Performance Analysis of OFDM Systems With Performance Analysis of OFDM Systems With Adaptive Sub...

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1118 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 4, APRIL 2008

different slots may be changed to generate different sub carrier

bandwidths. Users with similar requirement of sub carrier

bandwidth may share a time slot. The time frequency diagram

is as in Fig. 1. This model is used for analysis via simulation

in this work. Among other possible implementations, one

can be band division multiplexing based. The entire available

bandwidth may be divided into subbands with different sub

carrier bandwidth in each sub band. Each sub band can be

operated on by an IFFT with different number of sub carriers.

Since the end user equipment will operate on only one sub

band, therefore only one Programable FFT [10] is needed

for the user equipment. The number of sub carriers can be

adjusted in it, which can address the required sub carrier

bandwidth. The number of FFTs required at the base station

will be equal to the number of different types of sub carrier

bandwidths.

The guard interval duration (GI) is mostly fixed and its

length is dependent on the channel delay spread irrespective

of the sub carrier spacing. As long as the length of GI is

sufficiently large so that ISI is within tolerable limits the

choice of sub carrier bandwidth and choice of GI can be

treated independently. (In a separate work, the authors have

presented an analysis on the choice of GI in [11].)

As was mentioned earlier, adaptive bit loading is done along

with ASB. Though joint bit and power adaption is optimal,

it has been observed that the gain obtained in keeping the

power constant while varying the rate is very close to being

optimal [9]. Therefore the power per sub carrier is fixed and

equally distributed on all data sub carrier. The rate is varied on

each sub carrier by means of adaptive modulation. The study

is carried out without forward error control (FEC) coding, asin [9] to analyze the potential of the scheme.

III. ANALYTICAL MODEL

The time domain signal of the sth transmitted OFDMsymbol can be expressed as [2]

xs (t) =1Tf

Nf2 1

k=Nf2

X[s, k]ej2 kTf

(tsTsTgi)Ts(t sTs)

where Tf is the duration of OFDM symbol without the guard

interval, k is the sub carrier index, Nf denotes the numberof sub carriers, X[s, k] is the modulated data symbol on thesub carrier, Ts is the symbol duration which is the sum ofTf and the guard interval duration Tgi. Ts(t sTs) is thegate pulse of duration Ts starting from t = sTs, which canbe implemented in digital domain [12]. After passing through

the channel, the signal can be represented as,

r(t) =

max0

h()ej2fd (t)xs (t ) d+ (t) (1)

where h() represents the channel impulse response, maxis its maximum tail, (t) is the noise component and fdis the Doppler frequency for delay . With perfect timingsynchronization, but residual carrier frequency offset fc (Hz),the received OFDM symbol is

rs(t) = r(t)ej2fctTf(t sTs Tgi). (2)

The signal portion without the noise part ismax0

h()xs (t )ej2fd(t)ej2fct

Tf(t sTs Tgi) d. (3)Since h()ej2fd cannot be distinguished from h(), wecan represent h()ej2fd as h(). Therefore,

rs(t) =

max0

h()xs (t )ej2(fc+fd )t

Tf(t sTs Tgi t) d . (4)Considering that channel coefficients remain static over a

small period of time, which is less than the coherence time,

fc + fd can be termed as effective carrier offset and repre-sented as f. The relative offset, i.e. the ratio of the effectiveoffset to the sub carrier spacing can be defined as ffsc ,where fsc is the sub carrier bandwidth. The received subcarrier can be computed as [2], [13]

Rs,k = X[s, k

]H[s, k

]ej2(k

,k

,)S useful signal component

+

k,k=k

X[s, k]H[s, k]ej2(k,k

,)

inter carrier interference

S + [s, k

] (5)

where S = sinc{(k, k , )}, and, (k, k , ) = k k +, [s, k

] is the frequency domain noise component and

H[s, k

] is the channel coefficient for k th

sub carrier of sth

OFDM symbol. X[s, k

] is zero mean, which implies that the

interference term is also zero mean, therefore the power of theinterference term is the same as its variance, which becomes,

2ICIX[k

]

= E[|X[k]|2]Nf2 1

k=Nf2 ,

k=k

|H[k]|2

sinc2(

f

fsc+ k k)

(6)

where E is the expectation operator. It is assumed that thecoherence bandwidth is large enough, so that the channel

coefficients H[k] are the same for the most significant (neigh-

boring) sub carriers causing ICI [2]. Denoting E[|X[k]|2] byPX which is the average power per sub carriers, and, |H[k ]|2by PH[k ], the ICI power at the receiver on sub carrier k

is:

2ICIX[k

] PXPH[k ]

Nf2 1

k=Nf2 ,

k=k

sinc2(

f

fsc+ k k)

for small values off

fsc[2]

2ICIX[k

]

1

3

PXPH[k ](f

fsc

)2. (7)

Therefore signal to interference plus noise ratio (SINR) is

rx[k

] PX[k ]PH[k ]sinc(

fcfsc

)2

2 +13PXPH[k ](

ffsc

)2(8)

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 4, APRIL 2008 1119

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 104

4

2

0

2

4

6

8

10

12

14

16

Sub carrier bandwidth in Hz

SINRindB

10 kmph

80 kmph

160 kmph

240 kmph

Fig. 2. SINR vs sub carrier bandwidth at 15dB SNR.

Fig. 2 shows the plot of SINR vs sub carrier bandwidth

for different velocity conditions. Each curve is for a particular

velocity. It can be seen from this figure, that SINR improves

with increasing sub carrier bandwidth for a given Doppler

spread. But this does not ensure a monotonically increasing

throughput with increasing sub carrier bandwidth. This can

be seen from Fig. 3, where the throughput curves have been

obtained by considering adaptive bit loading per sub carrier.

This figure shows that the throughput, for a given Doppler

velocity, is maximum for a certain sub carrier bandwidth

only. The decrease in throughput after a certain sub carrier

bandwidth can be attributed to the increase of the sub car-rier bandwidth which causes the OFDM symbol duration to

decrease thereby increasing the percentage overhead duration,

as the guard interval has a fixed duration. For the above it can

be seen that by choosing the appropriate sub carrier bandwidth

there is potential for significant improvement in throughput.

IV. ADAPTIVE BANDWIDTH FOR SUB CARRIERS

We propose an algorithm here, to dynamically select the

appropriate sub carrier bandwidth and bit load per sub carrier

to maximize the throughput while satisfying a required BER

is presented here. The sub carrier bandwidth can be chosen asfchosen =arg max

fm

[Thpt(fm)] (9)

subject to

fm < Bcand (10)

Ts < Tc (11)

where Bc is the coherence bandwidth, and Tc is the coherencetime [14]. The index m, runs through the allowable sub carrier

bandwidths (out of a finite number of options) while meeting

the constraints of coherence bandwidth and coherence time

as mentioned above. The estimated throughput Thpt(fm) in

(9) can be written as

1

Bw(1

fm+ Tgi)

Nf2 1

k=Nf2

bL(k,fm)(1 bo(k,fm)) (12)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 104

0

0.5

1

1.5

2

2.5

Sub carrier bandwidth in Hz

Throughput(b/s/Hz)

10 kmph

80 kmph

160 kmph

240 kmph

Fig. 3. Throughput vs sub carrier bandwidth at 15dB SNR.

where Bw denotes the system bandwidth. The bit load estimateper sub carrier used in (12) can be expressed as [9]

bL(k,fm) = 212

log2

1 1.6

ln(bo req0.2 )

rx(k,fm)

(13)

The above expression is valid for square constellation, where

the operation . is the floor operation). In the above, bo reqis the target BER which is to be satisfied. The BER associated

with the chosen bit load is

bo(k,fm) = 0.2e

1.6rx(k,fm)

2bL(k,fm)1 . (14)

The above expression is tight for high SNR and has been

claimed to be valid within 1.5dB for 4-QAM to 1024-QAM

for bit error rate (BER) 103. The symbol rx(k,fm) usedin the above expression is taken as

rx(k,fm) PXPH[k]sinc(

ffm

)2

2 +12 (

ffm

)2PXPH[k](15)

The following steps are executed in sequence.

1) Select one sub carrier bandwidth from the availableoptions.

2) Evaluate (15), i.e. SINR at each sub carrier for the

selected sub carrier spacing. For the calculations, f hasto be estimated using advanced schemes as in [15].

3) Use the above in finding bit load for this chosen value

of sub carrier spacing following (13).

4) Calculate the associated BER for each sub carrier for

the chosen bit load using (14).

5) Use the above calculations of bit load and related BER

for each sub carrier in calculating the throughput for the

chosen sub carrier bandwidth following (12).

6) Store the value of the estimated throughput along withthe value of sub carrier bandwidth and associated bit

loads per sub carrier.

7) Repeat all the above steps for all possible values of sub

carrier bandwidth.



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8) Finally execute (9) to select the sub carrier bandwidth

and bit loads per sub carriers which has the highest

estimated throughput.

9) Since the rate of change of Doppler condition and

average channel quality is much slower compared to the

rate of change of channel coefficients, one may consider

to adapt the sub carrier spacing at a rate much less than

adapting the bit loading. The bit loading should be done

once per coherence time of the channel coefficients. i.e.

once a sub carrier spacing is selected, it may be used

until the Doppler condition or the average signal strength

changes significantly and hence step 1 and step 7 may be

skipped, and step 8 may be modified to Finally execute

(9) to select bit loads per sub carriers which has the

highest estimated throughput for the chosen sub carrier

bandwidth.

The overhead for signalling the chosen rate and sub carrier

bandwidth depends on the feedback mechanism. For simplicity

of analysis but to keep the results realistic, it is assumedthat BPSK is used for signalling the feedback information.

It is also assumed that number of bits per feedback is taken

as log2[n(X)], where n(X) denotes the number of signallinglevels for X where X is the feedback parameter. The rateof feedback of the bit loading parameter is done once per

coherence time per coherence bandwidth, while the maximum

Doppler frequency offset is fed back once per second. This

overhead is considered in the simulations.

V. RESULTS AND DISCUSSION

Each coefficient of the time domain channel impulse re-

sponse is taken as Rayleigh distributed with Jakes spec-trum [14]. Exponential power delay profile with rms delay

spread of 2 micro seconds was used. Bandwidth of 5 MHz

at carrier of 3.6 GHz is considered. The target bit error rate

is kept at 102. To implement ASB in our simulations theTDM mode (Fig. 1) was taken. Number of bits that can be

loaded on a sub carrier are 0,2,4,6,8 and 10, where 0 means

no transmission. The curves labeled 2048, up to 128, are for

fixed systems with as many sub carriers, which corresponds to

2.4 KHz to 39.063 KHz of sub carrier bandwidth respectively.

The options for number of sub carries for ASB system is

selected from this range. The curve labeled with ASB, is for

adaptive sub carrier bandwidth system.The variety of the combinations (distribution of active

users as a function of received signal strength and Doppler

condition) is very high. It is not possible to capture all of

them and any one particular scenario (ratio of users distributed

in different situations), will not be a complete representation.

Therefore, the performance is presented rather as a function

of received signal strength condition over a range of Doppler

conditions. It can be assumed that users can be in one of such

representative situations. Time based scheduling can be used

by some higher layer protocol, whereby one or many users in

similar condition of Doppler and signal strength are scheduled

together so that the sub carrier bandwidth in one time slot isthe same, while the bit loading per sub carrier can vary across

sub carriers depending on the channel condition of the sub

carrier, which has been described in the algorithm presented

earlier.

0 50 100 150 200 250 3000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2x 10

4

Doppler Velocity in Kmph

SubCarrierBan

dwidth(Hz)

ABS

Fig. 4. Sub carrier bandwidth selected by ASB systems.

0 50 100 150 200 250 3000

0.5

1

1.5

2

2.5


Throughputinbits/s/Hz

2048

1024

512

256

128

ABS

Fig. 5. Throughput comparison of ASB vs FSB OFDM systems, at SNR of15 dB.

Fig. 4 shows the average sub carrier spacing selected by

ASB scheme for different values of velocity at a received

signal to noise ratio (SNR) of 15 dB. The increase in av-

erage value of sub carrier bandwidth selected with increasingvelocity can be easily found.

Now the performance, in terms of throughput, of ASB

system is compared against fixed sub carrier bandwidth (FSB)

systems. The throughput of FSB system is computed consid-

ering adaptive bit loading per sub carrier as is considered for

the ASB system. If fsc is the sub carrier bandwidth, thenthe throughput is calculated using (12)(15) where fm isreplaced by fsc. Fig. 5 compares the throughput for ASBscheme against standard OFDM systems with fixed number

of sub carrier bandwidth for different values of the number of

sub carriers at SNR of 15 dB. It is clearly seen that a given

sub carrier bandwidth has the highest throughput over a smallrange of velocity, while ASB has the highest throughput over

the entire range. The system with 2048 sub carriers (fsc= 2.4KHz) is best for mobility less than 10kmph. Between

10 and 40 kmph, the system with 1024 sub carriers (fsc



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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 4, APRIL 2008 1121

0 50 100 150 200 250 3000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5


Throughputinbits/s/Hz

2048

1024

512

256

128

ABS

Fig. 6. Throughput comparison of ASB vs FSB OFDM systems, at SNR of25 dB.

= 4.88KHz) has the highest performance. Between 40 and

140 the one with 512 sub carriers (fsc = 9.77KHz) is themost efficient, beyond which the system with 256 sub carriers

(fsc = 19.531KHz) is the best. Interestingly the ASB systemrides the envelope, i.e. it has the highest throughput over all

velocities. In the low mobility region ASB is better than the

system using 512 sub carrier by about 12% and about 25%

better than one using 256 sub carriers. In the high mobility

region (near 200 kmph) ASB is better than the system with

512 sub carriers by about 25% and by more than 30% over

the system with 1024 sub carriers, when the one with 2048

sub carriers which is best for low mobility almost fails.

Fig. 6 shows similar curves as above but for a higher

SNR (25 dB). Comparing with the previous figure, it can

be observed that the difference in received signal strength

has caused the optimal range for each OFDM system (with

different but fixed sub carrier bandwidth) to change, but ASB

is still the most efficient in the entire range. The ASB scheme

is better than the fixed sub carrier bandwidth system by about

10% to 30% in different velocity regions.

Finally Fig. 7 shows the BER curves for all systems. It

can be observed that by use of adaptive bit loading (ABL),

the BER is maintained below the target level (10

2) for all

system and for all velocities. Schemes which do not use ABL

will show an increase in BER with increasing Doppler effect,

but with ABL, the BER requirement is met. Though the target

BER is satisfied by both schemes (ASB and FSB) using ABL,

it can be said that ABL alone is not sufficient for FSB OFDM

system to be efficient for all Doppler conditions. ASB with

ABL improves the throughput by a significant amount.

VI. CONCLUSION

It can be concluded that the novel ASB system, presented in

this work, for Doppler frequency spread scenario with varying

received signal strength conditions has the potential to improvethe throughput performance of FSB OFDM system by 10%

to 30% in different situations. While the latter system with a

chosen but fixed sub carrier bandwidth is optimum only over

a small range of velocities and received signal strength, the

50 100 150 200 250 300

2

3

4

5

6

7

8

9

10x 10

3


BER

2048

1024

512256

128

ASB

Fig. 7. BER of ASB and FSB OFDM systems at SNR of 15 dB, whentarget BER is kept at 0.01.

former system, i.e. the adaptive sub carrier bandwidth system,

has optimum performance over all conditions. ASB avoids the

complex compensation or interference cancelation mechanism

at the receiver, thereby allowing lower complexity receivers.

Thus the advantage of increased throughput with possibility of

low complexity receivers makes the proposed ASB a potential

candidate for consideration in future systems. The promising

results pave the path for further investigation with realistic

impairments such as channel information feedback delay,

channel estimation error, synchronization error along with the

use of forward error control coding.

ACKNOWLEDGEMENT

The authors are grateful to Tata Consultancy Services for

funding the work and to the RATE section member of Aalborg

University, especially to Muhammad Imadur Rahman and

Daniel Vaz Pato Figueiredo for excellent discussions.

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[4] G. L. Stuber, Principles of Mobile Communication. Kluwer Academic,2001.

[5] P. Robertson and S. Kaiser, The effects of Doppler spreads inOFDM(A) mobile radio systems, in Proc. IEEE VTC 50th VehicularTechnology Conference, vol. 1, Sept. 1999, pp. 1922.

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[9] S. T. Chung and A. J. Goldsmith, Degrees of freedom in adaptivemodulation: a unified view, IEEE Trans. Commun., vol. 49, no. 1, pp.15611571, Sept. 2001.

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