IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 9, SEPTEMBER 2007 605

Performance Bounds of Orthogonal Space-TimeBlock Codes Over Keyhole Nakagami-m ChannelsNghi H. Tran, Student Member, IEEE, Ha H. Nguyen, Senior Member, IEEE, and Tho Le-Ngoc, Fellow, IEEE

Abstract—The application of Bonferroni-type bounds is in-vestigated for orthogonal space-time block codes over a keyholeNakagami- fading channel, which includes a cascaded Rayleighfading channel as a special case. In particular, upper and lowerbounds on the symbol error rate and bit error rate are derived andshown to be very tight at any signal-to-noise ratio. The developedbounds are applicable for arbitrary signal constellations andmappings and can be accurately computed with single finite-rangeintegrals.

Index Terms—Bit error rate, Bonferroni-type bounds, keyholeNakagami- fading, orthogonal space-time block codes, symbolerror rate.

I. INTRODUCTION

SPACE-TIME trellis codes and space-time block codes areeffective methods to obtain diversity gains in multiple-input

and multiple-output (MIMO) systems. Due to its orthogonalstructure, the maximum-likelihood (ML) decoding of a or-thogonal space-time block code (OSTBC) is very simple [1].Considerable attention has been paid to the analysis of symbolerror rate (SER) and bit error rate (BER) of OSTBCs underconventional fading channels, such as the Rayleigh, Rician,and Nakagami- fading (see, e.g., [2]–[5] and the referencestherein). Recently, both theoretical analysis and experimentalresults [6]–[8] indicate that in many propagation scenarios, key-hole fading is a more suitable channel model. These facts thusbring new challenges in performance analysis under keyholefading channels, especially in the case of space-time codes.

To date, only a few efforts have been devoted to performanceanalysis of space-time coding over keyhole fading channels [4],[9]–[11]. For example, the exact SER of OSTBCs was studied in[9] over a cascaded Rayleigh fading channel. The method in [9],however, can only be applied for standard -PSK and square

-QAM constellations. The work in [9] was later extended in[4] to compute the exact SER of OSTBCs over a keyhole Nak-agami- fading but it has the same drawbacks as in [9]. Ref-erences [10], [11] only concentrate on the pairwise error anal-ysis to examine the performance behavior of space-time codes.To the best of our knowledge, the BER analysis of space-timecodes over keyhole fading channels is not available.

Manuscript received November 17, 2006; revised January 25, 2007. Thiswork was supported by an NSERC Discovery Grant. The associate editor co-ordinating the review of this manuscript and approving it for publication wasProf. Zhengdao Wang.

N. H. Tran and H. H. Nguyen are with the Department of Electrical andComputer Engineering, University of Saskatchewan, Saskatoon, SK S7N 5A9,Canada (e-mail: nghi.tran@usask.ca; ha.nguyen@usask.ca).

T. Le-Ngoc is with the Department of Electrical and Computer Engineering,McGill University, Montreal, QC H3A 2A7, Canada (e-mail: tho@ece.mcgill.ca).

Digital Object Identifier 10.1109/LSP.2007.896167

This letter applies Bonferroni-type bounds [5], [12] to derivethe tight lower and upper bounds on the SER and BER for OS-TBCs over a keyhole Nakagami- fading channel. In partic-ular, the stepwise algorithm of the Kounias lower bound and thegreedy algorithm of the Hunter upper bound introduced in [12]are applied to bound the SER and BER of the systems underconsideration. To this end, the pair-wise error probability (PEP)and the two-dimensional PEP are obtained. It is shown that thesePEPs can be efficiently computed by single finite-range inte-grals. Numerical results show that the developed bounds arevery tight over a wide range of SNR.

II. SYSTEM MODEL

In a space-time block-coded system, each block of infor-mation bits from the information sequence is first mapped toa sequence of symbols , where each canbe one of signal points in a -ary constella-tion . The mapping rule defines the correspondence between

bits and a signal point in . The sequenceis then encoded into an space-time block code ma-trix . Each element , and ,of is a linear combination of and their conju-gates. Here, is the number of transmit antennas and isthe number of rows of the space-time code matrix. The coderate is therefore symbol per channel use. In thecase of OSTBCs, the columns of are orthogonal. Assumethat the receiver is equipped with receive antennas. Overthe th symbol period of each space-time block matrix

, the received signal at the th an-tenna is given as , where

is circularly symmetric complex Gaussian random vari-able of unit variance. The complex coefficient is the pathgain from transmit antenna to receive antenna . Under the as-sumption of keyhole Nakagami- fading, is represented as

[4], where and are

i.i.d. random variables over , and andare i.i.d. Nakagami- random variables with fading severity pa-rameters and , respectively. The density functions ofand are given as follows [4]:

(1)

(2)

with , ; , , and and. Also assume that each variable has unit variance,

i.e., . Note that the cascaded Rayleighfading channel corresponds to the special case of .It is also assumed that the channel matrix withcoefficients remain constant during the transmission ofone space-time matrix, i.e., the channel is quasi-static fading.

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606 IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 9, SEPTEMBER 2007

III. BONFERRONI-TYPE BOUNDS OF THE SER AND BER

Bonferroni-type bounds are basically the bounds on the prob-ability of a union. Let be events definedon a given probability space. Then the probability of the unionevent is , where

and the sum is taken over all. The interesting result is that by truncating the

sum in at any , a lower or upper bound is obtained, de-pending on the sign of the last term. Specifically, ,

, and so on. Two specific ver-sions of the Bonferroni-type bounds are briefly described next.

The Kounias lower bound is given by

(3)

where is any arbitrary subset of . Thebest subset that gives the tightest bound can only be found byan exhaustive search to examine all subsets. To reducethe complexity of the exhaustive search, one can apply an effi-cient stepwise algorithm introduced in [12], whose complexityis linear in the number of events.

For , the tightest Hunter upper bound is

(4)

where is any tree spanning the indices of the set ,is an edge in and is the set of all spanning trees of

indexes. Since the set contains spanning trees, a directsearch to compute (4) is impossible for a medium to large .Instead, the greedy algorithm was applied in [12] to constructthe optimal spanning tree for (4). The detailed steps of thegreedy algorithm can be found in [12].

A. Bonferroni-Type Bounds of the SER

Due to the orthogonal structure of OSTBCs, it can be shownthat a MIMO fading channel can be equivalently represented bya single-input single-output (SISO) channel [13], which makesthe analysis of SER tractable. More specifically, the SER can beexpressed as [5]

(5)

where is the conditional probability of error given thatwas sent and indicates the event that has a larger

metric than [5]. It can be seen from (5) that to obtain the twoBonferrni-type bounds for the SER, one needs to compute thePEP, , and the two dimensional PEP, ,for given , , and .

Given the channel matrix , the conditional PEPfor the case of a OSTBC is given as [5]

(6)

where and . The pa-rameter is the squared Frobenius norm of , whose densityfunction for the case of keyhole Nakagami- fading is [4]

(7)

where is the modified Bessel function of thesecond kind [14]. Invoking the identity

, the conditional PEP is

(8)

By averaging over the pdf of , the unconditional PEP is:

(9)

where is given in (7). Using [15, 6.643.3] and [14,13.1.33], the inner integral is computed as

(10)

where is the confluent hypergeometric function [14].Finally, by substituting (10) into (9), one obtains the followingexpression for the unconditional PEP:

(11)

It can be seen that the integral in (11) is a single finite-range in-tegral. Therefore, the unconditional PEP can be easilycomputed.

The remaining challenge is to compute . Itis well known from [5] and [12] that the conditional two-di-mensional PEP can be expressed in terms of 2-D joint Gaussianfunction. In particular, one has [5]

(12)

where andis the 2-D joint Gaussian function, defined as

(13)

TRAN et al.: PERFORMANCE BOUNDS OF OSTBCS OVER KEYHOLE NAKAGAMI- CHANNELS 607

Similar to [5], by using the single integral representation of (13)in [16] when and are non-negative, one has

(14)

where . By averaging(14) over the pdf of , one obtains the following:

(15)

Observe that the above two-dimensional PEP can also be effec-tively computed by single finite-range integrals. Apply (11) and(15) to the Kounias and Hunter bounds, one obtains the lowerand upper bounds, respectively, for the SER, which shall be re-ferred to simply as the stepwise lower bound and the greedyupper bound.

B. Bonferroni-Type Bounds of the BER

The BER is computed as [5], [12], where is the bit error

probability when is sent. It is determined as follows:

(16)

In the above expression is the Hamming distance be-tween the label of and , which depends on the mappingrule . For the single PEP, , using the same techniqueas in the previous section, one has

(17)where . It should be mentioned that(17) is only valid when . When , one can

Fig. 1. Comparison of greedy upper bound (UB), stepwise lower bound (LB),and the exact analysis of the SER for 2� 2 Alamouti scheme with 8-PSK. Here,m = m = m.

Fig. 2. Star 8-ary and optimal 8-ary constellations. (a) Star 8-ary constellation.(b) Optimal 8-ary constellation.

use the relationship to obtain a similarexpression.

The 2-D PEP, , is given as

(18)

Again, the expression of in (14) is only valid for non-negative and . When at least one of and is negative,

can also be expressed in the form of other func-tions with non-negative arguments [5]. Hence, a similar expres-sion of can still be obtained.

Apply (17) and (18) to the Kounias and Hunter bounds, onehas the upper and lower bounds, respectively, for (16). They leadto the stepwise upper bound and the greedy lower bound on theBER.

608 IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 9, SEPTEMBER 2007

Fig. 3. Comparison of the greedy upper bounds and the stepwise lower boundson the SER for star 8-ary, 8-PSK, and optimal 8-ary constellations.

Fig. 4. Greedy lower bounds and the stepwise upper bounds on the BER for8-PSK with Gray and natural mappings.

IV. ILLUSTRATIVE RESULTS

Consider rate-1 Alamouti OSTBC scheme with

, namely . The number of

receive antennas is assumed to be . The SNR is simplydefined as (dB).

Fig. 1 shows the greedy upper bounds and the stepwise lowerbounds for the SER with 8-PSK constellation over different key-hole Nakagami- fading channels with . Forcomparison, the exact SER computed in [4] is also provided. Itcan be seen that the bounds are very tight at any SNR value. Infact, the bounds basically coincide with the exact SER.

Besides 8-PSK, the star 8-ary and optimal 8-ary constella-tions shown in Fig. 2 are also considered. The optimal 8-aryconstellation is taken from [17] and it is optimal in the sense thatit maximizes the minimum Euclidean distance among all 8-aryconstellations. As mentioned before, the analysis provided in [4]cannot be used to compute the SER for these two constellations.

The upper and the lower bounds of the SER for these twoconstellations and the 8-PSK are plotted in Fig. 3 for differentvalues of . It can be seen again that the derivedupper and lower bounds cannot be visually distinguished at anySNR and for any signal constellation. This means that both theupper and lower bounds can practically provide the exact SER

performance. Based on these tight bounds, it is also observedthat the optimal 8-ary constellation outperforms both 8-PSK andthe star 8-ary constellations. For example, with , thegains at the SER level of over the 8-PSK and star 8-aryconstellations are about 1.4 and 2.2 dB, respectively.

Finally, Fig. 4 presents the lower and upper bounds for theBER with 8-PSK employing Gray and natural mappings. It canbe seen that the two bounds are also very tight and they arepractically the same. The superiority of the Gray mapping overthe natural mapping can also be clearly observed.

V. CONCLUSIONS

Bonferroni-types bounds on the SER and BER were devel-oped for OSTBCs over keyhole Nakagami- fading channels.With simple representations involving single finite-range inte-grals, the bounds can be easily computed with a high accuracy.Numerical results showed that both the lower and upper boundsof the SER and the BER are very tight over a wide range of SNR.In fact, either the upper or the lower bounds can practically yieldthe exact SER and BER performance. The derived bounds areapplicable for arbitrary constellation and mapping.

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