Iterative Bit- and Power Allocation in Correlated MIMO Systems

Francisco Cano-Broncano

, Andreas Ahrens

and C´esar Benavente-Peces

Universidad Polit´ecnica de Madrid, Ctra. Valencia. km. 7, 28031 Madrid, Spain

Hochschule Wismar, University of Technology, Business and Design, Faculty of Engineering, Department of Electrical

Engineering and Computer Science, Philipp-M¨uller-Straße, PO box 1210, 23952 Wismar, Germany

Keywords:

Antennas Correlation, Iterative Bit Allocation, Multiple-input Multiple-output System, Power Allocation,

Singular-value Decomposition, Wireless Transmission.

Abstract:

In this contribution a novel iterative bit- and power allocation (IBPA) approach has been developed when trans-

mitting a given bit/s/Hz data rate over a correlated frequency non-selective (4 ×4) Multiple-Input Multiple-

Output (MIMO) channel. The iterative resources allocation algorithm developed in this investigation is aimed

at the achievement of the minimum bit-error rate (BER) in a correlated MIMO communication system. In

order to achieve this goal, the available bits are iteratively allocated in the MIMO active layers which present

the minimum transmit power requirement per time slot.

1 INTRODUCTION

The use of multiple antennas at the transmitter and

multiple antennas at the receiver side, which is

well-known MIMO system, improves channel per-

formance and link reliability of wireless communica-

tions. MIMO systems can considerably increase data

rate through spatial multiplexing and signiﬁcantly im-

prove robustness and coverage through diversity com-

bining (Yang et al., 2011).

Adaptive Modulation (AM) is a promising tech-

nique able to increase the spectral efﬁciency of wire-

less transmission systems by adapting the signal pa-

rameters, such as modulation constellation or trans-

mit power, based on the uncertain channel conditions

(Zhou et al., 2005). In order to achieve a better

system performance given a ﬁxed data rate an adap-

tive spatial modulation transmission scheme was pro-

posed in (Yang et al., 2011). The performance of

MIMO systems using spatial multiplexing is analysed

under bit- and power allocation techniques. Exist-

ing bit loading and transmit power allocation tech-

niques are often optimized for maintaining both a

ﬁxed power and a ﬁxed target bit-error rate while

attempting to maximize the overall data-rate. How-

ever, delay-critical real-time interactive applications,

such as voice or video transmission, may require a

ﬁxed data rate (Ahrens et al., 2008). Provided perfect

channel state information (PCSI) is available at the

transmitter side two major optimization problems are

considered to be solved. First, the optimal bit load-

ing. Second, the power allocation optimization prob-

lem (Weng et al., 2010). Given PCSI at the transmit-

ter, power and bits can be allocated to different lay-

ers. Adaptive bit and power allocation algorithms,

which can signiﬁcantly improve the MIMO system

performance, have received a huge research activity

lately, and can be divided into two groups according

to their performance: optimal and suboptimal algo-

rithms. Optimal allocation algorithms usually have

high computational complexity, making them difﬁcult

to apply to practical communication systems (Zheng

et al., 2013). In order to implement bit and power

allocation in practical communication systems many

computationally efﬁcient suboptimal allocation algo-

rithms have been proposed where most of them are

iterative. Krongold proposed a Lagrange-multiplier-

based integer-bits power allocation algorithm (Kron-

gold et al., 2000). The algorithm in (Goldfeld, 2002)

minimizes the BER subject to a requested data rate

and total transmit power by using adaptive power

loading and uniform bit allocation over all subchan-

nels. Zheng proposed a dynamic bound restriction

iterative algorithm framework to reduce the compu-

tational complexity (Zheng et al., 2013).

In this paper an optimized scheme with ﬁxed

transmission modes per time slot are ﬁrstly analysed.

Furthermore, the proposed algorithm performance

has been compared to already developed strategies.

The novel contribution of this research is the proposal

Cano-Broncano F., Ahrens A. and Benavente-Peces C..

Iterative Bit- and Power Allocation in Correlated MIMO Systems.

DOI: 10.5220/0004708400830093

In Proceedings of the 4th International Conference on Pervasive and Embedded Computing and Communication Systems (PECCS-2014), pages 83-93

ISBN: 978-989-758-000-0

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

of a new iterative bit and power allocation (IBPA) ap-

proach per MIMO layer based on unequal power dis-

tribution per MIMO active layer. The remaining part

of this paper is structured as follows: Section 2 intro-

duces the system model and the quality criteria con-

siderations. The proposed solutions of bit and power

allocation are discussed in section 3, while the associ-

ated performance results are presented and interpreted

in section 4. Finally, section 5 provides some con-

cluding remarks.

2 MIMO SYSTEM MODEL

According to (Ahrens et al., 2008) a frequency non-

selective MIMO communication link with n

anten-

nas in transmission and n

in reception can be de-

scribed as

u = H·c+ n , (1)

where u corresponds to the (n

×1) received data

vector, H is the (n

×n

) channel matrix, c is the

×1) transmitted data vector and n is the Addi-

tive White Gaussian Noise (AWGN) vector. Further-

more, it is assumed that the coefﬁcients of the channel

matrix H are independently Rayleigh distributed with

equal variance and that the number of transmit anten-

nas n

equals the number of receive antennas n

. In

order to avoid the inter-antenna interferences, Singu-

lar Value Decomposition (SVD) is used to transform

the MIMO channel into independent layers. By ap-

plying the SVD technique to the MIMO channel ma-

trix H, it is possible to rewrite the channel matrix as

H = S ·V·D

, where S and D

are unitary matrices

and V is a real-valued diagonal matrix of the posi-

tive square roots of the eigenvaluesof the matrix H

sorted in descending order. By using the SVD tech-

nique in pre- and post-processing, the MIMO chan-

nel can be described as multiple independent SISO

(single-input single-output) channels (so called lay-

ers) with different gains given by the singular values

in V,

y = S

·u = S

(H·D·c+ n) = V·c+ w . (2)

where H·D·c is the pre-processed transmit data vec-

tor and S

·u is the post-processed data vector at the

receiver side.

The considered quality criteria for end-to-end

wireless communication system performance is given

in terms of the bit-error-rate (BER), which quantiﬁes

the reliability of the entire wireless system from input

to output. The signal-to-noise ratio (SNR) per quadra-

ture component is deﬁned by

ρ =

(Half vertical eye opening)

Noise Power

)

, (3)

whereU

is the half vertical eye opening andU

is the

noise power per quadrature component taken at the

detector input. The relationship between the signal-

to-noise ratio ρ and the bit-error probability evaluated

for AWGN channels and M-ary Quadrature Ampli-

tude Modulation (QAM) is given by

log

(M)



1−

√



·erfc





. (4)

The application of the aforementioned SVD pre- and

ℓ,k

Figure 1: System model per MIMO layer ℓ and transmitted

data block k after SVD pre- and post-processing.

post-processing leads to the diagram in Fig. 1 with

different eye openings per activated MIMO layer ℓ

and per transmitted symbol block k according to

(ℓ,k)

ℓ,k

·U

s ℓ

. (5)

Considering QAM constellations, the average trans-

mit power P

s ℓ

per MIMO layer may be expressed as

s ℓ

ℓ

−1) . (6)

By taking L ≤ min(n

) MIMO activated layers

into account, the overall transmit power results in

∑

ℓ=1

s ℓ

. (7)

The layer-speciﬁc bit-error probability at the time

slot k is obtained from combining (3), (4), and (5) as

(ℓ,k)

log

ℓ

)



1−

√

ℓ



erfc

(ℓ,k)

√

(8)

The aggregate bit-error probability at the time slot k,

taking L activated MIMO-layers into account, results

(k)

∑

ν=1

log

)

∑

ℓ=1

log

ℓ

(ℓ,k)

. (9)

Finally, the BER of the whole system is obtained by

considering the different transmission block SNRs.

In order to balance the bit error probability along

the MIMO system activated layers bit and powerload-

ing can be helpful. The bit error probability at a given

time k is inﬂuenced by both the chosen QAM con-

stellation and the layer-speciﬁc weighting factors. In

PECCS2014-InternationalConferenceonPervasiveandEmbeddedComputingandCommunicationSystems

particular the layer-speciﬁc weighting factors inﬂu-

ence the overall performance. Therein, the ratio be-

tween the largest and the smallest weighting factor is

an unique indicator of the unequal weighting of the

MIMO layers at a given time k. Furthermore, the ra-

tio between the largest and the smallest singular val-

ues is signiﬁcantly inﬂuenced by the antennas corre-

lation effect due to the proximity of the multiple an-

tennas available at the transmitter and receiver sides.

In consequence, the transmit-to-receive antenna paths

become similar affecting the channel behaviour by in-

creasing the overall BER in the wireless communica-

tion link (Cano-Broncano et al., 2012).

In this work we focus on the inﬂuence of

transmitter-side antennas correlation on the MIMO

system performance. The correlation effect is de-

scribed by the correlation coefﬁcients. Transmitter-

side antennas correlation coefﬁcients describe the

similitude between paths corresponding to a pair of

antennas (at the transmitter side) with respect to a ref-

erence antenna (at the receiver side). Fig. 2 describes

the basic set-up for obtaining the correlation coefﬁ-

cient, where d is the transmitter-side antennas spac-

ing, d

is the distance between transmit antenna #1

and the receiver-side antenna (taken as reference) and

is the distance from transmit antenna #2 to the ref-

erence receive antenna (it is assumed d << d

); φ

is the departure angle. In consequence two paths are

established and the correlation coefﬁcient describes

how like they are.

d/2

antenna #1

Transmitter side

Receiver side

antenna #1

antenna #2

Figure 2: Antennas’ physical disposition: two transmit and

one receive antennas.

As shown in (Ahrens et al., 2013), the transmitter-

side correlation coefﬁcient is given by

(TX)

1,2

= e

j2πd

cos(φ)

. (10)

The antennas path correlation coefﬁcient for line of

sight (LOS) trajectories depends on the antennas sep-

aration d

and the transmit antennas reference axis

rotation angle φ (or signals angle of departure). By

taking the scattered environment of wireless chan-

nels into consideration, the transmit antennas refer-

ence axis rotation angle φ becomes time-variant and

(10) results in:

(TX)

1,2

(φ,σ

) = e

j2πd

cos(φ)

−

(2πd

sin(φ)σ

)

, (11)

where σ

corresponds to the standard deviation of the

scatters’ angle ξ. The parameters of the investigated

channel constellations are shown in Tab. 1. According

Table 1: Parameters of the investigated channel constella-

tions.

Description φ σ

Weak correlation 30

◦

1 1

Strong correlation 30

◦

1 0,25

to (Ahrens et al., 2013) the (n

) correlated matrix

MIMO system model H

is given by

vec(H

) = R

·vec(H) , (12)

where H is a (n

) uncorrelated channel matrix

with independent,identically distributed complex val-

ued Rayleigh elements and vec(·) is the operator

stacking the matrix H into a vector column-wise. Fol-

lowing the quite common assumption that the corre-

lation between the antenna elements at the transmitter

side is independent from the correlation between the

antenna elements at the receiver side, the correlation

matrix R

can be decomposed into a transmitter side

correlation matrix R

and a receiver side correlation

matrix R

by using the Kronecker product.

= R

⊗R

. (13)

In order to investigate the potential of bit- and power

loading, two different correlated channel proﬁles

(weak and strong) are depicted in Fig. 3 and Fig. 4.

Therein, the probability density function (PDF) of the

ratio ϑ between the smallest

4,k

and the largest

1,k

layer speciﬁc singular value for weakly and

strongly correlated frequency non-selective (4×4)

MIMO systems are shown. An uncorrelated one is

considered for comparison reasons.

From Fig. 3 it is observed that the difference be-

tween layers increases (i.e. the unequal weighting)

as the correlation does. This means that the ratio be-

tween the largest and the smallest singular value in-

creases, and due to that, the probability of having pre-

dominant layers increases. A similar conclusion can

be obtained by analysing the PDF curves from Fig. 4

where it is shown that the unequal weighting ratio be-

tween the largest and the smallest layer is even higher.

IterativeBit-andPowerAllocationinCorrelatedMIMOSystems

0 0.1 0.2 0.3 0.4 0.5

0.005

0.01

0.015

0.02

uncorrelated

correlated

pdf →

ϑ →

Figure 3: PDF (probability density function) of the ratio

ϑ between the smallest and the largest singular value for

weakly correlated (solid line) as well as uncorrelated (dot-

ted line) frequency non-selective (4×4) MIMO channel.

0 0.1 0.2 0.3 0.4 0.5

0.005

0.01

0.015

0.02

uncorrelated

correlated

pdf →

ϑ →

Figure 4: PDF (probability density function) of the ratio

ϑ between the smallest and the largest singular value for

strongly correlated (solid line) as well as uncorrelated (dot-

ted line) frequency non-selective (4×4) MIMO channel.

3 BIT AND POWER

ALLOCATION

Thanks to bit and power allocation techniques it is

possible to use wireless channel in an optimised way,

e.g. minimizing the BER performance at a ﬁxed data

rate under the constraint of a limited total MIMO

transmit power. In general, regarding the channel

quality, the BER performance is affected by both

the layer-speciﬁc weighting factors

ℓ,k

and the

QAM-constellation sizes M

ℓ

. Assuming a ﬁxed data

rate regardless the channel quality the resulting layer-

speciﬁc QAM constellations for a ﬁxed data through-

put are highlighted in Tab. 2.

Following the allocation of the bits per layer,

Table 2: Investigated QAM transmission modes.

throughput layer 1 layer 2 layer 3 layer 4

8 bit/s/Hz 256 0 0 0

8 bit/s/Hz 64 4 0 0

8 bit/s/Hz 16 16 0 0

8 bit/s/Hz 16 4 4 0

8 bit/s/Hz 4 4 4 4

ℓ,k

√

ℓ,k

Figure 5: Resulting layer-speciﬁc system model including

MIMO-layer PA.

Power Allocation (PA) can be added to optimize the

overall BER. The layer-speciﬁc power allocation fac-

tors

√

ℓ,k

adjust the half-vertical eye opening per

symbol block to

(ℓ,k)

APA

√

ℓ,k

·U

sℓ

. (14)

This results in the layer-speciﬁc transmit power

per symbol block k

(ℓ,k)

s PA

= p

ℓ,k

·P

sℓ

, (15)

where P

sℓ

denotes the allocated power per MIMO

layer without PA e.g. P

sℓ

= P

/L. Therein the parame-

ter L describes the number of activated MIMO layers.

Taking all activated MIMO layers into account, the

overall transmit power per symbol block k is obtained

(k)

s PA

∑

ℓ=1

(ℓ,k)

sPA

. (16)

With (14) the layer-speciﬁc bit-error probability at the

time k changed to

(ℓ,k)

bPA

log

ℓ

)



1−

√

ℓ



erfc

(ℓ,k)

APA

√

(17)

In order to ﬁnd the optimal set of PA parameters min-

imizing the overall BER, i.e.,

√

ℓ,k

, the Lagrange

multiplier method is used. Using the Lagrange multi-

plier method, the cost function J(p

,.. . , p

−1

) may

be expressed as

J(···) =

∑

ν=1

log

)

∑

ℓ=1

log

ℓ

(ℓ,k)

+ λ ·B ,

(18)

PECCS2014-InternationalConferenceonPervasiveandEmbeddedComputingandCommunicationSystems

where λ denotes the Lagrange multiplier. The param-

eter B in (18) describes the boundary condition

B =

∑

ℓ=1



sℓ

−P

(ℓ,k)

sPA



= 0 (19)

∑

ℓ=1

sℓ

(1− p

ℓ,k

) = 0 . (20)

A natural choice is again to opt for a scheme that uni-

formly distributes the overall transmit power along

the number of activated MIMO layers, i.e. P

sℓ

/L. In this case, the boundary condition simpliﬁes

B =

∑

ℓ=1

(1− p

ℓ,k

) = 0 . (21)

Following this equation the transmit power co-

efﬁcients have to fulﬁll the following equation

∑

ℓ=1

ℓ,k

= L. Differentiating the Lagrangian cost

function J(p

1,k

, p

2,k

,.. . , p

L,k

) with respect to the p

ℓ,k

and setting it to zero, leads to the optimal set of PA

parameters. In order to study the effect of PA thor-

Table 3: Investigated channel proﬁles assuming a (4×4)

MIMO system.

Proﬁle layer 1 layer 2 layer 3 layer 4

CM-1 1,7500 0,8750 0,4375 0,2188

CM-2 1,9000 0,6333 0,2111 0,0704

oughly, two different ﬁxed channel proﬁles as shown

in Tab. 3 are investigated. For comparison reason, the

channel proﬁle CM-1 describes a MIMO channelwith

low degree of correlation with ϑ = 0,125 whereas the

channel CM-2 introduces a high degree of correlation,

with ϑ = 0,037. Since the optimal PA solution is no-

tably computationally complex to implement, a sub-

optimal solution is investigated, which concentrates

on the argument of the complementary error function.

In this particular case the argument of the comple-

mentary error function

(ℓ,k)



(ℓ,k)

A PA



(22)

is assumed to be equal for all activated MIMO lay-

ers per data block k, i. e., ρ

(ℓ,k)

= constant ℓ =

1,2, .. .,L. Assuming that the transmit power per

layer is uniformly distributed, the power to be allo-

cated to each activated MIMO layer ℓ and transmitted

data block k can be simpliﬁed as follows:

ℓ,k

ℓ

−1)

ℓ,k

∑

ν=1

−1)

ν,k

. (23)

10 12 14 16 18 20

−4

−3

−2

−1

10 ·lg(E

) (in dB) →

bit-error rate →

(256,0,0, 0) QAM

(64,4,0,0) QAM

(16,16,0, 0) QAM

(16,4,4,0) QAM

(4,4,4,4) QAM

Figure 6: BER with optimal PA (dotted line) and without PA

(solid line) when using the transmission modes introduced

in Tab. 2 and transmitting 8 bit/s/Hz over channel CM-1.

10 12 14 16 18 20

−4

−3

−2

−1

10 ·lg(E

) (in dB) →

bit-error rate →

(256,0,0, 0) QAM

(64,4,0,0) QAM

(16,16,0, 0) QAM

(16,4,4,0) QAM

(4,4,4,4) QAM

Figure 7: BER with optimal PA (dotted line) and without PA

(solid line) when using the transmission modes introduced

in Tab. 2 and transmitting 8 bit/s/Hz over channel CM-2.

Here, for each symbol of the transmitted MIMO sym-

bol vector the same half vertical eye opening of

(ℓ,k)

A PA

√

ℓ,k

·U

sℓ

(24)

can be guaranteed (ℓ = 1,.. .,L), i.e.,

(ℓ,k)

A PA

= constant ℓ = 1, 2,. .. ,L . (25)

When assuming an identical detector input noise vari-

ance for each channel output symbol, the above-

mentioned equal quality scenario is encountered. The

BER curves for channel proﬁles CM-1 and CM-2 are

shown in Fig. 6 and Fig. 7. In order to use the MIMO

channel in an optimized way not all the MIMO layers

should be necessarily activated. Furthermore, PA in

combination with an appropriate number of activated

MIMO layers guarantee the best BER performance

when transmitting a ﬁxed data rate of 8 bit/s/Hz over

uncorrelated non-frequencyselective MIMO channel.

IterativeBit-andPowerAllocationinCorrelatedMIMOSystems

10 12 14 16 18 20

−3

−2

−1

10 ·lg(E

) (in dB) →

bit-error rate →

(16,16,0, 0) QAM

(16,4,4,0) QAM

Figure 8: BER with optimal PA (dotted line), equal-SNR

PA (dashed line) and without PA (solid line) when using the

transmission modes introduced in Tab. 2 and transmitting 8

bit/s/Hz over channel CM-2.

From the simulation results it can be seen that not all

the MIMO layers should be necessarily activated in

order to get the best BER.

In Fig. 8 the obtained BER curves with the opti-

mal PA are composed with the abovementioned equal

quality criteria. As demonstrated by computer sim-

ulations the loss in the overall BER with the equal

quality criteria is quite acceptable when using an op-

timized bit loading. Fig. 9 shows a comparison of

the BER curves among the listed QAM transmission

modes in Tab. 2 with and without PA and justiﬁes the

beforehand drawn conclusions. So far the efﬁciency

of ﬁxed transmission modes was studied focusing on

minimizing the overall BER.

An optimized scheme would now use the speciﬁc

transmission mode, guaranteeing the minimum BER

for each symbol block k. As highlighting in Fig. 10 by

using the best transmission mode per time slot a fur-

ther minimizing of the overall BER can be obtained.

The drawback is the higher signalling overhead.

Following the investigated ﬁxed transmission

modes, iterative bit and power allocation strategies

seem to be the most challenging solutions when min-

imizing the overall BER. Here the efﬁciency of bit

auctioning algorithms is investigated. Such solutions

test the amount of transmit power needed to transmit

a given numbers of bits per layer before allocating the

bit to the layer and deciding for a constellation which

requires the minimum power. For the iterative bit and

power allocation solution investigated in this work it

is ﬁrst tested in which layer the allocation of an extra

bit requires the minimum transmit power taking all

activated layers into account (ℓ = 1,2,..., L).

In order to ﬁnd out the best bit allocation perfor-

mance per time slot, it is required to ﬁnd the appro-

12 14 16 18 20 22 24

−8

−6

−4

−2

10 ·lg(E

) (in dB) →

bit-error rate →

(256,0,0,0) QAM

(64,4,0,0) QAM

(16,16,0,0) QAM

(16,4,4,0) QAM

(4,4,4, 4) QAM

Figure 9: BER with PA (dotted line) and without PA (solid

line) when using the transmission modes introduced in

Tab. 2 and transmitting 8 bit/s/Hz over non-frequency un-

correlated selective MIMO channels.

12 14 16 18 20 22 24

−8

−6

−4

−2

10 ·lg(E

) (in dB) →

bit-error rate →

(16,16,0,0) QAM

(16,4,4,0) QAM

Adaptive TM

Figure 10: BER with PA (dotted line) and without PA (solid

line) when using the transmission modes (TM) introduced

in Tab. 2 and transmitting 8 bit/s/Hz over non-frequency se-

lective channels (▽ , adaptive choice of the transmission

mode).

priate QAM constellation with the minimum transmit

power. For that, ﬁrstly from equation (4) it is required

to solve the SNR parameter resulting in

(ℓ,k)

IBPA

= 2 ·inverfc





(ℓ,k)

·log

ℓ

)

2·



1−

√

ℓ







. (26)

The half-level transmit amplitude on each MIMO

layer results together with (3) and (5) in

(ℓ)

s IBPA

)

(ℓ,k)

IBPA

ℓ,k

·U

. (27)

Consequently, using (26) and (27), the transmit power

PECCS2014-InternationalConferenceonPervasiveandEmbeddedComputingandCommunicationSystems

per MIMO layer (6) can be obtained as

(ℓ,k)

s IBPA

ℓ

−1)

ℓ,k

inverfc





(ℓ,k)

·log

ℓ

)

2·



1−

√

ℓ







(28)

With a given noise power per quadrature component

, the layer-speciﬁc weighting ξ

ℓ,k

and a reference

bit-error rate P

(ℓ,k)

, the transmit power per layer can

be calculated for a given M

ℓ

In a ﬁrst approach the proposed IBPA solution fo-

cuses on minimizing the overall BER at a ﬁxed data

rate. Fig. 12 illustrates the working principle at a ﬁxed

data rate of 8 bit/s/Hz using channel CM-2. Accord-

ing to Fig. 11 ﬁrstly the bit-error rate level is ﬁxed.

Next the bit allocation process is started with two bits.

The QAM constellation is transmitted over the layer

with the minimum power. Here layer ℓ = 1 and the

(4,0, 0,0) transmission mode is selected. After that,

as the overall transmit power P

(k)

s IBPA

∑

ℓ=1

(ℓ,k)

lower than one, another two bits are allocated. Result-

ing in the (16,0,0,0) transmission mode depicted in

the second row from Fig. 12. In the third row, as an-

other two more bits are allocated, the P

(k)

s IBPA

exceeds

the power limitation so the BER level is increased to

−6

, being (16, 4,0, 0) the updated QAM transmis-

sion mode. In the fourth row, the last two bits are allo-

cated due to the 8 bit/s/Hz limitation. In this last iter-

ation the BER level is increased to 10

−3

. This results

in the (64,4, 0,0) transmission mode. As the available

8 bits have been allocated, the bit allocation process

is ﬁnished. After the bit allocation process, power al-

location can be used to share the remaining power if

(ℓ,k)

s IBPA

< 1 and to balance the bit error probabilities

in the number of activated MIMO layers. Since the

transmit power is not uniformly distributed, the be-

forehand developed power allocation schemes can not

be used any longer, with P

(ℓ,k)

s IBPA

6= P

(k)

/L. In this case

the transmit power allocation parameter has to fullﬁll

the following boundary condition

(k)

∑

ℓ=1

(ℓ,k)

IBPA

·P

(ℓ,k)

s IBPA

. (29)

Again, a suboptimal power allocation can be found

which concentrates on the argument of the comple-

mentary error function resulting in

(ℓ,k)

IBPA

= ξ

ℓ,k

· p

(ℓ,k)

IBPA

s ℓ

)

, (30)

which is assumed to be equal for all activated

MIMO layers per data block k, i.e., ρ

(ℓ,k)

IBPA

constant (ℓ = 1,2,... ,L). The new power parameter

(ℓ,k)

IBPA

is calculated for each activated MIMO layer ℓ

and transmitted data block k, taking into consideration

the transmit power P

(ℓ,k)

s IBPA

per MIMO layer (28), the

boundary condition (29) and the SNR per quadrature

component (30). The new power parameter p

(ℓ,k)

IBPA

can

be shown to be calculated as follows

(ℓ,k)

IBPA

(ℓ,k)

s IBPA

ℓ

−1

ℓ,k

∑

ν=1

−1)

ν,k

. (31)

Using the power parameter p

(ℓ,k)

IBPA

from equation (31)

to be allocated in each transmitted MIMO symbol, the

half vertical eye opening changes to

(ℓ,k)

A IBPA

(ℓ,k)

IBPA

ℓ,k

·U

(ℓ)

s IBPA

, (32)

and in consequence the MIMO SNR per layer be-

comes

(ℓ,k)

IBPA

= p

(ℓ,k)

IBPA

·ξ

ℓ,k

(ℓ)

s IBPA

)

. (33)

With the obtained transmit power per MIMO layer ℓ

and the symbol block k, the half level transmit ampli-

tude results in

(ℓ)

s IBPA

(ℓ,k)

s IBPA

ℓ

−1)

(34)

Consequently, using (33) and (34), the bit-error rate

per MIMO layer P

(ℓ,k)

is modiﬁed to

(ℓ,k)

b IBPA

2·



1−

√

ℓ



log

ℓ

)

erfc







3· p

(ℓ,k)

IBPA

·ξ

ℓ,k

·P

(ℓ,k)

s IBPA

4·U

·(M

ℓ

−1)







(35)

Afterwards, the PA algorithm is applied according to

Fig. 11 and Fig. 12. Firstly the remaining non allo-

cated power is equally distributed over the MIMO ac-

tive layers. Secondly, an equal-SNR PA technique for

Non-Equal Power Distribution is introduced.

The drawback of the so far investigated IBPA so-

lution, the ﬁxed data rate per time slot, can be avoided

for improving the overall performance. First four

bits are equally allocated like the ﬁrst four bits in

the IBPA with ﬁxed data rate method. Within this

research the 8-QAM is excluded since the transmit

power per MIMO layer expression do not consider the

transmit power of an 8-QAM accurate enough. The

updated QAM constellation size is (16, 0,0, 0). The

QAM constellation with the minimum transmit power

is selected among the different transmission modes

achieved due to the bit allocation in the different lay-

ers. After that, bits are individually allocated as long

as the overall transmit powerP

(k)

s IBPA

is lower than one.

IterativeBit-andPowerAllocationinCorrelatedMIMOSystems

START

END

Fix BER level

BER level +

Bits > 8

Power

Allocation

YES NO

Bit

Allocation

< 1

Figure 11: Iterative Bit-and Power Allocation scheme at a

ﬁxed data rate.

IBPA fixed data rate

Bit Allocation

layers

Bit Allocation

BER level +

Bit Allocation

0.5

Equal Remaining

Equal-SNR

0.5

s μ

Power Distribution

Power Allocation

BER 10

-15

BER 10

-15

BER 10

-6

BER level +

BER 10

-3

layers

Figure 12: Iterative Bit-and Power Allocation at a ﬁxed data

rate.

Because of that the QAM transmission mode becomes

(16,2,0,0) in the third row of the Fig. 14, (32,2,0,0)

in the fourth row, (32,4,0,0) in the ﬁfth, (64,4,0,0)

in the sixth and (128,4,0,0) in the seventh one. In

this moment, because of allocating an extra bit means

exceed the transmit power limit, the deﬁnitive QAM

constellation results (128,4, 0,0). Besides, according

to the Fig. 13 and the Fig. 14, PA techniques are ap-

plied. The remaining not allocated power is equally

distributed over the MIMO active layers. And addi-

tionally, an equal-SNR Power Allocation (SNR-PA)

technique for Non-Equal Power Distribution is intro-

duced.

START

END

Fix BER level

Bit

Allocation

< 1

Power

Allocation

YES NO

Bit level +

Figure 13: Iterative Bit-and Power Allocation scheme with

variable data rate.

4 RESULTS

The investigation described in this paper has analysed

different resource allocation techniques aimed at in-

creasing the degree of design freedom, which largely

affects the bit-error probability performance on cor-

related MIMO communication systems. In general,

thanks to the selection of the most favourable QAM

transmission mode and the optimal transmit power al-

location per active layer the best BER performance

is achieved. Firstly, ﬁxed QAM constellations with

and without equal SNR-PA are analysed and com-

pared to frequency non-selective (4×4) MIMO chan-

nels with weak and strong correlation. Secondly, iter-

ative bit and power allocation set-ups are investigated

and compared with ﬁxed QAM constellation sizes in

order to achieve the optimal resources allocation per-

formance with the minimum bit-error probability.

4.1 Fixed QAM Constellation Sizes

Combined With Equal-SNR PA

Pre-establishing the QAM constellation, depicted on

Tab 2, a ﬁxed data rate is assured per data block (8

PECCS2014-InternationalConferenceonPervasiveandEmbeddedComputingandCommunicationSystems

4 1

5 1

5 2

6 2

BER level

Bit Allocation

Equal Remaining

Power Distribution

Equal-SNR

Power Allocation

Bit Allocation

Layers

Figure 14: Iterative Bit-and Power Allocation with variable

data rate.

bit/s/Hz). Fig. 15 shows the PDF of the transmis-

sion modes in bold introduced in Tab 2 and trans-

mitting 8 bit/s/Hz over uncorrelated, weak and strong

(channels proﬁles listed on Tab. 1) frequency non-

selective (4×4) MIMO channels. Here is highlighted

how the correlation effect modiﬁes the QAM trans-

mission mode chosen. The resultant BER curves with

ﬁxed QAM constellation are depicted in Fig. 16 and

Fig. 18 for the two different investigated channel pro-

ﬁles from Tab. 1, when transmitting at a bandwidth

efﬁciency of 8 bit/s/Hz. Here it is highlighted that

not all MIMO layers have to be activated in order

to achieve the best BERs. The target is ﬁnd out the

best combination of the QAM mode and the number

of MIMO layers, which gives the best possible BER

performance at 8 bit/s/Hz bandwidth efﬁciency. Sub-

sequently, PA techniques were applied for minimizing

the bit-error probability.

On the other hand, by comparing Fig. 16 and

Fig. 18 the BER increase due to antennas correlation

is noticed.

0,2

0,4

0,6

0,8

1,0

Transmission Mode

pdf →

MIMO

WC MIMO

SC MIMO

(64,4,0,0)

(16,16,0, 0)

(16,4,4,0)

Figure 15: PDF (probability density function) of choosing

different transmission modes when using the transmission

modes introduced in Tab. 2 and transmitting 8 bit/s/Hz over

uncorrelated, weak correlated (WC) and strong correlated

(SC) frequency non-selective (4×4) MIMO channels.

12 14 16 18 20 22 24

−8

−6

−4

−2

10 ·lg(E

) (in dB) →

bit-error rate →

(256,0,0, 0) QAM

(64,4,0,0) QAM

(16,16,0, 0) QAM

(16,4,4,0) QAM

(4,4,4,4) QAM

Figure 16: BER with equal-SNR PA (dotted line) and with-

out PA (solid line) when using the transmission modes in-

troduced in Tab. 2 and transmitting 8 bit/s/Hz over fre-

quency non-selective (4 ×4) MIMO channels under weak

antenna correlation.

IterativeBit-andPowerAllocationinCorrelatedMIMOSystems

12 14 16 18 20 22 24

−8

−6

−4

−2

10 ·lg(E

) (in dB) →

bit-error rate →

(256,0,0, 0) QAM

(64,4,0,0) QAM

(16,16,0, 0) QAM

(16,4,4,0) QAM

(4,4,4,4) QAM

IBPA, ﬁxed data rate

Figure 17: BER with equal-SNR PA (dotted line) and with-

out PA (solid line) when using the transmission modes in-

troduced in Tab. 2 and transmitting 8 bit/s/Hz over fre-

quency non-selective (4×4) MIMO channels under weak

antenna correlation.

12 14 16 18 20 22 24

−8

−6

−4

−2

10 ·lg(E

) (in dB) →

bit-error rate →

(256,0,0, 0) QAM

(64,4,0,0) QAM

(16,16,0, 0) QAM

(16,4,4,0) QAM

(4,4,4,4) QAM

Figure 18: BER with equal-SNR PA (dotted line) and with-

out PA (solid line) when using the transmission modes in-

troduced in Tab. 2 and transmitting 8 bit/s/Hz over fre-

quency non-selective (4×4) MIMO channels under strong

antenna correlation.

4.2 IBPA Combined With Equal-SNR

Due to the QAM transmission modes are previously

deﬁned in the ﬁxed QAM constellation approach,

some kind of rigidity in terms of resource allocation is

introduced. In order to increase the degree of freedom

in the MIMO performance design, iterative bit alloca-

tion with adaptive QAM constellation sizes per time

slot is developed in this contribution. Furthermore,

a optimal transmit power allocation with equal-SNR

not uniformly distributed per active layers is used for

minimizing the overall BER.

From Tab. 4 it is possible to compare the effect of

12 14 16 18 20 22 24

−8

−6

−4

−2

10 ·lg(E

) (in dB) →

bit-error rate →

(256,0,0, 0) QAM

(64,4,0,0) QAM

(16,16,0, 0) QAM

(16,4,4,0) QAM

(4,4,4,4) QAM

IBPA, ﬁxed data rate

Figure 19: BER with equal-SNR PA (dotted line) and with-

out PA (solid line) when using the transmission modes in-

troduced in Tab. 2 and transmitting 8 bit/s/Hz over fre-

quency non-selective (4×4) MIMO channels under strong

antenna correlation.

10 12 14 16 18 20

−8

−6

−4

−2

10 ·log

) (in dB) →

bit-error rate →

uncorrelated MIMO

strongly correlated MIMO

Figure 20: BER based IBPA at a ﬁxed data rate (solid line)

and at variable data rate (dotted line) when transmitting an

average data throughput of 8 bit/s/Hz over frequency non-

selective (4×4) MIMO channels with and without antenna

correlation.

the investigated correlated channel proﬁles listed in

Tab. 3 over BER performance in (4 ×4) MIMO sys-

tem with IBPA ﬁxed data rate. Due to the increasing

correlation effect from CM-1 to CM-2, the bit-error

probability increases.

The resultant BER curves are depicted in Fig. 17

and Fig. 19 when transmitting at 8 bit/s/Hz using

the iterative bit and power allocation algorithm pre-

viously described. Comparing the IBPA ﬁxed data

rate BER curves, in both investigated channels pro-

ﬁles weak and strong correlated, it is possible to show

the signiﬁcant BER performance improvement com-

pared with the ﬁxed QAM constellation sizes after ap-

plying optimal resources allocation techniques.

The constraint in the investigated IBPA algorithm

PECCS2014-InternationalConferenceonPervasiveandEmbeddedComputingandCommunicationSystems

Table 4: Investigated channel proﬁles assuming a

(4 × 4) MIMO system with IBPA ﬁxed data rate at

10log

) = 20 dB.

Proﬁle bit/s/Hz QAM TM BER

CM-1 8 (64,4,0,0) 1.6·10

−4

CM-2 8 (64,4,0,0) 1.8·10

−4

concerning to the requirement of ﬁxed data rate at

a given bit/s/Hz per time slot can be modiﬁed for a

variable data rate approach where the given bit/s/Hz

in average is assured with the aim to improve the

overall BER performance. From Tab. 5 it is possi-

ble to compare the effect of the investigated correlated

channel proﬁles, listed on Tab. 3, over BER perfor-

mance in (4 ×4) MIMO system with IBPA variable

data rate. Depending on the transmitting number of

bits the QAM transmission mode is modiﬁed. The

higher the transmit data rate the higher the BER per-

formance.

Fig. 20 shows the comparison between MIMO

system performances when using ﬁxed and variable

data rate when transmitting an average data through-

put of 8 bit/s/Hz over frequency non-selective (4×4)

MIMO channels with and without antenna correla-

tion. As highlighted by the BER curves, from the

variable data rate schema a noticeable better BER per-

formance is obtained.

Table 5: Investigated channel proﬁles assuming a (4 ×

4) MIMO system with IBPA variable data rate at

10log

) = 20 dB.

Proﬁle bit/s/Hz QAM TM BER

CM-1 7 (32,4,0,0) 1.4 ·10

−6

CM-1 8 (32,4,2,0) 3.2 ·10

−5

CM-1 12 (64,16,4, 0) 7.2·10

−3

CM-2 7 (32,4,0,0) 5.8 ·10

−6

CM-2 8 (64,4,0,0) 1.8 ·10

−4

CM-2 9 (128,4, 0,0) 2.5 ·10

−3

5 CONCLUSIONS

The chance to design and determine the number of ac-

tive MIMO layers, the number of the overall data rate,

the QAM modulation schema per layer and per time

slot and all combined with the appropriate allocation

technique of the transmit power can remarkably im-

provethe performanceof a correlated MIMO commu-

nication system. In this work a noteworthy BER per-

formance improvement has been accomplished by us-

ing optimal bit and power allocation techniques based

on iterative allocation methods over frequency non-

selective (4 × 4) MIMO channels with and without

antenna correlation. Additionally, in order to develop

these iterative power allocation algorithms (due to the

transmit power is not uniformly distributed over ac-

tive layers any longer) a new approach of suboptimal

power allocation parameter p

(ℓ,k)

IBPA

has been perfected

from the previous uniformly distributed power param-

eter p

(ℓ,k)

REFERENCES

Ahrens, A., Cano-Broncano, F., and Benavente-Peces, C.

(2013). Power Distribution for SVD-aided MIMO

Transmission with Receiver-Side Antennas Correla-

tion. In 10th IASTED International Conference on

Signal Processing, Pattern Recognition, and Applica-

tions (SPPRA), Innsbruck (Austria).

Ahrens, A., K¨uhn, V., and Weber, T. (2008). Iterative De-

tection for Spatial Multiplexing with Adaptive Power

Allocation. In 7th International Conference on Source

and Channel Coding (SCC), Ulm (Germany).

Cano-Broncano, F., Benavente-Peces, C., Ahrens, A., Or-

tega, F., and Pardo, J. (2012). Analysis of MIMO

Systems with Antennas Correlation with Linear and

Non-linear Spatial Distribution. In International Con-

ference on Pervasive and Embedded Computing and

Communication Systems (PECCS), pages 278–283,

Rome (Italy).

Goldfeld (2002). Systems BER Power Loading for OFDM

in Fading Channel. IEEE Transactions on Communi-

cations, 50:1729–1733.

Krongold, B., Ramchandran, K., and Jones, D. (2000).

Computationally Efﬁcient Optimal Power Allocation

Algorithm for Multicarrier Communication Systems.

IEEE Transactions on Communications, 48:23–27.

Weng, C.-C., Chen, C.-Y., and Vaidyanathan, P. (2010).

MIMO Tranceivers with Decision Feedback and Bit

Loading: Theory and Optimization. IEEE Transac-

tions on Signal Processing, 58:1334–1346.

Yang, P., Xiao, Y., Yu, Y., and Li, S. (2011). Adaptive

Spatial Modulation for Wireless MIMO Transmission

Systems. IEEE Communications Letters, 15:602–604.

Zheng, L., Stuber, G., Wang, D., and Cao, Y. (2013). Dy-

namic Bound Restriction Iterative Algorithm Frame-

work and Application to Power and Bit Allocation in

OFDM Systems. IEEE Transactions on Communica-

tions, 61:1160–1170.

Zhou, Z., Vucetic, B., Dohler, M., and Li, Y. (2005). MIMO

Systems with Adaptive Modulation. IEEE Transac-

tions on Vehicular Technology, 54:1828 – 1842.

IterativeBit-andPowerAllocationinCorrelatedMIMOSystems