IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 62, NO. 9, NOVEMBER 2013 4701

where

Mγeq(s)=

N∑l=1

(N

l

)P lλP

N−lλ l

l−1∑m=0

(−1)m(l−1m

)m+1

Am

s+Am

(26)

MγSD(s) =

∞∫0

e−sγfγSD(γ)dγ =

1/γSD

s+ 1/γSD

(27)

with fγSD(γ) = (1/γSD)e−(γ/γSD).

By performing inverse Laplace transform on Mγtotal(s), the joint

pdf of γtotal and l can be derived as

fγtotal,l �=0(γ, l �= 0) =N∑l=1

(N

l

)P lλP

N−lλ l

l−1∑m=0

(−1)m(l−1m

)m+ 1

× Am

AmγSD − 1{e−γ/γSD − e−Amγ}. (28)

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers for theircritical comments, which greatly improved this paper.

REFERENCES

[1] Z. Zhang, W. Zhang, and C. Tellambura, “OFDMA uplink frequency off-set estimation via cooperative relaying,” IEEE Trans. Wireless Commun.,vol. 8, no. 9, pp. 4450–4456, Sep. 2009.

[2] R. U. Nabar, H. Bolcskei, and F. W. Kneubuhler, “Fading relay channels:Performance limits and space-time signal design,” IEEE J. Sel. AreasCommun., vol. 22, no. 6, pp. 1099–1109, Aug. 2004.

[3] W. Zhuang and M. Ismail, “Cooperation in wireless communication net-works,” IEEE Wireless Commun., vol. 19, no. 2, pp. 10–20, Apr. 2012.

[4] Z. Zhang, W. Zhang, and C. Tellambura, “Cooperative OFDM channel es-timation in the presence of frequency offsets,” IEEE Trans. Veh. Technol.,vol. 58, no. 7, pp. 3447–3459, Sep. 2009.

[5] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity inwireless networks: Efficient protocols and outage behavior,” IEEE Trans.Inf. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004.

[6] P. L. Yeoh, M. Elkashlan, Z. Chen, and I. B. Collings, “SER of mul-tiple amplify-and-forward relays with selection diversity,” IEEE Trans.Commun., vol. 59, no. 8, pp. 2078–2083, Aug. 2011.

[7] Z. Zhang, C. Tellambura, and R. Schober, “Improved OFDMA up-link transmission via cooperative relaying in the presence of fre-quency offsets—Part I: Ergodic information rate analysis,” Eur. Trans.Telecommun., vol. 21, no. 3, pp. 224–240, Apr. 2010.

[8] M. Torabi, D. Haccoun, and J.-F. Frigon, “Impact of outdated relay selec-tion on the capacity of AF opportunistic relaying systems with adaptivetransmission over non-identically distributed links,” IEEE Trans. WirelessCommun., vol. 10, no. 11, pp. 3626–3631, Nov. 2011.

[9] K. B. Fredj and S. Aïssa, “Performance of amplify-and-forward systemswith partial relay selection under spectrum-sharing constraints,” IEEETrans. Wireless Commun., vol. 11, no. 2, pp. 500–504, Feb. 2012.

[10] D. S. Michalopoulos, H. A. Suraweera, G. K. Karagiannidis, andR. Schober, “Amplify-and-forward relay selection with outdated channelestimates,” IEEE Trans. Commun., vol. 60, no. 5, pp. 1278–1290,May 2012.

[11] D. Li and A. W. Long, “Outage probability of cognitive radio networkswith relay selection,” IET Commun., vol. 5, no. 18, pp. 2730–2735,Dec. 2011.

[12] Z. Zhang, K. Long, and J. Wang, “Self-organization paradigms and op-timization approaches for cognitive radio technologies: A survey,” IEEEWireless Commun., vol. 20, no. 2, pp. 36–42, Apr. 2013.

[13] Z. Zhang, K. Long, J. Wang, and F. Dressler, “On swarm intelligenceinspired self-organized networking: Its bionic mechanisms, designingprinciples and optimization approaches,” IEEE Commun. Surveys Tuts.,to appear.

[14] S. I. Hussian, M.-S. Alouini, M. Hasna, and K. Qarage, “Partial relayselection in underlay cognitive networks with fixed gain relays,” in Proc.IEEE VTC, May 2012, pp. 1–5.

[15] Z. Zhang, C. Tellambura, and R. Schober, “Improved OFDMA up-link transmission via cooperative relaying in the presence of frequencyoffsets—Part II: Outage information rate analysis,” Eur. Trans. Telecom-mun., vol. 21, no. 3, pp. 241–250, Apr. 2010.

[16] M. O. Hasna and M.-S. Alouini, “A performance study of dual-hop trans-missions with fixe gain relays,” IEEE Trans. Wireless Commun., vol. 3,no. 6, pp. 1963–1968, Nov. 2004.

[17] J. G. Proakis and M. Salehi, Digital Communications. New York, NY,USA: McGraw-Hill, 2008.

[18] Y. Zhao, R. Adve, and T. J. Lim, “Symbol error rate of selectionamplify-and-forward relay systems,” IEEE Commun. Lett., vol. 10, no. 11,pp. 757–759, Nov. 2006.

[19] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, andProducts, 7th ed. New York, NY, USA: Academic, 2007.

A Differential Feedback Scheme Exploiting theTemporal and Spectral Correlation

Mingxin Zhou, Leiming Zhang, Member, IEEE,Lingyang Song, Senior Member, IEEE, andMerouane Debbah, Senior Member, IEEE

Abstract—Channel state information (CSI) provided by a limited feed-back channel can be utilized to increase system throughput. However, inmultiple-input–multiple-output (MIMO) systems, the signaling overheadrealizing this CSI feedback can be quite large, whereas the capacity ofthe uplink feedback channel is typically limited. Hence, it is crucial toreduce the amount of feedback bits. Prior work on limited feedbackcompression commonly adopted the block-fading channel model, whereonly temporal or spectral correlation in a wireless channel is considered.In this paper, we propose a differential feedback scheme with full use ofthe temporal and spectral correlations to reduce the feedback load. Then,the minimal differential feedback rate over a MIMO time–frequency (ordoubly)-selective fading channel is investigated. Finally, the analysis isverified by simulation results.

Index Terms—Correlation, differential feedback, multiple-inputmultiple-output (MIMO).

I. INTRODUCTION

In multiple-input–multiple-output (MIMO) systems, channel adap-tive techniques (e.g., water-filling, interference alignment, beamform-ing, etc.) can enhance the spectral efficiency or the capacity of the

Manuscript received November 25, 2012; revised March 18, 2013; ac-cepted May 3, 2013. Date of publication June 5, 2013; date of currentversion November 6, 2013. This work was supported in part by the National973 project under Grant 2013CB336700, by the National Natural ScienceFoundation of China under Grant 61222104 and Grant 61061130561, by thePh.D. Programs Foundation of the Ministry of Education of China underGrant 20110001110102, and by the Opening Project of the Key Laboratory ofCognitive Radio and Information Processing (Guilin University of ElectronicTechnology). The review of this paper was coordinated by Prof. X. Wang.

M. Zhou and L. Song are with the State Key Laboratory of Advanced OpticalCommunication Systems and Networks, School of Electronics Engineeringand Computer Science, Peking University, Beijing 100871, China (e-mail:mingxin.zhou@pku.edu.cn; lingyang.song@pku.edu.cn).

L. Zhang is with Huawei Technologies Company Ltd., Beijing 100095,China (e-mail: leiming.zhang@pku.edu.cn).

M. Debbah is with SUPELEC, Alcatel-Lucent Chair in Flexible Radio,91192 Gif-Sur-Yvette, France (e-mail: merouane.debbah@supelec.fr).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2013.2266379

0018-9545 © 2013 IEEE

4702 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 62, NO. 9, NOVEMBER 2013

system. However, these channel adaptive techniques require accuratechannel conditions, which are often referred to channel state informa-tion (CSI). Oftentimes, in a frequency-division-duplex setting, CSI isestimated at the receiver and conveyed to the transmitter via a feedbackchannel. In recent years, CSI feedback problems have been intensivelystudied, due to its potential benefits to the MIMO systems [1], [2]. Itis significant to explore how to reduce the feedback load, due to theuplink feedback channel limitation.

In [3], four feedback rate reduction approaches were reviewed,where lossy compression using the properties of the fading processwas considered best. When the wireless channel experiences temporal-correlated fading, which is modeled as a finite-state Markov chain,the amount of CSI feedback bits can be reduced by ignoring thestates occurring with small probabilities [4]–[8]. The feedback ratein frequency-selective fading channels was studied in [9] and [10] byexploiting the frequency correlation.

In summary, all the given works mainly focus on feedback ratecompression considering either temporal correlation or spectral corre-lation. However, doubly selective fading channels are more frequentlyencountered in wireless communications as the desired data rate andmobility simultaneously grow. To the best of the authors’ knowledge,the scheme of making full use of the 2-D correlations has yet to bewell studied. Using both of the orthogonal dimensional correlationsin a cooperated way, the feedback overhead can be further reducedin the doubly selective fading channels. Thus, in this paper, wederive the minimal feedback rate using both the temporal and spectralcorrelations.

The main contributions of this paper can be briefly summarized asfollows: 1) We discuss the minimal feedback rate without differentialfeedback; 2) we propose a differential feedback scheme by exploitingthe temporal and spectral correlations; and 3) we derive the minimaldifferential feedback rate expression over a MIMO doubly selectivefading channel.

The rest of this paper is organized as follows: In Section II, wedescribe the differential feedback model and the statistics of the doublyselective fading channel. In Section III, we propose a differentialfeedback scheme by exploiting the 2-D correlations and derive theminimal feedback rate. In Section IV, we provide some simulationresults showing the performance of the proposed scheme.

II. SYSTEM MODEL

In this paper, we assume that the downlink channel is a mobilewireless channel, which is always correlated in time and frequencydomains, whereas the uplink channel is a limited feedback channel.

A. Statistics of the Downlink Channel

Since the channel corresponding to each antenna is independent andwith the same statistics, we can describe the separation property ofchannel frequency response H(t, f) at time t for an arbitrary transmitand receive antenna pair [11], i.e.,

rH(Δt,Δf) =E {H(t+Δt, f +Δf)H∗(t, f)}

=σ2Hrt(Δt)rf (Δf) (1)

where E{·} denotes the expectation function, and superscript (·)∗denotes the complex conjugate. σ2

H is the power of the channelfrequency response. rt(Δt) and rf (Δf) denotes the temporal andspectral correlation functions, respectively.

Assuming that the channel frequency response stays constant withinsymbol period ts and subchannel spacing fs, the correlation function

Fig. 1. System model of the differential feedback over MIMO doubly selec-tive fading channels.

for different periods and subchannels is written as

rH [Δm,Δn] = σ2Hrt[Δm]rf [Δn] (2)

where rt[Δm] = rt(Δmts), and rf [Δn] = r(Δnfs).Furthermore, if we just consider the time domain, the correlated

channel can be modeled as a time-domain first-order autoregressiveprocess (AR1) [4], i.e.,

Hm,n = αtHm−1,n +√

1 − α2tWt (3)

where Hm,n denotes the channel coefficient of the mth symbol inter-val and the nth subchannel, and Wt is a complex white noise variable,which is independent of Hm−1,n, with variance σ2

H . Parameter αt isthe time autocorrelation coefficient, which is given by the zero-orderBessel function of the first kind αt = rt[1] = J0(2πfdts), where fdis the Doppler frequency [12].

Similarly, if we just consider the frequency domain, the correlatedchannel can also be represented as a frequency-domain AR1 [9], i.e.,

Hm,n = αfHm,n−1 +√

1 − α2fWf (4)

where Wf is a complex white noise variable, which is independentof Hm,n−1, with variance σ2

H . Parameter αf determines the cor-relation between the subchannels, which is given by αf = rf [1] =

1/(√

1 + (2πfsΔ)2), where Δ is the root-mean-square delayspread [12].

B. Differential Feedback Model

The system model with differential feedback is shown in Fig. 1. Byusing the differential feedback scheme, the receiver just feeds back thedifferential CSI.

We suppose that there are Nt and Nr antennas at the transmitter andthe receiver, respectively. The received signal vector at the mth symbolinterval and the nth subchannel is given by

ym,n = Hm,nxm,n + nm,n. (5)

In the given expression, ym,n denotes the Nr × 1 received vectorat the mth symbol interval and the nth subchannel. Hm,n, whichis an Nr ×Nt channel fading matrix, is the frequency response ofthe channel. The entries are assumed independent and identicallydistributed (i.i.d.), obeying a complex Gaussian distribution with zeromean and variance σ2

H . Different antennas have the same characteristicin temporal and spectral correlations, i.e., αt and αf , respectively.

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 62, NO. 9, NOVEMBER 2013 4703

Moreover, there is no spatial correlation between different antennas.xm,n denotes the Nt × 1 transmitter signal vector and is assumed tohave unit variance. nm,n is an Nr × 1 additive white Gaussian noisevector with zero mean and variance σ2

0 . Both xm,n and nm,n areindependent for different m’s and n’s.

Through CSI quantization, the feedback channel output is written as[13]–[15]

Hm,n = Hm,n +Em,n (6)

where Hm,n denotes the channel quantization matrix, and Em,n isthe independent additive quantization distortion matrix whose entriesare zero mean and variance D/NrNt, where D represents the channelquantization distortion constraint.

The differential feedback is under consideration, as shown in Fig. 1.We can use the previous CSI to forecast the present CSI Hm,n at thetransmitter, i.e.,

Hm,n = a1Hm−1,n + a2Hm,n−1 (7)

where a1 and a2 are the coefficients of the channel predictor, whichwill be calculated by using the minimum-mean-square-error (MMSE)principle in the following section. Meanwhile, the receiver calculatesthe differential CSI, given the previous CSI. The differential CSI canbe formulated as

Hd = Diff (Hm,n|Hm−1,n,Hm,n−1) (8)

where Hd represents the differential CSI, which is obviously the pre-diction error, and Diff(·) is the differential function. Then, throughthe limited feedback channel, Hd should be quantized and fed back.

Finally, the CSI reconstructed by combining the differential CSI andthe channel prediction is utilized by the channel adaptive techniques. Inthis paper, we adopt the water-filling precoder; however, the analysisand conclusions given in this paper are also valid for other adaptivetechniques.

The channel quantization matrix is decomposed as Hm,n =UΣV+ using singular value decomposition at the transmitter. Uand V are unitary matrices, and Σ is a nonnegative diagonal matrixcomposed of eigenvalues of Hm,n.

With the water-filling precoder, the closed-loop capacity can beobtained as [13]–[15]

Cerg = E

[log det

(INr + J · J+(F−1)

)](9)

where J = Hm,nVZ, Je = Em,nVZ, and F = (1/A2)INr +E[JeJ

+e |J], in which A represents the amplitude of the signal sym-

bol, and Z denotes a diagonal matrix determined by water-filling[13]–[15], i.e.,

⎧⎪⎪⎨⎪⎪⎩

z2i =

{μ−

(γ2i,iA

2)−1

, γ2i,iA

2 ≥ μ−1

0, otherwiseNt∑i=1

z2i A2 = NtA

2, power constraint

(10)

where γi,i, i = 1, 2, . . . , Nt are the entries of Σ, and μ denotes acutoff value chosen to meet the power constraint.

It is obvious from (9) that the closed-loop ergodic capacity isdetermined by Hm,n and Hm,n, and the loss of capacity is mainlycaused by the quantization error. Therefore, given the limited feedbackchannel, the capacity can be enhanced by exploiting the channelcorrelations to reduce the quantization error.

III. MINIMAL DIFFERENTIAL FEEDBACK RATE

Here, exploiting the temporal and spectral correlations, we study theminimal feedback rate that denotes the minimal feedback bits requiredper block to preserve the given channel quantization distortion.

We first describe the feedback rate using normal quantization.Without the differential feedback scheme, the receiver feeds backHm,n to the transmitter. The information entropy of Gaussian variableX with variance σ2 is represented as [16]

h(X) =12log 2πeσ2. (11)

Thus, the feedback load has a positive relation with σ2H .

Furthermore, taking the quantization of the channel matrix intoconsideration, the feedback rate is determined by the rate distortiontheory of continuous-amplitude sources [16], i.e.,

R = inf{I(Hm,n; Hm,n) : E

[d(Hm,n; Hm,n)

]≤ D

}(12)

where inf{·} denotes the infimum function, I(Hm,n; Hm,n) de-notes the mutual information between Hm,n and Hm,n, andd(Hm,n; Hm,n) = ‖Hm,n − Hm,n‖2 denotes the channel quantiza-tion distortion, which is constrained by D.

Since the entries of H and H are i.i.d. complex Gaussian variables,the feedback rate can be written as

R = inf{NtNrI(Hm,n; Hm,n) : E

[d(Hm,n, Hm,n)

]≤ d

}(13)

where d = D/(NtNr) is the 1-D average channel quantizationdistortion. Hm,n and Hm,n represent the entries of Hm,n andHm,nrespectively. In addition, from (6), the 1-D channel quantizationis written as

Hm,n = Hm,n +Em,n. (14)

The mutual information can be written as

I(Hm,n; Hm,n) = h(Hm,n)− h(Hm,n|Hm,n, ). (15)

Combining (14), (15) can be rewritten as

I(Hm,n; Hm,n) ≥ h(Hm,n)− h(Em,n). (16)

Substituting (11) and (16) into (13), we obtain

R = NrNt log

(σ2H

d

). (17)

From (17), the feedback rate required for the nondifferential feed-back is very large. Nevertheless, by employing the temporal andspectral correlations, we can use the differential feedback scheme toreduce the feedback bits significantly. The transmitter can predict thepresent CSI Hm,n depending on the previous ones in time domainHm−1,n and frequency domain Hm−1,n. Then, the receiver quantizesHd or, equivalently, the error of the channel prediction, and feedsback to the transmitter. Finally, the transmitter reconstructs the CSIby both the channel prediction and the differential CSI. It is obviousthat the more accurate the channel is predicted, the fewer bits thatwill be fed back from the receiver. As Hm−1,n, Hm,n−1, and Hm,n

are correlated, an MMSE channel predictor can be constructed as (7),where the coefficients a1 and a2 are selected to minimize

MSE(a1, a2) = E|Hm,n −Hm,n|2. (18)

The mean square error (MSE) represents the statistical differencebetween the predicted value and the true value. We can obtain theminimized quantization bits by minimizing the MSE.

4704 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 62, NO. 9, NOVEMBER 2013

We can rewrite Hm,n as

Hm,n = Hm,n +Hd = a1Hm−1,n + a2Hm,n−1 +Hd (19)

where Hd is the differential feedback load to minimize. By theorthogonality principle [17], a1 and a2 are determined by{

E [(Hm,n − a1Hm−1,n − a2Hm,n−1)Hm−1,n] = 0

E [(Hm,n − a1Hm−1,n − a2Hm,n−1)Hm,n−1] = 0.(20)

Since the entries of Hm,n, Hm−1,n, and Hm,n−1 are i.i.d.complex Gaussian variables, the orthogonality principle can berewritten as{

E [(Hm,n − a1Hm−1,n − a2Hm,n−1)Hm−1,n] = 0

E [(Hm,n − a1Hm−1,n − a2Hm,n−1)Hm,n−1] = 0.(21)

Moreover, the 1-D frequency response of the channel can be repre-sented as

Hm,n = Hm,n +Hd = a1Hm−1,n + a2Hm,n−1 +Hd (22)

where Hm,n, Hm,n, Hm−1,n, Hm,n−1, and Hd represent the corre-sponding entries.

Direct calculation shows that (21) is equivalent to{rH [1, 0]− a1rH [0, 0]− a2rH [1, 1] = 0

rH [0, 1]− a1rH [1, 1]− a2rH [0, 0] = 0.(23)

With the separation property of the correlations of the channel fre-quency response (2) and combining rt[0] = rf [0] = 1 and rt[1] = αt,rf [1] = αf , (23) can be simplified by{

a1σ2H + a2αtαfσ

2H − αtσ

2H = 0

a1αtαfσ2H + a2σ

2H − αfσ

2H = 0.

(24)

From (24), a1 and a2 are given by

⎧⎪⎨⎪⎩

a1 =αt

(1−α2

f

)1−α2

tα2f

a2 =αf (1−α2

t )1−α2

tα2f

.

(25)

Combing (25) and (22), the 1-D MSE of the channel estimator is

MSE = Var(Hd) = σ2H

(1 − a2

1 − a22 − 2a1a2αtαf

). (26)

Finally, channel estimator Hm,n is given by

Hm,n =αt

(1 − α2

f

)1 − α2

tα2f

Hm−1,n +αf (1 − α2

t )

1 − α2tα

2f

Hm,n−1. (27)

Moreover, combining (19) and (27), Hm,n is given by

Hm,n=αt

(1−α2

f

)1−α2

tα2f

Hm−1,n+αf (1−α2

t )

1−α2tα

2f

Hm,n−1+Hd. (28)

Then, through the feedback channel, the error of the channelpredictor Hd can be fed back from the transmitter to the receiver.Similarly, from (11), the feedback load is positive related withVar(Hd) = σ2

H(1 − a21 − a2

2 − 2a1a2αtαf . Because ∂MSE/∂αt <0, ∂MSE/∂αf < 0, the feedback load can be much smaller than σ2

H ,which is the nondifferential feedback, particularly when the channelis highly correlated. For example, given αt > 0.75, αf > 0.75, thenMSE|αt>0.75,αf>0.75 < MSE|αt=0.75,αf=0.75 = 0.28σ2

H .

Fig. 2. MSE of the predictor at the transmitter for Nr = 2, Nt = 2, σ2H = 1,

and D = 0.1.

From (28), taking quantization impact into consideration, the min-imal differential feedback rate over doubly selective fading channelscan be calculated by the rate distortion theory of continuous-amplitudesources in a similar way, i.e.,

R = NrNt log

{a21 + a2

2 +2a1a2αtαfd

σ2H

+Var(Hd)

d

}(29)

where channel predictor coefficients a1 and a2 are determinedby a1 = (αt(1 − α2

f ))/(1 − α2tα

2f ) and a2 = (αf (1 − α2

t ))/(1 −α2tα

2f ). The average power of Hd is Var(Hd) = σ2

H(1 − a21 − a2

2 −2a1a2αtαf ). The detailed derivation is given in the Appendix.

The given expression gives the minimal differential feedback ratesimultaneously utilizing the temporal and spectral correlations. From(29), the minimal differential feedback rate is a function of αt, αf andchannel quantization distortion d and much smaller than that of thenondifferential feedback (17).

IV. SIMULATION RESULTS AND DISCUSSION

Here, we first provide the relationship between the MSE of thepredictor and the 2-D correlations in Fig. 2. The minimal differentialfeedback rate over MIMO doubly selective fading channels is givenin Fig. 3. Then, a longitudinal section in Fig. 3 is presented, wherewe assume that the temporal correlation and the spectral correlationare equal. Finally, we verify our theoretical results by a practical dif-ferential feedback system with the water-filling precoder and Lloyd’squantization algorithm [18].

A. MSE of the Predictor and Minimal Differential Feedback Rate

For simplicity and without loss of generality, we consider Nr =Nt = 2, and σ2

H = 1. Fig. 2 presents the MSE between the predictedvalue and the true value. As the temporal or spectral correlationincreases, the MSE decreases. Furthermore, when either αt or αf

comes to one, the MSE tends to zero.Fig. 3 plots the relationship between the minimal differential

feedback rate and the 2-D correlations with channel quantizationdistortion D = 0.1. It is very similar to the MSE shown in Fig. 2,because it presents the minimal bits required to quantize the differen-tial CSI.

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 62, NO. 9, NOVEMBER 2013 4705

Fig. 3. Minimal differential feedback rate for Nr = 2, Nt = 2, σ2H = 1, and

D = 0.1.

Fig. 4. Relationship between the minimal feedback rate and temporal andspectral correlations, when they are equal, for Nr = 2, Nt = 2, σ2

H = 1, andD = 0.1.

Additionally, because αt and αf could be any value, we provideone of the longitudinal section in Fig. 3 where the temporal correlationis equal to the spectral correlation in Fig. 4. For comparison, thedifferential feedback compression only using the 1-D correlation andthe nondifferential feedback scheme are also included in Fig. 4. It isobserved in Fig. 4 that the scheme using both temporal and spectralcorrelations is always better than the scheme using only the 1-Dcorrelation. As the correlations increase, the 2-D differential feedbackcompression exhibits a significant improvement compared with 1-Dcompression. This performance advantage even reaches up to 67%with αt = αf = 0.95.

B. Differential Feedback System With Lloyd’s Algorithm

Here, we consider the temporal correlation αt = 0.9 with 2-GHzcarrier frequency, the normalized Doppler shift fd = 100 Hz, andspectral correlation αf = 0.9, with Δ = 8 μs, which is a reason-

Fig. 5. Relationship between the ergodic capacity and the feedback rate withLloyd’s algorithm in AR1 model for Nr = 2, Nt = 2, σ2

H = 1, and SNR =5 dB.

able assumption [12]. We design a differential feedback system us-ing Lloyd’s quantization algorithm to verify our theoretical results[18]. We use Diff(Hm,n|Hm−1,nHm,n−1) = Hm,n − a1Hm−1,n −a2Hm,n−1 as a differential function, where a1 = αt(1 − α2

f )/1 −α2tα

2f and a2 = (αf (1 − α2

t ))/(1 − α2tα

2f ) in the 2-D differen-

tial feedback compression and a1 = αt and a2 = 0 in the 1-Dcompression.

The feedback steps can be summarized as follows. First, basedon Lloyd’s quantization algorithm, the channel codebook can begenerated according to the statistics of the corresponding differentialfeedback load at both the transmitter and the receiver. Second, the re-ceiver calculates the current differential CSI Hd. Third, the differentialCSI is quantized to the optimal codebook value Hd according to theEuclidean distance. Finally, the transmitter reconstructs the channelquantization matrix by Hm,n = a1Hm−1,n + a2Hm,n−1 + Hd.

In Fig. 5, we give the simulation results of the ergodic capacityemploying Lloyd’s algorithm. The theoretical capacity results are alsoprovided in Fig. 5. We can see in Fig. 5 that the performance of the 2-Dcompression is always better than the 1-D compression, which verifiesour theoretical analysis.

As shown in Fig. 5, with the increase in feedback rate b, the ergodiccapacities rapidly increase when b is small and then slow down inthe large b region, because when b is large enough, the quantizationerrors tend to zero. In addition, the capacities of Lloyd’s quantizationare lower than the theoretical results. The reasons are as follows.The Lloyd’s algorithm is optimal only in the sense of minimizing avariable’s quantization error but not in data sequence compression,while channel coefficient H is correlated in both temporal and spectraldomains. However, the imperfection reduces as b increases, becausethe quantization errors of both Lloyd’s algorithm and theoreticalresults tend to zero with sufficient feedback bits b.

V. CONCLUSION

In this paper, we have designed a differential feedback schememaking full use of both the temporal and spectral correlations andcompared the performance with the scheme without differentialfeedback. We have derived the minimal differential feedback ratefor our proposed scheme. The feedback rate to preserve the givenchannel quantization distortion is significantly small compared with

4706 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 62, NO. 9, NOVEMBER 2013

nondifferential feedback, as the channel is highly correlated in bothtemporal and spectral domains. Finally, we provide simulations toverify our analysis.

APPENDIX

DERIVATION OF THE MINIMAL DIFFERENTIAL FEEDBACK RATE

USING TEMPORAL AND SPECTRAL CORRELATIONS

The minimal differential feedback rate over MIMO doubly selectivefading channels can also be derived by the rate distortion theory. GivenHm−1,n and Hm,n−1 at the transmitter, the differential feedback ratecan be represented as

R = inf{I(Hm,n; Hm,n|Hm−1,n, Hm,n−1) :

E

[d(Hm,n; Hm,n)

]≤ D

}. (30)

Since the entries are i.i.d. complex Gaussian variables, (30) can bewritten as

R = inf{I(Hm,n; Hm,n|Hm−1,n, Hm,n−1) :

E

[d(Hm,n; Hm,n)

]≤ D

}. (31)

The 1-D channel quantization equality can be written as

Hm−1,n = Hm−1,n +Em−1,n

Hm,n−1 = Hm,n−1 +Em,n−1. (32)

Similarly, (28) yields

Hm,n = a1Hm−1,n + a2Hm,n−1 +Hd (33)

where a1= (αt(1− α2f ))/(1− α2

tα2f ), and a2= (αf (1−α2

t ))/(1 −α2tα

2f ). The conditional mutual information I(Hm,n; Hm,n|Hm−1,n,

Hm,n−1) can be written as

I(Hm,n; Hm,n|Hm−1,n, Hm,n−1)

= h(Hm,n|Hm−1,n, Hm,n−1)

− h(Hm,n|Hm,n, Hm−1,n, Hm,n−1). (34)

First, we calculate h(Hm,n|Hm−1,n, Hm,n−1). Substituting (32) into(33) yields

Hm,n = a1(Hm−1,n +Em−1,n) + a2(Hm,n−1 +Em,n−1) +Hd.(35)

Substituting (35) into (34), we obtain

I = h(a1Em−1,n + a2Em,n−1 +Hd)

−h(Em,n|Hm−1,n, Hm,n−1). (36)

Considering inequality h(Em,n|Hm−1,n, Hm,−1) ≤ h(Em,n), (36)can be written as

I ≥ h(a1Em−1,n + a2Em,n−1 +Hd)− h(Em,n). (37)

Since Em−1,n, Em,n−1, and Hd are complex Gaussian variables,and the information entropy of Gaussian variables with variance σ2

is h(X) = 1/2 log 2πeσ2, we calculate the variance of (a1Em−1,n +a2Em,n−1 +Hd), i.e.,

Var(a1Em−1,n + a2Em,n−1 +Hd) = a21d+ a2

2d

+Var(H2

d

)+ 2a1a2r(Em−1,n, Em,n−1). (38)

Now, we give the derivation of the correlation function of two noiseterms r(Em−1,n, Em,n−1). From (32), the quantization error can bedecomposed into two parts, i.e.,

Em−1,n =σ2H − σ2

H

σ2H

Hm−1,n + ψm−1,n

Em,n−1 =σ2H − σ2

H

σ2H

Hm,n−1 + ψm,n−1 (39)

where

ψm,n−1 = Hm,n−1 −σ2H

σ2H

Hm,n−1

ψm−1,n = Hm−1,n −σ2H

σ2H

Hm−1,n (40)

and ψ is a Gaussian variable with zero mean and variance (σ2H(σ2

H −σ2H))/σ2

H , independent of H .Then, the correlation function of Em−1,n and Em,n−1 can be

calculated as

r(Em−1,n, Em,n−1) =

(σ2H − σ2

H

)2σ2H

αtαf =d2

σ2H

αtαf . (41)

Substituting (41) into (38), we obtain

Var(a1Em−1,n + a2Em,n−1 +Hd)

= a21d+ a2

2d+ σ2Hd

+ 2a1a2d2

σ2H

αtαf . (42)

From (31), (37), and (42), we obtain

R = NrNt log

{a21 + a2

2 +2a1a2αtαfd

σ2H

+Var(Hd)

d

}(43)

where a1 = (αt(1 − α2f ))/(1 − α2

tα2f ), a2 = (αf (1 − α2

t ))/(1 −α2tα

2f ), and Var(Hd) = σ2

H(1 − a21 − a2

2 − 2a1a2αtαf ).

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