Channel Capacity Loss Reduction with ChannelState Detections

Dongdong Li Mazin Al-ShalashWireless R&D Department, Huawei Technologies

1700 Alma Drive, Suite 500, Plano, Texas 75075, USAEmails: dli@huawei.com, mshalash@huawei.com

Abstract—Adaptive modulation and coding schemes (MCS) arekey components in the next generation wireless system to improvechannel throughput by adaptively choosing the MCS accordingto channel states. However, due to channel feedback delay,processing delay, and scheduling delay, the channel state maychange at the time of transmission and results in channel capacityloss. Simply using previous channel state without any detectionapproach to choose the MCS results in a large channel capacityloss. In this paper, we propose to use minimum mean-squarederror (MMSE) and maximum a posteriori (MAP) approaches todetect the MCS at any given time and schedule the transmissionusing the detected MCS. Simulation results show that ourproposed MCS detection approaches reduce channel capacityloss, compared with the case when no detection is used.

I. I

In the next generation wireless system, e.g., worldwide in-teroperability for microwave access (WiMax), the time-varyingwireless channel is usually quantized into multiple states. Thenappropriate modulation and coding schemes (MCS) are chosenaccording to these quantized channel states to maximize thechannel capacity [1]. In a frequency division multiplexing(FDD) system, the mobile station (MS) sends the channelinformation back to the base station (BS), and the BS usesthe feedback channel information to schedule the transmissionto the MS. Due to the channel feedback delay, processingdelay, and scheduling delay, the channel state may changeat the time of transmission. Simply using previous channelstate to schedule future traffic transmission leads to a largechannel capacity loss [2]. It will be beneficial to us if wecan predict future channel state and schedule transmissionaccordingly. The channel state transition in the future can beestimated by using the finite state Markov channel (FSMC)model [3] [4]. However, the drawback of previous FSMCmodel is that in order to determine the FSMC transitionprobability for a long time period T , previous FSMC approachneeds to divide T into K smaller time intervals T/K, calculatethe transition probability at time T/K, and apply a K-steptransition probability. This K-step Markov approach tends toaccumulate a large predication error and may have misleadingresults to predict the channel state for a long time period [2]. Inaddition, detection approach exploring channel statistics, suchas the minimum mean-squared error (MMSE), can not be usedin this scenario, since the exact channel state distribution cannot be derived by using previous FSMC approach. PreviousFSMC approach can use our proposed maximum a posteriori

(MAP) to decide the MCS but with a worse performanceas shown later in this paper. We propose a novel approachto use both MMSE and MAP to detect the channel stateand choose the appropriate MCS. Simulation results showthat our approach leads to a larger channel capacity lossreduction, compared to previous FSMC approach and the nodetection approach. Furthermore, traditional FSMC channelmodel limits the channel state transitions by assuming thecurrent channel state is related to only one previous channelstate and the channel model is simplified to a birth-and-deathMarkov model [4]. Since this assumption neglects the histori-cal channel state information, it can not accurately model theactual channel transitions. We proposed an analytical approachto incorporate MCS history information to accurately modelthe actual channel transitions.

This paper is organized as follows: section II describes thesystem model and the approach to derive the closed-formexpression of channel transition probability and distribution.Then we describe how to use the MMSE and MAP approachesto detect the MCS. Section III gives the numerical resultsshowing the advantages of our proposed approach to calculatechannel capacity loss. Section IV summarizes the results.

II. S

The wireless channel can be quantized into multiple MCS’saccording to the channel SIR level. Denote the set of MCS’sas {MCS 1, · · · ,MCS i, · · · ,MCS N}, where 1 ≤ i ≤ N is theMCS index. Also assume MCS with a higher index providesa higher channel capacity but also needs a higher SIR. Furtherdenote the channel response at time tm as hm. We use ageneral channel model, i.e., the Nakagami-m channel model,to characterize the channel SIR distribution. By consideringan L order diversity system, the channel SIR γm = |hm|2/2follows an i.i.d Gamma distribution given by [5] [6]:

f (γm) =

(mt

γ̄

)mt γmt−1m

Γ(mt)exp(−mtγm

γ̄), γ ≥ 0 (1)

where mt = m × L, m is the fading severance and L isthe diversity order. A larger m means a less severe fadingchannel. m = 1 means the channel is Rayleigh distributed anda large m approximates the Rician distribution [7]. Γ(mt) =∫ ∞

0xmt−1 exp(−x)dx is the Gamma function, γ̄ = mtγ̄0 is the

average SIR of the system. γ̄0 is the average SIR of a Rayleighdiversity branch.

978-1-4244-1708-7/08/$25.00 ©2008 IEEE 754

The channel SIR γm at time tm can be further expressed ina summation form as [7]:

γm =|hm|2

2=

mt∑i=1

|hi,m|22=

mt∑i=1

(x2i,m + y2

i,m)

2(2)

where hi,m = xi,m + jyi,m. xi,m’s and yi,m’s are i.i.d Gaussianvariables with zero mean and variance γ̄0. The power γi,m =

|hi,m|2/2 follows the Rayleigh distribution.Now, let us look at the the channel transition model. Denote

the SIR quantization interval for MCS i as [Γi,Γi+1]. Accordingto equation (1), the channel state probability at MCS i is

P(MCS i) =∫ Γi+1

Γi

f (γ)dγ = P(mt,mtΓi+1/γ̄) − P(mt,mtΓi/γ̄)

(3)

where P(a, b) =∫ b

0xa−1 exp(−x)dx/Γ(a) is the incomplete

Gamma function [8].

A. Transition Probability and Distribution with One PreviousState

In the following, we will describe the approach to findthe closed-from expressions of the channel state transitionprobability and distribution. The channel response xi,m andyi,m at time tm is related to previous channel response at timetk as [1]

xi,m = a1xi,k + b1wx,i,m,

yi,m = a1yi,k + b1wy,i,m,(4)

where a21 + b2

1 = 1. wi,m = wx,i,m + jwy,i,m is a circularsymmetric complex Gaussian noise, wx,i,m and wy,i,m are i.i.dnormal distributed variables with zero mean and varianceγ̄0, and wi,m is independent of xi,k’s and yi,k’s. The channelcorrelation is assumed to be the same and can be modeledby using the Jakes’ model as [9] a1 = J0(2π fdT ), whereT is the time interval between current and future MCS’s.J0(z) = 1/π

∫ π0

cos(z sin θ)dθ is the Bessel function of the first

kind of zero order [9, pg. 69]. b1 =

√1 − a2

1.Denote current MCS as MCS i1 and the future MCS as

MCS i2 . Further denote the quantization interval of MCS i1 as[Γi1 ,Γi1+1]. From the results in [2], the transition probabilitythat the system will transit from MCS i1 at time t1 to MCS i2at time t2 = t1 + T is

P(MCS i2 |MCS i1 )

=P(Γi2 ≤ γi2 ≤ Γi2+1,Γi1 ≤ γi1 ≤ Γi1+1)/P(Γi1 ≤ γi1 ≤ Γi1+1)

=

∫ Γi2+1

Γi2

P(Γi2 ≤ γi2 ≤ Γi2+1|γi1 ) f (γi1 )dγi1/P(Γi1 ≤ γi1 ≤ Γi1+1)

=b2mt

1

[P(mt,mtΓi1+1/γ̄) − P(mt,mtΓi1/γ̄)]Γ(mt)

∞∑j=0

a2 j1 Γ( j + mt)

Γ( j + 1)×

[P( j + mt, si1+1) − P( j + mt, si1 )][P( j + mt, si2+1) − P( j + mt, si2 )](5)

where si1 = Γi1/(b21γ̄0) and si1+1 = Γi1+1/(b2

1γ̄0). γi1 follows theNakagami-m distribution in equation (1).

Similarly, from the results in [2], the probability densityfunction (PDF) of the channel SIR at time ti2 given MCS i1 is

f (γi2 |MCS i1 )

=b2mt

1

[P(mt,mtΓi1+1/γ̄) − P(mt,mtΓi1/γ̄)]Γ(mt)

∞∑j=0

a2 j1 Γ( j + mt)

Γ( j + 1)×

[P( j + mt, si1+1) − P( j + mt, si1 )] × (γi2 ) j+mt−1 exp(−γi2/b21γ̄0)

(b21γ̄0) j+mtΓ( j + mt)

(6)

B. MAP and MMSE Detection with One Previous State

Given current MCS as MCS i1 , the MAP detection criteriais to select future MCS i2 with the largest posterior probability,i.e., choosing MCS with max{P(MCS i2 |MCS i1 )P(MCS i1 )} byusing equation (3) and (5). For previous FSMC approach,P(MCS i2 |MCS i1 ) is obtained by using the approach outlinedin our previous study [2].

The MMSE estimation uses the expectation value of γi2 , i.e,E(γi2 |MCS i1 ). From equation (6), the expectation value of γi2is derived as

E[γi2 |MCS i1 ]

=b2mt

1

[P(mt,mtΓi1+1/γ̄) − P(mt,mtΓi1/γ̄)]Γ(mt)

∞∑j=0

a2 j1 Γ( j + mt)

Γ( j + 1)×

[P( j + mt, si1+1) − P( j + mt, si1 )]( j + mt)(b21γ̄0)

(7)

After estimating γi2 by using MMSE, the future MCS isselected by quantizing E[γi2 |MCS i1 ].

C. MAP and MMSE Detection with More than One PreviousState

We can extend previous MMSE and MAP approach byconsidering more than one previous channel states. The detec-tion decision is made conditional on more than one previouschannel states. Given previous MCS’s as MCS M , · · · ,MCS 1

with SIR γiM , · · · , γi1 and future MCS as MCS iM+1 withSIR γiM+1 , the MMSE estimation first finds out the condi-tional expectation value of γiM+1 , E(γiM+1 |MCS iM , · · · ,MCS 1);Then the MMSE selects the future MCS by quantizingE(γiM+1 |MCS iM , · · · ,MCS 1). For the MAP, we choose MCSwith max{P(MCS iM+1 ,MCS iM , · · · ,MCS 1)}.

For the MMSE detection, we can find a closed-form ex-pression of E(γiM+1 |MCS iM , · · · ,MCS 1). As an extension ofequation (4), the channel response xi,M+1 and yi,M+1 at timetM+1 can be expressed as a linear combination of previouschannel responses as [1]

xi,M+1 =

M∑k=1

aM+1,k xi,M−k+1 + bM+1wx,i,M+1,

yi,M+1 =

M∑k=1

aM+1,kyi,M−k+1 + bM+1wy,i,M+1,

(8)

where∑M

k=1 a2M+1,k + 2

∑Mk=1

∑Mp=k+1 aM+1,kaM+1,pβp−k + b2

M+1 =

1. The channel correlation βi can be modeled by using the

755

Jakes’ model as βi = J0(2πi fdT ). wx,i,M+1 and wy,i,M+1 are i.i.dnormally distributed variables with zero mean and variance γ̄0,and independent of xi,k’s and yi,k’s.

The channel weight coefficient aM+1,k’s can be derived byusing the Yule-Walker equation as AM+1 = BM+1RM+1

−1

[10], where AM+1 = [aM+1,1, aM+1,2, · · · , aM+1,M], BM+1 =

[β1, β2, · · · , βM], and RM+1 is a M×M circular matrix with itsfirst row as [1, β1, · · · , βM−1].

According to equations (2) and (8), the expected channelSIR can be derived as

E(γiM+1 |MCS iM , · · · ,MCS 1)

=E

mt∑i=1

(x2i,M+1 + y2

i,M+1)

2

=12

mt∑i=1

E

M∑

k=1

aM+1,k xi,M−k+1 + bM+1wx,i,M+1

2

+

12

mt∑i=1

E

M∑

k=1

aM+1,kyi,M−k+1 + bM+1wy,i,M+1

2

=

M∑k=1

a2M+1,kE

[γiM−k+1

]+ b2

M+1γ̄+

M∑k=1

M∑p=k+1

2aM+1,kaM+1,pβp−kE[γM−p+1

],

(9)

and

E[γM−k+1

]= mtγ̄0

P(mt + 1,ΓiM−k+1+1

γ̄0) − P(mt + 1,

ΓiM−k+1γ̄0

)

P(mt,ΓiM−k+1+1

γ̄0) − P(mt,

ΓiM−k+1γ̄0

),

(10)

where the SIR quantization interval for MCS i1 is denoted as[Γi1 ,Γi1+1]. The first term in equation (9) is the contributionof the history MCS’s. The second term in equation (9) is thecontribution of channel statistical average. The third term isa cross correlation term. Therefore, we can interpret equation(9) as a weighted sum of average SIR and history MCS’s. Ifthe channel history MCS information is recent, i.e., a smallerdelay T , the weight aM+1,k of MCS history is larger. If thechannel history MCS information is obsolete, i.e., a largerdelay T , the weight aM+1,k is smaller. In addition, by settingM = 1, equation (9) is equal to equation (7) but with a muchsimplified expression.

D. The Channel Capacity Loss

Assume that the BS uses the detected MCS j to schedulethe transmission to the MS, while the actual channel conditionwith a delay T is MCS i. If j = i, there is no capacity loss. Ifj > i, the system uses a MCS requiring a higher SIR, resultsin a high error probability and channel capacity loss. If j < i,the system uses a MCS that can not fully explore the actualhigher channel SIR and also results in channel capacity loss.Therefore, the channel capacity loss is given by [2]

δCi(T ) =|C j −Ci|, j � i (11)

By considering all MCS’s, the average channel capacity lossof the system is

δC(T ) =N∑

i=1

δCi(T )Pi (12)

where Pi is the state probability of the system in MCS i. Wecan calculate the channel capacity loss |Ci − C j| by using theShannon capacity formula under the assumption of constanttransmitting power [2], where

Ci =

∫ Γi+1

Γilog2(1 + γ)

(mt

γ̄

)mt γmt−1

Γ(mt)exp(−mtγ

γ̄)dγ

P(mt,mtΓi+1/γ̄) − P(mt,mtΓi/γ̄). (13)

III. N

We quantize the channel into 6 MCS’s [1] and the quantiza-tion interval in dB is [3.5 6 8 12 14]. The Doppler frequencyshift fd is chosen to be 200 Hz. Time resolution is set tobe δT = 0.1 ms. The actual channel transition probabilitymatrix is generated from pure simulation, where the channelresponse is generated by using the Jakes’ model [9]. The K-step Markov MAP MCS detection approach uses the transitionprobability generated from the traditional FSMC approach out-lined in our recent research [2]. Our proposed MAP detectionapproach uses the channel transition probability in equation(5) multiplying with its corresponding state probability inequation (3). Our proposed MMSE detection approach firstcalculates the expected value of the channel conditional SIRin equation (7), quantize it, and then finds the correspondingMCS index according to the MCS quantization interval. Theno detection approach uses current channel state as the futurechannel state. The capacity loss of using MCS i is calculatedby using equation (8). The average channel capacity loss iscalculated by using equation (9).

We first consider a fading scenario. Let L = 1 and mt = m,where m is the fading severance. A larger m means the fadingis less severe. The average SIR is chosen to be fixed at γ̄ =11 dB. For the Rayleigh fading scenario, i.e., mt = 1, the stateprobability in MCS 1 − 6 is 0.16, 0.11, 0.12, 0.32, 0.15, and0.14, respectively. By using equation (10), the correspondingcapacity Ci in bps/Hz is 1.0, 2.0, 2.6, 3.5, 4.4, and 5.4. Formt = 8, which approximates the Rician fading scenario, theprobability in the MCS is 0, 0.004, 0.05, 0.73, 0.20, and 0.01.The corresponding capacity Ci in bps/Hz is 1.6, 2.2, 2.7, 3.6,4.3, and 5.8.

Fig. 1 shows the average channel capacity with a Rayleighfading channel mt = 1. We could see that with no detection,i.e., using current channel state as the future channel state,has a slightly better performance than using MAP detectionup to 0.16 normalized delay. This is understandable, since thechannel does not change much within 0.16 normalized delay.Using MAP detection only introduces detection errors. It isalso obvious that our proposed MAP detection has a muchbetter performance than both no detection and previous K-step Markov MAP detection at normalized delay greater than0.16. At normalized delay 0.4, both MAP detection approachesobserve a 27% channel capacity loss reduction from 1.45

756

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.5

1

1.5

Normalized Delay (T × fd)

Capa

city L

oss (

bps/H

z)

No DetectionMAP Detection − ProposedMAP Detection − K−step Markov

Fig. 1. Average channel capacity loss for Rayleigh Fading (mt = 1).

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Normalized Delay (T × fd)

Capa

city L

oss (

bps/H

z)

No DetectionMAP Detection − ProposedMAP Detection − K−step Markov

Fig. 2. Average channel capacity loss for less severe fading channel mt = 8,the diversity order L = 1.

bps/Hz to 1.05 bps/Hz. We also see that our proposed MAP de-tection is better than the K-step Markov MAP detection, sinceour closed-from approach calculating the transition probabilitydoes not have the drawback of accumulated error from the K-step Markov approach. At normalized delay T × fd = 0.4,the channel correlation J0(2πT fd) = 0, we loss the channeltracking, and both our proposed MAP approach and the K-step Markov MAP detection will select MCS with the largeststate probability and their capacity loss curves begin to merge.

Fig. 2 shows the average channel capacity for a more severefading channel mt = 8. The results show that our proposedMAP detection approach has a less channel capacity loss thanboth no detection and the K-step Markov MAP detection,especially for normalized channel fading greater than 0.08. Atnormalized delay 0.4, both MAP detection approaches observea 40% channel capacity loss reduction from 0.33 bps/Hz to0.196 bps/Hz.

Next, we will take a close-up look at the channel capacity ofindividual MCS’s and see which MCS benefits most from ourproposed MAP detection. Fig. 3 shows the channel capacityloss of MCS in a Rayleigh fading channel with no detection,while Fig. 4 shows the channel capacity loss of MCS in aRayleigh fading channel using our proposed MAP detection.Comparing Fig. 4 to Fig. 3, we see that MCS 1 and MCS 6

has the largest channel capacity reduction by using the MAP

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.5

1

1.5

2

2.5

Normalized Delay (T × fd)

Capa

city

Loss

(bps

/Hz)

MCS1

MCS2

MCS3

MCS4

MCS5

MCS6

Fig. 3. MCS channel capacity loss for Rayleigh Fading with no detection(mt = 1).

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.5

1

1.5

2

2.5

Normalized Delay (T × fd)

Capa

city

Loss

(bps

/Hz)

MCS1

MCS2

MCS3

MCS4

MCS5

MCS6

Fig. 4. MCS channel capacity loss for Rayleigh Fading with our proposedMAP detection (mt = 1).

detection. In addition, MCS 2, MCS 3, and MCS 5 with MAPdetection introduces a larger detection error and tends tohave a larger channel capacity loss at a smaller normalizeddelay than that of no detection. There is insignificant channelcapacity loss for MCS 4 using the MAP detection. Not shownhere, we observe similar results for a less severe fading, e.g.,mt = 8, except that the MAP detection error is very smalland the performance of MAP detection is very close to thatof no detection for MCS 2, MCS 3, and MCS 5 with a smallnormalized delay.

Fig. 5 compares our proposed MMSE detection with theMAP detection and no detection approaches for various fadingseverance. We could see that the MMSE detection is slightlybetter than MAP detection as expected. The MMSE detectionperformance is almost the same as that of no detectionapproach at a smaller normalized delay, e.g., less than 0.11,and gradually surpass that of no detection approach at a highernormalized delay.

Next, we consider the diversity reception scenario with m =1 and mt = L. The average SIR of each diversity branch is setto be γ̄0 = 11 dB and the average SIR of the diversity systemis γ̄ = 11 + 10 log10(L) dB. The results are shown in Fig. 6.We could see that the MMSE detection is better than MAPdetection, especially for the case when mt = L = 2. Compared

757

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.5

1

1.5

Normalized Delay (T × fd)

Capa

city

Loss

(bps

/Hz)

No DetectionMAP Detection − ProposedMMSE Detection − Proposed

mt=1

mt=2

mt=4

mt=8

Fig. 5. Average channel capacity loss for various fading severance withMMSE and MAP detection, mt = m is the fading severance, the diversityorder L = 1.

with Fig. 5, we have less capacity loss, since the channel aremore likely staying with one MCS with the increase of averageSIR of each diversity branch.

Fig. 7 compares the performance of MMSE detections withMCS history. We notice that at a lower normalized delay, e.g,less than 0.15 for Rayleigh fading, the channel capacity lossof MMSE estimation with only one previous MCS historyis slightly higher than those with two or more MCS historyinformation. At a larger normalized delay, e.g., greater than0.15 for Rayleigh fading, the channel capacity loss of MMSEestimation with only one previous MCS history is lower thanthose with two or more MCS history information. For a lesssevere fading channel, e.g., mt = 8, the performance of MMSEestimation with only one previous MCS history is almostalways better than those of MMSE estimation with two ormore MCS history information. We further observe that theperformance of MMSE estimation by using two and threehistory information is almost the same. This is mainly dueto the facts that our channel model with one MCS historyhas already been able to model the transition between MCS.In practical system, when the transition between MCS’s cannot be accurately modeled, we may see much larger gain byincorporating more MCS history information.

IV. C

In this paper, we proposed to use MMSE and MAPapproaches to detect the MCS according to channel statetransition. Compared with no detection approach, our proposeddetection has the advantage of reducing the channel capacityloss by 27%−40% at a large channel feedback delay. Simula-tion results also suggest that with a smaller channel feedbackdelay, e.g., less than 0.16 normalized delay in a Raleigh fadingchannel, no detection has almost the same capacity loss as thatof MAP and MMSE detection. Actually, compared with nodetection, MAP detection even results in slightly more channelcapacity loss when the channel feedback delay is small. Wealso show that comparing to the MAP detection approach,the reduction of the channel capacity loss by using MMSEdetection approach is marginal.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.5

1

1.5

Normalized Delay (T × fd)

Cap

acity

Los

s (b

ps/H

z)

No DetectionMAP Detection − ProposedMMSE Detection − Proposed

mt=1

mt=2

mt=4

mt=8

Fig. 6. Average channel capacity loss for various diversity orders with MMSEand MAP detections for Rayleigh fading m = 1, mt = L is the number ofdiversity branches.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.5

1

1.5

Normalized Delay (T × fd)

Cap

acity

Los

s (b

ps/H

z)

Without EstimationMMSE Estimation − History=1MMSE Estimation − History=2MMSE Estimation − History=3

mt=1

mt=2

mt=8

mt=4

Fig. 7. Average channel capacity loss for various fading severance withMMSE detections using MCS history, mt = m is the fading severance, thediversity order L = 1.

R

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