Cross-layer Design Combining Adaptive Modulation

and Coding with ARQ on Frequency Domain

Subcarrier of OFDM

Zeng ju-ling , Xie bing, Zhou wen-an Song jun-de

School of Electronic Engineering,

Beijing University of Posts and Telecommunications , Beijing, China, 100876

Email:julingzeng@gmail.com

Abstract—The system model and the algorithm for a cross-layer

design combining the adaptive modulation and coding with ARQ

based on frequency domain subcarrier of OFDM is developed at

first. And then the effect of high speed move to the performance

of that is analyzed. In order to compensate the degraded

performance of wireless link due to high-speed move, a novel

Finite-State Markov channel model is formed to represent the

frequency domain subcarrier of OFDM modeled as random

process with Nakagami-m fading distribution. A correction of the

threshold of modulation and coding scheme and the channel

quality information as well as the method for link prediction in

Chase combining is investigated. On this foundation, an

improved cross-layer design combining of adaptive modulation

and coding with H-ARQ is proposed. The computer simulation

results reveal that the improved design and modified algorithm

has higher average throughput and lower FER than the original one when the terminal moves in high speed.

Keywords-cross-layer combining AMC with H-ARQ;

improvement of cross-layer design; FSMM; threshold of MCS;CQI

and Chase combining; throughput; FER

I. INTRODUCTION

The AMC/ARQ cross-layer design combine adaptive modulation and coding techniques (AMC) in physical layer with the selective automatic repeat request protocol (ARQ) in data link layer judiciously so that the stringent error control requirement is alleviated for the AMC in physical layer depending on ARQ’s ability to correct occasional packet errors at the data link layer ,which enable the higher order MCS is selected , resulting in the increased spectral efficiency and high data rate service .Another advantage of the AMC/ARQ cross-layer design is that it also combines the delay determined by ARQ in data link layer with the bit rate and the frame error rate by AMC in physical layer organically, which provides grantee of quality of service .So the cross-layer designs have been studied extensively with the demand for high data rates and quality of service (QoS) being growing at a rapid pace.

In OFDM system, due to signal’s transmitting para1lely in frequency domain and subcarrier’s experiencing flat-fading, AMC/ARQ cross-layer design on frequency domain subcarrier is more efficient than overall frequency, which increased

efficiency by about 20%[6] . Because of the controlling of the frequency and the time resource simultaneously, the AMC/ARQ cross-layer design based on frequency domain subcarrier make the dynamic resource allocation and QoS guarantee more efficient. So it is the focus for study at present.

The performance of the AMC/ARQ cross-layer design is degraded severely when the terminal move in high speed. The time-selective fading by high speed move will result in received SNR’s exceeding of interval of thresholds of the MCS in a frame duration, which lead to an unfit scheme of AMC selected. The time-selective fading also lead to the error CQI along with error update of MCS.[4][5]. The phenomenon above will be significant due to the time-selective fading of subcarrier of OFDM sever than the singer-carrier in the environment of high speed move.

With describing the state transition of SNR in variable channel between adjoining time, able to track the variation of the channel, the Finite-state Markov Model(FSMM) is very effective in enhancing the performance of the AMC/ARQ cross-layer design. However, the processing unit is frame or packet in AMC/ARQ cross-layer design, which require the same state duration of FSMM, the present FSMM with equal probability rather than same duration unable meet the demand. So the novel FSMM with equal duration for each state is formed to improve the AMC/ARQ cross-layer design.

The system model for the cross-layer combining of AMC with ARQ in singer-carrier system is formed in [2], where the maximum retransmission number is decided in data link layer according to the delay and the MCS in physical layer to frame error rate (FER). The spectral efficiency of that relative to AMC alone improves by about 0.25bits/symbol, using only one retransmission over Rayleigh-fading channels. A more perfect model is put forward in [3], where a finite state Markov model is formed to represent the channel state-transiting characteristic and an embedded Markov chain is used to analyze the queuing behavior induced by both the truncated ARQ protocol and the AMC scheme along with the optimizing of the cross-layer design. Because of assuming the channel’s constant in a frame duration, the design in [2] or [3] should only be applied in static or quasi-static system with singer-carrier. A model combining the queuing in data link layer with AMC in physical

This work was supported by the National 863 planning project of China under Grant 152007AA01Z204

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layer based on frequency domain subcarrier is proposed in [7], where the disadvantage neither combining ARQ nor considerations of the variation of the channel make it ineffective in terminal’s high-speed move. A method compensating the threshold migration in high-speed move with the relation between the offset of the threshold of MCS and the Doppler frequency and the variable SNR is suggested in [8], which cannot improve the system greatly because of the limited situation contained in it and not combining with ARQ.

Some improvement has been made in this paper. We develop the system model and the algorithm for a cross-layer design combining adaptive modulation and coding with H-ARQ on frequency domain subcarrier of OFDM and analyze the effect of high speed move to the performance of this design in Section II. In Section III, aiming at the degraded performance of wireless link due to high speed move, we first form a Finite-State Markov channel model with the same state duration of OFDM subcarrier, modeled as a random process with Nakagami-m distribution. And then investigate a correction of the threshold of MCS and the channel quality information(CQI) as well as the link prediction method in Chase combining based on FSMM. On this foundation, an improved cross-layer design combining AMC with H-ARQ is proposed. Our simulation results for spectral efficiency are presented in Section VI, which reveal that the improved design and modified algorithm has higher average throughput and lower FER than the former in terminal’s high-speed move.

II. CROSS-LAYER DESIGN ON FREQUENCY DOMAIN

SUBCARRIER OF OFDM AND PERFORMANCE ANALYSIS

A. Statistic Characteristic of Frequency Domain Subcarrier

The transmission of the signal in OFDM is derived in [10]

0,1, 1k k k kR H S N k N= + = − (1)

where N is the total number of the subcarrier and ,k kR S is the

transmitted data and received data on the kth subcarrier

respectively. The kH is frequency-domain channel response

samples on the kth subcarrier and expressed as:

( )1

0

exp 2L

k

l

lkH h l j

Nπ

−

== − (2)

where L is the ratio of the maximal delay to data rate

,sm

LTT = and the ( )h l is time-domain channel response

samples of the l-th path and ( ) 0, , 1 1h n n L L N= = + − .

Because the ( ) 'h l s are independent and the items of (2)

are orthogonal with each other, the 'kH s are independent with

each other.The each frequency-domain subcarrier is approximately to be Nakagami-m distributed, provided that the

every ( )h n with Nakagami-m distribution [10]. The kN in (1)

is the additive white noise power spectrum density on the kth subcarrier:

1

0

exp 2N

k n

n

knN j

Nη π

−

== − (3)

where 'n sη are white noise sequence with zero mean and one

variance. The ratio of the signal to noise (SNR) on the k th subcarrier is defined as:

( ) ( )2 2 2

2

0 0

k k k

k k

k k

E H s E sH

N Nγ = = (4)

SNR are independent identically distributed (i.i.d) due to

kH ’s i.i.d. The probably density function of the SNR on the

k th subcarrier is:

( ) ( )1

k

m mmmp e

m

γγ

γγγ

γ

− −=

Γ (5)

Where ( )Eγ γ= and 1

0

( )m tm t e dt

∞− −Γ = and m is the

Nakagami fading parameter and 1/ 2m ≥ . Bigger m is, the

more sever channel fading is. When 1m = ,the Nakagami-m distribution become Rayleigh’s.

B. Cross-layer Design Combining AMC with ARQ

For every subcarrier’s in frequency flat, the channel quality can be captured by a single parameter, the received SNR. Generally, the subcarriers with equal SNR or in the same state, are selected to form the clustered subchannel for AMC, which achieve link level SNR gain due to the reduced channel selectivity. Because each subcarrier in subchannel has homology characteristic, the cross layer design in this paper replacing on overall frequency with on the frequency subcarrier of OFDM doesn’t lose generality. So the method provided in [2] can be adopted and the layer structure of the system is shown in Fig.(1).The selective ARQ protocol is adopted in data link layer where the processing unit is a packet and AMC is adopted in physical layer where the processing unit is a frame. Each frame includes the different quantity of the packet because of the different MCS. The channel quality information (CQI) and ACK/NACK are transmitted via feedback channel and are used for AMC and ARQ respectively. We assume that the subcarrier of the physical channel is in frequency flat and is Nakagami-m distributed along with error-free and zero latency in feedback. The algorithm of the cross layer design is as follows:

1) The maximum number of retransmission allowed for a

packet:

max max

rNRTT

τ= (6)

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where maxτ is the longest delay and RTT is the time for a

recycle of transmission and receiving.

2) Decision of threshold of the MCS Provided that the probability of packet loss after

retransmissions is no larger than lossP , the instantaneous frame

error rate (FER) is guaranteed to be no greater than 0P for each

chosen AMC mode at the physical layer and it should be as:

max1

0rN

lossP P+ ≤ (7)

From (7), we obtain:

( )max1/ 1

0 arg

rN

loss t etP P P+≤ ≈ (8)

From [2], we can obtain that the relation between the argt etP

and the ratio of the signal to noise is approximately as:

( )1 0

( )exp

n

n

P

n

n n P

PERa g

γ γγ

γ γ γ< <

=− ≥

(9)

Where the parameters , ,nn n Pa g γ vary with the MCS and can be

obtained by looking up the table (1) from [2].The corresponding thresholds of each MCS are obtained from the following equations:

0

1

0

1ln , 1,2,n

n

n n

n

FERn N

g a

+

Γ =

Γ = − =

Γ = +∞

(10)

When 1n k nγ γ γ +≤ ≤ , we think the kth subcarrier is in the nth

state and the mode n is selected.(In this paper, the threshold of

MCS is denoted by kΓ while the threshold of received SNR by

kγ .For convenience, the FER of each frame is assumed equal

to the average FER)

3) The operating stages of the cross-layer design are as

following:

a) Deciding the parameters such as max

arg, ,t et r nP N Γ

b) The transmitter receive the CQI and ACK/NACK before the frame transmission and update the MCS for the new

data according to the following: when1n nCQI +Γ ≤ ≤ Γ , we

think the k-th subcarrier is in the nth state and the mode n is

selected.

c) While for the retransmitted packet, prediction for link level performance in Chase combining is carried at first and

then update the MCS by the predicted performance in (14).

d) For the new data, the receiver receive them and correct error to send ACK or NACK to the transmitter; for the retransmitted data, the receiver uses Chase combining to

combine them. If a packet is not received correctly after max

rN

retransmissions, we will drop it and declare packet loss.

Figure 1. AMC/ARQ cross-layer design layer construction

III. THE FSMM WITH EQUAL DURATION OF THE STATE IN

FREQUENCY DOMAIN SUBCARRIER

A. General principle for FSMM:

It is verified in [12] that the first-order stationary finite-state Markov model can represent the transition of SNR from one interval to the adjacent ones in time-varying channel. The basic principle to form a FSMM is as follows: partition the range of

the received SNR (represented byγ ) into a finite number of

intervals, whose threshold represented by the gradually

increased sequence 1 2 1, Kγ γ γ + with

10γ = and

1Kγ + = ∞ .When 1

[ , ]k kγ γ γ +∈ , we say that the channel is in the

k -th state, which constructs the stationary state space

{ }1 2, KS S S S= . Each state is correspond to a special BER

and the probability ,i jP of transition from state i to j is defined

as (11) for any time and any { }, ,1,2,i j K∈

( ), 1|i j n j n iP P S s S s−= = = (11)

,0i jP = , if | i - j| > 1, { },

1

1 1, 2K

i j

j

P i K=

= ∀ ∈ (12)

The state probability of the kth steady state is defined as:

( ) ( )1, 2,k n kP S s k kπ = = ∈ (13)

{ },

1

1,2,K

j j k k

j

p k Kπ π=

= ∀ ∈ (14)

For any given state k, the input stream is equal to output ones:

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, ,

1 1

K K

j j k k k l

j l

p pπ π= =

= (15)

The method to decide the thresholds of the state of FSMM is as follows:

1) Compute the thresholds by the equal probability of the

state, which is named equal probabilistic method.

2) Compute the thresholds by the equal duration of each

state, which is named equal duration method.

B. FSMM with Equal Duration of The State

The FSMM of fading channel based on equal probability is formed in [13], where it cannot be guaranteed that the duration of state is longer than that of a frame. Because the processing unit of the cross layer design is frame or packet and the signal is parallel transmitted in frequency domain subcarriers in OFDM, the duration of the state should be equal and longer than the time of a frame. So we form the FSMM by the equal duration .The basic principle is as follows: according to (13),

compute the thresholds of SNR when kτ is given and then

partition the range of the received SNR into some non-overlapping interval by such thresholds, that is to partition channel into different state.

In order to guarantee the duration of state longer than the time of a frame, we let

k k fc Tτ = (16)

1

1

1 1

2 2

1

, ,1

2k k

k k

km mm m

m f

k k

m mm m

cf T

m me e

γ γγ γ

γ γγ γ

πγ γ

γ γ+

+

− −− −

+

Γ − Γ=

+

(17)

Once kc is decided and let 1

0γ = and 1Kγ + = ∞ at the

same time, each threshold can be obtained by (17). (Note: For

convenience, assume that 1m = and 1γ = in this paper).

Determine kc by following method in cross-layer design:

Letting 1Kγ + = ∞ and Kγ is equal to the threshold of the

highest order of the MCS, the kc can be obtained by (17)

after mf and fT is given. The space between Kγ and 1Kγ + is

the biggest of all intervals and k Kc c≤ for any k. To guarantee

the duration longer than the time of a frame, we can let kc c=and 1 Kc c≤ ≤ for 1,2,k K= .

From [6], we can see that the bigger c is, the smaller K and

the bigger interval are with m Pf T given in Rayleigh fading

channel, which is also suitable for Nakagami-m channel. It is

suggested that kτ or kc shouldn’t be too big when FSMM is

formed, otherwise the too big interval will include multiple

thresholds of MCS, thus it results in the unfit MCS is selected.

Meanwhile, it is should be kept that k fTτ ≥ in order to keep

channel invariant during the time of a frame.

Considering 3 fRTT T≈ , it is proper that 1 3kc≤ ≤ .

The change of the threshold and interval of the state is with

m given in table ( ). From table ( ), we can see that on Nakagami-m channel, the length of the interval is gradually increased when m change from 1/2 to 1and 2.This shows that the smaller m is ,which mean the more sever fading is, the bigger the interval is under the same duration.

C. The Transition Probability of FSMM with Equal Duration of The State

The transition probability from the kth state to the (k+1)th state of the FSMM defined above is as:

( )1

1

2

1

11

, 1

1

2

, ,

k

m m

k m f

k pk

k k

k k

k k

me f T

N TNp

R m mm m

γπγ

πγ γ

γ γ

+−

− Γ

+++

+

+

ΓΓ

= = =Γ − Γ

( )1, 2 1k K= − (18)

Also:

1

2

. 1

1

2

, ,

k

m m

k m f

k k

k k

me f T

pm m

m m

γπγ

γ γγ γ

−− Γ

−

+

Γ=

Γ − Γ (19)

0,0 0,11p p= −

, , 11K K K Kp p −= − (20)

, , 1 , 1,

1 2,3 1k k k k k kp p p k K− += − − = − (21)

IV. SOME MODIFICATIONS IN CROSS-LAYER DESIGN

The determination of kc and the number of state as well as

the interval of the state are affected by mf . When the mf is

increased with higher move speed, according to (17), the number of the state must be reduced and the interval must be

lengthened ifkc be kept invariant. On the contrary, if keep

interval invariant, the duration of the state must be decreased. Terminal’s high-speed moving will degrade the cross-layer design. Specially, because the data rate in subcarrier is only 1/N to the data link and the time selective fading is more sever than the singer-carrier, the effect that high-speed move degrade the performance is more obvious in OFDM system.

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A. Modification on threshold of MCS based on FSMM

In order to select proper MCS in high-sped move channel,

we calculate the duration of each interval 1[ , ]k k +Γ Γ of the

present MCS by (16) and (17) with the given mf .Because kτ is

decreased with the increase of mf provided the thresholds and

the total number of them is constant, it may be shorter than

fT Once fTτ ≤ , we should cancel the threshold 1k +Γ , let

1[ , ]k k +Γ Γ denote one state. And the mode k+1 should be cast.

B. Modification on CQI

If 3 fRTT T≈ and 1 3kc≤ ≤ , that is to say that the duration

of the state is shorter than RTT, the channel will have changed when the CQI delivered by the receiver via feedback channel reach the transmitter, when the CQI should be corrected by transition probability of FSMM. We should first decide the channel in the nth state at a certain time k by the expression

that 1

( ) [ , ]n nCQI k γ γ +∈ and then let dτ denote the feedback

delay time. When 2n d nτ τ τ≤ ≤ ,

( ) ( ) ( ), 1 , , 11 n n n n n nCQI k CQI k P P P− ++ = ∗ + + (22)

When ( 1)k d kn nτ τ τ≤ ≤ + , compute the ( 1)CQI k + by n

times iteration in (22) .If the modified CQI satisfied

1[ , ]n nCQI +∈ Γ Γ , the mode n is selected.

C. Modification on Chase combining

The basic method of Chase combining is as (23) [11]:

,

1

( )n

c n k

k

dBγ ε γ=

= × (23)

where ,c nγ is the SNR resulted of the nth combining and kγ is

the SNR of the kth retransmission as well as ε the efficiency factor of combining according to the maximum ratio. It is required that the MCS be kept identical in every transmission or retransmission. Because of the effect of high-speed move to quality of channel, the constant MCS may result in not adapting to the channel. We should adopt AMC and make

some modification on ε . Meanwhile,in (23), each kγ must be

replaced by ( )CQI k modified in (22).

V. IMPROVED CROSS LAYER DESIGN COMBINING AMCWITH ARQ

After the parameters such as max

arg, ,t et r nP N γ are decided,

the steps of cross layer design AMC with ARQ are as follows:

(1).At the beginning of every frame, the transmitter received the feedback signal CQI and ACK/NACK and compute the new CQI by the feedback delay using (22) .For the

new data, if 1n nCQI +Γ ≤ ≤ Γ , the mode n is selected. If the

CQI is lower than the threshold of the lowest order MCS, we should select other subchannel.

(2).While for the retransmitted packet, predict link level performance using (23) by the modified CQI and compute the

sum of SNR of all max

rN times retransmission. If the sum meet

the request of the mode in last time transmission, the same mode is selected in retransmission, or we should wait or select the other subchannel.

(3).At receiver, the new data is received and error-detected. After that, ACK or NACK and CQI are send to the transmitter .For the retransmitted data, the receiver uses Chase combining to combine them using the modified CQI. If a packet is not

received correctly after max

rN times retransmissions, we will

drop it and declare the packet is lost.

VI. NUMERIC RESULTS

An OFDM with 512 subcariers is adopted to base for the cross-layer design combining AMC with ARQ on frequency domain subcarrier., with parameters for the simulation are as follows: bit rate 20Mbits/s,bit rate 39.1kbits/s in subcarrier, the frame time 1ms,and 40 OFDM symbols per frame. Other parameters are as follows: carrier frequency 2 GHz, speed 18.36km/h or 3*18.36km/h.Thus the maximum Doppler

frequency is 33.8 Hz or 3*33.8 Hz, and m ff T is 0.0338 or

3*0.0338. A channel generator based on Jakes’ model with 11 paths is developed to simulate Nakagami-m channel with m=1

in time domain. With m=1 and 1γ = and FER=0.01 and the

number for retransmission is 1.The thresholds of MCS in

additive white noise channel is in kΓ row in table . The

fading characteristics of channel having been taken into consideration in computing the duration of the state, the thresholds of MCS in fading can be thought as the same as in

additive white noise channel. In another hand, letting 1Kγ + = ∞and 15.9784K dbγ = (from table ), c is computed as 1.87 by

(17). For convenience, letting c=1.8, the thresholds of FSMM

is computed by (17) as shown in table . Comparing the

thresholds of MCS in table with those in table , any interval partitioned by thresholds of FSMM does not include more than two thresholds of MCS along with that

1.8 3f k f fτ τ τ τ≤ = ≤ , which is proper for FSMM as above

mentioned. On the other hand, because of the bigger thresholds interval of MCS than that of FSMM resulting the longer duration than the frame time, the thresholds of MCS need not

be modified. In this case, for 2n d nτ τ τ≤ ≤ , the ( 1)CQI k +computed by (22) is in the same interval as ( )CQI k be. Thus

for one time retransmission, MCS need not updated. So the middle or low speed move affect little to AMC/ARQ cross-layer design.

Table show the thresholds of FSMM and the corresponding transition probability when speed change to 18.6*3km/h while the duration of the state is 1.8 times of the frame time. It is clearly that number of the state is reduced and the interval is increased, which result in more than two

thresholds of MCS in table are included in one interval of

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FSMM. The duration of these included intervals of MCS must be shorter than that of FSMM, thus the mode 4, 5, and 6 in

table should be canceled. For the reason that the duration of mode 3 must be longer than that of FSMM ,the mode 3 should be selected when SNR is bigger than 5.8656dB.For mode 1

and 2, the durations are 1.8 fτ and 1.85 fτ respectively, which

meet the requirement of the duration of FSMM. So when the terminal move in high speed, only three scheme (1/2BPSK, 1/2QPSK, 3/4QPSK) exist by modifying MCS. Meanwhile,

because of the increased interval of FSMM, ( 1)CQI k +computed by (26) is always in different interval of FSMM with

( )CQI k .So high-speed move affect more to AMC/ARQ cross-

layer design.

When terminal moves at 3*18.36km/h, the assumptions are as follows: free error in feedback,CRC_16 for error-detection ,thresholds of MCS as shown in the first three column of table

, state and corresponding transition probability of FSMM as

shown in table and 3RTT for longest delay, namely 2 for biggest number of retransmission, considering the spectral efficiency decreasing to the same as that of the next lower-order MCS when retransmission in 1/2QPSK or 3/4QPSK for two times,1for the biggest number for retransmission. Other parameters are as mentioned above. Spectral efficiency is

computed as,c physical

c

SS

N= . We simulate the AMC/ARQ cross

layer design according to the project in section and respectively, where the configure is as shown in Fig1.The changes of average spectral efficiency and FER with average SNR are depicted in Fig2 and 3 respectively. We can see that when terminal move in high speed, the improved AMC/ARQ cross-layer design has higher spectral efficiency and lower FER compared to that former.

VII. CONCLUSION

In this paper, we developed an improved AMC/ARQ cross-layer design for terminal’s high –speed moving based on the frequency domain subcarrier of OFDM .For that, a novel FSMM with equal duration of the state is formed to modify the threshold of MCS and the CQI as well as the method for link prediction for high-speed move. The numeric results reveal the superior performance of the improved project .FSMM supply a basis for tracking the variation of the channel thus for the accurate cross-layer design combining AMC with ARQ. And then this design provide a base for guaranteeing QoS traffic and efficiently using of radio resource. With demand’s increased, the accurate AMC/ARQ cross-layer design must be the research focus in the future.

TABLE I. MCS AND CORRESPONDING THRESHOLDS

Mode1 Mode2 Mode3 Mode4 Mode5 Mode6

Mod. BPSK QPSK QPSK 16QAM 16QAM 64QAM

Code rate 1/2 1/2 3/4 9/16 3/4 3/4

Efficiency 0.50 1.00 1.50 2.25 3.00 4.50

na 274.7229 90.2514 67.6181 50.1222 53.3987 35.3508

ng 7.9932 3.4998 1.6883 0.6644 0.3756 0.0900

pnγ -1.5331 1.0942 3.9722 7.7021 10.2488 15.9784

kΓ -0.0410 2.874 5.8656 9.7115 12.2326 18.1424

TABLE II. THRESHOLDS OF FSMM WITH MOVE SPEED 18.36KM/H

Index Thresholds

(dB)

Index Thresholds

(dB)

Index Thresholds

(dB)

1 -3.0543 12 7.5905 23 12.5092

2 -1.3607 13 8.1462 24 12.8704

3 0.0586 14 8.6721 25 13.2239

4 1.2808 15 9.1715 26 13.5709

5 2.3548 16 9.6477 27 13.9127

6 3.3133 17 10.1031 28 14.2509

7 4.1794 18 10.5399 29 14.5872

8 4.9698 19 10.9603 30 14.9239

9 5.6973 20 11.3658 31 15.2640

10 6.3717 21 11.7582 32 15.6124

11 7.0007 22 12.1389 33 15.9784

TABLE III. THRESHOLDS AND CORRESPONDING TRANSITION

PROBABILITY OF FSMM WITH MOVE SPEED 3*18.36KM/H

Index Threshold(db) , 1k k

p + , 1k kp − ,k k

p

1 -Inf 0.7916 0.2084

2 -10.2774 0.3365 0.2462 0.4173

3 -2.7330 0.2278 0.3685 0.4037

4 1.3611 0.1508 0.4551 0.4041

5 4.3043 0.0790 0.5358 0.3852

6 6.7920 0.6246 0.3754

7 Inf

TABLE IV. THRESHOLDS AND INTERVALS OF FSMM FOR EQUAL C AND

DIFFERENT M

m=1/2 m=1 m=2

thresholds/db interval/db Thr. Int. Thr. Int.

-20.0000 -12.0474 -13.0495

-13.9794 6.0206 -6.0158 6.0316 -7.4880 5.5615

-4.4370 7.5424 -2.4754 3.5404 -4.3782 3.0098

0 4.4370 0.0490 2.5244 -2.1292 2.2490

2.9226 2.9226 2.0232 1.9742 -0.3450 1.6842

5.1055 2.1829 3.6514 1.6282 1.1452 1.4902

6.8485 1.7430 5.0454 1.3940 2.4330 1.1878

8.2995 1.4510 6.2726 1.2272 3.5746 1.1416

9.5424 1.2529 7.3777 1.1051 4.6075 1.0329

8.3934 1.0157 5.5590 0.9515

6.4510 0.8920

0 2 4 6 8 100.2

0.4

0.6

0.8

1

1.2

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Figure 2. Average spectral efficiency Vs different SNR

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Figure 3. Average frame error ratio Vs different SNR

REFERENCES

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