cutoff rate of mimo systems in rayleigh fading channels with imperfect csir and finite frame error...

3292 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 7, SEPTEMBER 2009

Cutoff Rate of MIMO Systems in Rayleigh FadingChannels With Imperfect CSIR and

Finite Frame Error ProbabilityPaul K. M. Ho, Member, IEEE, Kar-Peo Yar, Member, IEEE, and Pooi Yuen Kam, Senior Member, IEEE

Abstract—Using the random coding argument, in this paper,we derive the effective cutoff rate RE of multiple-input–multiple-output (MIMO) space–time (ST) codes in Rayleigh fast-fadingchannels with imperfect channel state information at the receiver(CSIR). In contrast to conventional cutoff rate analysis, where thecodeword length N is infinity and the frame error probabilityPf is implicitly zero, we loosen the definition to include the morerealistic case of finite N and finite Pf . Using the tight upperand lower bounds on the pairwise error probability (PEP) inthe work of Li and Kam, we are able to derive in turn tightlower and upper bounds on the cutoff rate of MIMO ST cod-ing systems. The results are very general and can be appliedto any linear modulation scheme, like M -ary phase-shift keying(MPSK) and M -ary quadrature-amplitude modulation (MQAM).Numerically, we found that for the two-transmit and two-receiveantenna configuration, the nonasymptotic segments of our cutoffrate upper bounds for four, 16, and 64 MQAM actually coincidewith the ergodic capacity curve. Furthermore, we found that theperformance of the Smart-Greedy codes proposed in the seminalwork of Tarokh et al. falls within the range predicted by our cutoffrate bounds. Finally, we show in this paper that for MIMO systemsemploying pilot-symbol-assisted channel estimation, the asymp-totic cutoff rate no longer linearly increases with the number oftransmit antennas. Instead, for a normalized fade rate of fD and asignal constellation of size M , the maximum effective cutoff rate is(8fD)−1 log2 M , which is achieved when the number of transmitantennas is the integer closest to 1/(4fD). The result suggeststhat with a large antenna array and high user mobility, a morebandwidth-efficient channel-estimation strategy is desired.

Index Terms—Cutoff rate, multiple-input–multiple-output(MIMO) systems, pilot-symbol-assisted modulation (PSAM),Rayleigh fast fading channel, space–time (ST) codes.

I. INTRODUCTION

MULTIPLE INPUT–multiple output (MIMO) is a tech-nology that uses multiple antennas at the transmitter and

Manuscript received April 21, 2008; revised November 13, 2008. Firstpublished February 24, 2009; current version published August 14, 2009. Thiswork was supported in part by Singapore MoE AcRF Tier 2 under GrantT206B2101. This paper was presented in part at the IEEE Conference onMilitary Communications, San Diego, CA, November 17–19, 2008. The reviewof this paper was coordinated by Prof. H. H. Nguyen.

P. K. M. Ho is with the School of Engineering Science, Simon FraserUniversity, Burnaby, BC V5A 1S6, Canada (e-mail: [email protected]).

K.-P. Yar is with the Institute for Infocomm Research, Agency forScience, Technology and Research, Singapore 138632 (e-mail: [email protected]).

P. Y. Kam is with the Department of Electrical and Computer Engineer-ing, National University of Singapore, Singapore 117576 (e-mail: [email protected]).

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.2009.2015669

receiver to achieve space–time (ST) diversity, spatial multiplex-ing, or beamforming (precoding) in wireless communicationchannels. As demonstrated in [1]–[3], MIMO can providedramatic capacity improvement over conventional single-input–single-output (SISO) transmission in a fading environ-ment without increasing the transmission bandwidth. Since thereport of these seminal works, there has been an explosion ofresearch activities in the field, many of which address MIMOfrom an information-theoretic perspective, namely, determiningthe capacity of MIMO under different modes and operatingconditions; see, for example, [4]–[10]. While much insight intothe design of practical MIMO systems can be gained from theseperformance limits, there are also some limitations associatedwith capacity analysis. First, with imperfect channel state in-formation (CSI), the capacity of MIMO in fading channels isstill not precisely known. Only bounds are available; see, forexample, [11] and [12]. Because of fading, noise, and limitedbandwidth available for channel sounding and feedback, onewould expect that the CSI estimates available at the transmitterand/or receiver to be imperfect all the time. Second, even whenthe CSI is perfect, the capacity-achieving modulation followsa Gaussian distribution, which is challenging to implement inpractice. In the actual deployment of MIMO, fixed constel-lations will always be used. Third, in any capacity analysis,the block length of the code is implicitly taken to be infinityalthough, in actual systems, the codeword length is alwaysfinite. For example, in the seminal work of Tarokh et al. [13],the authors focus on the frame error rates of ST trellis codeswith a frame size of 130 symbols.

Because of the aforementioned issues, we consider the adop-tion of a cutoff-rate-like parameter RE as an alternative mea-sure of the performance limit of MIMO with finite codewordlength and fixed frame error probability (FEP). To the best ofour knowledge, there are only a limited number of cutoff-rateanalyses for MIMO systems in the literature; see, for example,[14] and [15]. Furthermore, given that the modulation is fixedin any cutoff rate analysis, the incorporation of imperfectCSI in the analysis is straight forward; see, for example, theanalyses in [16] and [17] for SISO systems. In contrast, thecapacity of MIMO with imperfect CSI is still not known.The cutoff rate that we derive is based on the new boundsfor the pairwise error probability (PEP) of ST codes in [18].We prefer these bounds over the exact PEP [19] in the cutoffrate analysis because they are tight, simple, and intuitive. Moreimportantly, the results are implicit functions of the codewordlength N .

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HO et al.: CUTOFF RATE OF MIMO SYSTEMS IN RAYLEIGH FADING CHANNELS 3293

This paper is organized as follows. The MIMO system modeland the corresponding upper and lower bounds on PEP aresummarized in Section II. Based on these PEP bounds, theupper and lower bounds on the effective cutoff rate, as functionsof the codeword length and fixed FEP, are derived in Section III.These bounds on the effective cutoff rate are expressed inanalytical as well as integral forms, where the latter is moresuitable for long codeword lengths. An application of the cutoffrate analysis to pilot-symbol-assisted modulation (PSAM) isprovided in Section IV, where we highlight the potential ratelosses caused by a lowering of the effective channel signal-to-noise ratio (SNR) because of imperfect channel estimation andby the insertion of pilot symbols. Numerical results for differentMIMO configurations and CSI assumptions are provided inSection V. Comparisons with the cutoff rates obtained fromthe Chernoff bound on the PEP and with capacity are madewhen applicable. Finally, the conclusions of this investigationare provided in Section VI.

II. SYSTEM MODEL AND PEP

In this paper, we consider an ST-coded MIMO system usingT antennas to transmit modulation symbols from multiple usersover a Rayleigh fading channel using time-division multiplex-ing (TDM) [18]. The symbol transmitted by antenna i of anyuser during the kth TDM frame is denoted by vi(k) and is takeneither from a unit-energy M -ary phase-shift keying (MPSK)constellation SMPSK = {exp(j2πm/M)}M−1

m=0 or from a unit-energy M -ary quadrature-amplitude modulation (MQAM) con-stellation, i.e.,

SMQAM ={(

2m−1−√

M)

μ−j(2n−1−

√M

)μ}√

M

m,n=1

where μ =√

1.5/(M − 1). Each receiver is equipped withR receive antennas, and the fading gains in the T × R linksbetween the transmitter and receiver are independent andidentically distributed (i.i.d.) zero-mean complex Gaussianprocesses with variance σ2

h in both real and imaginary com-ponents. Mathematically, the kth transmitted ST codewordsegment v(k) = (v1(k), v2(k), . . . , vT (k))T , k = 1, 2, . . . , N ,of each user is related to its received version r(k) =(r1(k), r2(k), . . . , rR(k))T according to

r(k) =√

EsH(k)v(k) + n(k) (1)

where N is the codeword length;√

Es is the average sym-bol energy; H(k) = [hij(k)] is the channel fading matrix,where, for any given k, hij(k)’s are all i.i.d., and CN(0, 2σ2

h);and n(k) is an additive white Gaussian noise vector and isCN(0, N0IR) distributed. Since we assume that the TDMframe is longer than the coherent time of the channel, differentcodeword segments of the same user experience independentfading.

In [18], the authors derived upper and lower bounds on thePEP for the ST-coded system in (1) with maximum-likelihood(ML) detection and pilot-symbol-assisted channel estimation.Specifically, for an erroneous codeword {v(k)}N

k=1 having dis-tances {d2

j (k); j = 1, 2, . . . , T}Nk=1 with the transmitted code-

word {v(k)}Nk=1, the PEP P2 is lower and upper bounded as

follows [18, eqs. (9) and (11)]:

P2 ≥ (2q − 1)!!2(2q)!!

∏k∈κ

⎛⎝1 +

Es

∑Tj=1 σ2

j d2j (k)

2(N0 + 2Es

∑Tj=1 σ2

j

)⎞⎠

−R

P2 ≤ (2q − 1)!!2(2q)!!

∏k∈κ

⎛⎝ Es

∑Tj=1 σ2

j d2j (k)

2(N0 + 2Es

∑Tj=1 σ2

j

)⎞⎠

−R

(2)

where N is the codeword length, κ is the set of symbol positionswhere the two codewords are different, q = |κ|R is the totaldiversity order, 2q!! = 2q · (2q − 2) · · · 2, (2q − 1)!! = (2q −1) · (2q − 3) · · · 1, 2σ2

j , j = 1, 2, . . . , T are the variances of thechannel estimates, and 2σ2

j , j = 1, 2, . . . , T are the variancesof the estimation errors. Since we assume identical links,2σ2

j = 2σ2h

, and 2σ2j = 2σ2

e ; j = 1, 2, . . . , T . Furthermore, bydefining the transmit SNR to be Γ = TEs/N0 and the effectiveSNR to be

γ =2EsTσ2

h

N0 + 2EsTσ2e

=

(2σ2

h

1 + 2σ2eΓ

)Γ (3)

the upper and lower bounds in (2) can be simplified, respec-tively, to

PUB =(2q − 1)!!2(2q)!!

∏k∈κ

⎛⎝ γ

4T

T∑j=1

d2j (k)

⎞⎠

−R

(4)

PLB =(2q − 1)!!2(2q)!!

∏k∈κ

⎛⎝1 +

γ

4T

T∑j=1

d2j (k)

⎞⎠

−R

. (5)

It is evident from these equations that the upper and lowerbounds are tight at large SNR, as γ/(4T )

∑Tj=1 d2

j (k) ≈ (1 +γ/(4T )

∑Tj=1 d2

j (k)) when γ → ∞.

III. CUTOFF RATE

As stated earlier, we are interested in determining the cutoffrate of the MIMO system described in the previous sectionunder the conditions of a finite codeword length N and a finitetargeted FEP Pf . Since, traditionally, the cutoff rate R0 is onlydefined for systems with infinite N and implicitly zero Pf [20],our result is, thus, not exactly the cutoff rate in the conventionalsense. In the absence of a better terminology, we simply referto it as the effective cutoff rate RE .

First, consider the PEP P2 in (2). The average of this pa-rameter, over all random choices of codeword pairs, can bewritten as

E[P2] = 2−RCN (6)

where E[·] is the expectation operator, and

RC = − 1N

log2 (E[P2]) (7)


TABLE IRATE LOSS AS A FUNCTION OF CODEWORD LENGTH AND TARGETED FEP

is the error exponent of the average PEP. It follows that theaverage word/FEP takes on the form [21]

E[PS ] = 2RNE[P2] = 2−(RC−R)N (8)

where R is the actual coding rate (in bits per symbol), andE(R,N) = RC − R is the corresponding error exponent [22].If N is infinite, then any coding rate up to RC will yield azero E[PS ]. Consequently

R0 = limN→∞

RC

is the conventional definition of the cutoff rate [20]. On theother hand, when N is finite, there is no guarantee thatE[PS ] = 0 even when the actual coding rate R is less thanRC . This means that the error exponent RC alone cannot beregarded as a rate limit. However, it can be converted into onethrough the following two-step procedure.

As pointed out in [13], the coded systems are typicallydesigned based on a targeted frame FEP Pf . By setting theright-hand side (RHS) of (8) to be less than Pf , we can seethat this FEP target is met when the transmission rate R isless than

R∗C = RC − ε; ε = − 1

N

log(Pf )log 2

(9)

where ε represents the rate reduction due to the finite wordlength effect. We list in Table I the value of this reductionfor various combinations of N and Pf . As an example, witha relatively stringent requirement of Pf = 10−3, the rate reduc-tion is 0.1557 bits/symbol for a short codeword length of 64.To put this rate loss in perspective, we compare it against theasymptotic value of RC . As shown later in this section, at alarge SNR, RC approaches T · log2(M) + 1/N . This meansfor T = 2 and M = 2 that the rate loss previously mentionedis 7.8% of RC . The loss as a percentage becomes even less sig-nificant when more antennas and/or a larger signal constellationare used. Note also that when N → ∞, the absolute value of therate loss itself approaches zero for any practical FEP of interest.Consequently, R∗

C approaches RC , which itself approaches thetrue cut off rate R0.

The second and final step required to convert RC into ameaningful rate limit is the quantization of R∗

C . Assume anM -ary modulation over each of the T transmit antennas. Then,there are altogether MTN code patterns, and we can chooseI out of these MTN patterns to form a code of rate R =log2(I)/N bits/symbol, 1 < I ≤ MTN . For example, whenM = 2, T = 1, and N = 3, there are altogether eight binary

patterns of length 3, and the only possible code rates are0.33, 0.53, 0.67, 0.77, 0.86, 0.94, and 1. With this observationin mind, it becomes clear that for a finite N , the maximumtransmission rate in (9) should be modified to

RE =1N

log2

⌊2NR∗

C

⌋(finite N, non-zero FEP) (10)

where ·� is the floor operator. We will refer to (10) as theeffective cutoff rate, as it takes into account all the finite wordlength and finite FEP effects. We next proceed to determine thelower and upper bounds on RC in (7), which, in turn, will beused to obtain the corresponding bounds on RE .

First, consider the upper bound on the PEP in (4). Using arandom coding argument, the expected value of this bound is

E[PUB] =N∑

k=0

Pr[q = kR]E [PUB|q = kR]

=pN

2

(1 +

N∑k=1

(N

k

)(1p− 1

)k (2kR − 1)!!(2kR)!!

αk

)

(11)

where

p = M−T (12)

is the probability that∑T

j=1 d2j (k) = 0, and

α = E

⎡⎢⎣⎛⎝ γ

4T

T∑j=1

d2j (k)

⎞⎠

−R ∣∣∣∣∣∣T∑

j=1

d2j (k) �= 0

⎤⎥⎦ . (13)

In the Appendix, we provide analytical expressions for theterm α for MPSK and MQAM modulations [refer to (A3), (A6),and (A11)]. Now, since the “N chooses k” function in (11)rapidly increases with N , directly calculating the sum in (11)is problematic when N is large (e.g., N > 170 in Matlab 6running on an Intel Core 2 CPU-based PC). A more “robust”approach is to view the sum in (11) as the z-transform of theproduct of two signals in the time domain. Specifically, let usdefine

a[n] ={

pN

2

(N

n/R

), n = 0, R, 2R, . . . , NR

0, otherwise

A(v) =NR∑n=0

a[n]v−n =12pN (1 + v−R)N (14)

b[n] =(2n − 1)!!

(2n)!!, n = 0, 1, . . .

B(v) =∞∑

n=0

b[n]v−n =1√

1 − v−1, |v| > 1 (15)

z−1 =((

1p− 1

)α

)1/R

=((MT − 1)α

)1/R(16)


where the transform in (15) is taken from [23]. This means(11) can be written as E[PUB] =

∑∞n=0 a[n]b[n]z−n, which

is simply the z-transform C(z) of the product signal c[n] =a[n]b[n]. This transform can be written as a contour integral

E[PUB] =C(z) =1

2πj

∮C

A(v)B(z/v)v−1dv

=12

12πj

∮C

pN (1 + v−R)N 1√1−v

z

v−1dv (17)

where the contour must lie within the region of convergence ofB(z/v), i.e., |v| < z. Once E[PUB] is obtained, it can readilybe used in place of E[P2] in (7) to obtain a lower bound onRC . The latter bound is further used in (10) to provide a lowerbound on the effective cutoff rate RE .

Similarly, we can express the average of the lower bound onPEP (5) over random codes in integral form as

E[PLB] =12

12πj

∮C

pN(1 + v−R

)N 1√1−vZ

v−1dv, |v| < Z

(18)

where

Z−1 =((

1p− 1

)β

) 1R

(19)

β = E

⎡⎢⎣⎛⎝1 +

γ

4T

T∑j=1

d2j (k)

⎞⎠

−R ∣∣∣∣∣∣T∑

j=1

d2j (k) �= 0

⎤⎥⎦ . (20)

The reader is again referred to the Appendix for the analyticalexpressions of β for various modulations. Finally, an upperbound on RE can be obtained by first replacing the averagePEP in (7) by E[PLB] and then substituting the resultant upperbound on RC into (9) and (10).

IV. PSAM AND RATE REDUCTION

The cutoff rate analysis presented in the previous sectionsis based on pilot-symbol-assisted channel estimation, whereimperfection in the channel estimates is reflected in the ef-fective SNR γ obtained from the variances of the CSI 2σ2

h

and the estimation error 2σ2e [refer to (3)]. The pilot-assisted

estimator itself can operate based on the least square [19] orthe minimum mean-square error (MMSE) principle [24]. In thefollowing, we derive the effective SNR of an MMSE-basedMIMO channel estimator, which is similar to the estimatorused in the SISO PSAM system in [24]. Channel estimation inthe system is performed prior to deinterleaving, which requiresthe transmitted data stream to be organized into frames of Ksymbols, T of which are pilot, and the rest being data. Thepilot insertion frequency 1/K must be at least 2fD, where fD isthe Doppler frequency of the fading channel normalized by thesymbol rate. For simplicity, we assume that the T pilot symbolsin each frame take on the values of

√Tu1,

√Tu2, . . . ,

√TuT ,

where ui is a unit-basis vector with its ith component equal

to unity and zero otherwise. In other words, the transmitteronly engages one transmit antenna at a time during the pilotinterval, but each antenna is to transmit at T times the powerlevel it normally does during data transmission. This pilot-encoding scheme ensures proper energy normalization acrossthe pilot and data transmission phases, and more importantly,it guarantees that the pilot signals arriving at the receiver fromdifferent transmit antennas are separable. The T pilot symbolswithin each frame are transmitted in succession, and the fad-ing variation within each T -length pilot interval is assumednegligible. Once the channel estimates at these pilot insertioninstants are obtained, they are fed to a bank of interpolationfilters for generating the channel estimates at the data positions.Wiener filters designed based on a Jakes power spectrum andthe normalized Doppler frequency fD are used for this purpose.For convenience, we use the mean-square value of the channelestimate and the mean-square estimation error at the mid-frameposition as representative values of 2σ2

hand 2σ2

e , respectively.In practice, there will be a small degree of variation in the CSIquality across a frame (on the order of 1/4 dB). As demonstratedin [24] for PSAM, 2σ2

h= 2σ2

h · ρ2, and 2σ2e = 2σ2

h · (1 − ρ2),

where ρ is the correlation coefficient between hij and hij (whenthe estimator input SNR is Γ = TES/N0). Consequently, theeffective SNR in (3) can be rewritten as

γ =ρ2 · Γ

1 + (1 − ρ2) · Γ . (PSAM-MIMO). (21)

See [24] for an expression of ρ. All the numerical resultspresented in the next section on PSAM-MIMO were generatedusing the foregoing definition of SNR. Furthermore, the rateRE in (10) is multiplied by the factor

η = 1 − T/K (22)

to reflect the drop in throughput due to the insertion of pilotsymbols.

The multiplication of the rate in (10) by the foregoingfactor has an interesting implication on the maximum rate thatMIMO-PSAM can achieve. Consider the following asymptoticanalysis. When Γ → ∞, then ρ → 1, γ = Γ, and the parametersα and β in (13) and (20) are both equal to zero. As a conse-quence, both upper and lower bounds on the PEP are equal topN/2, and (10) approaches − log2 p = T log2 M when N islarge. Multiplying T log2 M by the factor in (22) allows us toobtain the maximum effective rate of MIMO-PSAM as

R∗E,PSAM(T ) = (1 − T/K) · T log2 M. (23)

By treating the number of transmit antennas as a continuousvariable, (23) becomes a quadratic function in T and is largestat T = K/2. At this value of T , R∗

E,PSAM(T ) = K · log2 M/4.Since 1/K ≥ 2fD because of the Nyquist sampling require-ment in PSAM, this means that the effective rate of MIMO-PSAM is always upper bounded by

R∗E,PSAM = log2 M/(8fD) (24)


independent of the number of transmit antenna. In comparison,the capacity and the asymptotic cutoff rate of MIMO withperfect CSI indefinitely increase with the number of transmitantennas. This implies that PSAM is not an efficient channelestimation strategy for high-mobility applications involving alarge antenna array. However, for low mobility and a smallnumber of antennas, we may still get a substantial increase inthroughput by adding an extra antenna. We are further remindedthat in PSAM, we are trading transmission efficiency for lowimplementation complexity. If we are willing to sacrifice im-plementation complexity instead, then it is possible to performblind channel estimation without resorting to pilot symbols,which allow the rate to (theoretically) increase with the numberof transmit antennas. However, these blind channel estimatorsusually iterate between data detection and channel estimationand are, thus, computationally intensive. Furthermore, any er-roneous decision on the data may severely degrade the qualityof the subsequent channel estimates. The result is a drop inthe effective SNR γ and, hence, a lowering of the maximumachievable rate from the theoretical limit.

V. NUMERICAL RESULTS

In this section, we present the numerical results for theeffective cutoff rate RE of ST-coded MIMO systems operatingin fast-fading channels. Both MPSK and MQAM modulationsare considered. The cutoff rates of these MIMO systems areplotted against the transmit SNR Γ. The received SNR γ usedin the numerical calculations is obtained under the conditionthat 2σ2

h = 1. As can be seen from (3), when there is nochannel-estimation error, i.e., when 2σ2

h= 2σ2

h and 2σ2e = 0,

then γ = Γ. We would also like to point out that the numericalintegrations in (17) and (18) were carried out using circularcontours of radius 0.99z and 0.99Z, respectively. These choicesprovide numerical stability over all N and SNR considered. Ingeneral, the smaller N is, the less stringent it is in choosing thecontour’s radius.

Included in all the figures of this section are the ergodiccapacity curves of MIMO with perfect CSI at the receiver(CSIR). These curves were obtained from the simulation of[1] as

C = EH

[− log2

∣∣∣IR +γ

THH†

∣∣∣] (25)

where H is the channel matrix in (1), IR is an identity matrixof size R, and | · | and (·)† denote, respectively, the determinantand conjugate transpose of a matrix. We are cautioned thatwhen comparing the cutoff rate results against capacity, one hasto bear in mind that the former is based on a fixed FEP, whilethe latter implicitly assumes zero error probability.

A. SISO-BPSK

We first consider the cutoff rate of the simplest system: SISO,BPSK signaling, and perfect channel estimation. Our upperand lower bounds on RE for N = 16 and 1024 (“short” and“long” codewords) are shown in Fig. 1 for a targeted FEP of

Fig. 1. Our upper/lower bounds and the Chernoff lower bound on the effectivecutoff rate for SISO-BPSK at an FEP of Pf = 10−3 and perfect CSIR.

Fig. 2. Upper bounds on the effective cutoff rate for SISO-BPSK withdifferent codeword lengths at an FEP of Pf = 10−3 and perfect CSIR.

Pf = 10−3. Also shown in Fig. 1 is the Chernoff lower boundon RE , which is obtained from [15] as

RC = 1 − log2

(1 +

11 + γ

); Chernoff-based SISO-BPSK

(26)

into (9) and (10). It is clear from the figures that our bounds aremuch tighter than the Chernoff bound when N is small. On theother hand, for a larger N , like N = 1024, the Chernoff lowerbound almost coincides with or is equal to our upper bound atthe practical channel SNR of interest. Therefore, both are tight!At a rate of 0.5 bits/symbol, the gap between the cutoff rateof the N = 1024 system and capacity is approximately 1.5 dB.The gap is widened to 5 dB when the rate is 0.8 bits/symbol.

Fig. 2 plots our upper bound on RE with N as a parameter. Itis evident from the figure that the cutoff rate quickly increaseswith N . At a large enough N , the rate loss due to a finitecodeword length becomes insignificant.


Fig. 3. Effective cutoff rate of MIMO-QPSK with two transmit and tworeceive antennas, perfect CSIR, and an FEP of Pf = 10−3.

Fig. 4. Effective cutoff rate of MIMO-QPSK with two transmit antennas, acodeword length of N = 128, perfect CSIR, and an FEP of Pf = 10−3.

B. Effect of MIMO Configuration

We next consider the cases of two transmit antennas withfirst- and second-order receive diversity. These configurationsare denoted as 2 × 1 and 2 × 2 MIMO, respectively. Theupper bounds on the cutoff rate for the 2 × 2 configurationwith QPSK modulation, perfect CSIR, and an FEP of Pf =10−3 are shown in Fig. 3 with the codeword length N as aparameter. For a rate of less than 2.5 bits/symbol, these cutoffrate curves run parallel to the capacity curve. In particular, whenN = 1024, the cutoff rate upper bound and the capacity curveare almost indistinguishable. On the other hand, when N = 16,the difference between capacity and cutoff rate is approximately2 dB in the SNR.

Focusing on a codeword length of N = 128, in Fig. 4, weshow the capacity and cutoff rates of 2 × 1 and 2 × 2 MIMO-QPSK with perfect CSIR and Pf = 10−3. In addition, theperformance of the smart-greedy codes from [13, p. 763] withthe same FEP and approximately the same N are shown. It is

Fig. 5. Effective cutoff rate of MIMO-MQAM with two transmit and tworeceive antennas, a codeword length of N = 64, perfect CSIR, and an FEPof Pf = 10−3.

observed that in the 2 × 2 case, the performance of the smart-greedy code lies within our upper and lower bounds on theeffective cutoff rate. However, in the 2 × 1 case, the smart-greedy code is 2.5 dB worse than that predicted by our lowerbound. We do not consider this 2.5-dB deviation a discrepancy;it just alludes to the fact that this simple two-state 2 × 1 Smart-Greedy is not “optimal,” and more efficient codes do exist andare waiting to be found.

C. Effect of Constellation Size

We show in Fig. 5 the cutoff rates of 2 × 2 MIMO with4QAM, 16QAM, and 64QAM codes of length N = 64 andperfect CSIR. It is observed that the nonasymptotic segments ofthe cutoff rate upper bounds are superimposed on one anotherand coincide with the capacity curve over ranges of SNRs.While the superposition suggests that 64QAM is always thebest choice among the three modulations, the intersection ofthe lower bounds hints that a smaller constellation is favoredover a larger constellation at lower SNR. Specifically, when theSNR is below 12 dB, then 4QAM is preferred over the two othermodulations. On the other hand, when the SNR is above 19 dB,then 64QAM should be used. Finally, when the SNR is betweenthe two SNR thresholds, 16QAM is the best choice.

D. Effect of Imperfect CSIR

The numerical results provided in the last three sections arefor the ideal case of perfect CSI at the receiver. We next con-sider the effect of imperfect channel estimation using PSAMwith interpolators of size 12 and pilot frames of size 1/(2.5fD).Fig. 6 shows the effects a normalized Doppler frequency offD = 0.03 has on the 2 × 2 MIMO-MQAM systems in Fig. 5.In comparing the two figures, we notice that, asymptotically,there is a 10%–12% drop in rate because of imperfect CSI andthe insertion of pilot symbols. In the nonasymptotic operatingregions, the gap between perfect (see Fig. 5) and imperfect CSI(see Fig. 6) is in the range of 3–5 dB.


Fig. 6. Effective cutoff rate of MIMO-PSAM with MQAM, two transmit andtwo receive antennas, a codeword length of N = 64, an FEP of Pf = 10−3,and a normalize Doppler frequency of fD = 0.03.

Fig. 7. Effective cutoff rate of MIMO-PSAM with 64QAM, two transmit andtwo receive antennas, a codeword length of N = 64, and an FEP of Pf =

10−3, with the normalized Doppler frequency fD as a parameter.

Finally, in Fig. 7, we vary the normalized Doppler frequencyand observe the corresponding rate changes in a 2 × 2 MIMO64QAM PSAM system. For simplicity, only upper bounds onRE and the capacity curve (with perfect CSIR) are shown.Using the cutoff rate for perfect CSI as a reference, it isobserved that, asymptotically, a 5% normalized Doppler causesalmost a 25% drop in rate. The drop caused by a 1% Doppler,on the other hand, is only about 6%.

VI. CONCLUSION

As opposed to most of the information-theoretic work onMIMO, which focus on the system capacity, we have consid-ered in this paper an alternative measure, namely, the cutoff rateof the system. The main rationales of adopting the cutoff rateas the performance limit are the following: 1) In any practicalsystem, the modulation is fixed, and 2) incorporating imperfect

CSI in the analysis is straight forward. In terms of cutoff rateanalysis, our work deviates from those in the literature throughloosening of the cutoff rate definition to include finite codewordlength and finite FEP. Through using the tight upper and lowerbounds on the PEP in [18], we are able to derive in turn tightlower and upper bounds on the cutoff rate of MIMO ST codingsystems. Specifically, we found that segments of the cutoffrate upper bounds of MIMO-MQAM coincide with the ergodiccapacity curve. Furthermore, we found that the performance ofthe Smart-Greedy codes proposed by Tarokh et al. [13] lieswithin the range predicted by our cutoff rate bounds. Finally,we have shown in this paper that for MIMO systems employingpilot-symbol-assisted channel estimation, the asymptotic cutoffrate no longer indefinitely increases with the number of transmitantennas. Instead, for a normalized fade rate of fD and a signalconstellation of size M , the maximum effective cutoff rate islog2 M/(8fD), which is achieved when the number of transmitantennas is the integer closest to 1/(4fD). The result suggeststhat with a large antenna array and high user mobility, a morebandwidth-efficient channel estimation strategy should be usedinstead.

APPENDIX

In this Appendix, we derive the terms α and β in (13) and(20), respectively. First, consider

Δ =T∑

j=1

d2j (k) (A1)

the square Euclidean distance between a pair of coded symbols.With BPSK signaling and the random coding argument, eachd2

j (k) is a binary random variable with a probability densityfunction (pdf) of pd2

j(x) = (1/2)(δ(x) + δ(x − 4)), where δ(·)

is the discrete-time unit impulse function. The characteristicfunction (CF) of d2

j (k) can be described in terms of the z-transform of its pdf, which yields Pd2

j(z) = (1/2)(1 + z−4).

Since Δ is the sum of T number of i.i.d. d2j (k)’s, its CF is

simply PΔ(z) = 2−T (1 + z−4)T . In other words, the parameterΔ is binomial distributed with a pdf

pΔ(x) =12T

T∑t=0

(T

t

)δ(x − 4t) (BPSK). (A2)

Since α and β are the averages of (γΔ/4)−R and (1 +γΔ/4)−R given that Δ �= 0, respectively, we can deduce from(A2) that

α =1

2T − 1

T∑t=1

(T

t

)(γt

T

)−R

(BPSK) (A3)

β =1

2T − 1

T∑t=1

(T

t

)(1 +

γt

T

)−R

(BPSK). (A4)

Next, for QPSK, the random-coding argument leads to a pdfof pd2

j(x) = (1/4)(δ(x) + 2δ(x − 2) + δ(x − 4)) and a CF of

Pd2j(z) = (1/4)(1 + z−2)2 for d2

j (k). The CF of Δ is, thus,


PΔ(z) = 2−2T (1 + z−2)2T . Taking the inverse z-transform ofPΔ(z) yields the pdf of Δ as

pΔ(x) =1

22T

T∑t=0

(2T

t

)δ(x − 2t) (QPSK). (A5)

As a result, the average values of (γΔ/4)−R and (1 +γΔ/4)−R, respectively, given that Δ �= 0, are

α =1

22T − 1

2T∑t=1

(2T

t

)(γt

2T

)−R

(QPSK) (A6)

β =1

22T − 1

2T∑t=1

(2T

t

)(1 +

γt

2T

)−R

(QPSK). (A7)

In comparing (A6) and (A7) with (A3) and (A4), we concludethat the α and β parameters for QPSK are those of BPSK withtwice as many transmit antennas.

What about MQAM? This signal constellation is theCartesian product of two identical Q-ary pulse-amplitude mod-ulation (PAM) constellations {Ai}Q

i=1 and {Bj}Qj=1, where

Q =√

M . The ith signal levels in the two Q-ary PAM con-stellations are Ai = Bi = (2i − 1 − Q)μ, i = 1, 2, . . . , Q. Forthe MQAM constellation to have unit average energy, it isrequired that 2E[A2

i ] = 1, or μ =√

1.5/(M − 1). Since thesquare distance between the ith and jth signal levels in {Ai}Q

i=1

is 4μ2(i − j)2, we can deduce that the square distance betweensymbol pairs randomly drawn from the constellation, afternormalization by (2μ)2, has a pdf of

pA(x) =1

Q2

[Q · δ(x) + 2

Q−1∑d=1

(Q − d) · δ(x − d2)

](A8)

and a CF of

PA(z)=1

Q2

[Q+2

Q−1∑d=1

(Q−d)z−d2

]≡

(Q−1)2∑k=0

akz−k (A9)

where the ak’s can be obtained by equating the coefficients withthe same powers of z on both sides of the second equality. Theresult means that the CF of Δ/(2μ)2 is simply

P 2TA (z) =

2T (Q−1)2∑k=0

ckz−k (A10)

where the sequence (c0, c1, . . . , c2T (Q−1)2) is the sequence(a0, a1, . . . , c(Q−1)2) convolving with itself 2T times. Once theck’s are found, the terms α and β in (13) and (20) become

α =1

1−c0

2T (Q−1)2∑k=1

ck

( γ

4T(4kμ2)

)−R

=1

1−c0

2T (√

M−1)2∑k=1

ck ·(

3γ

2(M−1)Tk

)−R

(MQAM)

(A11)

β =1

1−c0

2T (Q−1)2∑k=1

ck

(1+

γ

4T(4kμ2)

)−R

=1

1−c0

2T (√

M−1)2∑k=1

ck ·(

1+3γ

2(M−1)Tk

)−R

(MQAM).

(A12)

Finally, for any signal constellation with square distancesbetween symbol pairs that cannot be expressed as integermultiples of a basic quantity, the term Δ has a multinomialdistribution. Suppose there are J distinct square distances,D1,D2, . . . , DJ , between symbol pairs in the signal constel-lation. If under the random coding argument these squaredistances occur with probabilities p1, p2, . . . , pJ , then the prob-ability that Δ =

∑Tj=1 d2

j (k) =∑J

i=1 niDi is

Pr

[Δ =

J∑i=1

niDi

]

={

T !n1!,n2!,...,nJ !p

n11 , pn2

2 , . . . , pnJ

J ,∑J

i=1 ni = T0, otherwise

(A13)

for nonnegative integers n1, n2, . . . , nJ . The result can then bereadily used in (13) and (20) to compute α and β.

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Paul K. M. Ho (S’84–M’85) received the B.A.Sc.degree in electrical engineering from the Universityof Saskatchewan, Saskatoon, SK, Canada, in 1981and the Ph.D. degree in electrical engineering fromQueen’s University, Kingston, ON, Canada, in 1985.

Since 1985, he has been with the School of En-gineering Science, Simon Fraser University (SFU),Burnaby, BC, Canada, where he is currently a Pro-fessor. He had taken leaves from SFU and was withGlenayre Electronics, Vancouver, BC, between May1992 and September 1993 and with the Electrical

and Computer Engineering Department, National University of Singapore,Singapore, in 2000 and 2001. He is the holder of a number of U.S. patentsand has been a consultant to a number of companies in Canada and abroad. Hisresearch interests include noncoherent detection in fading channels, coding andmodulation, space–time coding and signal processing, channel estimation, andperformance analysis.

Dr. Ho is a Registered Professional Engineer in the province of BritishColumbia.

Kar-Peo Yar (M’08) received the B.Eng. (Hons)and M.Sc. degrees in electrical engineering from theNational University of Singapore, Singapore, in 1998and 2001, respectively, and the Ph.D. degree in elec-trical and computer engineering from the Universityof Michigan, Ann Arbor, in 2007.

Since 2007, he has been with the Institute for Info-comm Research, Singapore, where he is currently aResearch Engineer. His research interests consist ofcoding and modulation, energy analysis, and cross-layer design.

Dr. Yar was awarded the National Science Scholarship by the Agency forScience, Technology, and Research to study at the University of Michiganin 2001.

Pooi Yuen Kam (M’83–SM’87) was born in Ipoh,Malaysia, in 1951. He received the B.S., M.S., andPh.D. degrees in electrical engineering from theMassachusetts Institute of Technology, Cambridge,in 1972, 1973, and 1976, respectively.

From 1976 to 1978, he was a Member of TechnicalStaff with the Bell Telephone Laboratories, Holmdel,NJ, where he was engaged in packet network stud-ies. Since 1978, he has been with the Departmentof Electrical and Computer Engineering, NationalUniversity of Singapore, Singapore, where he is

currently a Professor. He served as the Deputy Dean of Engineering and theVice Dean for Academic Affairs with the Faculty of Engineering, NationalUniversity of Singapore, from 2000 to 2003. He spent the sabbatical year1987 to 1988 with the Tokyo Institute of Technology, Tokyo, Japan, under thesponsorship of the Hitachi Scholarship Foundation. In 2006, he was invitedto the School of Engineering Science, Simon Fraser University, Burnaby, BC,Canada, as the David Bested Fellow. His research interests are in detection andestimation theory, digital communications, and coding.

Dr. Kam is a member of Eta Kappa Nu, Tau Beta Pi, and Sigma Xi.Since 1996, he has been the Editor for the modulation and detection forwireless systems of the IEEE TRANSACTIONS ON COMMUNICATIONS. Hewon the Best Paper Award at the 2004 IEEE Vehicular Technology SocietyFall Conference.

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