Telecom Article a 12.2011

1504 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 5, MAY 2011

Does Fast Adaptive ModulationAlways Outperform Slow Adaptive Modulation?

Laura Toni, Member, IEEE, and Andrea Conti, Senior Member, IEEE

AbstractLink adaptation techniques are important modernand future wireless communication systems to cope with qualityof service fluctuations in fading channels. These techniquesrequire the knowledge of the channel state obtained with aportion of resources devoted to channel estimation instead of dataand updated every coherence time of the process to be tracked. Inthis paper, we analyze fast and slow adaptive modulation systemswith diversity and non-ideal channel estimation under energyconstraints. The framework enables to address the followingquestions: (i) What is the impact of non-ideal channel estimationon fast and slow adaptive modulation systems? (ii) How todefine a proper figure of merit which considers both resourcesdedicated to data and those to channel estimation? (iii) Doesfast adaptive always outperform slow adaptive techniques? Ouranalysis shows that, despite the lower complexity and feedbackrate, slow adaptive modulation (SAM) can achieve higher spectralefficiency than fast adaptive modulation (FAM) in the presenceof energy constraint, diversity, and non-ideal channel estimation.In addition, SAM satisfies bit error outage requirements also inFAM-denied region.

Index TermsAdaptive modulation, multichannel reception,channel estimation, fading channels, performance evaluation.

I. INTRODUCTION

THE diffusion of high speed digital wireless communi-cations has increased the need for reliable high datarate communications in variable channel conditions. Adap-tive modulation techniques allow to maximize the spectralefficiency (SE) in fading channels without compromising theperformance in terms of bit error probability (BEP) and biterror outage (BEO) (see, e.g., [1][7]). The -ary quadratureamplitude modulation ( -QAM) achieves high SE and it iswidely considered in adaptive modulation systems. In [3],for example, power and rate were both adapted to channelconditions for a -QAM uncoded system. The gain derivedfrom an adaptive rather than a fixed transmission scheme isreported, and it was shown that the channel capacity adaptingonly the data rate is almost the same of adapting both rate andpower. The fast adaptive modulation (FAM) technique tracksinstantaneous channel variations due to small-scale fading;the receiver estimates the instantaneous signal-to-noise ratio(SNR) and send a feedback to the transmitter with the optimal

Manuscript received April 19, 2010; revised November 26, 2010; acceptedJanuary 8, 2011. The associate editor coordinating the review of this paperand approving it for publication was A. Yener.

This research was supported in part by the FP7 European Network ofExcellence NEWCom++ grant.

L. Toni is with TERA, Italian Institute of Technology (IIT), Via Morego,30 16163 Genova, Italy, and with WiLab (e-mail: [email protected]).

A. Conti is with ENDIF, University of Ferrara, Via Saragat, 1 44100 Ferrara,Italy, and with WiLab at University of Bologna, Viale Risorgimento 2, 40136Bologna, Italy (e-mail: [email protected]).

Digital Object Identifier 10.1109/TWC.2011.020111.100643

constellation signaling to be used, [1][4]. Through adap-tation, good channel conditions are exploited increasing thetransmitted throughput and, at the same time, preserving theperformance in case of bad channel conditions. In [5], a slowadaptive modulation (SAM) technique has been proposed, inwhich the constellation signaling is adapted tracking large-scale channel variations (averaged over the small-scale fading)and compared to FAM. The analysis applies for systemsemploying diversity and operating in small-scale and large-scale fading channels with ideal channel estimation.

Typical performance metrics for adaptive communicationsystems are the BEP, the BEO (i.e., the BEP-based outageprobability [8]), and the SE. It is worth noting that FAMleads to best performance at the cost of a frequent channelestimation or prediction and high feedback rate. For a giventarget BEP, the SE and BEO achieved by SAM are closeto that of FAM and show a significant improvement withrespect to a fixed modulation scheme. With respect to FAM,the SAM technique requires a reduced feedback rate and haslower complexity.

For both FAM and SAM techniques, an important role isplayed by the channel estimation. The effects of outdatedchannel estimation are investigated for adaptive modulationsystems in [3]. Adaptive modulation systems with non-idealchannel estimation for single- and multi-carrier systems withFAM are analyzed in [3], [6], [9][11]. Channel estimationtechniques typically utilize resources that would be devoted todata transmission (e.g., pilot symbols can be inserted duringthe transmission of data symbols) thus sacrificing the SE.Hence, it is important to define proper figure of merit able tocapture the trade-off between quality of service and resourceutilization depending on the amount of energy devoted to dataand pilots.

In this paper, we analyze slow and fast adaptive -QAMsystems with diversity accounting for resources dedicated tonon-ideal channel estimation. The contribution is three-fold:(i) to define the achieved SE (ASE) which enables to takeinto account the portion of transmitted frame dedicated todata and pilot symbols; (ii) to analyze SAM with diversityin the presence of non-ideal CSI under energy constraints;(iii) to compare FAM and SAM under various conditions andconstraints. The framework developed enables the design ofadaptive diversity systems under energy constraints and non-ideal channel estimation.

The reminder of this paper is organized as follows. In Sec-tion II the system model and assumptions are presented and inSection III the performance is derived and a new metric whichconsiders also the resources utilized for channel estimation is

1536-1276/11$25.00 c 2011 IEEE

TONI and CONTI: DOES FAST ADAPTIVE MODULATION ALWAYS OUTPERFORM SLOW ADAPTIVE MODULATION? 1505

(Np,

CombinerDiversity

ChannelEstimator

Channel withDiversity N

Demodulator Modulator Adaptive

Insertion Pilot

)

Fig. 1. Block scheme of the considered adaptive communication system ( assumes different meanings depending on the adaptive modulation technique).

defined. In Section IV the performance is evaluated in termsof BEP, BEO, and the new definition of SE under constraints.In Section V numerical results are given with indications onhow they can be utilized by a system designer. Finally, ourconclusions are given in Section VI.

II. SYSTEM MODEL AND ASSUMPTIONS

We consider an adaptive modulation system (see Fig. 1)with -QAM squared constellation signaling in compositeRayleigh fading and log-normal shadowing over d-branchmultichannel reception with maximal ratio diversity (MRD).Independent, identically distributed (i.i.d.) fading and sameshadowing level over all branches, (i.e., microdiversity) isconsidered.1 We denote by the small-scale fading gain onthe -th branch which is distributed as a complex Gaussianrandom variable (RV) with mean zero and variance 2h perdimension for all branches.

We consider discrete variable-rate modulationschemes where a set of + 1 constellation signaling{0,1, . . . ,} can be adopted. As an example, digitalvideo broadcasting system employs {4, 16, 64, 256},then = 4+1 and = 3. The constellation signalingis chosen opportunistically depending on the value of aquantity which is, respectively, the instantaneous SNR in the case of FAM and the mean SNR , averagedover small-scale fading, for SAM. Given a target BEP b ,the required SNR for the modulation is such thatb(

) =

b . The opportunistic modulation is chosen by

comparing the estimated SNR value with the SNRs requiredfor each modulation to satisfy the target BEP. When the SNRvalue falls in the region

[ ,

+1

), the -th constellation

size is adopted.We consider a pilot symbols assisted modulation (PSAM)

scheme (see, e.g., [12][16]) where the transmitted frameis composed by s data symbols (each with mean energys) and p pilot symbols for channel estimation (each withmean energy p = s).2 The mean SNR on each diversitybranch is =

{2}s/0, where 0/2 is the two-sidedspectral density of the additive white Gaussian noise (AWGN)

1Since we consider microdiversity, in the following we will omit the branchsubscript in the notation of the mean SNR, averaged over small-scale fading.

2The frame is structured to transmit p pilots within a coherence time ofthe channel.

and {} denotes the statistical expectation (here evaluatedover the small-scale fading). The shadowing is assumed log-normal distributed where dB = 10 log10 is a Gaussian RVhaving mean dB and variance 2dB. The channel estimator ismaximum-likelihood and fading channels over branches aresuch that h = [12 . . . d ] is constant over a frame, andindependent frame by frame (block fading channel [17][19]).The estimated fading gain on the -th branch is

= + (1)

where is a zero mean complex Gaussian RV with varianceper dimension 2e . In [13] the

2e is derived as a function of

the pilot symbols energy and noise spectral density, as givenby

2e =0

2pp. (2)

For -QAM adaptive modulation systems, the non-idealchannel estimation affects both the transmitter side (in thechoice of the opportunistic constellation signaling) and thereceiver side (in diversity combining and bit reconstructing).To adapt the constellation size to the most updated chan-nel estimation, tight delay constraints should be met in theevaluation of the channel state information (CSI) used atthe transmitter side. Hence, for each frame the p receivedpilot symbols are processed for the channel estimation, whichis sent over an error-free feedback channel. Since the CSIemployed at the receiver side does not have such a tight delayconstraint, a longer interpolation involving a larger set of pilotsymbols can be harnessed at the receiver, leading to a morereliable CSI than the one at the transmitter side. It follows thatthe channel estimate at the transmitter can be less accurate thanthat at the receiver (see, e.g., [3]). In the following we assumenon-ideal CSI at the transmitter side and an ideal CSI at thereceiver side. Note that, for the effects of delayed channelestimates (outdated channels) the reader may refer to, e.g., [3],[11], here we focus on updated but erroneous CSI feedback.

In addition to the analytical results, evaluated based on theblock fading model discussed above, we also show simulativeresults based on both the block fading (to verify the analysis)and the time-varying models. In particular, to simulate thetime-varying small-scale fading, we assume Rayleigh fadingand the modified Jakes model [20], [21].


III. PERFORMANCE METRICS WITH NON-IDEALCHANNEL ESTIMATION

For a given target BEP, typical performance metrics foradaptive modulation systems are the SE and the BEO. Weevaluate them for multichannel communications with non-ideal channel estimation. For each possible constellation sig-naling, the SNR value required to reach the target BEP isevaluated and compared to the estimated SNR value (i.e.,FAM for FAM and SAM for SAM). The mean SE and theBEO for systems affected by channel estimation errors canbe evaluated from the probability density function (PDF) orthe cumulative distribution function (CDF) of , that we nowevaluate for FAM and SAM systems.

The performance of FAM systems depends on the estimatedinstantaneous SNR at the combiner output T. For MRD FAMis then given by

FAM = T =

d=1

=

d=1

2 s0

. (3)

Both real and imaginary parts of are zero-mean Gaussiandistributed with variance 2h +

2e . Therefore, the PDF of the

estimated instantaneous SNR T, conditioned to the mean SNRper branch , is a chi-square distribution

T() =d1

d2d (d)exp

[ 2t

](4)

for 0 and 0 otherwise, where 2t = 1/2 and

=

(

{

} {}

{

})

{ {}2

}

{ {}2}

=

{

}

{2

}

{2}

=

2h

2h + 2e=

p

p+1

(5)

is the envelope of the complex correlation between and [13]. In (5), = p/s and the second equality follows

from {} = {

}= {} = 0. From (4), the

marginal PDF of the estimated instantaneous SNR is derived

as T() = ()(), and the CDF can be easily

obtained. For log-normal shadowing the PDF of the meanbranch SNR is given by

() =

2dBexp

[ (10 log10 dB)

2

22dB

](6)

for 0 and 0 otherwise, with = 10/ ln 10.The performance of SAM systems depends on the estimated

mean SNR SAM = which is given by3

SAM = = {} = 2(2e +2h )s0

= 2t = +1

p. (7)

From (6) and (7), the CDF of the estimated mean SNR resultsin

() =

(dB 10log10( 1p )

dB

)(8)

for 1/(p) and 0 otherwise, where () =

2/2

is the Gaussian- function. For simplicity of notation, in thefollowing we will write to represent FAM or SAM for FAMor SAM systems, respectively.

A. Bit Error Outage

The BEO is an important performance metric for digitalcommunication systems which is defined as the probabilitythat the BEP is greater than the target BEP [8], [22][24],that is

o(b ) = {b() > b } . (9)

The system is in outage when even the constellation size0, which is the more robust against disturbances, does notfulfill the BEP requirement or when the increasing of isdictated by an erroneous channel estimation. For FAM theexact instantaneous BEP expression is given by (10), reportedat the bottom of this page [25], [26] whereas for SAM theexact mean BEP expression is given by (11), reported at thebottom of this page, where denotes the largest integer lessthan or equal to , and

d () =1

/20

sin2()sin2() + 3(2+1)

2

2(1)

d d . (12)3Note that = for ideal channel estimation.

b(T) =2

log2()

log2()

=1

(12)1=0

(1) 21

(21

21

+

1

2

)

((2+ 1)

3T

( 1)

)(10)

() =

0

b(T)T() =2

log2()

log2()

=1

(12)1=0

(1) 21

(21

21

+

1

2

) 0

((2+ 1)

3T

( 1)

)T()

d ()

(11)


In the case of ideal channel estimation, the BEO is given by

o(b ) = (

0) (13)

where () is the CDF of , being = T for FAM and = for SAM [5],

In systems with non-ideal CSI at the transmitter, the esti-mated SNR can be an underestimate or an overestimate ofthe true value. The former case leads to a reduction of theSE and the BEO, while the latter leads to an increase of theSE and BEO. In particular, when > , although the trueSNR would fall within the range for the -th modulation, themodulation +1 might be adopted. In this case, the BEP canbe greater than the target BEP and the system is in outage.

Thus, the system is in outage when < 0 or >

and for all . Therefore, for = +, the outageoccurs for < or < 0 and the conditionedBEO o results to be{(

0) +

=1

[(

) ( )

]with 0

(0) otherwise.

(14)For FAM systems, = T is a RV whose PDF () isderived in Appendix and the unconditioned BEO is given by

o =

o() . (15)

For SAM systems, we note from (7) that = 1/(p) isdeterministic and

o = (0) +

=1

[ () ()] , (16)

where =dB ,dB

dB

=dB 10 log10

( 1p

)dB

.

In the absence of energy constraint, it is expected that, byincreasing p, the channel estimation accuracy increases andthe BEO approaches the one for ideal CSI. We compare thecase of non-ideal channel estimation with the ideal one, byevaluating the penalty on the required median SNR for a giventarget BEO and BEP. We recall that adaptive systems withideal CSI at the receiver side are in outage when even the mostrobust constellation size (i.e., = 0) does not meet thetarget BEP. In the presence of small- and large-scale fading,the BEO is a function of the median SNR value dB. Givena target BEO o , dB,0 denotes the median SNR value whichreach o = o when the lowest constellation size is adopted,that is = 0. In the case of non-ideal CSI, the system isin outage not only when the lowest constellation size does notmeet the target BEP, but also when the SNR in the feedbackis overestimated. In this case, the system might switch to amodulation level higher than the best one, and the systemmight experience a BEP greater than the target one. Here wetake into account both the effects causing the BEO, and wedenote dB,0 as the median SNR value which reaches the targetBEO for non-ideal channel estimation. Then, the median SNRpenalty for a given BEO is defined as

dB dB,0 dB,0 (17)

which represents the increasing in required median SNR fornon-ideal CSI with respect to the case of ideal CSI.

B. Achieved Spectral Efficiency

The available SE is given by log2 which would bereached if channel conditions are such that the system isalways in service with the greatest constellation size. In thepresence of non-ideal channel estimation and outage events,the ASE (i.e., the SE achieved considering resources dedicatedto channel estimation) might be lower than the available SEand its characterization is important for system design. Themean SE [bps/Hz] with ideal channel estimation is given by[5]

=

1=0

[(

+1) ()

]+ [1 ( )] ,

(18)where = log2 .

In the case of non-ideal channel estimation, the SNR is replaced by the estimated one . The insertion of pilotsymbols occupies part of the resources that could otherwisebe utilized for data symbols. In each frame of length totsymbols, p pilot symbols and s = tot p data symbolsare transmitted. By denoting the fraction dedicated to pilotsand data as p = p/tot and s = s/tot, respectively, wedefine the ASE as

A s = (tot p)

tot. (19)

For MRD the ASE implicitly depends on the number ofbranches, d. By increasing d, the BEP is reduced, and theSNR thresholds decreases, leading in general to a greater SE.In the case of subset diversity (SSD), where branches outof d are selected, d/p pilots should be transmitted toguarantee p received symbols per branch [27][30], where denotes the largest integer higher than or equal to . Then,the ASE would result in A (tot d/p) /(tot),which explicitly depends on and d.

We also define the mean spectral efficiency penalty as theratio of the SE evaluated with ideal CSI (ideal) and the ASEevaluated in the non-ideal CSI case A as

(ideal)

A. (20)

Obviously, with ideal channel estimation = 1. We recallthat for SAM systems, the coherence time of tracked channelvariation is greater than the one for FAM, and for a givenamount of resources dedicated to the channel estimation, theportion of data symbols per frame is greater than the one forFAM, thus FAMs SAMs . Conversely, for a given value of p,the number of pilot symbols dedicated to channel estimationis greater for SAM than for FAM.

IV. PERFORMANCE ANALYSIS UNDER CONSTRAINTS

In adaptive modulation systems with non-ideal CSI, theperformance strictly depends on the pilot scheme adopted.In the following, we analyze the effects of both the imposed


constraints and the pilot scheme design on the system per-formance. In both FAM and SAM systems, the followingconstraints are imposed over each frame p +s = tot

pp +ss = tot .(21)

Then, the energy dedicated to data depends on the adoptedpilot scheme as

s =

ptot

( 1) + 1(22)

where tot/tot. Note that the increasing of p leads toa better channel estimation and to a lowering of s, the twosituations have opposite effects on the system performance.Consequently, even the exact SNR is a function of the pilotscheme. This energy partition problem was investigated in [16]for cellular systems.

A. FAM Systems

From the above constraints, the SNR = T can beexpressed as

T =

d=1

20

ptot

( 1) + 1=

Tptot

( 1) + 1(23)

where T d

=1 2/0. By substituting (23) in theBEP expression b() reported in (10), the b results afunction of two parameters: p which characterizes the pilotscheme design and T which represents the SNR per generic(pilot or data) symbol. Thus, through (10) the instantaneousBEP is

b (T, p,) = b (T) , with T =T

ptot

( 1) + 1for = (24)

Unlike the SNR T, the T does not depend on the pilotscheme, but only on the mean energy over the frame, the0, and the channel conditions. Therefore, we will comparesystems with different pilot schemes for a given T. Whenconstraints are imposed, T is the SNR variable in the BEPexpression based on which the constellation size is chosen (for = 1, then T T). It means that the -th SNR threshold( = ) is defined as

T, such that b(T, , p,

)= b . (25)

Due to the imposed constraints, a double effect of the pilotassisted channel estimation is present: i) parameters p and affect the data energy, leading to an increase of the thresholdslevels T,; ii) the accuracy of the estimation in the feedbackchannel depends on p and . In particular, the feedbackestimated SNR is

T =

d=1

2 0

.

where only depends on the pilot scheme. When idealsystems (ideal CSI without constraints) are considered, Tcorresponds to T.

In FAM systems with energy and symbols constraints, theBEO can be evaluated from (9) and (14) for ideal and non-ideal CSI respectively, with = T, =

T, , = T,

and () = T (), being T () the CDF of T. TheASE can still be evaluated from (18) and (19), with () =T (). The CDF of T can be derived from the marginalPDF of T() =

T()(), where the conditional

PDF is

T() =d1

d2dt (d)exp

[ 2t

],

for 0, and 0 otherwise.

B. SAM Systems

In SAM systems, the same considerations of the FAM casehold, but based on quantities averaged over the small-scalefading. The mean SNR is given by

={2}

0

ptot

( 1) + 1=

ptot

( 1) + 1(26)

where {2}/0. In the degenerative case of = 1,we have that corresponds to . Through (11), the mean BEPexpression is given by

b (,p,) = b () = ptot

(1)+1(27)

and the -th SNR threshold ( = ) is

such that b( , p,

)= b .

In the case of non-ideal channel estimation, the mean SNR is

{2} 0

= +1

tot+

1 pp

. (28)

When ideal systems (ideal CSI without constraints) are con-sidered, corresponds to .

For ideal CSI systems, the BEO is evaluated from (9)and (13), with = , =

, and () = (), where

() is the CDF of . For log-normal shadowing, dB isGaussian distributed with mean dB and variance

2dB, with

CDF given by4

() =

(dB 10 log10()

dB

). (29)

4When = 1, dB = dB, while for = 1, the median SNR penaltybecomes dB dB,0 dB,0 .


For non-ideal CSI systems, the BEO can be derived from (14)and (28) as

o = (0) +

=1

[ () ()] , (30)

where =dB ,dB

dB

=dB 10 log10

( 1tot

1pp

)dB

.

The ASE is evaluated from (18) and (19), with = , = and () = () is given by

() =

(dB 10 log10( 1tot

1pp

)

dB

)for (1/tot) + (1 p)/p and 0 otherwise.

V. NUMERICAL RESULTS

We now present analytical, mainly, and simulative resultsin terms of ASE and BEO for both FAM and SAM systems.The simulative results are intended to verify the analysisin block fading and compare SAM and FAM also in time-varying channels. We assume coherent detection of -QAMwith {4, 16, 64, and 256}, d-branch MRD, and Graycode mapping. Composite Rayleigh fading and log-normalshadowing channels are considered with both ideal and non-ideal channel estimation.5 For non-ideal channel estimation,the ASE is evaluated by (19), with a target BEP of 102 anda maximum BEO of 5% (typical values for uncoded systems).We denote by the ratio between the frame lengths withSAM and FAM (i.e., the ratio of the coherence time, for large-and small-scale fading), as

=SAMtotFAMtot

.

We assume FAMtot = 100 symbols and = 1000 [31][33] which provide a conservative comparison since for re-alistic shadowing and fading channels [32], [34][37], canbe greater than 1000 and consequently the gain of SAMcompared to FAM can be even higher than what shown inthe results. We now provide an example. For time-varyingchannels, the coherence time of the small-scale fading isinversely proportional to the maximum Doppler frequency:at 900MHz the coherence time is about 72ms and 4ms for amobile speed of 3 km/h and 50 km/h, respectively. On the otherhand, the coherence time of the shadowing is proportionalto the coherence distance (e.g., 100-200m in a suburbanarea and tens of meters in an urban area [38]). Assuminga coherence distance of 100m, this results in a coherencetime of about 120 103 ms and 7.2 103 ms at 3 km/h and50 km/h, respectively. This would lead to = 1600 1800.Assuming a symbol period of 66s as for universal mobiletelecommunications system (UMTS)[39], FAMtot = 60 1000symbols in a coherence time.6 In the following figures, we

5Without loss of generality, we assume 2h = 1/2.6For time-varying fading, the coherence time of the small-scale fading has

been derived as c = 9/(16D) [37], where D is the Doppler frequency.

20 30 40

dB

4

6

8

A

[b/s

/Hz]

FAM, Np=2

FAM, Np=6

SAM, Np=2

SAM, Np=6

Fixed Mod, M=16Fixed Mod, M=64Fixed Mod, M=256FAM, Simulation SAM, Simulation

BE

O>

5%

Fig. 2. ASE for SAM and FAM systems with non-ideal channel estimation:max = 256, dual-branch MRD, maximum BEO 5%, b = 10

2, = 1,FAMtot = 100, = 1000 , and dB = 8. Comparison with fixed modulationsystems ( = 16, 64 and 256). Both analytical and simulative results areprovided.

first illustrate a comparison between analytical and simulativeresults and we consider the block fading channels describedin Section II.

In Fig. 2 the ASE of FAM, SAM, and fixed modulationsystems is reported as a function of the mean SNR withd = 2 branches MRD, max = 256, = 1 (i.e., p = s),and dB = 8.7 For non-ideal channel estimation, SAM canoutperform FAM, for both p = 2 and 6, which confirm theimportance of the analysis for design in practical systems.In particular it can be noticed that the lowest median SNR,for which the BEO requirement of 5% is satisfied, becomesadvantageous for SAM as the channel estimation accuracyincreases (i.e., p increases). Thus, SAM satisfies BEO re-quirements also in FAM-denied region (i.e., the dB regionin which FAM systems experience a BEO greater than thetarget one). Then, by increasing p within the frame, thecrossing point beyond which SAM outperforms FAM is forlower median SNR. Moreover, it should be noticed that bothFAM and SAM provide a considerable gain in terms of ASEcompared to the fixed modulation schemes. In this figure, bothanalytical and simulative results are provided, and their goodmatch validates the analytical framework.

In Fig. 3, the comparison between FAM and SAM schemesis shown in terms of both ASE and BEO for ideal and non-ideal channel estimation. The two systems with dual-branchMRD are compared for several maximum constellation values(max = 4, 16, 64, and 256), pilot schemes (p = 2, and 6,and = 1), and dB = 20, and 30, with dB = 8. For idealchannel estimation, by increasing the max value, the ASEincreases while the BEO is constant to the value obtained for = 4; here, the FAM slightly outperforms SAM in termsof ASE. Conversely, when non-ideal channel estimation isconsidered, the BEO increases accordingly with the maximumconstellation parameter due to the overestimation of the SNRin the feedback. In particular, the FAM systems have a non

7For fixed modulation systems, that does not require a feedback, ideal CSIis assumed.


10-3

10-2

10-1

100

BEO

0

2

4

6

8 A

[b/s

/Hz]

FAM, Np=2

FAM, Np=6

FAM, Ideal CSISAM, N

p=2

SAM, Np=6

SAM, Ideal CSIM

max=4, 16, 64, 256

dB

=30 dB

=20

Fig. 3. ASE vs. BEO for FAM and SAM systems with max =4, 16, 64, and 256, dual-branch MRD, = 1, dB = 8, dB = 20, and30, FAMtot = 100, and = 1000 . Results are evaluated for both ideal andnon-ideal CSI (with p = 1, 2, and 6).

15 20 25 30 35 40 45

dB

3

4

5

6

7

8

A[b

/s/H

z]

Ideal CSINon-Ideal CSI, N

p=2

Non-Ideal CSI, Np=4

Non-Ideal CSI, Np=6

Nd=4 Nd=2

BE

O>

5%

Fig. 4. ASE vs. dB for SAM systems with MRD (d = 2 and 4), max =256, maximum BEO 5%, b = 10

2, = 1, dB = 8, and SAMtot = 200.Results are evaluated for both ideal and non-ideal channel estimation.

negligible increasing of the BEO value; for example, fordB = 30 dB, max = 16, and p = 6, a FAM systemexperiences a BEO of 102, despite the 3.5103 experiencedby the SAM system. Also, for dB = 30 dB, the ASE of SAMis greater than that of FAM for both p = 2 and 6.

The ASE as a function of dB is reported in Fig. 4 forSAM systems with MRD (d = 2 and 4 branches), max =256, = 1, dB = 8, and several values of p. Here, weconsider SAMtot = 200 symbols, which emphasizes more thanin practice the effect of non-ideal CSI and pilot insertion on thesystem performance. The tradeoff between channel estimationquality and ASE can be observed in the figure; note that forthe considered system, p = 2 provides a sufficient qualityestimation, with ASE greater than that for p = 4 and 6.

The effect of pilot energy (i.e., the effect of ) is evaluatedin Fig. 5. In this figure, the median SNR penalty dB asa function of is reported for SAM systems with dual-branch MRD receivers, max = 256, SAMtot = 200 symbols,target BEO of 5% and various p values. The numerical

0 1 2 3 4 5 6 7 8 9 10

0

0.5

1

1.5

2

dB

Np = 1

Np = 2

Np = 4

Np = 6A=5.56

A

=5.36

A

=5.21

A=5.12

Fig. 5. dB vs. for SAM systems with dual-branch MRD, max = 256,target BEO 5%, b = 10

2, dB = 8, p = 1, 2, and 6, and SAMtot =200.

values reported represent the ASE evaluated for = 0.5(i.e., p = s/2) and for those values of median SNRsatisfying BEO requirements. Low median SNR penalty leadsto performance close to that of ideal case. In particular,for each p, an value that minimizes the penalty can beobtained. By increasing the channel estimation accuracyincreases, while the symbol energy might decrease.

In Fig. 6 we provide the performance of SAM systems forvarious tot values (i.e., several frame lengths). In particu-lar, we compare the systems performance in terms of ASE(Fig. 6(a)) and in terms of BEO (Fig. 6(b)) as function of pfor SAM systems with max = 256, four-branch MRD, = 1,dB = 20 and 25, and dB = 8. Fig. 6(a) shows the ASE vs. pwhen tot is equal to 102, 103, and 104. For low p the systemwith the lowest number of pilot symbols (i.e., the system withtot = 10

2) outperforms the others. From (7), it can be noticedthat, the lower is p, the higher is . The overestimationof the mean SNR can achieve an ASE higher than the idealsystems, but it leads to an increasing of the BEO, as drivenby (16). This drawback is depicted in Fig. 6(b), where theBEO as a function of p is provided. The greater the p, thegreater the and thus the greater is the BEO. For example,when dB = 25 and p = 0.01, the system with tot = 102

(and thus p = 1) achieves a BEO of 2 102 despite the5 103 and 3.5 103 experienced by the systems withtot = 10

3 and 104, respectively.A comparison between FAM and SAM performance in

terms of the ASE penalty in (20) is provided in Fig. 7, forsystems with dual-branch MRD, max = 256, target BEO5%, b = 10

2, dB = 8, FAMtot = 100 and = 1000.Remember that the ASE penalty needs to be minimized and,in particular, approaches 1 for systems with A that tendsto (ideal). For both FAM and SAM, the penalty increasesaccordingly with the number of pilot symbols within eachframe. Also, the penalty on the ASE for SAM is slightly lowerthan that of FAM.

The case of time-varying channel is considered in Fig. 8,where the ASE as a function of the median SNR is providedfor FAM and SAM for dual-branch systems with non-ideal


0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1n

p

5

6

7

8 A

[b/s

/Hz]

Ntot

=102

Ntot

=103

Ntot

=104

dB

=25

dB

=20

(a) ASE

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1n

p

10-3

10-2

10-1

BE

O

Ntot

=102

Ntot

=103

Ntot

=104

dB

=20

dB

=25

(b) BEO

Fig. 6. ASE and BEO vs. p for SAM systems with max = 256, four-branch MRD, = 1, dB = 8, dB = 20 and 25, and tot = 102, 103, and 104.

0 5 10 15 20 25 30N

p

1

1.1

1.2

1.3

1.4

1.5

FAM, dB

= 15

FAM, dB

= 45

SAM, dB

= 15

SAM, dB

= 45

Fig. 7. vs. p for FAM and SAM systems with dual-branch MRD,max = 256, b = 10

2, dB = 8, FAMtot = 100, = 1000, andnon-ideal CSI.

CSI, max = 256, = 1, p = 6 , and dB = 8. Time-varying channels are considered for several coherence times( = 3, 50, and 110 km/h), and the simulative results arereported. As observed in the figure, also for time-varyingfading channels, the SAM can outperform the FAM. Moreover,in the slow fading case, the performance degradation due tochannel estimation errors is reduced, respect to fast fadingchannels. For example, at 50 km/h SAM achieves an ASE of7.5 [b/s/Hz] with dB = 34, despite the dB = 38 dB requiredby the FAM. In addition to the ASE comparison, the SAMcan outperform FAM in terms of BEO. The region of dB overwhich the SAM experiences an outage probability lower thanor equal to the target BEO (in-service region) is wider thanfor FAM. This means that SAM systems are defined also inFAM-denied regions. Note also that, the higher is the velocity,the greater is the gap between the FAM and SAM in-serviceregions.

From the above results, the system designer can obtain theminimum value of the median SNR for specified target BEP,

20 25 30 35 40 45

dB

3

4

5

6

7

8

A [

b/s/

Hz]

FAM, 3 km/hFAM, 50 km/hFAM, 110 km/hSAM, 3 km/hSAM, 50 km/hSAM, 110 km/h

BE

O>

5%

Fig. 8. ASE for SAM and FAM in systems with time-varying channel modelwith mobility of 3, 50, and 110 km/h and with non-ideal channel estimation:max = 256, dual-branch MRD, maximum BEO 5%, b = 10

2, = 1, = 1000 , p = 6 and dB = 8. Simulative results are given.

BEO, and ASE in various channel conditions and diversitysettings. Since the median SNR is tied to the propagation lawand location of the user, one can design the wireless system(e.g., cell size, and power levels for cellular systems) thatsatisfies the requirements. Moreover, based on the imposedconstraints, it is possible to understand the performance ofboth FAM and SAM in terms of ASE and BEO, and selectthe adaptive technique which is more suitable to fulfill therequirements.

VI. CONCLUSIONS

We analyzed fast adaptive modulation (FAM) and slowadaptive modulation (SAM) systems with multichannel re-ception and non-ideal channel estimation under energy con-straints. An appropriate figure of merit for the evaluation of theachieved SE (ASE) is defined. It takes into account the tradeoffbetween channel estimation and data reconstruction for agiven total amount of energy per frame. The mathematical


framework enables a system designer to evaluate the amountof energy and resources to be devoted to channel estimation forgiven target bit error probability and outage. Numerical resultsshow that SAM systems, despite the lower feedback rate,can outperform the FAM systems and operate in FAM-deniedregions. This gives a different prospective for the design ofadaptive communication systems.

ACKNOWLEDGMENT

The authors wish to thank the editor and anonymousreviewers for helpful suggestions. The authors wish also tothank M. Z. Win, M. Chiani, O. Andrisano, and colleagues ofNEWCom++ WPR.3 for helpful discussions.

APPENDIXBIT ERROR OUTAGE IN FAM SYSTEMS

For FAM systems the = T is the RV resulting fromthe difference of two correlated chi-square distributed RVs

T = T T = s0

where

=

d=1

(2 2

).

and 2 = 2h/(2h +

2e ) = /(+(p)

1) is the normalizedcorrelation. As reported in [40], the PDF of can beexpressed as (31) at the bottom of the page, where 21 =

2h ,

22 = 2h +

2e , and

=

[(22 21 + 42122

(1 2))2]1/2

2122 (1 2)

= 22 21

2122 (1 2)

.

Thus, the PDF of T, conditioned to results

T() = ()=T22h /22h

. (32)

The unconditioned PDF of the mean SNR can be written as

T() =

T()() . (33)

Knowing the distribution of T, the unconditioned BEO forFAM systems can be evaluated.

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Laura Toni (S06-M09) received M.S. degree (withhonors) in electrical engineering and the Ph.D. de-gree in electronics, computer science and telecom-munications engineering from the University ofBologna, Italy, in 2005 and 2009, respectively..

In 2005, she joined the Department of Electron-ics, Informatics and Systems at the University ofBologna, to develop her research activity in the areaof wireless communications. During 2007, she hadbeen a visiting scholar at the University of Californiaat San Diego, CA, working on video processing over

wireless systems. Since 2009, she has been a frequent visitor to the UCSD,working on joint source and channel coding on wireless communicationsystems.

In June 2009, she joined the TeleRobotics and Applications (TERA)department at the Italian Institute of Technology (IIT), Genova, Italy. Herresearch interests are in the areas of image and video processing, wirelesscommunications, and underwater communications.

Andrea Conti (S99-M01-SM11) received theLaurea degree (summa cum laude) in telecommu-nications engineering and the Ph.D. degree in elec-tronic engineering and computer science from theUniversity of Bologna, Italy, in 1997 and 2001,respectively.

In 2005, he joined the University of Ferrara,Italy, where he is an Assistant Professor. He wasResearcher with Consorzio Nazionale Interuniversi-tario per le Telecomunicazioni (CNIT) from 1999 to2002 and with Istituto di Elettronica e di Ingegne-

ria dellInformazione e delle Telecomunicazioni, Consiglio Nazionale delleRicerche (IEIIT/CNR) from 2002 to 2005 at the Research Unit of Bologna.In Summer 2001, he was with the Wireless Systems Research Department atAT&T Research Laboratories. Since 2003, he has been a frequent visitorto the Wireless Communication and Network Sciences Laboratory at theMassachusetts Institute of Technology (MIT), where he presently holdsthe Research Affiliate appointment. He is a coauthor of Wireless Sensorand Actuator Networks: Enabling Technologies, Information Processing andProtocol Design (Elsevier, 2008). His research interests involve theory andexperimentation of wireless systems and networks including network local-ization, adaptive diversity communications, cooperative relaying techniques,interference management, and sensor networks.

Dr. Conti is currently an Editor for the IEEE COMMUNICATIONS LET-TERS. He served as an Editor for the IEEE TRANSACTIONS ON WIRELESSCOMMUNICATIONS from 2003 to 2009 and Guest-Editor for the EURASIPJournal on Advances in Signal Processing (Special Issue on Wireless Co-operative Networks) in 2008. He is the Technical Program Chair for theIEEE International Conference on Ultra-Wideband 2011; Technical ProgramCo-Chair for the IEEE Vehicular Technology Conference 2011-Spring andthe Wireless Communications Symposium of IEEE Global CommunicationsConference 2010; and Technical Program Vice-Chair for the IEEE WirelessCommunications and Networking Conference 2009. He served as reviewerfor IEEE and IET journals and as Technical Program Committee Memberfor numerous IEEE conferences. He is elected Vice-Chair and Secretaryof the IEEE Communications Societys Radio Communications TechnicalCommittee for the terms 2011-2012 and 2009-2010, respectively.

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