Power Adaptation in OFDM Systems Based on Velocity ... - IEEE Xplore

IEEE COMMUNICATIONS LETTERS, VOL. 19, NO. 4, APRIL 2015

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Power Adaptation in OFDM Systems Based on Velocity Variation Under Rapidly Time-Varying Channels Zhicheng Dong, Pingzhi Fan, and Xianfu Lei

Abstract—A novel power adaptation scheme, based on the terminal velocity instead of channel state information (CSI), in OFDM systems over rapidly time-varying channels, is proposed. Simulation results show that the proposed power adaptation scheme based on the velocity variation is effective for improving the performance of OFDM systems in fast fading channels. Index Terms—Average spectral efficiency (ASE), intercarrier interference (ICI), orthogonal frequency division multiplexing (OFDM), rapidly time-varying channels.

I. I NTRODUCTION

B

ECAUSE of its remarkable ability to mitigate the intersymbol interference (ISI) brought on by frequency selective fading channels, OFDM has received much attention and become as a leading transmission technique [1]. However, OFDM systems are sensitive to the inter-carrier interference (ICI) due to the time-selective fading. In high speed train applications, the performance of OFDM systems will be significantly deteriorated due to the loss of orthogonality among subchannels under rapidly time-varying channels [2]–[5]. Adaptive transmission techniques can be used to improve the performance of OFDM systems because of its ability to counteract the impact of ICI. A lot of research has shown that adjusting the system parameters (transmit power, data rate, subcarrier bandwidth, etc.) based on channel state information (CSI) is beneficial [2]–[7]. However, because of the channel estimation error or feedback delay, it is not always possible to have accurate instantaneous CSI at the transmitter [2], [6], [8]. Previous studies have indicated that conventional adaptive schemes based on CSI may not be able to improve the system performance in rapidly time-varying channels [2], [6]. Especially for the high speed trains, the velocity can be up to 500 km/h [9]. Therefore, it may be unfeasible to apply conventional adaptive transmission techniques in such rapidly time-varying environments.

Manuscript received July 25, 2014; revised January 13, 2015; accepted January 16, 2015. Date of publication January 23, 2015; date of current version April 8, 2015. This work was supported by the National Basic Research Program of China (973 Program No. 2012CB316100), the National Science Foundation of China (NSFC, No. 61471302), the 111 Project (No. 111-2-14), the Fundamental Research Funds for the Central Universities (No. SWJTU12ZT02/2682014ZT11), and the 2013 Doctoral Innovation Funds of Southwest Jiaotong University and the Fundamental Research Funds for the Central Universities. The associate editor coordinating the review of this paper and approving it for publication was K. J. Kim. Z. Dong is with Key Laboratory of Information Coding & Transmission, Southwest Jiaotong University, Chengdu 610031, China. Zhicheng Dong is also with the School of Engineering, Tibet University, Lhasa 850000, China (e-mail: [email protected]). P. Fan and X. Lei are with Key Laboratory of Information Coding & Transmission, Southwest Jiaotong University, Chengdu 610031, China (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/LCOMM.2015.2396070

In this letter, we propose a novel power adaptation scheme for OFDM systems over very fast fading channels, which adjusts transmit power based on the terminal velocity1 instead of CSI at the transmitter.2 Simulation results show that the proposed power adaptation scheme is effective for OFDM systems over rapidly time-varying channels, compared with the non-adaptive scheme. The rest of the letter is organized as follows. The system model is introduced in Section II. The new power adaptation scheme will be presented in Section III. Some representative simulation results and discussions are provided in Section IV. Finally, Section V concludes this work. II. S YSTEM M ODEL In this paper, downlink transmission is considered where the receiver is on the high speed train and the transmitter is ground base station. Both the channel fading and the velocity are assumed to be time-varying with certain distributions. Specifically, the velocity is assumed to be constant during a data block, but to change from block to block following the truncated normal distribution [11]. Each data block consists of a large number of OFDM symbols which undergo very fast channel fading [12], [13]. The transmitter (base station) gets information about the train velocity3 to achieve power adaptation. Notation v is the instantaneous terminal speed (km/h) in a data block. B denotes the total system bandwidth for an OFDM system. K denotes the total number of subcarriers for an OFDM system. L denotes the number of multipaths of time-varying channel. c is the speed of light. LCP is the number of cyclic prefix (CP) on an OFDM symbol. fd is the system carrier frequency for an OFDM system. d(n) denotes the transmission data for the nth subcarrier. S is the average transmit power. w(m) is complex additive white Gaussian noise (AWGN) with mean zero and variance σ2w . h(m, l) is the time-varying channel gain of the lth multipath at time m. 1 The velocity can serves as an indirect metric of CSI, which is much easier to obtain. Besides, in our proposed scheme, we also need to know the complete knowledge of the distribution of the velocity. 2 The proposed scheme is a kind of mobility adaptation [10]. 3 The velocity can be estimated using different techniques, such as [14], [15]. The velocity information is fed back through dedicated feedback channel [9] to the transmitter (base station) for making handover or power adaptation.

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denotes the Doppler frequency, defined as fmax = fd v/c. TOFDM is the OFDM system period, defined as TOFDM = K/B + LCP /B. In the frequency domain, the signal of nth subcarrier for OFDM system can be expressed as [13] 1 K−1 L−1 − j2πmn/K Y (n) = √ ∑ ∑ h(m, l)x(m−l)+w(m) e K m=0 l=0 = d(n)H(n) + I(n) +W (n), (1) fmax

where

I(n) K−1

1/K

∑

k=0,k=n

is K−1

d(k) ∑ Hk

the

ICI,

defined

(m)e j2πm(k−n)/K ,

as

I(n) =

H(n)

=

m=0

(1/K) ∑K−1 m=0 Hn (m), x(m) =

K−1 √1 ∑ d(n)e j2πmn/K , K n=0

−LCP ≤

m ≤ K − 1. We assume that L is less than the number of cyclic prefixes LCP ; Hn (m) is the frequency response of the channel at time m, defined as Hn (m) = ∑L−1 l=0 h(m, l) exp(− j2πln/K) K−1 1 √ and W (n)= K ∑m=0 w(m) exp(−j2πmn/K). The autocorrelation of the frequency response of the channel is E{Hn (m)Hn∗ (m )} = J0 (2π fmax TOFDM (m − m )/K) where J0 denotes the zerothorder Bessel function of the first kind [2], [13]. According to [2], [13], the ICI power, which is independent of the subchannel index, can be given as [2], [13], (2) PICI = E |I(n)|2 = PN γ, 0 ≤ n ≤ K − 1. where γ = S/σ2w , PN is the normalized ICI power defined as α1 (2π fd vTOFDM )2 with α1 = 1/2 [16]. In the next section, 12c2 we will examine the power adaptation issue. III. P OWER A DAPTATION BASED ON V ELOCITY For the nth subcarrier, the signal to interference plus noise ratio (SINR) with power allocation SPv , can be presented as SINR(n) =

S |H(n)|2 Pv γ |H(n)|2 Pv , = PN γPv + 1 PN SPv + σ2w

(3)

where Pv is the normalized transmit power for a speed v, α1 2 PN = 12c 2 (2π f d vTOFDM ) . In the following, we will obtain the optimal value of Pv so as to optimize the overall spectral efficiency. Based on (3), the instantaneous bit error rate (BER) can then be approximated as [2]–[8] −C2 γγPv Ω ≈ C1 exp , (4) (2Rv − 1)(PN γPv + 1) where C1 = 0.2, C2 = 1.6, γ = |H(n)|2 , and Rv is the data rate in each subcarrier. In a practical traffic system, the velocity of a vehicle is timevarying. A truncated normal distribution N(µ, σ2v ) was proposed in [11] to model the velocity variation, where µ and σ2v denote the mean and the variance respectively. The probability density function (pdf) of the velocity, f (v), can be given as [11] 2 2 exp − (v−µ) 2σ2v √ √ , (5) f (v) = 2(vmax −µ) 2(vmin −µ) σv (2π) erf − erf 2σv 2σv

√ where erf(.) is the error function defined as erf(x) = 2/ 2π 0x exp(−t 2 )dt, vmin = 0 and vmax = 500 km/h. Since lim f (v) = σ2v →+∞

the pdf approaches to a uniform distribution as σ2v becomes large. Assume that each subcarrier of the OFDM system transmits M-ary QAM symbols. The data rate on each subcarrier is Rv = log2 M. During each data block (the fading is very fast and ergodic), the average BER across the pdf of the channel gain can be expressed as [8] ∞ −C2 γγPv Ω= C1 exp pγ (γ)dγ (2Rv − 1)(PN γPv + 1) 0 C2 ργPv =C1 +1 , (6) (2Rv − 1) (PN γPv + 1) 1 vmax −vmin ,

where pγ (γ) = (1/ρ) exp(−γ/ρ) and ρ = 1 − PN [2], [13], [16]. Assuming the target average BER is Ω = BER, the optimal rate at a specific velocity v (the data rate is constant in the data block with this velocity regardless of the channel variation), can be derived from (6) as C2 ργPv +1 . (7) Rv = log2 (C1 /BER − 1)(PN γPv + 1) The average spectral efficiency (ASE) over all realizations of the velocity can be therefore obtained as vmax C2 ργPv +1 f (v)dv. (8) CASE =C6 log2 (C1 /BER−1)(PN γPv +1) vmin

α1 K K 2 where C6 = BTOFDM = K+L , PN = 12c 2 (2π f d vTOFDM ) in (3) CP and ρ = 1 − PN in (6). To maximize the ASE, the constrained optimization problem is formulated as K max Pv K + LCP vmax C2 ργPv + 1 f (v)dv (9a) × log2 (C1 /BER − 1)(PN γPv + 1) vmin

subject to

vmax vmin

SPv f (v)dv = S

(9b)

Pv ≥ 0

(9c)

We have proved that (9a) is concave with respect to Pv , as shown in the Appendix. It is straightforward to show that (9b) and (9c) are convex with respect to Pv . Therefore, the optimal solution can be obtained by the KKT conditions. The Lagrangian for the problem can be expressed as vmax C2 ργPv +1 log2 La {Pv } = (C1 /BER − 1)(PN γPv + 1) vmin vmax × f (v)dv − λ Pv f (v)dv − 1 , (10) vmin

where λ denotes the Lagrangian multiplier. It can be found through a numerical search based on power constraint. Solving ∂La {Pv } = 0, the optimal normalized power adaptation based on ∂Pv velocity can be expressed as C3 + ζ Pv = (11) C4

DONG et al.: POWER ADAPTATION IN OFDM SYSTEMS BASED ON VELOCITY VARIATION

Fig. 1.

691

The pdf of the velocity f (v). Fig. 2. The normalized power distribution vs. velocity (SNR = 30 dB).

where C3 = 2 ln(2)BERPN − ln(2)C2 ρBER − 2 ln(2)C1 PN , C4 = 2 2 ln(2)γPN (C2 ρBER+PN C1 −PN BER), ζ = ln(2)2 ρ2 BER C22 + 2 4 ln(2)PN C22 ρ2 BER γ/λ + 4 ln(2)PN2 C2 ργBER(C1 − BER)/λ. It can be observed that √when Pv ≥ 0, there exists a cutoff velocity c

6C BERγ(BERC γ+λ ln(2)BER−λ ln(2)C )

2 2 1 vo f f = min{vmax , }, above C2 BERγTOFDM π fd which no data is transmitted. The corresponding optimal rate adaptation according to the velocity is obtained as

C5C3 +C5 ζ+PN γC3 +PN γ ζ+C4 , (12) Rv = max 0, log2 PN γC3 +PN γ ζ+C4

C2 ργ where C5 = (C /BER−1) . When Pv = 1, the power adaptation 1 becomes the non-adaptive scheme [8].

IV. S IMULATION R ESULTS We consider an OFDM system with system bandwidth B = 5 MHz and carrier frequency fd = 2.5 GHz. The number of subcarriers K is 512. The target BER is 10−3 . An exponential power delay profile with maximum excess delay of 2 µs is applied. The pdfs of the velocity for different parameters are presented in Fig. 1. We can see that the distribution of the velocity is significantly different for different parameters. In Fig. 2, the normalized power allocation Pv is plotted as a function of the velocity. It can be seen that there exists a cutoff velocity above which no power will be allocated at the transmitter. The low velocity region plays an important role in determining the overall system performance since the ICI in this region is low. From the figure, one can see that both the mean and the variance have significant influence on the power allocation. More specifically, for a fixed variance, it is obvious that more power should allocated to the low velocity region as the mean increases since the the system has less chance to work in this region. For a fixed mean, the effect of the variance on the power allocation can be illustrated as the following two cases: 1) When the mean is small, larger variance results in higher allocated power because the system has less chance to work in the low velocity region. 2) When the mean is large, larger

Fig. 3. The corresponding rate distribution vs. velocity (SNR = 30 dB).

variance results in lower allocated power because the system has more chance to work in the low velocity region. We can also see that the power distribution is close to that of uniform distribution (σ2v = ∞) as the velocity variance increases. In Fig. 3, the corresponding rate adaptation Rv is illustrated as a function of the velocity. It is shown that the data rate is lower as the velocity increases. Again, the low velocity region determines the overall system performance. When the mean is larger, the data rate is higher in the low velocity region. The reason is that the probability that the system works in the low velocity region is smaller so that the system needs to use more power to fully utilize this region. The ASE vs. σ2v for different means of the velocity is shown in Fig. 4. We can see that the ASE decreases with µ increases as expected. It is also shown that the proposed power adaptation scheme outperforms the non-adaptive one, especially when σ2v is large. This is because that larger σ2v leads to larger velocity variation. Also, the ASE becomes close to each other for

692


A PPENDIX P ROOF OF THE C ONCAVITY OF (9a) C2 ργPv K K+LCP log2 [ (C1 /BER−1)(PN γPv +1) + 1]. We Kψ v (Pv ) can obtain that ∂C∂P = ((K+L )(ηPv +1)(ψP > v v +χψPv +χ) ln(2)) CP 2 2 ∂ Cv (Pv ) Kψ(2ψPv η+2χη Pv +2ηχ+ψ) = − ((K+L )(ηP +1)2 (ψP +χηP +χ)2 ln(2)) < 0. 0 and ∂Pv2 v v v CP

Assume Cv (Pv ) =

Where ψ = C2 ργBER, η = PN γ and χ = C1 − BER. Therefore, Cv (Pv ) is concave and it can be shown that vmax C2 ργPv K K+LCP vmin log2 [ (C1 /BER−1)(PN γPv +1) + 1] f (v)dv is concave with respect to Pv [17]. ACKNOWLEDGMENT

Fig. 4. ASE vs. σ2v for different speeds (SNR = 30 dB).

The authors would like to thank Prof. Norm Beaulieu, Prof. Erdal Panayirci and the anonymous reviewers for their comments and suggestions, which helped improving the quality of the manuscript. R EFERENCES

Fig. 5. ASE vs. SNR.

different values of µ when σ2v is very large, since the pdf of the velocity approaches to the uniform distribution. In Fig. 5, the ASE vs. different SNR for different µ and σ2v is illustrated. The figure shows that the proposed scheme achieves better performance than the non-adaptive scheme, especially when SNR and σ2v are large. The performance of adaptive scheme is improved by 10% when the variance increases to 200 km/h for large SNR. As the variance increases, the performance is close to the performance under uniform distribution (σ2v = ∞). V. C ONCLUSION In this letter, a power adaptation method based on the velocity was proposed. It has been shown that there is a cutoff velocity above which on power will be allocated. The proposed scheme will be useful for wireless communication systems over very rapidly fast fading channels where it is not feasible to obtain the accurate CSI at the transmitter. The proposed scheme shows significant performance gain compared to the non-adaptive scheme, especially when SNR and σ2v are large.

[1] R. Prasad, OFDM for Wireless Communications Systems. Norwood, MA, USA: Artech House, 2004. [2] Z. Dong, P. Fan, E. Panayirci, and P. Mathiopoulos, “Effect of power and rate adaptation on the spectral efficiency of mqam/ofdm system under very fast fading channels,” EURASIP J. Wireless Commun. Netw., vol. 2012, no. 1, p. 208, Jul. 2012. [3] Z. Dong, P. Fan, E. Panayirci, and X. Lei, “Conditional power and rate adaptation for mqam/ofdm systems under cfo with perfect and imperfect channel estimation errors,” IEEE Trans. Veh. Technol., vol. 63, Dec. 2014, DOI: 10.1109/TVT.2014.2377153, to be published. [4] S. Das, E. De Carvalho, and R. Prasad, “Performance analysis of ofdm systems with adaptive sub carrier bandwidth,” IEEE Trans. Wireless Commun., vol. 7, no. 4, pp. 1117–1122, Apr. 2008. [5] J. Wen, C. Chiang, T. Hsu, and H. Hung, “Resource management techniques for ofdm systems with the presence of inter-carrier interference,” Wireless Personal Commun., vol. 65, no. 3, pp. 515–535, Aug. 2012. [6] A. Goldsmith and S. Chua, “Variable-rate variable-power mqam for fading channels,” IEEE Trans. Commun., vol. 45, no. 10, pp. 1218–1230, Oct. 1997. [7] S. Chung and A. Goldsmith, “Degrees of freedom in adaptive modulation: A unified view,” IEEE Trans. Commun., vol. 49, no. 9, pp. 1561–1571, Sep. 2001. [8] S. Ye, R. Blum, and L. Cimini, “Adaptive OFDM systems with imperfect channel state information,” IEEE Trans. Wireless Commun., vol. 5, no. 11, pp. 3255–3265, Nov. 2006. [9] O. B. Karimi, J. Liu, and C. Wang, “Seamless wireless connectivity for multimedia services in high speed trains,” IEEE J. Sel. Areas Commun., vol. 30, no. 4, pp. 729–739, May 2012. [10] Z. Dong, P. Fan, and X. Lei, “Mobility adaptation in ofdm systems over rapidly time-varying fading channels,” in Proc. IEEE Int. Commun. Syst., Nov. 2014, pp. 4042–4056. [11] S. Yousefi, E. Altman, R. El-Azouzi, and M. Fathy, “Analytical model for connectivity in vehicular Ad Hoc networks,” IEEE Trans. Veh. Technol., vol. 57, no. 6, pp. 3341–3356, Nov. 2008. [12] E. Panayirci, H. Senol, and H. Poor, “Joint channel estimation, equalization, data detection for ofdm systems in the presence of very high mobility,” IEEE Trans. Signal Process., vol. 58, no. 8, pp. 4225–4238, Aug. 2010. [13] Y. Choi, P. Voltz, and F. Cassara, “On channel estimation and detection for multicarrier signals in fast and selective rayleigh fading channels,” IEEE Trans. Commun., vol. 49, no. 8, pp. 1375–1387, Aug. 2001. [14] S. Mohanty, “Vepsd: A novel velocity estimation algorithm for nextgeneration wireless systems,” IEEE Trans. Wireless Commun., vol. 4, no. 6, pp. 2655–2660, Nov. 2005. [15] W. Lee and D.-H. Cho, “Mean velocity estimation of mobile stations by spatial correlation of channels in cellular systems,” IEEE Commun. Lett., vol. 13, no. 9, pp. 670–672, Sep. 2009. [16] Y. Li and L. Cimini Jr, “Bounds on the interchannel interference of ofdm in time-varying impairments,” IEEE Trans. Commun., vol. 49, no. 3, pp. 401–404, Mar. 2001. [17] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004.