : P (i) = 0, (F.11)

αi(1+αiP(i))−1− r

Tpay

β(1+βP (i))−1 ≡ 1

ρ, for all i such that : P (i) > 0, (F.12)

where the expression at r.h.s. of (F.11), (F.12) is the derivative of the objective

function (F.8) with respect to i -th variable P (i). Now, eq.(F.11) directly leads to

(5.15), while (F.12) may be rewritten in the following equivalent form:

βP ∗(i)2 +

1 − β

[(1 − r

Tpay

)ρ − 1

αi

]P ∗(i) − ρ

αi

[αi − 1

ρ− βr

Tpay

]= 0,

for P ∗(i) > 0. (F.13)

Therefore, after resolving (F.13) with respect to P (i) and, then, retaining only

the corresponding non negative root, eq. (5.16) is directly obtained.

177

5.9 Appendix G - Derivation of the throughput formula

(5.22)

After rewriting the maximal throughput TIU(H) in (5.11) conveyed by the

channel (5.4) for Isotropic and Unitary matrix input signals Φ as in

TIU

(H)

=1

Tpay

sup‖Φ‖2

Et

h(Y|H

)− h

(Y|Φ, H

), (G.1)

it can be noted that the conditional r.v.s yj|Φ, hj , 1 j r, are zero-mean

independent proper complex Gaussian r.v.s sharing the covariance matrix

Cov(y

j|Φ, hj

)=

σ2εγTpay

tΦΦ† + ITpay , 1 j r. (G.2)

Therefore, for h(Y|Φ, H

)in (G.1) the following developments holds:

h(Y|Φ, H

) −EY|Φ,

ˆH

lg p

(Y|Φ, H

)(a)= r lg

((πe)Tpay det

[ITpay +

σ2εγTpay

tΦΦ†

])=

=(b) r lg

((πe)Tpay det

[It +

σ2εγTpay

tΦ†Φ

])(c)= rTpay lg (πe)+rt lg

(1 +

σ2εγTpay

t

), (G.3)

where (a) follows from (G.1) and above stated statistical properties of r.v.s yj|Φ, hj

, (b) is a consequence of the property det[I + AB] = det[I + BA] and, finally, (c)

accounts for the unitary property of Φ.

About h(Y|H

) −E

Y| ˆH

lg p

(Y|H

)in (G.1), recent contributions [6]

remark that this last resists closed-form analytical evaluation also in the particular

case of σ2ε = 1, so that in the following the analysis is limited to compute it under

assumption of large t. Toward this end, after indicating as vect(Y) the column

vector obtained via the ordered stacking of the r columns of the observed matrix

Y in (5.4), the corresponding conditional r.v. vect(Y)|H is zero-mean and its

covariance matrix may be computed from (5.4) as in

Cov(vect (Y) |H

) E

vect (Y) (vect (Y))†|H

178

= (γTpay/t)(H

T ⊗ ITpay

)EΦ

vect (Φ) (vect (Φ))†

×

×(H

∗ ⊗ ITpay

)+ (γTpay/t)σ

2ε

(Ir ⊗ EΦ

ΦΦ†

)+ IrTpay (G.4)

Since the (equally-distributed) elements of the isotropic random matrix Φ in (G.4)

are zero-mean and rows of Φ are (equally-distributed) random vectors satisfying

the power constraint in (5.3.a), from the Law of Large Numbers [18, p.265] one

may conclude that for large t the sample average (1/t)ΦΦ† converges (in the mean

square sense) to 1/TpayITpay , it can be written

1

tΦΦ† =

1

Tpay

ITpay , (large t) (G.5)

Therefore, after introducing in (G.4) the SVD of H in (5.13), for large t the deter-

minant of the covariance matrix (G.4) converges to the following form:

det[Cov

(vect (Y) |H

)]=(1 + γσ2

ε

)rTpay

det

[ItTpay +

γTpay

t (1 + γσ2ε)

EΦ

vect (Y) (vect (Y))†

(Λ

2 ⊗ ITpay

)]. (G.6)

Now, argumentations based on the Central Limit Theorem assure that for large

t the random variable vect(Y)|H becomes proper and Gaussian so that its differ-

ential entropy approaches the limit form [30]

h(vect (Y) |H

)= lg det

[(πe)rTpay Cov

(vect (Y) |H

)], (large t). (G.7)

Therefore, for large t the exploitation of (G.3),(G.4),(G.6) and (G.7) allows to

rewrite eq.(G.1) as in

CIU

(H)

= r lg

[(1 + γσ2

ε

)/

(1 +

γσ2εTpay

t

)t/Tpay]

+

+(1/Tpay) sup‖Φ‖2

Et

lg det

[ItTpay +

γTpay

t (1 + γσ2ε)

EΦ

vect (Φ) (vect (Φ))†(

Λ2 ⊗ ITpay

)].(G.8)

179

Since Λ in (G.8) is a diagonal matrix (see(5.13.a)), an application of the

Hadamard inequality [19, p.502] guarantees that sup in (G.8) is attained only

when the signal covariance matrix EΦ

vect (Φ) (vect (Φ))†

is diagonal. As a

consequence, it can be possible to rewrite (G.8) in the following form:

CIU

(H)

= r lg

[(1 + γσ2

ε

)/

(1 +

γσ2εTpay

t

)t/Tpay]

+

+(1/Tpay) sup‖Φ‖2

Et

s∑

j=1

T∑n=Ttr+1

lg

[1 +

γTpay

t (1 + γσ2ε)

λ2jE

‖φj(n)‖2] , (G.9)

where φj(n) is the signal-sample radiated by j-th transmit antenna during n-th slot

of the payload phase. Since the channel matrix H is constant during the payload

phase, values assumed by expectations E‖φj(n)‖2 in (G.9) must be independent

from the payload-index n. In addition, the unitary property in (5.10) constraints

the summation∑T

n=Ttr+1 E‖φj(n)‖2 to be unit for any index j. Therefore, as

a direct consequence of the above two constraints, each expectation E‖φj(n)‖2

in (G.9) must equate 1/Tpay, regardless of n and j. Hence, after replacing the

expectations in (G.9) by 1/Tpay, (5.22) is directly derived.

180

5.10 Appendix H - Derivation of the outage bounds (5.37),(5.38)

After assuming t ≤ r, µ1 ≥ µ2 ≥ . . . ≥ µt indicate the magnitude-ordered

eigenvalues of the (t × t) complex Wishart-matrix

W 2

1 − σ2ε

HH†. (H.1)

Therefore, since αi in (5.14.a) may be rewritten as αi = γµiTpay(1 − σ2ε)/(2t(1 +

γσ2ε)), a direct exploitation of the relationship (5.19) allows to express the proba-

bility P (TG(H) ≤ η) as in

P (TG(H) ≤ η) = P (χ ≤ η−r lg(1+γσ2ε)), (H.2)

where the real-values r.v. χ is defined as

χ supt∑

i=1P (i)t/Tpay

t∑i=1

lg

(1 +

(1 − σ2ε) γTpay

2t (1 + γσ2ε)

P (i)µi

)

− (r/Tpay) lg

(1 +

γσ2ε

tT 2

payP (i)

). (H.3)

Although the joint pdf of the eigenvalues µ(i), 1 ≤ i ≤ 1 of the Wishart

matrix in (H.1) is well-know [21], nevertheless the computation of the pdf of the

r.v. (H.3) resists closed-form expression even in the limit cases of σ2ε → 0 and

σ2ε → 1 [10, 16 and references therein]. Therefore, in the remaining part of this

Appendix upper and lower bounds the r.v. χ in (H.3) are derived.

H.1 The Upper-Bound (5.37)

By generalizing a simple power allocation strategy originally pursued in [35,

p.213] for ADSL applications, it has been recently remarked in [23] that for large

181

t and at medium-high SNRs an effective (although sub-optimal) power allocation

policy consists to assign equal powers to the first τ ≥ 1 transmit antennas and

thus turn off the last (t − τ) ones, according to the following ON-OFF like law:

P (i) t

τTpay

, 1 ≤ i ≤ τ and P (i) 0, τ + 1 ≤ i ≤ t. (H.4)

Therefore, after replacing the above ON-OFF power allocation (H.4) in (H.3),

the resulting r.v.

Θ t∑

i=1

lg

(1 +

(1 − σ2ε) γTpay

2t (1 + γσ2ε)

µiP (i)

)− r

Tpay

lg

(1 +

γσ2ε

tT 2

payP (i)

)≡

≡ − (rτ/Tpay) lg

(1 +

γσ2εTpay

τ

)+

τ∑i=1

lg

(1 +

(1 − σ2ε) γ

2τ (1 + γσ2ε)

µi

), (H.5)

obviously lower bounds the χ one in (H.3). Thus, the probability in (H.2) can be

upper bounded as

P(TG

(H)

η)

P(Θ η − r lg

(1 + γσ2

ε

)) ≡≡ P

(lg det

[Iτ +

(1 − σ2ε) γ

2τ (1 + γσ2ε)

diag µ1, ..., µτ]

k

), (H.6)

where

k η + r lg

⎡⎢⎣(1 + γσ2

εTpay

τ

)τ/Tpay

1 + γσ2ε

⎤⎥⎦ . (M.7)

Afterwards, after exploiting the inequality det[I+A] ≥ 1+Tra[A] holding for any

covariance matrix, (H.6) may be upper bounded as

P(TG

(H)

η)

P (v Th1 (η, τ)) , (H.8)

where Th1(η, τ) is given by (5.37.a) and v ∑τ

i=1 µi is the summation of first

τ eigenvalues of Wishart matrix in (H.1). Since v is a centered chi-square r.v.

182

with 2rτ degrees of freedom [2,21], for any assigned τ the probability at the r.h.s.

of (H.8) may be directly computed by resorting to the cumulative distribution

function of v, reported, for example in [34.p.42]. The resulting limit upper bounds

the outage P (TG(H) ≤ η) for any τ and then the focus is on the choice τ so to

minimize this bound. So doing, relationship (5.37) is directly obtained.

H.2 The Lower-Bound (5.38)

For lower-bounding P (TG(H) ≤ η) in (H.2), it can be possible to proceed

to upper bound the r.v. χ defined in (H.3). For this purpose, under constraint∑ti=1 P (i) = t/Tpay, the second summation in (H.3) achieves its minimum when

only one P (i) is strictly positive, so that the following lower bound takes place:

mint∑

i=1P (i)=t/Tpay

t∑

i=1

lg

(1 +

γσ2ε

tT 2

payP (i)

)≡ lg

(1 + γσ2

εTpay

). (H.9)

Therefore, after indicating as µmax the largest eigenvalue of the Wishart matrix

W in (H.1), it can be possible to proceed to upper bound χ in (H.3) according to

the following developments:

χ(a)

supt∑

i=1P (i)=t/Tpay

t∑

i=1

lg

(1 +

(1 − σ2ε)

2t (1 + γσ2ε)

µmaxP (i)

)− r

Tpay

lg(1 + γσ2

εTpay

)(b)

t lg

(1 +

(1 − σ2ε) γ

2t (1 + γσ2ε)

µmax

)− r

Tpay

lg(1 + γσ2

εTpay

), (H.10)

where (a) follows from (H.9) and definition of µmax, while (b) arises from an appli-

cation of the Jensen’s inequality. Although no closed-form expression is available

for the pdf of µmax [21], nevertheless µmax is upper-bounded by the Euclidean norm

‖W‖E of the corresponding Wishart matrix W [20, p.359], so that directly from

(H.1) it is possible to obtain the following chain of inequalities:

µmax ‖W‖E 2

1 − σ2ε

∥∥∥H∥∥∥2

E≡ 2

1 − σ2ε

r∑j=1

t∑i=1

∥∥∥hji

∥∥∥2

Ψ. (H.11)

183

Hence, after introducing (H.11) in to (H.10), it is possible to lower bound (H.2) as

in

P(TG

(H)

η)

P (Ψ Th2 (η)) , (H.12)

where Th2(·) is given by (5.38.a). Finally, after noting that the r.v. Ψ in (H.11)

is chi-square distributed with 2rt degrees of freedom [34, p.42], the lower bound

(5.38) directly arises from (H.12).

184

5.11 Appendix I - Derivation of the Outage Bounds (5.40),

(5.41)

After inserting (H.1) into (5.22), the outage probability P (TIU(H) ≤ η) may

be developed as in

P(TIU

(H)

η)

= P

⎛⎜⎝ζ η + r lg

⎡⎢⎣(1 + γσ2

εTpay

t

)t/Tpay

1 + γσ2ε

⎤⎥⎦⎞⎟⎠ , (I.1)

where ζ is the real nonnegative r.v. defined as (see (H.1))

ζ lg det

[It +

(1 − σ2ε) γ

2t (1 + γσ2ε)

W

]≡

t∑j=1

lg

(1 +

(1 − σ2ε) γ

2t (1 + γσ2ε)

µj

). (I.2)

I.1 The Upper Bound (5.40)

Thus, by resorting once a time to the inequality : det[I + A] ≥ 1 + Tra[A], it

may possible to lower bound the r.v. ζ in (I.2) as in

ζ lg

[1 +

(1 − σ2ε) γ

2t (1 + γσ2ε)

(t∑

j=1

µj

)]. (I.3)

Therefore, since the trace:t∑

j=1

µj of the Wishart matrix in (H.1) is a chi-square

distributed r.v. with 2rt degrees of freedom, eqs. (I.1), (I.2), (I.3) give arise to the

upper bound (40).

I.2 The Lower Bound (5.41)

By applying the Jensen’s inequality to (I.2), the r.v. ζ results upper bounded

as in

ζ t lg

[1 +

(1 − σ2ε) γ

2t2 (1 + γσ2ε)

(t∑

j=1

µj

)]. (I.4)

185

Therefore, after inserting (I.4) into (I.1) and, then, by resorting to the above

mentioned chi-square cumulative distribution function of the trace of the Wishart

matrix W, it is possible to directly arrive at the lower-bound (5.41).

186

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