Adaptive spatial multiplexing for time-varying correlated MIMO channels

EUROPEAN TRANSACTIONS ON TELECOMMUNICATIONSEur. Trans. Telecomms. 2010; 21:50–63Published online 9 February 2009 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/ett.1352

Information Theory

Adaptive spatial multiplexing for time-varying correlated MIMO channels

Yan Zhou∗

Department of Corporate R&D, Qualcomm Inc, San Diego, CA 92121, USA

SUMMARY

This work presents an adaptive spatial multiplexing scheme for time-varying correlated multiple-inputmultiple-output (MIMO) channels. Unlike the conventional spatially uncorrelated block fading channelmodel, both temporal channel variation in each data block and spatial correlation are considered in thedesign. The proposed system continuously transmits packets with identical structure. Each packet consistsof a training phase followed by a data transmission phase. The training signal is used to estimate instantaneouschannel state at the receiver. In each symbol period of data transmission phase, the transmitter simultaneouslytransmits multiple data streams, which are decoded at the receiver based on the estimated channel state. Thetransmitter is assumed to know only the channel statistics, with which the power and rate for each data streamas well as the number of data streams are adjusted in each symbol period to achieve a target bit error rate(BER). The results reveal that the throughput-maximising transmission strategy makes a judicious diversity-multiplexing tradeoff by allocating power to an optimum number of data streams. Moreover, the packetlength is also optimised based on the temporal channel variation rate. The simulation results demonstratethat the throughput can be significantly enhanced by the optimum packet length. Copyright © 2009 JohnWiley & Sons, Ltd.

1. INTRODUCTION

Multiple antennas at the transmitter and receiver have beenshown to dramatically improve the capacity and reliabilityof wireless communication links and are often referredto as multiple-input multiple-output (MIMO) systems [1,2]. However, design of communication strategies for time-varying channels is challenging due to temporal channelvariations that lead to outdated channel state informationat both ends. The impact of feedback delay on spatialmultiplexing in time-varying MIMO channels has beenstudied in References [3, 4], which assume that error-free but delayed instantaneous channel information isavailable at the transmitter. The use of correction matricesat the receiver is suggested by Reference [3] to partiallycompensate for the effect of delay, while Reference [4]incorporates the channel uncertainty due to time variations

* Correspondence to: Yan Zhou, Department of Corporate R&D, Qualcomm Inc, 5775 Morehouse Drive, San Diego, CA 92121, USA.E-mail: [email protected]

in the precoder design. In the space-time coded adaptivemodulation scheme [5], the transmitter optimises thesymbol modulation levels in orthogonal space-time codesbased on the feedback of instantaneous received SNR,and the effect of feedback delay has been investigated. InReference [6], Alamouti coded data are transmitted alongtwo orthogonal basis beams with optimum power allocationand modulation levels. In contrast to References [3–5],the transmitter is assumed to know only the instantaneousestimated channel information with feedback delay, whichbetter matches the realistic situation. In Reference [7],the proposed adaptive transmit beamformer exploits thepredicted channel information to mitigate the effect offeedback delay, and the results show that a critical valueof normalised prediction error is a good indicator of perfor-mance degradation. Compared with time-varying MIMOchannels, the time-varying Single-input Single-output

Received 7 July 2008Revised 10 December 2008

Copyright © 2009 John Wiley & Sons, Ltd. Accepted 24 December 2008

SPATIAL MULTIPLEXING FOR TIME-VARYING CORRELATED MIMO CHANNELS 51

(SISO) channels have been studied more extensively. Somerepresentative works are described here. The pilot-basedestimation of time-varying SISO channels and its impact onreceiver performance have been studied in Reference [8].A closed-form lower bound of average channel capacity isderived in Reference [9] in terms of channel estimation errorvariance, and the optimal pilot symbol spacing and powerallocation are determined by maximising this lower bound.The effects of feedback delay and error on the performanceof adaptive systems with instantaneous channel feedbackhave been investigated in References [10, 11].

This work is motivated by two reasons. First, mostexisting research on time-varying MIMO systems assumesa spatially uncorrelated block fading channel model, whichignores temporal channel variation in each data blockas well as the spatial correlation. This will in generallead to inaccurate performance predictions. Secondly,most existing works assume that the transmitter knowsthe instantaneous channel information, which can beeither fed back from the receiver in Frequency DivisionDuplex (FDD) systems [7, 12] or estimated at thetransmitter in Time Division Duplex (TDD) systems [3].However, both methods will induce delay and error inthe channel information, which will inevitably degradethe performance [12]. In this work, a statistically adaptivespatial multiplexing scheme is proposed for narrowbandtime-varying correlated MIMO channels and has thefollowing features: (1) both temporal channel variation ineach data block and spatial correlation are jointly consideredin the design; (2) instead of instantaneous channelinformation, the transmitter is assumed to only know thechannel statistics, which usually change at much slowertime scales and can thus be tracked more easily and reliablyand (3) unlike conventional adaptive modulation schemeswhose power and rate for each data stream are invariant ineach data block, the proposed adaptive spatial multiplexingscheme adjusts the power and rate for each data stream aswell as the number of data streams in every symbol periodbased on the channel statistics, and the data block length isfurther optimised to maximise the spectral efficiency.

The results reveal that the rate-maximising transmissionstrategy makes a judicious tradeoff between signal receptionquality and the number of data streams by allocatingpower to an optimum number of streams. Furthermore, thespectral efficiency can be greatly enhanced by the optimumdata block length, which is a function of the temporalchannel variation rate. The proposed statistics-based designis also compared to that with perfect instantaneous ChannelState Information at Transmitter (CSIT). The results implythat the former might be a good complexity-performance

tradeoff by avoiding the instantaneous feedback of CSITwith an acceptable performance loss.

The paper is organised as follows. Section 2 describesthe channel model for time-varying correlated MIMOchannels. Section 3 provides an overview of the proposedscheme. The design objective and procedure are statedin Section 4. The optimisation of individual systemcomponents is elaborated in Section 5. For comparison,the scheme with perfect instantaneous channel informationat transmitter is described in Section 6. Numerical resultsincluding the performance comparison of both schemes arepresented in Section 7 and Section 8 contains concludingremarks.

2. TIME-VARYING CORRELATED MIMOCHANNEL MODELLING

Consider a MIMO system with uniform linear arrays(ULAs) at both ends with NT transmit antennas and NR

receive antennas. For a narrowband† time-varying MIMOchannel, the NR × 1 received signal vector at time t can bewritten as

yc(t) = Hc(t)xc(t) + nc(t), 0 � t � T (1)

where xc(t) is the NT × 1 transmitted signal vector, nc(t) isthe i.i.d. Gaussian noise vector, and T presents the overallsignalling duration. The NR × NT channel matrix at time t

is modelled as

Hc(t) =L∑

l=1

βlaR(φR,l)aHT (φT,l)e

j2πνlt (2)

where L is the number of paths. For the lth path, the arraysteering and response vectors are given by

aT (φT,l) = [1, e−j2π sin(φT,l)d/λ, . . . , e−j2π(NT −1) sin(φT,l)d/λ

]T

aR(φR,l) = [1, e−j2π sin(φR,l)d/λ, . . . , e−j2π(NR−1) sin(φR,l)d/λ

]T

(3)

where the physical path angle φ ∈ [−π, π] is measuredw.r.t. the array broadside, λ denotes the wavelength and

† For simplicity, we focus on the narrowband single-carrier MIMOtransmission. It could be extended to the wideband multicarriertransmission.

Copyright © 2009 John Wiley & Sons, Ltd. Eur. Trans. Telecomms. 2010; 21:50–63DOI: 10.1002/ett

52 Y. ZHOU

d is the antenna spacing. WLOG, we focus on the criticalspacing d = λ/2 in this paper. The complex path amplitudeβl = αle

jϕl has envelope αl > 0 and phase ϕl uniformlydistributed in [0, 2π]. WLOG, the transmitter and receiverare assumed to move in the array-broadside directions atspeeds of vT and vR. The resultant Doppler frequencyshift of the lth path is given by νl = (vR cos(φR,l) +vT cos(φT,l))/λ with the maximum shift as fmax = (vR +vT )/λ. Moreover, we assume that {αl}, {φR,l}, {φT,l}, {νl}are fixed for a given scattering environment over the timescales of interest, while {ϕl} randomly change for differentchannel realisations [13, 14].

Instead of signalling in the spatial domain in Equation(1), we consider signalling in the eigen domain, whichconforms to the capacity-achieving signalling strategy forthe correlated MIMO channels [15]. The signal relation inthe eigen domain can be written as

y(t) = H(t)x(t) + n(t) (4)

where H(t) = UHR Hc(t)UT denotes the eigen-domain

channel matrix with UR = [uR,1, . . . ,uR,NR ] and UT =[uT,1, . . . ,uT,NT ] comprising the eigenvectors of thereceive and transmit channel correlation matrices Rrx =E[Hc(t)HH

c (t)] and Rtx = E[HHc (t)Hc(t)], respectively.

The vectors y(t) = UHR yc(t), x(t) = UH

T xc(t), n(t) =UH

R nc(t) represent the received, transmitted and noisevectors in the eigen domain. It has been shown in References[16, 17] that the eigen-domain channel coefficients{H(q, p, t)} can be accurately modelled as zero-meancomplex Gaussian entries uncorrelated for different (q, p)′s.

Mathematically, the channel coefficient correlation can beexpressed as

E[H(q, p, t)H∗(q′, p′, t′)] = rq,p(t − t′)δqq′δpp′ (5)

where δij represents the Kronecker delta function and thetemporal correlation can be specified as

rq,p(t − t′) � E[H(q, p, t)H∗(q, p, t′)]

=L∑

l=1

α2l

∣∣∣uHR,qaR(φR,l)

∣∣∣2

∣∣aHT (φT,l)uT,p

∣∣2ej2πνl(t−t′). (6)

Based on the uncorrelated nature, it has been shown inReference [15] that the capacity-achieving input covariancematrix has a diagonal structure in the eigen domain, i.e.E[x(t)xH (t)] is diagonal. This motivates the considerationof transmitting independent data streams along differenteigen directions in the proposed system, as described inthe next section.

3. DESCRIPTION OF PROPOSED SYSTEM

The system continuously transmits and receives packetswith identical structure. WLOG, the structure of the firstpacket is shown in Figure 1. It has a training phase with Ntr

symbol periods followed by a data transmission phase withND symbol periods. The fixed symbol period is denoted

Figure 1. Structure of a transmission packet in eigen domain.



Figure 2. Block diagram of the proposed system.

as Ts, and the time 0 represents the end of training phase.The packet is transmitted and received in the eigen domain.WLOG, the NT eigen-domain channel columns are sortedin descending order according to their statistical strengths.‡

Following the optimal training scheme in Reference [18],a training symbol is transmitted in the pth training symbolperiod with full power ρ from the pth eigenbeam, which issteered by the pth transmit eigenvector uT,p and illuminatesthe pth column of H(t). This also implies 1 � Ntr �NT . Accordingly, the transmitted training signal matrixin the eigen-time domain is D = √

ρINtr with INtr as anidentity matrix. In the nth data symbol period, a k(n) × 1signal vector x(n) is launched from the first k(n) transmiteigenbeams with 1 � k(n) � Ntr, and no power is allocatedto the remaining beams.

Next, we describe the signal relation in the training phase.For simplicity, the length of training phase is first fixed asNtr = NT and, hence, D = √

ρ INT . We also assume thatthe eigen-domain channel matrix H(t) in the training phaseis the same§ as that at time 0. Accordingly, the NR × NT

received training signal matrix in the eigen domain can bewritten as

Z = H(0)D + W = √ρ H(0) + W (7)

‡ which means E‖H(:, 1, t)‖2F �, . . . , � E‖H(:, NT , t)‖2

F , whereH(:, p, t) represents the pth column of H(t) and ‖·‖2

F denotes theFrobenius norm. It can be shown that the order is independent of t for agiven scattering environment.§ Extension to the case considering channel variation in the training phaseis feasible.

where the NR × NT noise matrix W has i.i.d. complexGaussian entries with zero mean and variance o2. Forthe convenience of linear processing, we further definez = vec(Z) = [zT

1 , . . . , zTNT

]T with zp as the pth columnof Z. It is obvious that

z = √ρh(0) + w (8)

where h(0) = vec(H(0)) and w = vec(W).The signal relation in the nth data symbol period is

illustrated in Figure 2. At the transmitter side, a datavector s(n) = [s1(n), . . . , sk(n)(n)]T is formed with k(n)independent QAM symbols, which have zero mean andcovariance matrix E[s(n)sH (n)] = Ik(n). The number ofbits in sp(n) is denoted as rp(n) and is determined by theconstellation size of sp(n) [19]. The k(n) × 1 transmittedsignal vector in the eigen domain is given by

x(n) = �1/2(n)s(n) (9)

where �(n) represents a diagonal power-shaping matrix.Accordingly, the covariance matrix of x(n) has acapacity-achieving diagonal structure [15]. The vectorx(n) is further zero-padded and eigen transformed intothe spatial (antenna) domain for transmission: xc(n) =UT [xT (n)

... 0T ]T, where 0 is a (NT − k(n)) × 1 all-zerovector. Due to the eigen transformation, the k(n) datasymbols in x(n) are respectively launched in the directionsof the first k(n) transmit eigenbeams. To simplify the


54 Y. ZHOU

notation, k(n) and its shortening k are used interchangeablyin the following.

At the receiver side, the signal vector on the NR receiveantennas can be written according to Equation (1) asyc(n) = Hc(n)xc(n) + nc(n) where H(t) in the nth datasymbol period is assumed to be the same as H(nTs), whichis denoted as H(n) hereafter. The yc(n) is then transformedinto the eigen domain via

y(n) = UHR yc(n) = H(n)[xT (n)

... 0T ]T + n(n)

= Hk(n)x(n) + n(n) (10)

where y(n) is the received signal vector in the eigen domain,Hk(n) contains the first k columns of H(n), and n(n) hasi.i.d. complex Gaussian noise entries with zero mean andvariance o2. To decode the data vector, we need to knowH(n), which is estimated by feeding the received trainingsignal vector z in Equation (8) to a NRNT × NRNT linearchannel predictor L(n)

h(n) = [hT

1 (n), . . . , hT

NT(n)]T = L(n)z

H(n) = [h1(n), . . . , hNT (n)] (11)

where the NR × 1 vector hp(n) denotes the pth subvectorof the NRNT × 1 predicted eigen-domain channel vectorh(n), and H(n) represents the predicted H(n). Now, H(n)can be decomposed as a sum of H(n) and the associatedprediction error matrix E(n)

H(n) = H(n) + E(n) (12)

The received signal vector in Equation (10) can also beexpressed in terms of H(n) as

y(n) = Hk(n)x(n) + Ek(n)x(n) + n(n)

� Hk(n)x(n) + v(n) (13)

where v(n) represents the effective noise vector. With theknowledge of H(n), a k × NR linear decoder is then formedto estimated s(n) from y(n)

s(n) = G(n)y(n) (14)

4. DESIGN OBJECTIVE AND PROCEDURE

The overall design objective is to maximise the average rateper packet subject to the bit error rate (BER) requirementfor each data stream with a given transmit power ρ

max

(AR =

∑ND

n=1∑k(n)

p=1 rp(n)

Ntr + ND

),

s.t. BERp(n) � BERtar, tr(�(n)) = ρ (15)

where BERp(n) represents the average BER of the pthQAM symbol at time n, BERtar is the target BER‖, andtr(·) denotes the matrix trace. The optimisation is over thesystem components L(n), G(n), �(n), {rp(n)}, k(n) and ND

with channel statistics known at both sides¶. However, thejoint optimisation of Equation (15) is difficult. Therefore,we split the general problem into multiple individualoptimisation problems. Each problem only involves onesystem component, and they are solved sequentially. Theprocedure is described below.

Step 1. In each data symbol period, L(n) is optimised byminimising the mean square error (MSE) betweenthe true eigen-domain channel vector and itsprediction

Lo(n) = arg minL(n)

E‖h(n) − L(n)z‖2F (16)

where h(n) = vec(H(n)), z is given by Equation(8), and the expectation is over both h and w.Therefore, L(n) is determined by channel statisticsand hence is the same for different packets, since thechannel statistics do not change across the packetsfor a given scattering environment.

Step 2. For a given number of data streams k, wesequentially optimise G(n), �(n) and {rp(n)}.

(1) The decoder G(n) is optimised by minimising theMSE between s(n) and s(n)

Go(n, k) = arg minG(n)

E‖s(n) − G(n)y(n)‖2F

(17)

‖ Extension to different target BERs for different data streams is feasible.¶ Both receiver and transmitter know the temporal channel correlations{rq,p(nTs), n = 1, . . . , Nmax} for all (q, p)’s with Nmax as an upper boundof ND.



where the variable k emphasises that the solutionis conditioned on k, y(n) is given by Equation (13),and the expectation is over s(n), Ek(n), and n(n).As shown later, Go(n, k) is a function of Hk(n) and�(n). Therefore, it varies in different packets withdifferent realisations of Hk(n).

(2) After the optimisation of decoder, the power-shaping matrix �(n) is aimed at further minimisingthe MSE between s(n) and s(n) with the giventransmit power

�o(n, k) = arg min�(n)

MSE(n), s.t. tr(�(n)) = ρ

MSE(n) = E‖s(n) − Go(n, k)y(n)‖2F (18)

where the expectation is over s(n), n(n), Ek(n),and Hk(n), since the transmitter only knows thechannel statistics. This also implies that �o(n, k)is determined by the channel statistics and hence isthe same for different packets.

(3) Next, we use p(n) to represent the Signal toInterference plus Noise Ratio (SINR) of the pthstream at the decoder output for a given H(n). Theassociated BER is denoted as BER(rp(n), p(n))where rp(n) is the number of bits in the QAMsymbol sp(n). The expressions of p(n) andBER(rp(n), p(n)) will be specified later. Thetransmission rate of the pth stream is optimised bymaximising rp(n) while keeping the average BERunder the target BER

Rp(n, k) = maxrp(n)∈{0,1,2,...}

rp(n),

s.t. BERp(n) � E[BER(rp(n), p(n))] � BERtar

(19)

where the expectation is over H(n), since thetransmitter only knows the channel statistics.Therefore, Rp(n, k) is determined by the channelstatistics and hence does not change across thepackets. Applying Equation (19) to all k streams,we obtain the optimised rates {Rp(n, k)}kp=1.

Step 3. The number of data streams k is further optimisedby maximising the total rate in the nth data symbolperiod

ko(n) = arg maxk∈S1

k∑p=1

Rp(n, k),

S1 = {1, . . . , min{NT , NR}} (20)

where the upper bound of k is chosen asmin{NT , NR}, which is the maximum rank ofH(n). Apparently, ko(n) does not change acrossthe packets. Furthermore, the optimised decoder,power-shaping matrix and rates associated withko(n) are selected as the final designs

Go(n) = Go(n, ko(n)), �o(n) = �o(n, ko(n)),

Ro(n) = [R1(n, ko(n)), . . . , Rko(n)(n, ko(n))]T

(21)

Step 4. With those optimised components in Equations(16), (20) and (21), the average rate per packet inEquation (15) becomes only a function of the datablock length ND

AR(ND) =∑ND

n=1 Ro(n)

ND + Ntr

,

Ro(n) =ko(n)∑p=1

Rp(n, ko(n)),

Ntr = maxn∈{1,...,ND}

ko(n) (22)

where Ro(n) denotes the instantaneous total ratein the nth data symbol period, and the numberof training symbol periods Ntr is equal to thenumber of transmit eigenbeams used for datatransmission**. Note that Ntr could be less thanNT initially assumed in Section 3. This is becauseif the number of used transmit beams is less thanNT , we only need to transmit training signals fromthem instead of from all the NT transmit beamsto reduce the training overhead. Furthermore, theoptimum data block length is the one maximisingthe average rate

Nopt = arg maxND∈S2

AR(ND), S2 = {1, . . . , Nmax}

(23)

where Nmax denotes an upper bound of ND.

The proposed statistics-based design has the followingtwo features. (1) It has low complexity compared with the

** Note that only the first ko(n) eigenbeams are excited in the nth datasymbol period and, hence, max

n∈{1,...,ND}ko(n) represents the number of

eigenbeams excited in the whole data transmission phase.


56 Y. ZHOU

CSIT-based design [3–7]: instantaneous channel feedbackis not required, and the same optimised packet structure isapplied over different packets when the channel statistics arestatic; (2) The temporal and spatial channel correlations areexploited by optimising the packet length and the numberof data streams, which significantly improves the averagerate, as shown in the results.

5. OPTIMISATION OF SYSTEM COMPONENTS

5.1. Optimisation of channel predictor

The solution to Equation (16) can be derived from theorthogonality principle [20] as

Lo(n) = E[h(n)zH

] (E

[zzH

])−1

= √ρ�(n)

(ρ�(0) + o2INRNT

)−1(24)

where h(n) = vec(H(n)) with H(n) as the shorteningfor H(nTs) and vec(·) defined above Equation (8).According to Equation (5), �(n) = E[h(n)hH (0)] hasa diagonal structure and, hence, Lo(n) is diagonalas well. It is straightforward to show that the ((p −1)NR + q)th diagonal element of Lo(n) is equal to√

ρ rq,p(nTs)/(ρ rq,p(0) + o2). Furthermore, due to z =vec(Z) and h(n) = vec(H(n)), the ((p − 1)NR + q)thelements of z and h(n) correspond to Z(q, p) andH(q, p, n), respectively. The above relations together withh(n) = Lo(n)z give the expressions of H(q, p, n) and theassociated prediction error as

H(q, p, n) =√

ρrq,p(nTs)

ρrq,p(0) + o2 Z(q, p)

=√

ρrq,p(nTs)

ρrq,p(0) + o2

(√ρH(q, p, 0) + W(q, p)

)E(q, p, n) = H(q, p, n) − H(q, p, n),

q = 1, . . . , NR, p = 1, . . . , NT (25)

where Z(q, p) is specified via Equation (7). SinceH(q, p, t) has zero-mean complex Gaussian distribution,both H(q, p, n) and E(q, p, n) also have zero-meancomplex Gaussian distribution

H(q, p, n) ∼ CN(0, σ2(q, p, n)),

σ2(q, p, n) = ρ|rq,p(nTs)|2/(ρ rq,p(0) + o2)

E(q, p, n) ∼ CN(0, ε2(q, p, n)),

ε2(q, p, n) = rq,p(0) − σ2(q, p, n) (26)

Furthermore, based on Equations (25) and (5), we can derivethe following properties

E[H(q, p, n)H∗(q′, p′, n)

]= σ2(q, p, n)δqq′δpp′

E[E(q, p, n)E∗(q′, p′, n)

] = ε2(q, p, n)δqq′δpp′ ,

E[E(q, p, n)H∗(q′, p′, n)

]= 0 (27)

Note that Equation (25) corresponds to its matrix version(12), while Equations (26) and (27) indicate that H(n) andE(n) in Equation (12) have uncorrelated complex Gaussianentries, and they are also mutually uncorrelated.

5.2. Optimisation of decoder and power-shapingmatrix

The solution to Equation (17) can be derived according toorthogonality principle as

Go(n, k) = E[s(n)yH (n)

] (E

[y(n)yH (n)

])−1

= �1/2(n)HHk (n)

(Hk(n)�(n)HH

k (n) + �(n))−1

(28)

where y(n) is given by Equation (13) and the effective noisecovariance matrix can be specified as

�(n)=E[v(n)vH (n)

] = E[Ek(n)�(n)EH

k (n)] + o2INR

=diag(tr (�1(n)�(n)) , . . . , tr

(�NR (n)�(n)

)) + o2INR

(29)

where �q(n) = E[EHk (q, :, n)Ek(q, :, n)] = diag(ε2(q, 1,

n), . . . , ε2(q, k, n)) with Ek(q, :, n) as the qth row ofEk(n) and ε2(q, p, n) given by Equation (26). Note thatthe 2nd step is due to the identity E[tr(BABH )] =tr(E[BHB]A) as well as the uncorrelated entries in Ek(n).After obtaining Go(n, k), we can specify the MSE(n) inEquation (18) as

MSE(n)

= Etr[(

s(n) − Go(n, k)y(n))(

s(n) − Go(n, k)y(n))H

]Copyright © 2009 John Wiley & Sons, Ltd. Eur. Trans. Telecomms. 2010; 21:50–63

DOI: 10.1002/ett


= Etr(Ik + �1/2(n)HH

k (n)�−1(n)Hk(n)�1/2(n))−1

� Etr(Ik + �1/2(n)HH

k (n)�−1(n)Hk(n)�1/2(n))−1

� MSE(n) (30)

�(n) = tr (�(n)) diag(ε2(1, n), . . . , ε2(NR, n)

)+ o2INR,

ε2(q, n) = maxp=1,...,k

ε2(q, p, n)

where the 2nd step is due to the matrix inverse lemma††,and the inequality stems from �(n) � �(n) as well as theproperties of matrix ordering‡‡ [21].

The optimum �(n) is aimed at minimising MSE(n).However, the optimisation may not be convex due to theterm tr(�q(n)�(n)) in Equation (29) and the inverse on�(n) in Equation (30). Therefore, �(n) is optimised byminimising the upper bound of MSE(n)

�o(n, k) = arg min�(n)

MSE(n), s.t. tr(�(n)) = ρ (31)

which is convex optimisation as shown below. Under theconstraint tr(�(n)) = ρ, the �(n) is only a function of ρ

and is independent of �(n). Furthermore, the upper boundcan be rewritten§§ as

MSE(n) = k − NR

+ Etr(INR + �

−1/2(n)Hk(n)�(n)HHk (n)�−1/2(n)

)−1

(32)

By applying the result in Reference [22], it can beshown that Equation (32) is a convex function of �(n).Therefore, Equation (31) is convex programming and,hence, the global minimiser can be reliably obtained viaoptimisation routines. In practice, since MSE(n) has noclosed-form expression, the expectation is replaced byan arithmetic mean over a large number of realisationsof H(n) generated according to Equation (26), and thecorresponding MSE(n) is plugged into Equation (31) foroptimisation.

†† (B−1 +CA−1CH )−1 = B−BC(A+CHBC)−1CHB.‡‡A � Bmeans thatA−B is positive semi-definite.A � B⇒ A−1 �B−1 ⇒ (I+CA−1CH )−1 � (I+CB−1CH )−1.§§ This is based on the identity tr(IM +AHA)−1 = M − N + tr(IN +AAH )−1 withA as a N × M matrix.

5.3. Optimisation of rates, stream number and datablock length

With Go(n, k) and �o(n, k), the SINR corresponding tothe pth received stream of s(n) = Go(n, k)y(n) can bestraightforwardly shown as

p(n) = λp(n)HHk (:, p, n)

×∑

j �=p

λj(n)Hk(:, j, n)HHk (:, j, n) + �(n)

−1

× Hk(:, p, n) (33)

where Hk(:, p, n) represents the pth column of Hk(n), andλp(n) denotes the pth diagonal element of �o(n, k). To geta tractable BER expression, the interference plus noise seenby each stream at the decoder output is assumed to have aGaussian distribution, which has been justified in Reference[23]. With this assumption, a tight BER approximation forQAM constellation with 2i points is given by Reference[24]: BER(i, p(n)) = 0.2exp(−1.5p(n)/(2i − 1)). Therate of the pth stream is optimised via Equation (19) bysequentially searching over {rp(n) = 0, 1, 2, . . .}. For eachrp(n), we first compute both p(n) and BER(rp(n), p(n))for each realisation of H(n) and then calculate BERp(n)by averaging BER(rp(n), p(n)) over different H(n)realisations. The maximum rp(n) satisfying the constraintin Equation (19) is chosen as the optimum rate Rp(n, k).Deriving a closed-form solution to Rp(n, k) is difficult,since the distribution ofp(n) for arbitrary channel statisticsis unknown. The asymptotic distribution for i.i.d. H(n) withlarge number of antennas can be found in Reference [25],and the distribution for i.i.d. H(n) with arbitrary numberof antennas is given by Reference [26] but is complicatedfor numerical evaluation. However, closed-form solutioncan be obtained for SISO and Single-input multiple-output(SIMO) channels with arbitrary correlations, as shown inAppendix.

Next, the optimum number of streams k is searchedaccording to Equation (20) by maximising the total ratein the nth data symbol period, and the correspondingsystem components are selected as the final designs inEquation (21). With those optimised components, theaverage rate can be computed as a function of the datablock length ND, and the optimum ND is then searched viaEquation (23).


58 Y. ZHOU

6. ADAPTATION WITH INSTANTANEOUSFEEDBACK

The statistics-based design in Section 3 only requireschannel statistics at the transmitter. However, most existingworks [3–7] assume the availability of CSIT. In this section,we describe the design with CSIT to compare with theproposed statistics-based design. In the CSIT-based design,the transmitter is assumed to know the received trainingsignal matrix Z, which is perfectly fed back from thereceiver without delay and error. Therefore, the transmitterknows the predicted eigen-domain channel matrix H(n) asdefined in Equation (11).

The transceiver with CSIT has the same structure asthat in Figure 2 except that the power-shaping matrix andthe zero-padding block are replaced by a general linearprecoder F(n). The optimisation is also the same as thestatistics-based design except that the precoder and ratesare optimised for each realisation of H(n). For a fixednumber of streams k, the NR × 1 received signal vector isgiven by

y(n) = H(n)F(n)s(n) + n(n)

= H(n)F(n)s(n) + v(n) (34)

where F(n) is the NT × k precoder, and v(n) =E(n)F(n)s(n) + n(n). Equation (34) will reduce toEquation (13) if F(n) = [�1/2(n)

... 0k×(NT −k)]T. Theprecoder is aimed at minimising the following MSE

MSE(H(n)

)= tr

(Ik + FH (n)HH (n)�−1(n)H(n)F(n)

)−1

� tr(Ik + FH (n)HH (n)�−1(n)H(n)F(n)

)−1(35)

which is derived in the same way as Equation (30)except that the expectation is not on H(n), since thetransmitter now knows H(n). Let V�VH be the eigendecomposition of HH (n)�−1(n)H(n). The upper boundof MSE(H(n)) can be minimised by choosing F(n) =V[�1/2

...0k×(NT −k)]T with the optimum power-shapingmatrix � = diag( 1, . . . , k) specified in Reference [27].Furthermore, the SINR of the pth stream at the MMSE

decoder output can be straightforwardly shown as

p(n) = FH (:, p, n)HH (n)

×(∑

j �=p

H(n)F(:, j, n)FH (:, j, n)HH (n) + �(n)

)−1

× H(n)F(:, p, n) (36)

where F(:, p, n) denotes the pth column of F(n).The maximum rate of the pth stream under the BERrequirement‖‖ can be shown as Rp(n, k) = log2(1 −1.5p(n)/ ln(BERtar/0.2)), and the optimum numberof data streams is searched according to ko(n) =arg max

k∈S1

∑kp=1 Rp(n, k). In contrast to the statistics-based

design, F(n), {Rp(n, k)} and ko(n) are optimised for a givenH(n) instead of its statistics. The corresponding average ratecan be expressed as

AR(ND) =∑ND

n=1 Ro(n)

ND + Ntr

,

Ro(n) = E

ko(n)∑p=1

Rp(n, ko(n))

(37)

where the expectation is on H(n), and the training blocklength Ntr is equal to the number of transmit eigen directionswith non-zero channel strength¶¶. This is because unlike thestatistics-based design, the precoder and rates are optimisedbased on each estimated H(n). To estimate H(n), the pilotsymbols have to be sent from all non-vanishing transmiteigen directions.

7. RESULTS AND DISCUSSIONS

7.1. Simulation parameters and procedure

We consider a multipath channel with L = 103 pathswhose arrival and departure angles {φR,l, φT,l} arerandomly uniformly distributed within a 2D angular region:[−φmax, φmax] × [−φmax, φmax], where φmax determinesthe angular spread. Each path has the same strengthα2

l = 1/L, which implies the channel normalisation

‖‖ Mathematically, Rp(n, k) = maxi∈{0,1,2,...}

i, s.t.BER(i, p(n)) � BERtar

with BER(i, p(n)) defined below Equation (33).¶¶ Which is equal to the number of columns of H(n) whose Frobeniousnorms have non-zero expectation.



E‖Hc(t)‖2F = NRNT . Both the transmitter and receiver

have the same speed vT = vR = 10 km/h. The carrierfrequency is 1.8 GHz, and the resultant maximum Dopplershift is fmax = 33.3 Hz. The symbol period Ts is specifiedvia the product fmaxTs, which determines the fading rate.The BER target is 10−3, the noise power per receive antennao2 is 1, and the other parameters NR, NT , φmax, fmaxTs, ρ

are specified in the results.Based on the above physical parameters, rq,p(t − t′)

is calculated via Equation (6) for all (q, p)’s, and both{σ2(q, p, n)} and {ε2(q, p, n)} are computed by Equation(26), which determines the statistics of H(n) and E(n). Thestatistics-based design follows the procedure in Section 4,where the power-shaping matrix and transmission rates areoptimised with 104 realisations of H(n). For the designwith CSIT, the precoder and rates are optimised for eachrealisation of H(n).

7.2. Instantaneous and average rates

The results of instantaneous rate Ro(n) in Equation (22)for the statistics-based design are shown in Figure 3,where ρ = 30 dB, fmaxTs = 10−2 and φmax = π. The‘MIMO, �o(n)’ and ‘MIMO, �o(n, NT )’ represent theperformance for �o(n) with the optimum stream numberko(n) and �o(n, NT ) with the maximum stream numberNT , respectively. It is obvious that Ro(n) decays asthe time index n increases in each case. This can beunderstood from Equation (26), which indicates that

Figure 3. Instantaneous rate as a function of symbol index forangular spread φmax = π, transmit SNR ρ = 30 dB and fading ratefactor fmaxTs = 10−2.

Figure 4. Average rate in each packet for angular spread φmax =π, transmit SNR ρ = 30 dB and fading rate factor fmaxTs = 10−2.

larger n impairs the power*** of each estimated channelcoefficient σ2(q, p, n) while boosts that of the associatedestimation error ε2(q, p, n) due to the vanishing temporalcorrelation rq,p(nTs). Accordingly, the SINR of each datastream in Equation (33) and hence Ro(n) diminish astime progresses. The corresponding average rate AR(ND)is plotted in Figure 4. As ND increases, AR(ND) ineach case goes up before reaching the peak and thengradually decreases. This is because smaller ND reducesthe portion of data transmission time in each packet andhence degrades transmission efficiency, while larger ND

extends data transmission time at the cost of lower Ro(n)in the extended period, which eventually brings downAR(ND).

Next, we study the impact of number of data streamson the MIMO performance, which is best revealed by thecomparison of �o(n, NT ) and �o(n). The former distributespower over all NT transmit eigenbeams, while the latterallocates power only to the ko(n) strongest beams. In eachcase, the number of excited data streams is equal to therank of the power-shaping matrix. Compared with �o(n),�o(n, NT ) induces a 55% to 100% reduction of Ro(n) inFigure 3 and a 64% to 71% reduction of AR(ND) in Figure 4.The ranks of both power-shaping matrices are plotted inFigure 5, which shows that �o(n, NT ) has a fixed rank of 11,while the rank of �o(n) ranges between 4 and 8. Therefore,activating all 11 data streams with less rate per stream is

*** Which is the variance of the estimable part and represents theestimation quality.


60 Y. ZHOU

Figure 5. Ranks of power-shaping matrices for angularspread φmax = π, transmit SNR ρ = 30 dB and fading rate factorfmaxTs = 10−2.

inferior to concentrating power on fewer streams with morepower and rate in each.

The performance of non-MIMO configurations isinvestigated next. The instantaneous and average ratesRo(n) and AR(ND) for SISO and SIMO are plottedin Figures 3 and 4, respectively. Compared with SISO,SIMO achieves significant improvement in both Ro(n) andAR(ND). This is because SIMO captures more channelpower by the NR receive antennas, and the received SINRis further stabilised by the antenna diversity. In addition,SIMO has a lower training cost than MIMO due to theuse of single transmit antenna. Both antenna diversity andlow training cost make the maximum AR(ND) of SIMOroughly match that of MIMO with �o(n, NT ) in Figure 4.In fact, MIMO with �o(n, NT ) and SIMO represent the full-multiplexing and full-diversity schemes†††, respectively,while MIMO with �o(n) makes a judicious diversity-multiplexing tradeoff by exciting the optimum number ofdata streams. As demonstrated in Figures 3 and 4, thetradeoff brings significant improvement over SIMO andMIMO with �o(n, NT ).

Finally, it would be instructive to compare the statistics-based design to that with CSIT, whose performance isrepresented by ‘MIMO, CSIT’. It can be observed inFigure 3 that MIMO with CSIT on average achieves a34% improvement in Ro(n) over MIMO with �o(n), while

††† They correspond to transmitting the maximum 11 streams and 1 streamwith 11-branch diversity, respectively.

Figure 6. Average rate in each packet for angular spreadθmax = 0.5, transmit SNR ρ = 30 dB and fading rate factorfmaxTs = 10−2.

the improvement in the maximum AR(ND) is 26% asshown in Figure 4. These results indicate that the loss maybe acceptable if there is no instantaneous channel statefeedback.

7.3. Optimum data block length

As shown in Figure 4, the average rate AR(ND) ismaximised by an optimum data block length ND, whichis defined as Nopt in Equation (23). It can be observed thatNopt for MIMO with �o(n), SIMO and SISO is 22, 6 and2, respectively. Moreover, Nopt for each configuration isalso affected by the number of antennas. Figure 6 showsthe average rate when the maximum number of antennasat one side reduces to 7 for MIMO and SIMO. The Nopt isfound to be 15 for MIMO and 6 for SIMO. As a summaryof the above results, Nopt decreases according to the orderMIMO > SIMO > SISO and decreases for less antennasin each case.

The fading rate affects Nopt as well. Figure 7 shows theaverage rate for fmaxTs = 10−1. Compared with Figure 4,the higher fading rate significantly reduces Nopt, which‡‡‡ is8 and 3 for MIMO and SIMO, since the training signal hasto be sent more frequently to track the channel variation.The higher fading rate also impairs the average rate in eachcase due to the larger channel estimation error.

‡‡‡ The average rate is always zero in case of SISO.



Figure 7. Average rate in each packet for angular spreadθmax = 0.5, transmit SNR ρ = 30 dB and fading rate factorfmaxTs = 10−1.

8. CONCLUSIONS

This work proposes an eigen-domain adaptive spatialmultiplexing scheme for correlated time-varying MIMOchannels. With the knowledge of channel statistics, thetransmitter adjusts the power and rate for each data streamas well as the number of data streams in each symbolperiod, and the data block length is further optimised tomaximise the average rate. Due to the difficulty of the jointoptimisation, the general problem is split into individualoptimisation problems corresponding to different systemcomponents, which are solved sequentially. Major resultsfor the performance of instantaneous and average rates aresummarised below.

� For each antenna configuration, the instantaneous rateRo(n) decreases for larger time index n, while theaverage rate AR(ND) is usually a hill-shape functionimplying an optimum ND;

� The rate-maximising power-shaping matrix �o(n)makes a judicious diversity-multiplexing tradeoff byexciting the optimum number of data streams, whichsignificantly improves the rate;

� Compared with the statistics-based design, the designwith CSIT improves the maximum average rate by 26%.Therefore, the former might be a good complexity-performance tradeoff by avoiding the instantaneousfeedback of CSIT, which will both reduce the resourceand suffer from the feedback imperfections in practice.

Major results on the behaviour of optimum data block lengthNopt are summarised below.

� Nopt decreases according to the order MIMO >

SIMO > SISO and decreases for less antennas in eachcase;

� Higher fading rate factor fmaxTs yields smaller Nopt andreduces both Ro(n) and AR(ND).

APPENDIX

In the Appendix, closed-form solution to the optimum ratein Equation (19) is derived for SISO and SIMO channels. Incase of SISO channel, H(n) reduces to a random variableH(n). The corresponding received SINR in Equation (33)is simplified to (n) = ρ|H(n)|2/�(n), which has chi-square distribution (14-5-22) in Reference [19] with meanof (n) = ρσ2(n)/�(n), where σ2(n) and �(n) are given byEquation (26) and (29) for the special case of NR = NT =1. By averaging BER(i, (n)) defined below Equation (33)over the distribution of (n), we obtain the average BER asBER(i) = 0.2/(1 + 1.5(n)/(2i − 1)), and the maximumnumber of bits satisfying the constraint BER(i) � BERtaris given by

RSISO(n) = ⌊log2

(1.5(n)/ (0.2/BERtar − 1) + 1

)⌋(38)

In case of SIMO channel, H(n) becomes aNR × 1 column vector [H(1, 1, n), . . . , H(NR, 1, n)]T.The received SINR of the single stream is(n) = ρ

∑NR

q=1 |H(q, 1, n)|2/�(q, q, n) with mean of

(n) =∑NR

q=1 ρσ2(q, 1, n)/�(q, q, n) �∑NR

q=1 q(n), where

σ2(q, 1, n) and �(q, q, n) can be specified via Equations(26) and (29) for the special case of NT = 1. Thedistribution of (n) is given by (14-5-26) in Reference[19], and the corresponding average BER can be shown as

BER(i) =NR∑q=1

πq(n)0.2/

(1 + 1.5q(n)

2i − 1

),

where πq(n) =∏j �=q

q(n)

q(n) − j(n)(39)

However, it is difficult to compute the maximum numberof bits satisfying BER(i) � BERtar, since BER(i) doesnot have a closed-form inverse function. Therefore, theoptimum rate still has to be searched numerically according


62 Y. ZHOU

to

RSIMO(n) = maxi∈{0,1,2,...}

i,

s.t. BER(i) = E[BER(i, (n))] � BERtar. (40)

To get a closed-form expression of RSIMO(n), we resort tothe method in Reference [28] and references therein, whichapproximates (n) as Z = aχ2(b), where a is a scalingfactor andχ2(b) denotes a chi-square variable withbdegreesof freedom§§§. The values ofa andb are obtained by equatingthe first two moments of (n) and Z

E[(n)] =NR∑q=1

q(n) = E[Z] = ab,

var[(n)] =NR∑q=1

2q(n) = var[Z] = a22b (41)

which yields a = ∑NR

q=1 2q(n)/

∑NR

q=1 2q(n) and b =2(

∑NR

q=1 q(n))2/∑NR

q=1 2q(n). The distribution of Z can

be derived from (2-1-110) in Reference [19] as

fZ(z) = 1

a2b/2 Gam(b/2)(z/a)b/2−1e−z/(2a), z � 0

(42)

where Gam(·) denotes the Gamma function. The averageBER can be straightforwardly shown as

BER(i) =∫ +∞

0fZ(z) BER(i, z) dz

= 0.2/(1 + 3a/

(2i − 1

))b/2(43)

and the maximum number of bits satisfying BER(i) �BERtar is given by

RSSIMO(n) =

⌊log2

(3a/

[(0.2/BERtar)

2/b − 1]

+ 1)⌋

(44)

where a and b are specified below Equation (41). In case ofSISO channel, a = (n)/2 and b = 2, and Equation (44)is identical to Equation (38). Good agreement between

§§§ Which is a sum of squares of b independent zero-mean unit-varianceGaussian variables.

Equations (44) and (40) have been confirmed by numericalresults, which are not reported due to the space limitation.

ACKNOWLEDGEMENT

This work has greatly benefited from the comments of ProfessorAkbar Sayeed in the Department of Electrical and ComputerEngineering, University of Wisconsin-Madison.

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AUTHOR’S BIOGRAPHY

Yan Zhou (S’00) received the B.E. degree in communications engineering from Beijing University of Posts and Telecommunications,China, in 1997, the Master degree from National University of Singapore, Singapore, in 2001, and the PhD from University of Wisconsin-Madison, USA, in 2006. Since August 2006, he has been working as a senior engineer on physical layer designs at Qualcomm Inc, SanDiego, USA. His research interests include wide-band CDMA and OFDM systems, smart antennas, power control, and multimediatransmission techniques.

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