Research

# Effect of power and rate adaptation on the spectral efficiency of MQAM/OFDM system under very fast fading channels

Zhicheng Dong1,2*, Pingzhi Fan1, Erdal Panayirci3 and P T Mathiopoulos4

Author Affiliations

1 Institute of Mobile Communications, Southwest Jiaotong University, Chengdu, 610031, China

2 School of Engineering, Tibet University, Lhasa, 850000, China

3 Department of Electronics Engineering, Kadir Has University, Istanbul, 34230, Turkey

4 Institute for Space Applications and Remote Sensing, National Observatory of Athens, Athens, 15236, Greece

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EURASIP Journal on Wireless Communications and Networking 2012, 2012:208 doi:10.1186/1687-1499-2012-208

 Received: 15 February 2012 Accepted: 25 May 2012 Published: 2 July 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

##### Keywords:
OFDM; Intercarrier interference (ICI); Power and rate adaptation; Spectral efficiency; High mobility

### Introduction

Orthogonal frequency-division multiplexing (OFDM) has been shown to be an effective technique to overcome the inter-symbol interference (ISI) caused by frequency-selective fading with a simple transceiver structure. It has emerged as the leading transmission technique for a wide range of wireless communication standards [1] such as the IEEE’s 802.16 family—better known as Mobile Worldwide Interoperability Microwave Systems for Next-Generation Wireless Communication Systems (WiMAX)—and the Third-Generation Partnership Project (3GPP) in the form of its long-term evolution (LTE) project. Both systems employ orthogonal frequency division multiplexing/multiple access (OFDMA) as well as a new single-carrier frequency-division multiple access (SC-FDMA) format. To promote the IEEE 802.16 and LTE standards, recently, a high mobility feature has been introduced (IEEE 802.16m, LTE Advanced (LTE-A)) to enable mobile broadband services at vehicular speeds beyond 120 km/h.

In order to study adaptive MQAM/OFDM systems, one should investigate first the forms of the SNR and BER expressions in the presence of very rapidly time varying channels. Much of the research on the performance analysis of OFDM in fast fading channels are based on an approximation of the ICI components as Gaussian random variables or by using the average ICI power. Such simple closed-form expressions and bounds for the total ICI power are presented in [11,20-22]. These have been employed to obtain approximate expressions for the average signal-to-interference plus noise ratio (SINR) [23-25], and error rates [3-5].

### System model

We consider an OFDM system with N subcarriers. The time-domain transmitted signal can be written as

(1)

where L is the length of the guard interval and dkstands for the frequency domain data symbol transmitted over the kth OFDM subchannel. We assume that all symbols have the same energy Es=E{∣dk2}per subcarrier, Then the received signal at the input of the FFT takes the known form

(2)

where h(nl) is the channel impulse response of the lth tap at time n. We assume Rayleigh fading channels, and a Jakes’ Doppler spectrum with maximum Doppler frequency fmax=fcv/c, where fc is the carrier frequency (Hz), v is the speed of the mobile terminal (km/h), and c is the speed of light. Assume that the maximum channel delay spread is less than or equal to the guard interval L, and w(n) is zero-mean, complex additive white Gaussian noise with variance . The kth subcarrier output from the FFT can be expressed as [26]

(3)

where

(4)

(5)

Hk(n)in (4) is the Fourier transform of the time-varying channel impulse response at time n, defined as and . The variance of Wk is . Note that and =J0(2ΠfmaxTOFDM(nn)/N), where TOFDM=1/Δf=N/B, and B is the total bandwidth (Hz) (we assume the overhead due to the guard interval is ignored in this article, but the guard interval would reduce the spectral efficiency in practical OFDM systems). fmaxTOFDM=fcvN/cB is the normalized Doppler frequency of the system. J0(.)denotes the zeroth-order Bessel function of the first kind.

The second term in (3) represents intercarrier interference (ICI) caused by the time-varying nature of the channel. From (5), the ICI power of the kth OFDM subcarrier can be obtained as

(6)

where

Since the data on each subcarrier are uncorrelated, and E {|dm|2}=Es, it follows easily from (6) that the ICI power, is independent of the subcarrier index k and has the following form [20,26]:

(7)

To simplify the issue, the normalized ICI power PNis adopted in this article, where PN=PICI/Es. When fc, N, and B are constant, PICIincreases with the speed v. Note that, in a time-invariant channel, the value of v is zero therefor PICIbecomes also zero. In line with the channel model of [10,17], we make two assumptions:

Each subchannel is narrow enough that it experiences frequency-flat fading.

Hkhas a complex Gaussian distribution with zero mean and variance ρ0 for all k at each OFDM interval [26], where

(8)

Assuming that the transmitted symbols are mutually uncorrelated, the instantaneous effective SINR for the kth subcarrier with fixed power allocation can be written as [15],

(9)

where is the instantaneous SINR for the kth subcarrier and is the average SNR when there is no ICI. The average effective SINR is then defined as

(10)

From (10), it follows that decreases as the speed increases when is fixed. Figure 1 shows the average effective SINR as a fuction of . As can be seen from Figure 1, the received average SINR increases with the average transmitted SNR. When the velocity equals 100 km/h, it changes almost linearly as function of the average SNR. However, when the velocity reaches 500 km/h, the trend of the increase in the SINR is no longer linear due to the fact that the ICI also increases with speed.

Figure 1. Average SINR versus average SNR.

We now consider, as a first case, a non-adaptive modulation scheme where β(γk)=β is a constant for all k. Assuming M-ary QAM symbols are transmitted over each subchannel of an OFDM system, the number of bits/symbol sent over the kth subchannel is β(γk)=log2M(γk). The instantaneous BER for the kth subcarrier is approximated by [28]:

(11)

Since (11) can not be easily invertible, we consider a different BER approximation formula with these properties [28].

(12)

where C1=0.2, C2=1.6.

Assuming that the subchannel Hk is a complex Gaussian random variable and all Hkhave identical distributions with E{|Hk|2}given by (8), the average BER can be approximated as

(13)

where pv(v)=1/ρ0exp(−v/ρ0)and ρ0 is defined by (8).

Assuming εis the target average BER, the maximum bit load, given the average BER constraint is

(14)

Figure 2 shows the tightness of the simplified average BER approximation for MQAM, according to (12), with different bit rates for velocities 100 and 500 km/h, respectively. It is evident that the simplified approximation nearly predicts the BER expression of (11). It also shows that the BER performance degrades with an increase in vehicle speed when the bit rate is fixed. In order to guarantee BER performance, the bit rate must be reduced under high mobility.

Figure 2. Average BER versus average SNR.

### Methods

For adaptive OFDM, different power levels and modulation bit load are allocated to different subchannels. The instantaneous BER of each subchannel is

(15)

where the transmit power s(γ[k])and β(γ[k]) are adapted with γ[k], subject to the average power , and the instantaneous BER constraints Ω([k])=ε. is the normalized ICI power (variance) of the kth subcarrier. From (6) it follows that

(16)

Note that it is very difficult to determine the exact when all the transmit powers carried by the subcarriers are not the same. However, we can obtain an upper bound in terms of the largest transmit normalized power , , for all the subcarriers, as will be explained shortly. Consequently, employing the upper bound on the normalized ICI variance of all subcarriers, determined by smax,the upper bound of the instantaneous BER for each subchannel can be expressed as

(17)

where PN is the normalized ICI power when all the subchannels carry the same average power, determined from (6) as . According to (17), the maximum bit load size for each subchannel is obtained as

(18)

The maximum bit load in (18) is a lower bound for each subchannel because of the largest ICI power. According to Assumption 2 for the channel model, it can be shown that the average capacity of an adaptive OFDM system does not depend on the system parameters or the specific power delay profile of the wide sense stationary uncorrelated scattering (WSSUS) mobile channels [17].

To maximize the average capacity of an adaptive OFDM system, the following constrained optimization problem is considered next:

(19a)

subject to

(19b)

(19c)

(19d)

where Ex{·}denotes expectation with respect to x. Then, the Lagrangian for the optimization problem can be defined as

(20)

where λ is the Lagrange multiplier. The optimal power adaptation must be nonnegative and satisfy

(21)

Solving (21) yields the optimal power adaptation

(22)

where the value of the Lagrange multiplier λ can be obtained from the average power constraint using numerical methods. The optimal rate adaptation is then given as

(23)

In practice, however, only an integer value of the rate is acceptable. Therefore, the rate is truncated to the nearest integer value.

The maximum average capacity of an adaptive OFDM system is determined by averaging β(γk) over the probability distribution of γk. Consequently, it is computed from

(24)

where is the optimized threshold for γkbelow which the channel is not used. Note that this threshold can be found from (22) so that . To find a closed-form expression for (24), we use the pdf of pγ[k](γk)given by

(25)

were, is the average received SNR when there is no ICI. As can be seen from (8), E{|Hk|2} is independent of the subcarrier index k. Therefore, can be substituted by . As shown in Appendix 1, (24) can be evaluated as

(26)

where Ei(·)denotes the exponential integral [29].

In order to evaluate smaxin (20), we employ the instantaneous BER expression (15) for each subchannel k where the normalized ICI power for that subchannel is computed. However, the resulting constrained optimization problem stated by (19a)–(19d) to find the optimal average capacity of the kth subchannel in an adaptive OFDM system needs the knowledge of the other subchannel powers, which makes the solution computationally intractable. One feasible way to solve the problem is to assume that the normalized ICI power of the kth subchannel is determined by its own power, that is, However, the BER performance, in this case, cannot meet the constraint (19d) since the transmit powers of the adjacent subcarriers are larger than that of the kth subcarrier. The fact that Hk being the best subchannel implies that as well as , determined by , take their maximum values resulting in the constraint (19d) satisfied. Consequently, smax can be determined as . We can obtain an upper bound from (22) as

(27)

For the worst case, the ICI is determined by the largest power, when γ[k]= and a lower bound on the maximum spectral efficiency in adaptive OFDM/MQAM systems is obtained.

#### Spectral efficiency of adaptive OFDM with imperfect CSI

In practice, it is impossible to obtain perfect channel information due to noisy channel estimation and the unavoidable delay between when channel estimation is achieved and when the estimation result is used for actual transmission. We will consider imperfect CSI in this section and study the performance degradation in spectral efficiency resulting from these CSI errors.

#### Effect of imperfect CSI

In a practical system, the channel estimator at the receiver provides the channel gains estimates , which are assumed as , where εk(n)represents the channel estimation error that is independent of the true channel gain Hk(n) and is distributed according to [30]. Consequently, the distribution of becomes where ρ0=E{∣Hk2} and is given by (8). Suppose the estimated channel gain is the only known information about the current CSI for the kth subcarrier. Since the instantaneous BER of each subcarrier, Ω(k), depends on the value of the true channel Hk, through which is assumed unknown, it is not possible to fix Ω(k)to be the target value. However, we can define the average BER given for the kth tone, as explained below.

As shown in Appendix 2, the instantaneous BER of each subchannel under imperfect CSI can be expressed as

(28)

where .

Consider the special case when Hkgiven is complex Gaussian with mean μand variance σ2. It follows that z=|Hk|conditioned on has a Ricean distribution [10]. Then the average BER for the kth subcarrier is defined as . Taking the expectation over the Ricean distribution, the average BER can be expressed as

(29)

where I0(·) is the zeroth-order modified Bessel function of the first kind. Taking the integral in (29), we obtain the the average BER:

(30)

where f(γk) is defined as

(31)

Let f(γ[k])be the appropriate function in (30) such that it will set an upper bound ε for the instantaneous BER, that is, . Consequently, from (30) we have

(32)

Note that in (30), f(.) cannot be solved in a closed mathematical form. However, f(γk)can be approximated by φ(γk) [31]:

(33)

Closeness of the approximation φ(γ[k])to f(γ[k]) is shown in Figure 3, over the entire range of |μ|2/σ2.

Figure 3. f(γ[k])andφ(γ[k])as a function of |μ|2/σ2.

The following relation between the power and rate of the kth subcarrier can be derived from (31)

(34)

To solve the optimization problem for power adaptation, its Lagrangian for the optimization problem is defined as

(35)

The value of Lagrangian multiplier λ1can be found from the average power constraints by using numerical methods, subject to the same constrained optimization problem presented in (19b). Following the same analytical derivations as that of (22), the optimal power adaptation is obtained as

(36)

from which the corresponding optimal rate adaptation is found as

(37)

Finally, the maximum average spectral efficiency can be expressed as

(38)

where is an optimized threshold below which the channel is not employed. It can be determined from

where fε(.)=f(.) which satisfies (32).

Using the Gaussian error assumption, it can be shown that Hk given is complex Gaussian with mean and variance given by [10],

(39)

Replacing μand σ2 in (38) with μ1and , respectively, we can obtain the average spectral efficiency,

(40)

where .

Evaluation of in (28) is similar to the derivations for the evaluations of smaxpresented in Adaptive MQAM/OFDM scheme section. For the worst case, the ICI is decided by the largest power. We can obtain the largest power if γ[k]→. A lower bound on the maximum spectral efficiency in adaptive OFDM/MQAM systems is then obtained as

(41)

#### Delay in CSI

We now investigate the performance degradation in power and rate adaptation caused by the delay transmitting CSI from the receiver to the transmitter. We assume perfect CSI. However, the channel estimate at the transmitter due to the transmission delay is expressed as , where τD=ΔnTOFDM is the time delay, in seconds, between the time at which the channel estimation is performed and the actual transmission. It follows from (5) that the correlation between Hk(n)and is

(42)

Based on the assumption that the channel has a WSSUS model, it can be shown that [26]

Then, (42) can be expressed as

(43)

Since Hk and , it follows from (4) that the conditional distribution of Hkgiven is complex Gaussian with mean and variance and , respectively. Similar to the previous derivations, the average BER for the kth subchannel can be determined from (32) with μ=μ2and . The instantaneous BER of each subchannel is

(44)

Going over the same steps starting from (28), as previously done for noisy channel estimation, we can obtain the maximum spectral efficiency as

(45)

where λ2 is a Lagrangian multiplier and its value is determined from the average power constraint.

The evaluation of in (44) is similar to the derivations for the evaluations of smaxpresented in Adaptive MQAM/OFDM scheme section. For the worst case, the ICI is decided by the largest power. We can obtain the largest power if γ[k]→. A lower bound on the maximum spectral efficiency in adaptive OFDM/MQAM systems is then obtained,

(46)

#### Numerical results and discussion

In this section, we examine the performance of the solution for the power and rate adaptation derived in the previous sections. In all simulations presented here, we assume the total bandwidth B is 15.36MHz, the numbers of subcarriers are N=1024, N=512 or N=256, and the corresponding subcarrier bandwidths are Δf=15 KHz, Δf=30 KHz or Δf=60 KHz, fc=2.5 GHz, v=100 and 500 km/h. The BER requirement is 10−3. In Figure 4, the adaptive power control scheme is plotted as a function of the average SNR. The figure shows similar trends with respect to γ[k]for different mobilities. It also indicates that higher speeds yield larger cutoffs and transmit powers as γ[k]→ + mainly due to fact that the ICI power also gets larger. Higher transmit power at higher speeds as γ[k]→ + is also due to the average power constraint.

Figure 4. for MQAM , .

In Figure 5, the bit rate adaptation β(γ[k])is plotted as a function of the average SNR. We conclude from Figure 5 that it is possible to transmit more bits per second as γ[k] increases and that the higher speed yields the larger cutoff. It also shows that the lower speed has a larger bit rate than its higher speed counterpart when γ[k]is fixed. As can be seen in Figure 4, the higher speed case yields a lower bit rate than that for the lower speed case, even with a larger transmit power, because of the larger ICI variance obtained at higher speeds.

Figure 5. β(γ[k])for MQAM (, ).

Figure 6 shows the average spectral efficiency of adaptive OFDM systems as well as of non-adaptive OFDM for different speeds. It also indicates that a higher speed result in lower average spectral efficiency, mainly due to the fact that it induces larger ICI power. As can be seen in Figure 6 the adaptive OFDM scheme can achieve higher gains in spectral efficiency compared with non-adaptive OFDM under different velocities. However, they both experience error floors at high SNR values in the average spectral efficiency. We also observed that the spectral efficiency of the system cannot be improved beyond a certain level; this is mainly due to the contribution of the average SNR to the ICI power. The average spectral efficiencies of truncated integer rate adaptive OFDM are also given in Figure 6. In spite of the fact that its spectral efficiency is close to that of adaptive OFDM, we observe that there is still a gap between them, since the optimal parameters are obtained in the rate-domain.

Figure 6. Average spectral efficiency versus average SNR.

The performance of adaptive OFDM systems with imperfect channel estimation at speeds 100 and 500 km/h are shown in Figures 7 and 8, respectively. The value of is a measure of the quality of the channel estimation technique employed. The smaller the value, the better the channel estimation; indicates that we have perfect CSI. From Figures 7 and 8, we can conclude that as the channel estimation becomes poor, the average spectral efficiency decreases, almost independently from the user mobility. The fact that the effect of noise resulting from imperfect CSI will be dominant compared to the ICI effect. For example, when = −20 dB, the performance is worse than the non-adaptive for some values of SNR, as seen from Figure 7. However, Figure 8 shows that the performance loss is moderate when the channel estimation error is less than −40 dB. For the same channel estimation error, the performance loss incurred for low speed systems are greater than for high speed systems, since the low speed systems rely on more accurate CSI to improve the system performance. Consequently, we conclude that good channel stimulation is very essential in adaptive modulation schemes to improve the average spectral efficiency performance. The results also show that the closed-form expressions for adaptive OFDM systems with imperfect channel estimation is valid.

Figure 7. Average spectral efficiency for adaptive OFDM with imperfect CSI for velocity equals to 100 km/h.

Figure 8. Average spectral efficiency for adaptive OFDM with imperfect CSI for velocity equals to 500 km/h.

There is no impact of delay on the non-adaptive systems. However, for adaptive schemes, an unavoidable delay occurs between the times at which the channel estimation takes place and when the estimation result is employed for actual transmission. It is very important and necessary to investigate this issue, especially for high mobility systems. In Figure 9, we assume N=1024, fc=2.5 GHz, Δf=15 KHz, v=100 and 500 km/h, the delay τD=1/Δfmeans a duration of one OFDM symbol. In this article, we assume that the overhead due to the guard interval is ignored by simply assuming the cyclic prefix (CP) is equal to zero. Figure 9 shows that the higher the velocity, the worse the performance. Because the higher mobility results in a larger variation of the channel within one OFDM symbol duration, the higher mobility results in decreasing channel correlation. We observe from Figure 9 that the adaptive scheme operating with a velocity equal to 100 km/h, it can resist the same delay (one OFDM duration) better than system with a velocity of 500 km/h. Consequently, under the high mobility condition, in order to mitigate the impact of delay, an effective and simple scheme is to reduce the OFDM symbol duration if the maximum path delay is not large.

Figure 9. Average spectral efficiency for adaptive OFDM with delay CSI for different velocities.

Figure 10 shows the average spectral efficiency of an adaptive OFDM system with different delays and with different numbers of subcarriers. The velocity is chosen to be 500 km/h. From this figure we conclude that in order to mitigate the impact of delay, we need to increase the OFDM subcarrier spacing when the total bandwidth B is constant. We assume the CP is ignored. Clearly, for a fixed total bandwidth B, the OFDM symbol duration with N=512 is half of that when N=1024 and consequently, the impact of delay would be reduced when N=512. Figure 10 shows that the OFDM system with N=256is less affected by the delay. Meanwhile, increasing the subcarrier spacing of the OFDM system will also reduce the ICI effect when the velocity is constant. In practice, if the subcarrier spacing of an OFDM system is increased, the average spectral efficiency would be reduced when the maximum path delay is constant. So there is a tradeoff.

Figure 10. Average spectral efficiency for adaptive OFDM with different subcarrier bandwidth for velocity equals to 500 km/h.

### Conclusion

In this article, a power and rate adaptation scheme has been studied to maximize average spectral efficiency of MQAM/OFDM systems in a fast fading channel with perfect and imperfect CSI. Especially when dealing with high mobility and imperfect CSI, the banded structure of the channel matrix in the frequency domain has been taken into account. Under imperfect CSI, the effect of channel estimation errors and the delay in CSI on the spectral efficiency of the system were investigated. In order to exploit the impact of the ICI due to high mobility, new BER expressions were obtained for adaptive modulation in the presence of fast fading channels with perfect and imperfect CSI.

Considering a BER constraint for every subcarrier, a lower bound was derived and closed form expressions were obtained for the average spectral efficiency of adaptive OFDM systems under fast fading channel with perfect and imperfect CSI. We concluded that the spectral efficiency of the proposed adaptive system under fast fading channels tends to have an average floor depending on the ICI and the average SNR. On the other hand, we observed that under imperfect CSI, the proposed adaptive system has an average floor for the spectral efficiency, depending on the imperfection in the CSI. Since very rapidly time varying channels lead to larger ICI, we have shown that the average spectral efficiency was seriously degraded under high mobility.

We have concluded from the theoretical results and the computer simulations that adaptive OFDM systems in fast fading channel have considerable gain over non-adaptive counterparts as a function of the channel estimation errors. Therefore, more accurate CSI has yielded better performance. Otherwise, the performance of the adaptive system would have degraded substantially and become worse than that of a non-adaptive system.

For adaptive OFDM system, high mobility causes inevitable delay between the channel estimation and the actual transmission at the transmitter. One of the important conclusions of this work was that increasing the spacing between the OFDM subcarriers mitigates the impact of delay on the spectral efficiency. Theoretical and numerical results have indicated that the impact of delay on the spectral efficiency of the system would be mitigated by increasing the spacing between the OFDM subcarriers, when the CP is neglected. Consequently, increasing the subcarrier bandwidth would decrease the duration of the OFDM symbol. Consequently, the time selectivity of the channel would be decreased as the frequency selectivity increasing. There exists an optimal subcarrier bandwidth when the CP is constant. Our future work is to find the optimal subcarrier bandwidth under some constraints.

### Appendices

#### Appendix 1: Derivation of (24)

Assume ξ = − C2/ln(ε/C1), , .

For t=γ[k]−γ0, we can express the above integral as follows

By means of the integral formula [29]

We obtain the final result as follows

(47)

Closed form expressions of (40) and (45) can be obtained by a similar approach.

#### Appendix 2: Derivation of (28)

Suppose the estimated channel is the only known information about the current CSI for the kth subcarrier. We will obtain the ICI power based on .

where Es = E{|dm|2}, .

We obtain the instantaneous BER of each subchannel under imperfect CSI as follows

(48)

where

### Competing interests

The authors declare that they have no competing interests.

### Acknowledgements

This work was supported by the National Basic Research Program of China (973 Program No. 2012CB316100), the National Science Foundation of China (NSFC, No. 61032002), and the 111 Project (No. 111-2-14). The authors would like thank the anonymous reviewers of this work whose valuable comments and corrections improved the quality of the article substantially. The authors would also like thank Weixi Zhou, Jing Zhao and Meng Chen for their constructive discussions and suggestions that very much improved this article.

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