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Accurate Error-Rate Performance Analysis of OFDM on Frequency-Selective NakagamiFading Channels
m
Zheng Du, Member, IEEE, Julian Cheng, Member, IEEE, and Norman C. Beaulieu, Fellow, IEEE
Abstract—Error rates of orthogonal frequency-division multiplexing (OFDM) signals in multipath slow fading Nakagamifading channels are considered. The exact probability density function of a sum of Nakagami- random phase vectors is used to derive a closed-form expression for the error rates of OFDM signals. The precise error-rate analysis is extended to a system using multichannel reception with maximal ratio combining. An asymptotic error-rate analysis is also provided. For a two-tap channel with finite values of Nakagami- fading parameters, our analysis and numerical results show that the asymptotic error-rate performance of an OFDM signal is similar to that of a single carrier signal transmitted over a Rayleigh fading channel. On the other hand, our analysis further shows that a frequency-selective channel that can be represented by two constant taps has similar asymptotic error-rate performance to that of a one-sided Gaussian fading channel. It is observed that, depending on the number of channel taps, the error-rate performance does not necessarily improve with increasing Nakagami- fading parameters. Index Terms—Error analysis, fading channel, orthogonal frequency-division multiplexing (OFDM).
I. INTRODUCTION
O
RTHOGONAL frequency-division multiplexing (OFDM) is an important wide-band transmission scheme for modern wireless communication systems. OFDM, which requires a relatively simple equalizer at the receiver, can reduce or eliminate intersymbol interference (ISI) and is particularly suitable for transmission over multipath fading channels. Both Rayleigh and Rician fading models are widely encountered in practical systems and have been used in wireless communications studies and, in particular, in studies of OFDM systems. For example, in [1], Glavieux et al. considered the performance of an OFDM binary frequency-shift keying (OFDM-BFSK) scheme in underwater communication scenarios assuming Rayleigh and Rician fading channels. More recently, Lu et al. studied the performance of an OFDM -ary
Paper approved by J. Wang, the Editor for Modulation, Detection, and Equalization of the IEEE Communications Society. Manuscript received August 18, 2004; revised January 26, 2005 and June 21, 2005. This paper was presented in part at the Vehicular Technology Fall Conference, Los Angeles, CA, September 2004, and at GLOBECOM 2004, Dallas, TX, November 2004. Z. Du and N. C. Beaulieu are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail: [email protected]; [email protected]). J. Cheng was with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada. He is now with the Department of Electrical Engineering, Lakehead University, Thunder Bay, ON P7B 5E1, Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2005.863729
differential phase-shift keying (OFDM-MDPSK) scheme in Rician fading channels [2]. The Nakagami- [3] distribution has been employed as another useful and important model for characterizing the amplitude of fading channels. Both the theoretical and the practical importance of the Nakagami- channel have motivated intensive research into studying the performance of various communication systems operating in such channels. For example, Eng and Milstein studied the performance of a direct-sequence code-division multiple-access (DS-CDMA) system in Nakagami- fading channels [4]. In [5], Alouini and Goldsmith used a moment generating function (MGF) technique to study the error performance of coherent modulations over Nakagami- channels. More recently, Yang and Hanzo [6] investigated the performance of multicarrier DS-CDMA in Nakagami- fading channels. Previous works that studied transmission of an OFDM signal over multipath Nakagamichannels assumed that the frequency-domain channel response samples are also Nakagami- -distributed with the same fading parameters as the time-domain channel [7]. However, there are no experimental results presented in the literature that support this assumption. Furthermore, intuitively, this assumption does not seem to be valid. Consider an -point fast Fourier transform (FFT) used to determine the sampled frequency-domain response from the sampled time-domain response. In the case of Rayleigh fading, the faded signal samples have a joint complex Gaussian distribution. Then, the application of the FFT represents a linear transformation of jointly Gaussian random variables (RVs) and yields jointly Gaussian RVs [8]. Thus, one expects a frequency response sample to have a Rayleigh fading distribution when the time-domain signal is Rayleigh faded. However, sums of Nakagami- random phase vectors1 do not, in general, have a Nakagami- -distributed envelope. Therefore, it is not expected that the frequency-domain samples will assume the Nakagami- distribution when the time-do, which main signal is Nakagami- faded, except for is the special case of Rayleigh fading. Recently, Kang et al. claimed that the distribution of samples of the frequency-domain channel impulse response can be approximated by another Nakagami- distribution with a new fading parameter different from the time-domain fading parameter [9]. In this paper, we will use an exact mathematical analysis to show that such Nakagami- approximations can be unreliable.
m m
1In this paper, a Nakagamirandom phase vector is defined to have its amplitude following a Nakagami- distribution and its phase following a uniform distribution.
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The contributions of this paper are the following. We first revisit the problem of determining the probability density function (pdf) of a sum of random phase vectors and recall an integral solution to the pdf of the sum of Nakagami- random phase vectors. We then use this pdf expression to evaluate the exact error-rate performance of an OFDM system in multipath Nakagami- fading channels for both single channel and multichannel diversity reception. An asymptotic error-rate performance analysis is provided for OFDM signals in single-tap and two-tap multipath Nakagami- channels. For a two-tap channel with finite values of Nakagami- fading parameters, our analysis and numerical results reveal that the asymptotic error-rate performance of an OFDM signal is similar to that of a single carrier signal transmitted over a Rayleigh fading channel. On the other hand, we find that a multipath channel model with two constant taps has similar asymptotic error-rate performance to that of a one-sided Gaussian channel. We observe, through both analysis and numerical examples, that the error-rate performance of an OFDM signal over a multipath Nakagami channel does not necessarily improve with increasing Nakagami fading parameters. This paper is organized as follows. In Section II, we describe our system model. In Section III, we present an integral solution for the pdf of a sum of Nakagami- random phase vectors. This pdf expression is then used in Section IV to obtain a closed-form error-rate expression for coherently modulated OFDM signals. Section V studies the asymptotic error performance of an OFDM system in multipath Nakagami channels. A precise error-rate analysis for multichannel reception using maximal ratio combining (MRC) is given in Section VI. Numerical results and discussion are provided in Section VII. Finally, in Section VIII, we make our concluding remarks.
, , after The received signal analog-to-digital (A/D) sampling and removal of the cyclic prefix, is input to a discrete Fourier transform (DFT) processor, and the output signal becomes
II. SYSTEM MODEL We consider an OFDM system with subcarriers. For each OFDM symbol, we denote the modulated data sequence as . After the inverse discrete Fourier transform (IDFT), the time-domain OFDM signal can be expressed as [10] (1) . Suppose that the channel impulse response of a where multipath fading channel is modeled as a finite impulse response , . We assume that (FIR) filter with taps the maximum delay of the multipath fading channel is with , i.e., for , and we express the frequency-domain channel impulse response as (2) We further assume that the maximum delay is less than the length of the cyclic prefix, and perfect timing and frequency synchronization are achieved at the receiver. Therefore, ISI is not considered in our analysis.
(3) are independent identically distributed (i.i.d.) comwhere plex Gaussian noise components with zero mean and unit variance. Equation (3) shows that each OFDM subcarrier undergoes , but the distribution of a flat fading channel described by is only Nakagami- if . Since there is no benefit from using OFDM on a flat fading channel, we consider frequency-selective fading channels where the total bandwidth occupied by the OFDM signal exhibits frequency-selective fading in the model. and In this paper, we assume that the channel tap coefficients in the multipath fading channel model , , are mutually independent complex RVs, which can be written . The amplitude (or the modulus) as is modeled as a Nakagami- RV with pdf
(4) where is the Gamma function, is the or the power of the th tap, and is expectation of the Nakagami fading parameter for the th tap. The Nakagami determines the severity of fading channels. fading parameter Two special values of are of particular interest. In the case , the Nakagami- fading specializes to Rayleigh of , the Nakagamifading. In the limiting case when fading channel approaches a static channel, and the pdf of becomes , where is the Dirac ’s are assumed to delta function. Finally, the fading phases be mutually independent and uniformly distributed over and independent of the fading amplitudes ’s. The frequency-domain representation of the channel impulse response can also be expressed as
(5) where it can be shown that the RVs are also uniformly distributed over for different values. We assume that the receiver has perfect knowledge of the can be perfectly esfading channel, that is, the phase of timated and compensated at the receiver. Therefore, to analyze the performance of our OFDM signal in multipath Nakagami-
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channels, it is sufficient to consider only the statistical charac, which is the modteristics of ulus of a sum of complex random vectors , .
is the zeroth-order Bessel function of the first kind. where Substitution of (10) into (11) yields [3, eqs. (94) and (95)]
III. PDF OF A SUM OF NAKAGAMI RVS AND ITS APPROXIMATION
(12) The integral representation of the pdf in (12) is exact. The result in (12) was stated in [3] without a detailed proof. In [14], Abdi et al. derived a more general result that is valid for sums of an arbitrary number of random vectors with arbitrary statistics. Our derivation here concisely gives a proof for the special case of Nakagami- random vectors and is included for completeness. We now check special cases of the pdf in (12). When , , using the Kummer’s transformation [13] and the integral identity [12, eq. (6.614.1)], one can show that
In this section, we present an integral expression for the pdf of the amplitude of a sum of Nakagami- random phase vectors using a characteristic function (CF) approach. and , which were defined in The joint CF of the RVs (5), is (6) and . where the expectation is taken over the joint pdf of , it can be shown that and are In general, when dependent, and their joint pdf is given by [11, eq. (6)]
(13) where When
and
, which is a Rayleigh pdf as one expects. , we have for all values (14)
(7) and For convenience, we define . Using these definitions and applying (7) to (6), one can show
When the channel has two taps ( (8.11.31)], (14) reduces to
), with the aid of [15, eq.
(15) and (8) (16) The integral identities [12, eqs. (3.915.2) and (6.631.1)] allow one to further simplify (8) to [3, eq. 43]
(9) is the confluent hypergeometric function [13]. where , where Now, we write and . Because the channel tap coefficients are independent, the joint CF of and becomes [3, eq. (95)]
(10) Using the inversion theorem [8], one can show that the pdf of is given by the amplitude (11)
which, as expected, agrees with the result obtained by Simon in [16, eq. (4)]. Nakagami [3] further approximated the pdf of with another Nakagami- distribution with new fading parameters and , where and
(17)
In the special case when for all values, it is seen from (17) that , as expected, and the approximation in (17) is exact. This is because, from (5), the sum of independent zero-mean complex Gaussian RVs gives a zero-mean complex Gaussian RV with its envelope following a Rayleigh distribution exactly, as shown in (13). In the limiting case when beapproaches unity. This is comes large, it can be shown that also expected since, by a central limit theorem, a sum of a large number of independent zero-mean complex RVs will give a resultant zero-mean complex RV that is approximately Gaussian . with its envelope following a Rayleigh distribution, i.e.,
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In [9], Kang et al. verified the goodness of the approximation in (17) by comparing cumulative distribution functions (cdfs) for the Nakagami- approximation with simulated data. The authors went on and used the Nakagami- approximation to study the error-rate performance of OFDM signals in multipath Nakagami- channels. The Nakagami- approximation to the , while it is a useful tool for rapid analysis, can, pdf of however, be inaccurate. In Section VII, by comparing the pdf of the Nakagami- approximation and the exact analytical pdf derived in (12), we will show that the Nakagami- approximation can be poor, which is a result that contradicts the previous study in [9]. Furthermore, in Section VII, we will also show that OFDM error-rate estimation based on the Nakagami- approximation can be unreliable.
Nakagami- -distributed. We found that the error rates obtained using (21) were numerically equivalent to those obtained from the closed-form expression derived by Eng and Milstein [4, eq. (A8)]. In terms of a hypergeometric function with multiple variables [12, eq. (9.19)], Appendix B further shows that the integral repin (21) can be written in closed-form as resentation of
IV. ERROR-RATE PERFORMANCE USING SINGLE-CHANNEL RECEPTION
(22)
The conditional bit-error rate (BER) and symbol-error rate (SER) for many coherent modulation schemes can be expressed , where , is a factor as relating to signal-to-noise ratio (SNR), and is the fading amplifor our system model. Using an alternative tude which is representation of the -function [5] (18) with respect to and averaging obtains the error rate as
in (11), one
where is called the Lauricella function of variables [17]. A useful computer algorithm for computing the Lauricella function is provided in [17, Appendix B, p. 293]. V. ASYMPTOTIC ERROR-RATE PERFORMANCE ANALYSIS In this section, using an asymptotic SNR analysis, we reveal some insights into the error-rate performance of OFDM signals on multipath Nakagami- fading channels. We first rewrite (19) as (23) A Taylor series expansion for
gives
(19) is the joint CF of the where, as we recall from (10), . Appendix A shows that the real and imaginary parts of error-rate expression in (19) can be simplified to
(24) where
and thus (25)
(20) When is the th-order modified Bessel function of the where first kind. It should be emphasized that the single integral representation of the error-rate expression in (20) is valid for arbitrary fading statistics. For our system model, one obtains from (10) and (20)
(21) Equation (21) can be used to numerically compute the error rates of a large class of modulation schemes for OFDM signals transmitted over Nakagami- multipath channels. In the special case of a single-tap channel ( ), it follows that is
becomes large, using (25) and (23), one can show2 (26)
where . for and . We now examine the power series expansion for (25) if assumes a normalized Nakagami- pdf with different parameters, i.e., a single-tap . The first four power-series expansion channel with 0.5, 1.0, 1.5, coefficients for Nakagami- fading with 2.0, and 2.5 are calculated and tabulated in Table I. Note that, when is large, only the first nonzero terms in the power-series expansion (25) will be significant in computing the asymptotic 2Rigorous development of (26) and exact expressions for the SER in similar series form (instead of just the first-term approximation) based on Taylor’s expansion of the fading pdf around zero has been used in [18].
DU et al.: ACCURATE ERROR-RATE PERFORMANCE ANALYSIS OF OFDM ON FREQUENCY-SELECTIVE NAKAGAMI-
TABLE I FIRST FOUR POWER EXPANSION COEFFICIENTS FOR (1=A)f (r=A), (r ) IS NAKAGAMI-m-DISTRIBUTED WITH FADING WHERE f PARAMETER m = 0.5, 1.0, 1.5, 2.0, AND 2.5
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To evaluate the integral in (30), we first construct an ancillary integral as
(31) Using [19, eq. 287(22)], we can deduce the following integral identity:
error rate . In particular, for a Nakagami- fading , from (26), the first nonzero term channel with decreases inversely with . On the other hand, for a Rayleigh fading ( ) channel, the first nonzero term decreases . Therefore, the asymptotic error-rate inversely with performance of OFDM signals with single-carrier transmission is worse (one tap) on the Nakagami- channel with than that in the Rayleigh fading channel, as expected (because the error-rate curve in the Rayleigh fading channel is steeper). We now consider the asymptotic error-rate performance of OFDM signals in a multipath Nakagami- channel with two taps ( ). From (12), the pdf of now becomes
(32) is the hypergeometric function [13]. Apwhere as plying (32) to (31), we can express
(33) Utilizing [13, eq. (15.3.6)] and letting braic manipulation, we have
, after some alge-
(27) Since [13, eq. (9.1.12)]
(34) (28)
and In the special case when from (34), the asymptotic error rate becomes
it is obvious that
,
(35) (29)
. Since and it will be shown in the ensuing analysis that the first coefficient in the series expansion (29) is nonzero, (29) and (26) imply that, for an OFDM signal transmitted over a two-tap multipath Nakagami- channel, the slope of the asymptotic error-rate curve is constant (which is the same as for the Rayleigh fading case) and does not change with changing values. This property of an OFDM system is different from the asymptotic error-rate performance for a single-carrier system in single-tap Nakagami- fading channels where the slope of the error-rate curve in the large SNR region changes with different values. in (29) and have We now determine the value of
(30)
where (36) Numerical computations show that attains a global minimum at and monotonically increases for , as shown in Fig. 1. Thus, larger values will yield higher error rates. Our numerical results in Section VII confirm this analysis. This observation is interesting because, in a single-carrier transmission on Nakagami- fading channels, larger values will lead to smaller error rates. , we have from (14) In the limiting case when (37) which implies that, similar to the asymptotic error-rate performance in a one-sided Gaussian channel for a single-carrier signal, the asymptotic error rate for an OFDM signal in a two-ray channel with constant taps will decay as the square root of the SNR.
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here is the MGF technique described in [5]. The conditional error-rate expression is modified to
for MRC and is given by (38) is the fading amplitude in the th branch. where Averaging (38) with respect to the joint pdf of the fading ampli, tudes, we obtain the average error rate, denoted by is the as (39), shown at the bottom of the page, where . joint CF of the real and imaginary parts of Following Section IV, we can show Fig. 1. Function P
= = 1:0.
( ) versus m
m
for
L
= 2 with
m
=
m
=
m
and
To this end, we have performed a precise asymptotic errorrate analysis for OFDM signals over taps Nakagamichannels. It is found that its asymptotic error-rate performance is similar to that in a single-tap Rayleigh channel. Exact asymptaps become intractable. totic error-rate analyses for However, when the number of channel taps is large, by a central limit theorem, one can show that the asymptotic error-rate performance is also similar to that of Rayleigh fading. Therefore, one may hypothesize that the asymptotic error-rate performance of an OFDM signal on multipath Nakagami- channels with more than one channel tap is similar to the error-rate performance in a single-tap Rayleigh fading channel. VI. ERROR-RATE PERFORMANCE USING DIVERSITY RECEPTION In this section, we study the error-rate performance of OFDM insignals in multipath Nakagami- fading channels using dependent diversity branches with MRC. The approach we use
(40) where (41) which is the MGF of . We show in Appendix B [cf. (57)] that, if the channel taps distributions with fading parameters assume Nakagami, one has
(42) Putting (42) and (41) into (40), we have (43), shown at the bottom of the page. In Section VII, (43) will be used to obtain the BER of a binary phase-shift keying (BPSK)-modulated OFDM
(39)
(43)
DU et al.: ACCURATE ERROR-RATE PERFORMANCE ANALYSIS OF OFDM ON FREQUENCY-SELECTIVE NAKAGAMI-
Fig. 2. Exact and approximate cdfs for
= = 1 :0 .
()
jH k j
with
m
=
m
= 5 0 and :
Fig. 3.
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Exact and approximate pdfs for
= = 1 0.
()
jH k j
with
m
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=
m
= 5 0 and :
:
signal over multipath Nakagami- channels using MRC recep) reception, tion. In the special case of single-branch ( numerical results for the BER obtained using (43) equal numerical results for the BER obtained from (22). In evaluation of the integrand in (43), the Lauricella transformation [17, p. 121]
(44) where can be used to guarantee the convergence of the Lauricella functions in (43) for large values of arguments. and For the special case where , i.e., the channel gains become two constant values with equal power, and the error-rate expression in (43) can be reduced to a more compact form. To show this, using (14), we have
(45) where the last equality follows from [12, eq. (6.633.2)], and, therefore, the error rate is given by
(46) VII. NUMERICAL RESULTS AND DISCUSSION Figs. 2–4 assume a multipath Nakagami- channel with two taps where and . The precise curves (solid lines) are obtained using the accurate approach proposed in this paper, and the approximation curves (dashed lines) are obtained from the Nakagami- approximation used in [9]. We observe that the theoretical results match the Monte
Fig. 4. Precise and approximate BER of OFDM-BPSK over a frequency-selective Nakagami-m channel when m = m = 5:0 and
= = 1:0. The precise results are obtained from the approach proposed in this paper, and the approximate results are obtained by the Nakagami-m approximation in [3], [9].
Carlo simulation results (stars) exactly. Both the cdf in Fig. 2 and the pdf in Fig. 3 indicate that the Nakagami- approximation in [9] can be a poor approximation in this case for the distribution of the amplitude of the sum of Nakagami- random vectors. Importantly, Fig. 4 shows that the BER values of OFDM-BPSK obtained using this approximation can severely underestimate the true BER values. For example, when the average SNR at each subcarrier is 25 dB, the true BER is 20 times the value of the BER obtained using the approximation. Similarly, the value of the SNR required to obtain a target value of BER is, in fact, 5 dB more than the value estimated by the approximation. We conclude that the Nakagami- approximation is unreliable. Fig. 5 plots the pdfs of with taps and different fading parameters. The shapes of the pdf for several values further verify that the distribution of the amplitude in the
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Fig. 5. m
=
Fig. 6. m
1 0. :
=
Exact pdfs of jH (k )j for m = = with = = 1:0.
m
1
m
= 5 0, :
BERs of OFDM-BPSK for m = m = = 8:0, m = m = 15:0, and m = m
m
m
=
m
= 8 0, and :
1 4, = = 5 0, = 1 with = = :
m
m
:
frequency domain is not a Nakagami- distribution (observe that, for example, the Nakagami- pdf contains only one mode, whereas two local maxima can be observed in Fig. 5). For the , the pdf agrees with the limiting case when one obtained by Simon [16, Fig. 3]. The corresponding BER curves for BPSK are given in Fig. 6. As predicted in Section V, the slopes of the error-rate curves in the large SNR region are identical for different values (finite) and the error-rate performance degrades with increasing . Fig. 6 also shows that the error-rate performance over the channel with and is worse than the error-rate performance over the channel under fading conditions (corresponding to finite values). This numerical result confirms our asymptotic analysis in Section V. Fig. 7 compares the exact and asymptotic BER performances with two taps and various fading parameters. In Fig. 7, it is seen that the asymptotic performance
Fig. 7. Precise and asymptotic BERs of OFDM-BPSK for m = m = 1:4, m = m = 5:0, m = m = 8:0, m = m = 15:0, and m = m = with = = 1:0.
1
Fig. 8. BERs of OFDM-BPSK for m = m = 8:0, and m = m = MRC reception.
m
=
= 2 0,
=
= 5 0,
1 for single-channel and dual-branch m
:
m
m
:
can be a good approximation to the exact performance (within a factor of about 2) for SNR values greater than 10 dB when . Fig. 8 indicates that the same behavior holds for a system using dual-branch reception. We observe in Fig. 8 that, with two-branch MRC, the slopes of the error-rate curves become larger and significant gains are obtained; for example, in the two-tap multipath Nakagami- channel with fading parameter , to achieve a target BER of 1.5 10 , two-branch reception requires 10 dB less SNR than single-channel reception. Finally, Fig. 9 gives the error-rate performance of a BPSK-modulated OFDM signal transmitted over a multipath Nakagami- channel with different numbers of taps at SNR dB. Contrary to the two-tap case, we observe that the BER rates in fact decrease with increasing parameters when the number of taps is three. In the limit when increases, the BER performance is less variable as changes and converges
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one has (50) Thus,
(51) Substituting (51) into (47) and using [12, eq. (3.915.2)] as well as the properties of the Bessel functions [13, eqs. (9.63), (9.6.30)], after some simplification, one obtains (20) as desired. APPENDIX B DERIVATION OF (22) Fig. 9. BER at SNR = 25 dB versus the number of taps L.
to that in a Rayleigh fading channel. This can be expected beapproaches cause, by a central limit theorem, the pdf of Rayleigh when the number of taps is large. VIII. CONCLUSION
In this appendix, we derive the closed-form solution in (22) for the error rate of OFDM signals operating in multipath Nakagami- channels. To show this, we first note that the modified Bessel function is related to the confluent hypergeometric function by [13, eq. (13.6.3)] (52) Therefore, we can express
In this paper, we have studied the characteristics of OFDM signals over multipath Nakagami- channels. The pdf of a sum of Nakagami- random phase vectors has been used to derive an exact error-rate expression for coherently modulated OFDM signals transmitted over multipath Nakagami- fading channels. Our asymptotic analysis and numerical results have shown some unique properties pertaining to the transmission of OFDM signals over such channel models.
and
as (53a) (53b)
Substitution of (53) into (20) yields
APPENDIX A DERIVATION OF (20)
(54)
Starting from (19), with an exchange of the order of integration, can be rewritten as
Now, the integral property [12, eq. (7.622.3)] gives
(47) Utilizing an integral identity [12, eq. (6.631.7)], one can express the inner integral in (47) as
(55a) (55b) (55c)
(48) where and Bessel function [13]. Since eq. (8.467)]
is the -order modified can be expanded as [12,
(49)
where is the Lauricella function defined in [17] and is also known as the hyperis geometric function with multiple variables [12], and the Whittaker function [13]. The Whittaker function is related to the confluent hypergeometric function by [12, eq. (9.220.2)]
(56)
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Using (55) and (56), we deduce the following integral identity:
(57)
Zheng Du (M’04) received the B.Sc. and Ph.D. degrees in electrical engineering from the University of Science and Technology of China, Hefei, China, in 1998 and 2003, respectively. He has been with the iCORE Wireless Communications Laboratory, University of Alberta, Edmonton, AB, Canada, as a Research Associate since 2003. His current research interests include broadband digital communications systems, orthogonal frequency-division multiplexing, and space–time coding.
Now, applying (57) to (54), we obtain (22).
REFERENCES [1] A. Glavieux, P. Y. Cochet, and A. Picart, “Orthogonal frequency division multiplexing with BFSK modulation in frequency selective Rayleigh and Rician fading channels,” IEEE Trans. Commun., vol. 42, no. 2–4, pp. 1919–1928, Feb./Mar./Apr. 1994. [2] J. Lu, T. T. Tjhung, F. Adachi, and C. L. Huang, “BER performance of OFDM-MDPSK system in frequency-selective Rician fading with diversity reception,” IEEE Trans. Veh. Technol., vol. 49, no. 7, pp. 1216–1225, Jul. 2000. [3] M. Nakagami, “The -distribution, a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. G. Hoffman, Ed. Oxford, U.K.: Pergamon, 1960. [4] T. Eng and L. B. Milstein, “Coherent DS-CDMA performance in Nakagami multipath fading,” IEEE Trans. Commun., vol. 43, no. 2–4, pp. 1134–1143, Feb./Mar./Apr. 1995. [5] M.-S. Alouini and A. J. Goldsmith, “A unified approach for calculating error rates of linearly modulated signals over generalized fading channels,” IEEE Trans. Commun., vol. 47, no. 9, pp. 1324–1334, Sep. 1998. [6] L.-L. Yang and L. Hanzo, “Performance of generalized multicarrier DS-CDMA over Nakagami- fading channels,” IEEE Trans. Commun., vol. 50, no. 6, pp. 956–966, Jun. 2002. [7] A. Scaglione, S. Barbarossa, and G. B. Giannakis, “Optimal adaptive precoding for frequency-selective Nakagami- fading channels,” in Proc. 52nd IEEE Veh. Technol. Conf., vol. 3, 2000, pp. 1291–1295. [8] A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. New York: McGraw-Hill, 1991. [9] Z. Kang, K. Yao, and F. Lorenzelli, “Nakagami- fading modeling in the frequency domain for OFDM system analysis,” IEEE Commun. Lett., vol. 7, no. 10, pp. 484–486, Oct. 2003. [10] T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun., vol. 45, no. 12, pp. 1613–1621, Dec. 1997. [11] J. Cheng, N. C. Beaulieu, and X. Zhang, “Precise BER analysis of dualchannel reception of QPSK in Nakagami fading and cochannel interference,” IEEE Commun. Lett., vol. 9, no. 4, pp. 316–318, Apr. 2005. [12] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic, 2000. [13] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. [14] A. Abdi, H. Hashemi, and S. Nader-Esfahani, “On the PDF of the sum of random vectors,” IEEE Trans. Commun., vol. 48, no. 1, pp. 7–12, Jan. 2000. [15] A. Erdélyi, Tables of Integral Transforms. New York: McGraw-Hill, 1954, vol. II. [16] M. K. Simon, “On the probability density function of the squared envelope of a sum of random phase vectors,” IEEE Trans. Commun., vol. COM-33, no. 9, pp. 993–996, Sep. 1985. [17] H. Exton, Multiple Hypergeometric Functions and Applications, G. M. Bell, Ed. Sussex, U.K.: Ellis Horwood, 1976. [18] Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun., vol. 51, no. 8, pp. 1389–1398, Aug. 2003. [19] A. Erdélyi, Higher Transcendental Functions. New York: McGrawHill, 1953, vol. I.
m
m
m
m
Julian Cheng (S’96–M’04) received the B. Eng. degree (First Class) in electrical engineering from the University of Victoria, Victoria, BC, Canada in 1995, the M.Sc. (Eng.) degree in mathematics and engineering from Queen’s University, Kingston, ON, Canada in 1997, and the Ph.D. degree in electrical engineering from the University of Alberta, Edmonton, AB, Canada, in 2003. He is currently an Assistant Professor with the Department of Electrical Engineering, Lakehead University, Thunder Bay, ON, Canada. Previously, he worked for Bell Northern Research (BNR) and Northern Telecom (now NORTEL Networks). His current research interests include digital communications over fading channels, orthogonal frequency division multiplexing, spread spectrum communications, and statistical signal processing for wireless applications. Dr. Cheng was the recipient of numerous scholarships during his undergraduate and graduate studies, which included a President Scholarship from the University of Victoria and a postgraduate scholarship from the Natural Sciences and Engineering Research Council of Canada (NSERC). He was also a winner of the 2002 NSERC Postdoctoral Fellowship competition.
Norman C. Beaulieu (S’82–M’86–SM’89–F’99) received the B.A.Sc. (honors), M.A.Sc., and Ph.D degrees in electrical engineering from the University of British Columbia, Vancouver, BC, Canada, in 1980, 1983, and 1986, respectively. He was a Queen’s National Scholar Assistant Professor with the Department of Electrical Engineering, Queen’s University, Kingston, ON, Canada, from September 1986 to June 1988, an Associate Professor from July 1988 to June 1993, and a Professor from July 1993 to August 2000. In September 2000, he became the iCORE Research Chair in Broadband Wireless Communications with the University of Alberta, Edmonton, AB, Canada, and, in January 2001, the Canada Research Chair in Broadband Wireless Communications. His current research interests include broad-band digital communications systems, ultrawide-bandwidth systems, fading channel modeling and simulation, diversity systems, interference prediction and cancellation, importance sampling and semianalytical methods, and space–time coding. Dr. Beaulieu is a member of the IEEE Communication Theory Committee and served as its Representative to the Technical Program Committee of the 1991 International Conference on Communications and as Co-Representative to the Technical Program Committee of the 1993 International Conference on Communications and the 1996 International Conference on Communications. He was General Chair of the Sixth Communication Theory Mini-Conference in association with GLOBECOM 97 and Co-Chair of the Canadian Workshop on Information Theory 1999. He has been an Editor for Wireless Communication Theory of the IEEE TRANSACTIONS ON COMMUNICATIONS since January 1992 and was Editor-in-Chief from January 2000 to December 2003. He served as an Associate Editor for Wireless Communication Theory of the IEEE COMMUNICATIONS LETTERS from November 1996 to August 2003. He has also served on the Editorial Board of the PROCEEDINGS OF THE IEEE since November 2000. He was the recipient of the Natural Science and Engineering Research Council of Canada (NSERC) E.W.R. Steacie Memorial Fellowship in 1999. He was elected a Fellow of the Engineering Institute of Canada in 2001 and was awarded the Médaille K.Y. Lo Medal of the Institute in 2004. He was elected Fellow of the Royal Society of Canada in 2002 and was awarded the Thomas W. Eadie Medal of the Society in 2005. He was awarded the University of British Columbia Special University Prize in Applied Science in 1980 as the highest standing graduate in the faculty of Applied Science.
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Accurate Error-Rate Performance Analysis of OFDM on Frequency-Selective NakagamiFading Channels
m
Zheng Du, Member, IEEE, Julian Cheng, Member, IEEE, and Norman C. Beaulieu, Fellow, IEEE
Abstract—Error rates of orthogonal frequency-division multiplexing (OFDM) signals in multipath slow fading Nakagamifading channels are considered. The exact probability density function of a sum of Nakagami- random phase vectors is used to derive a closed-form expression for the error rates of OFDM signals. The precise error-rate analysis is extended to a system using multichannel reception with maximal ratio combining. An asymptotic error-rate analysis is also provided. For a two-tap channel with finite values of Nakagami- fading parameters, our analysis and numerical results show that the asymptotic error-rate performance of an OFDM signal is similar to that of a single carrier signal transmitted over a Rayleigh fading channel. On the other hand, our analysis further shows that a frequency-selective channel that can be represented by two constant taps has similar asymptotic error-rate performance to that of a one-sided Gaussian fading channel. It is observed that, depending on the number of channel taps, the error-rate performance does not necessarily improve with increasing Nakagami- fading parameters. Index Terms—Error analysis, fading channel, orthogonal frequency-division multiplexing (OFDM).
I. INTRODUCTION
O
RTHOGONAL frequency-division multiplexing (OFDM) is an important wide-band transmission scheme for modern wireless communication systems. OFDM, which requires a relatively simple equalizer at the receiver, can reduce or eliminate intersymbol interference (ISI) and is particularly suitable for transmission over multipath fading channels. Both Rayleigh and Rician fading models are widely encountered in practical systems and have been used in wireless communications studies and, in particular, in studies of OFDM systems. For example, in [1], Glavieux et al. considered the performance of an OFDM binary frequency-shift keying (OFDM-BFSK) scheme in underwater communication scenarios assuming Rayleigh and Rician fading channels. More recently, Lu et al. studied the performance of an OFDM -ary
Paper approved by J. Wang, the Editor for Modulation, Detection, and Equalization of the IEEE Communications Society. Manuscript received August 18, 2004; revised January 26, 2005 and June 21, 2005. This paper was presented in part at the Vehicular Technology Fall Conference, Los Angeles, CA, September 2004, and at GLOBECOM 2004, Dallas, TX, November 2004. Z. Du and N. C. Beaulieu are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail: [email protected]; [email protected]). J. Cheng was with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada. He is now with the Department of Electrical Engineering, Lakehead University, Thunder Bay, ON P7B 5E1, Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2005.863729
differential phase-shift keying (OFDM-MDPSK) scheme in Rician fading channels [2]. The Nakagami- [3] distribution has been employed as another useful and important model for characterizing the amplitude of fading channels. Both the theoretical and the practical importance of the Nakagami- channel have motivated intensive research into studying the performance of various communication systems operating in such channels. For example, Eng and Milstein studied the performance of a direct-sequence code-division multiple-access (DS-CDMA) system in Nakagami- fading channels [4]. In [5], Alouini and Goldsmith used a moment generating function (MGF) technique to study the error performance of coherent modulations over Nakagami- channels. More recently, Yang and Hanzo [6] investigated the performance of multicarrier DS-CDMA in Nakagami- fading channels. Previous works that studied transmission of an OFDM signal over multipath Nakagamichannels assumed that the frequency-domain channel response samples are also Nakagami- -distributed with the same fading parameters as the time-domain channel [7]. However, there are no experimental results presented in the literature that support this assumption. Furthermore, intuitively, this assumption does not seem to be valid. Consider an -point fast Fourier transform (FFT) used to determine the sampled frequency-domain response from the sampled time-domain response. In the case of Rayleigh fading, the faded signal samples have a joint complex Gaussian distribution. Then, the application of the FFT represents a linear transformation of jointly Gaussian random variables (RVs) and yields jointly Gaussian RVs [8]. Thus, one expects a frequency response sample to have a Rayleigh fading distribution when the time-domain signal is Rayleigh faded. However, sums of Nakagami- random phase vectors1 do not, in general, have a Nakagami- -distributed envelope. Therefore, it is not expected that the frequency-domain samples will assume the Nakagami- distribution when the time-do, which main signal is Nakagami- faded, except for is the special case of Rayleigh fading. Recently, Kang et al. claimed that the distribution of samples of the frequency-domain channel impulse response can be approximated by another Nakagami- distribution with a new fading parameter different from the time-domain fading parameter [9]. In this paper, we will use an exact mathematical analysis to show that such Nakagami- approximations can be unreliable.
m m
1In this paper, a Nakagamirandom phase vector is defined to have its amplitude following a Nakagami- distribution and its phase following a uniform distribution.
0090-6778/$20.00 © 2006 IEEE
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The contributions of this paper are the following. We first revisit the problem of determining the probability density function (pdf) of a sum of random phase vectors and recall an integral solution to the pdf of the sum of Nakagami- random phase vectors. We then use this pdf expression to evaluate the exact error-rate performance of an OFDM system in multipath Nakagami- fading channels for both single channel and multichannel diversity reception. An asymptotic error-rate performance analysis is provided for OFDM signals in single-tap and two-tap multipath Nakagami- channels. For a two-tap channel with finite values of Nakagami- fading parameters, our analysis and numerical results reveal that the asymptotic error-rate performance of an OFDM signal is similar to that of a single carrier signal transmitted over a Rayleigh fading channel. On the other hand, we find that a multipath channel model with two constant taps has similar asymptotic error-rate performance to that of a one-sided Gaussian channel. We observe, through both analysis and numerical examples, that the error-rate performance of an OFDM signal over a multipath Nakagami channel does not necessarily improve with increasing Nakagami fading parameters. This paper is organized as follows. In Section II, we describe our system model. In Section III, we present an integral solution for the pdf of a sum of Nakagami- random phase vectors. This pdf expression is then used in Section IV to obtain a closed-form error-rate expression for coherently modulated OFDM signals. Section V studies the asymptotic error performance of an OFDM system in multipath Nakagami channels. A precise error-rate analysis for multichannel reception using maximal ratio combining (MRC) is given in Section VI. Numerical results and discussion are provided in Section VII. Finally, in Section VIII, we make our concluding remarks.
, , after The received signal analog-to-digital (A/D) sampling and removal of the cyclic prefix, is input to a discrete Fourier transform (DFT) processor, and the output signal becomes
II. SYSTEM MODEL We consider an OFDM system with subcarriers. For each OFDM symbol, we denote the modulated data sequence as . After the inverse discrete Fourier transform (IDFT), the time-domain OFDM signal can be expressed as [10] (1) . Suppose that the channel impulse response of a where multipath fading channel is modeled as a finite impulse response , . We assume that (FIR) filter with taps the maximum delay of the multipath fading channel is with , i.e., for , and we express the frequency-domain channel impulse response as (2) We further assume that the maximum delay is less than the length of the cyclic prefix, and perfect timing and frequency synchronization are achieved at the receiver. Therefore, ISI is not considered in our analysis.
(3) are independent identically distributed (i.i.d.) comwhere plex Gaussian noise components with zero mean and unit variance. Equation (3) shows that each OFDM subcarrier undergoes , but the distribution of a flat fading channel described by is only Nakagami- if . Since there is no benefit from using OFDM on a flat fading channel, we consider frequency-selective fading channels where the total bandwidth occupied by the OFDM signal exhibits frequency-selective fading in the model. and In this paper, we assume that the channel tap coefficients in the multipath fading channel model , , are mutually independent complex RVs, which can be written . The amplitude (or the modulus) as is modeled as a Nakagami- RV with pdf
(4) where is the Gamma function, is the or the power of the th tap, and is expectation of the Nakagami fading parameter for the th tap. The Nakagami determines the severity of fading channels. fading parameter Two special values of are of particular interest. In the case , the Nakagami- fading specializes to Rayleigh of , the Nakagamifading. In the limiting case when fading channel approaches a static channel, and the pdf of becomes , where is the Dirac ’s are assumed to delta function. Finally, the fading phases be mutually independent and uniformly distributed over and independent of the fading amplitudes ’s. The frequency-domain representation of the channel impulse response can also be expressed as
(5) where it can be shown that the RVs are also uniformly distributed over for different values. We assume that the receiver has perfect knowledge of the can be perfectly esfading channel, that is, the phase of timated and compensated at the receiver. Therefore, to analyze the performance of our OFDM signal in multipath Nakagami-
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channels, it is sufficient to consider only the statistical charac, which is the modteristics of ulus of a sum of complex random vectors , .
is the zeroth-order Bessel function of the first kind. where Substitution of (10) into (11) yields [3, eqs. (94) and (95)]
III. PDF OF A SUM OF NAKAGAMI RVS AND ITS APPROXIMATION
(12) The integral representation of the pdf in (12) is exact. The result in (12) was stated in [3] without a detailed proof. In [14], Abdi et al. derived a more general result that is valid for sums of an arbitrary number of random vectors with arbitrary statistics. Our derivation here concisely gives a proof for the special case of Nakagami- random vectors and is included for completeness. We now check special cases of the pdf in (12). When , , using the Kummer’s transformation [13] and the integral identity [12, eq. (6.614.1)], one can show that
In this section, we present an integral expression for the pdf of the amplitude of a sum of Nakagami- random phase vectors using a characteristic function (CF) approach. and , which were defined in The joint CF of the RVs (5), is (6) and . where the expectation is taken over the joint pdf of , it can be shown that and are In general, when dependent, and their joint pdf is given by [11, eq. (6)]
(13) where When
and
, which is a Rayleigh pdf as one expects. , we have for all values (14)
(7) and For convenience, we define . Using these definitions and applying (7) to (6), one can show
When the channel has two taps ( (8.11.31)], (14) reduces to
), with the aid of [15, eq.
(15) and (8) (16) The integral identities [12, eqs. (3.915.2) and (6.631.1)] allow one to further simplify (8) to [3, eq. 43]
(9) is the confluent hypergeometric function [13]. where , where Now, we write and . Because the channel tap coefficients are independent, the joint CF of and becomes [3, eq. (95)]
(10) Using the inversion theorem [8], one can show that the pdf of is given by the amplitude (11)
which, as expected, agrees with the result obtained by Simon in [16, eq. (4)]. Nakagami [3] further approximated the pdf of with another Nakagami- distribution with new fading parameters and , where and
(17)
In the special case when for all values, it is seen from (17) that , as expected, and the approximation in (17) is exact. This is because, from (5), the sum of independent zero-mean complex Gaussian RVs gives a zero-mean complex Gaussian RV with its envelope following a Rayleigh distribution exactly, as shown in (13). In the limiting case when beapproaches unity. This is comes large, it can be shown that also expected since, by a central limit theorem, a sum of a large number of independent zero-mean complex RVs will give a resultant zero-mean complex RV that is approximately Gaussian . with its envelope following a Rayleigh distribution, i.e.,
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In [9], Kang et al. verified the goodness of the approximation in (17) by comparing cumulative distribution functions (cdfs) for the Nakagami- approximation with simulated data. The authors went on and used the Nakagami- approximation to study the error-rate performance of OFDM signals in multipath Nakagami- channels. The Nakagami- approximation to the , while it is a useful tool for rapid analysis, can, pdf of however, be inaccurate. In Section VII, by comparing the pdf of the Nakagami- approximation and the exact analytical pdf derived in (12), we will show that the Nakagami- approximation can be poor, which is a result that contradicts the previous study in [9]. Furthermore, in Section VII, we will also show that OFDM error-rate estimation based on the Nakagami- approximation can be unreliable.
Nakagami- -distributed. We found that the error rates obtained using (21) were numerically equivalent to those obtained from the closed-form expression derived by Eng and Milstein [4, eq. (A8)]. In terms of a hypergeometric function with multiple variables [12, eq. (9.19)], Appendix B further shows that the integral repin (21) can be written in closed-form as resentation of
IV. ERROR-RATE PERFORMANCE USING SINGLE-CHANNEL RECEPTION
(22)
The conditional bit-error rate (BER) and symbol-error rate (SER) for many coherent modulation schemes can be expressed , where , is a factor as relating to signal-to-noise ratio (SNR), and is the fading amplifor our system model. Using an alternative tude which is representation of the -function [5] (18) with respect to and averaging obtains the error rate as
in (11), one
where is called the Lauricella function of variables [17]. A useful computer algorithm for computing the Lauricella function is provided in [17, Appendix B, p. 293]. V. ASYMPTOTIC ERROR-RATE PERFORMANCE ANALYSIS In this section, using an asymptotic SNR analysis, we reveal some insights into the error-rate performance of OFDM signals on multipath Nakagami- fading channels. We first rewrite (19) as (23) A Taylor series expansion for
gives
(19) is the joint CF of the where, as we recall from (10), . Appendix A shows that the real and imaginary parts of error-rate expression in (19) can be simplified to
(24) where
and thus (25)
(20) When is the th-order modified Bessel function of the where first kind. It should be emphasized that the single integral representation of the error-rate expression in (20) is valid for arbitrary fading statistics. For our system model, one obtains from (10) and (20)
(21) Equation (21) can be used to numerically compute the error rates of a large class of modulation schemes for OFDM signals transmitted over Nakagami- multipath channels. In the special case of a single-tap channel ( ), it follows that is
becomes large, using (25) and (23), one can show2 (26)
where . for and . We now examine the power series expansion for (25) if assumes a normalized Nakagami- pdf with different parameters, i.e., a single-tap . The first four power-series expansion channel with 0.5, 1.0, 1.5, coefficients for Nakagami- fading with 2.0, and 2.5 are calculated and tabulated in Table I. Note that, when is large, only the first nonzero terms in the power-series expansion (25) will be significant in computing the asymptotic 2Rigorous development of (26) and exact expressions for the SER in similar series form (instead of just the first-term approximation) based on Taylor’s expansion of the fading pdf around zero has been used in [18].
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TABLE I FIRST FOUR POWER EXPANSION COEFFICIENTS FOR (1=A)f (r=A), (r ) IS NAKAGAMI-m-DISTRIBUTED WITH FADING WHERE f PARAMETER m = 0.5, 1.0, 1.5, 2.0, AND 2.5
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To evaluate the integral in (30), we first construct an ancillary integral as
(31) Using [19, eq. 287(22)], we can deduce the following integral identity:
error rate . In particular, for a Nakagami- fading , from (26), the first nonzero term channel with decreases inversely with . On the other hand, for a Rayleigh fading ( ) channel, the first nonzero term decreases . Therefore, the asymptotic error-rate inversely with performance of OFDM signals with single-carrier transmission is worse (one tap) on the Nakagami- channel with than that in the Rayleigh fading channel, as expected (because the error-rate curve in the Rayleigh fading channel is steeper). We now consider the asymptotic error-rate performance of OFDM signals in a multipath Nakagami- channel with two taps ( ). From (12), the pdf of now becomes
(32) is the hypergeometric function [13]. Apwhere as plying (32) to (31), we can express
(33) Utilizing [13, eq. (15.3.6)] and letting braic manipulation, we have
, after some alge-
(27) Since [13, eq. (9.1.12)]
(34) (28)
and In the special case when from (34), the asymptotic error rate becomes
it is obvious that
,
(35) (29)
. Since and it will be shown in the ensuing analysis that the first coefficient in the series expansion (29) is nonzero, (29) and (26) imply that, for an OFDM signal transmitted over a two-tap multipath Nakagami- channel, the slope of the asymptotic error-rate curve is constant (which is the same as for the Rayleigh fading case) and does not change with changing values. This property of an OFDM system is different from the asymptotic error-rate performance for a single-carrier system in single-tap Nakagami- fading channels where the slope of the error-rate curve in the large SNR region changes with different values. in (29) and have We now determine the value of
(30)
where (36) Numerical computations show that attains a global minimum at and monotonically increases for , as shown in Fig. 1. Thus, larger values will yield higher error rates. Our numerical results in Section VII confirm this analysis. This observation is interesting because, in a single-carrier transmission on Nakagami- fading channels, larger values will lead to smaller error rates. , we have from (14) In the limiting case when (37) which implies that, similar to the asymptotic error-rate performance in a one-sided Gaussian channel for a single-carrier signal, the asymptotic error rate for an OFDM signal in a two-ray channel with constant taps will decay as the square root of the SNR.
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here is the MGF technique described in [5]. The conditional error-rate expression is modified to
for MRC and is given by (38) is the fading amplitude in the th branch. where Averaging (38) with respect to the joint pdf of the fading ampli, tudes, we obtain the average error rate, denoted by is the as (39), shown at the bottom of the page, where . joint CF of the real and imaginary parts of Following Section IV, we can show Fig. 1. Function P
= = 1:0.
( ) versus m
m
for
L
= 2 with
m
=
m
=
m
and
To this end, we have performed a precise asymptotic errorrate analysis for OFDM signals over taps Nakagamichannels. It is found that its asymptotic error-rate performance is similar to that in a single-tap Rayleigh channel. Exact asymptaps become intractable. totic error-rate analyses for However, when the number of channel taps is large, by a central limit theorem, one can show that the asymptotic error-rate performance is also similar to that of Rayleigh fading. Therefore, one may hypothesize that the asymptotic error-rate performance of an OFDM signal on multipath Nakagami- channels with more than one channel tap is similar to the error-rate performance in a single-tap Rayleigh fading channel. VI. ERROR-RATE PERFORMANCE USING DIVERSITY RECEPTION In this section, we study the error-rate performance of OFDM insignals in multipath Nakagami- fading channels using dependent diversity branches with MRC. The approach we use
(40) where (41) which is the MGF of . We show in Appendix B [cf. (57)] that, if the channel taps distributions with fading parameters assume Nakagami, one has
(42) Putting (42) and (41) into (40), we have (43), shown at the bottom of the page. In Section VII, (43) will be used to obtain the BER of a binary phase-shift keying (BPSK)-modulated OFDM
(39)
(43)
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Fig. 2. Exact and approximate cdfs for
= = 1 :0 .
()
jH k j
with
m
=
m
= 5 0 and :
Fig. 3.
FADING CHANNELS
Exact and approximate pdfs for
= = 1 0.
()
jH k j
with
m
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=
m
= 5 0 and :
:
signal over multipath Nakagami- channels using MRC recep) reception, tion. In the special case of single-branch ( numerical results for the BER obtained using (43) equal numerical results for the BER obtained from (22). In evaluation of the integrand in (43), the Lauricella transformation [17, p. 121]
(44) where can be used to guarantee the convergence of the Lauricella functions in (43) for large values of arguments. and For the special case where , i.e., the channel gains become two constant values with equal power, and the error-rate expression in (43) can be reduced to a more compact form. To show this, using (14), we have
(45) where the last equality follows from [12, eq. (6.633.2)], and, therefore, the error rate is given by
(46) VII. NUMERICAL RESULTS AND DISCUSSION Figs. 2–4 assume a multipath Nakagami- channel with two taps where and . The precise curves (solid lines) are obtained using the accurate approach proposed in this paper, and the approximation curves (dashed lines) are obtained from the Nakagami- approximation used in [9]. We observe that the theoretical results match the Monte
Fig. 4. Precise and approximate BER of OFDM-BPSK over a frequency-selective Nakagami-m channel when m = m = 5:0 and
= = 1:0. The precise results are obtained from the approach proposed in this paper, and the approximate results are obtained by the Nakagami-m approximation in [3], [9].
Carlo simulation results (stars) exactly. Both the cdf in Fig. 2 and the pdf in Fig. 3 indicate that the Nakagami- approximation in [9] can be a poor approximation in this case for the distribution of the amplitude of the sum of Nakagami- random vectors. Importantly, Fig. 4 shows that the BER values of OFDM-BPSK obtained using this approximation can severely underestimate the true BER values. For example, when the average SNR at each subcarrier is 25 dB, the true BER is 20 times the value of the BER obtained using the approximation. Similarly, the value of the SNR required to obtain a target value of BER is, in fact, 5 dB more than the value estimated by the approximation. We conclude that the Nakagami- approximation is unreliable. Fig. 5 plots the pdfs of with taps and different fading parameters. The shapes of the pdf for several values further verify that the distribution of the amplitude in the
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Fig. 5. m
=
Fig. 6. m
1 0. :
=
Exact pdfs of jH (k )j for m = = with = = 1:0.
m
1
m
= 5 0, :
BERs of OFDM-BPSK for m = m = = 8:0, m = m = 15:0, and m = m
m
m
=
m
= 8 0, and :
1 4, = = 5 0, = 1 with = = :
m
m
:
frequency domain is not a Nakagami- distribution (observe that, for example, the Nakagami- pdf contains only one mode, whereas two local maxima can be observed in Fig. 5). For the , the pdf agrees with the limiting case when one obtained by Simon [16, Fig. 3]. The corresponding BER curves for BPSK are given in Fig. 6. As predicted in Section V, the slopes of the error-rate curves in the large SNR region are identical for different values (finite) and the error-rate performance degrades with increasing . Fig. 6 also shows that the error-rate performance over the channel with and is worse than the error-rate performance over the channel under fading conditions (corresponding to finite values). This numerical result confirms our asymptotic analysis in Section V. Fig. 7 compares the exact and asymptotic BER performances with two taps and various fading parameters. In Fig. 7, it is seen that the asymptotic performance
Fig. 7. Precise and asymptotic BERs of OFDM-BPSK for m = m = 1:4, m = m = 5:0, m = m = 8:0, m = m = 15:0, and m = m = with = = 1:0.
1
Fig. 8. BERs of OFDM-BPSK for m = m = 8:0, and m = m = MRC reception.
m
=
= 2 0,
=
= 5 0,
1 for single-channel and dual-branch m
:
m
m
:
can be a good approximation to the exact performance (within a factor of about 2) for SNR values greater than 10 dB when . Fig. 8 indicates that the same behavior holds for a system using dual-branch reception. We observe in Fig. 8 that, with two-branch MRC, the slopes of the error-rate curves become larger and significant gains are obtained; for example, in the two-tap multipath Nakagami- channel with fading parameter , to achieve a target BER of 1.5 10 , two-branch reception requires 10 dB less SNR than single-channel reception. Finally, Fig. 9 gives the error-rate performance of a BPSK-modulated OFDM signal transmitted over a multipath Nakagami- channel with different numbers of taps at SNR dB. Contrary to the two-tap case, we observe that the BER rates in fact decrease with increasing parameters when the number of taps is three. In the limit when increases, the BER performance is less variable as changes and converges
DU et al.: ACCURATE ERROR-RATE PERFORMANCE ANALYSIS OF OFDM ON FREQUENCY-SELECTIVE NAKAGAMI-
FADING CHANNELS
327
one has (50) Thus,
(51) Substituting (51) into (47) and using [12, eq. (3.915.2)] as well as the properties of the Bessel functions [13, eqs. (9.63), (9.6.30)], after some simplification, one obtains (20) as desired. APPENDIX B DERIVATION OF (22) Fig. 9. BER at SNR = 25 dB versus the number of taps L.
to that in a Rayleigh fading channel. This can be expected beapproaches cause, by a central limit theorem, the pdf of Rayleigh when the number of taps is large. VIII. CONCLUSION
In this appendix, we derive the closed-form solution in (22) for the error rate of OFDM signals operating in multipath Nakagami- channels. To show this, we first note that the modified Bessel function is related to the confluent hypergeometric function by [13, eq. (13.6.3)] (52) Therefore, we can express
In this paper, we have studied the characteristics of OFDM signals over multipath Nakagami- channels. The pdf of a sum of Nakagami- random phase vectors has been used to derive an exact error-rate expression for coherently modulated OFDM signals transmitted over multipath Nakagami- fading channels. Our asymptotic analysis and numerical results have shown some unique properties pertaining to the transmission of OFDM signals over such channel models.
and
as (53a) (53b)
Substitution of (53) into (20) yields
APPENDIX A DERIVATION OF (20)
(54)
Starting from (19), with an exchange of the order of integration, can be rewritten as
Now, the integral property [12, eq. (7.622.3)] gives
(47) Utilizing an integral identity [12, eq. (6.631.7)], one can express the inner integral in (47) as
(55a) (55b) (55c)
(48) where and Bessel function [13]. Since eq. (8.467)]
is the -order modified can be expanded as [12,
(49)
where is the Lauricella function defined in [17] and is also known as the hyperis geometric function with multiple variables [12], and the Whittaker function [13]. The Whittaker function is related to the confluent hypergeometric function by [12, eq. (9.220.2)]
(56)
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 2, FEBRUARY 2006
Using (55) and (56), we deduce the following integral identity:
(57)
Zheng Du (M’04) received the B.Sc. and Ph.D. degrees in electrical engineering from the University of Science and Technology of China, Hefei, China, in 1998 and 2003, respectively. He has been with the iCORE Wireless Communications Laboratory, University of Alberta, Edmonton, AB, Canada, as a Research Associate since 2003. His current research interests include broadband digital communications systems, orthogonal frequency-division multiplexing, and space–time coding.
Now, applying (57) to (54), we obtain (22).
REFERENCES [1] A. Glavieux, P. Y. Cochet, and A. Picart, “Orthogonal frequency division multiplexing with BFSK modulation in frequency selective Rayleigh and Rician fading channels,” IEEE Trans. Commun., vol. 42, no. 2–4, pp. 1919–1928, Feb./Mar./Apr. 1994. [2] J. Lu, T. T. Tjhung, F. Adachi, and C. L. Huang, “BER performance of OFDM-MDPSK system in frequency-selective Rician fading with diversity reception,” IEEE Trans. Veh. Technol., vol. 49, no. 7, pp. 1216–1225, Jul. 2000. [3] M. Nakagami, “The -distribution, a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. G. Hoffman, Ed. Oxford, U.K.: Pergamon, 1960. [4] T. Eng and L. B. Milstein, “Coherent DS-CDMA performance in Nakagami multipath fading,” IEEE Trans. Commun., vol. 43, no. 2–4, pp. 1134–1143, Feb./Mar./Apr. 1995. [5] M.-S. Alouini and A. J. Goldsmith, “A unified approach for calculating error rates of linearly modulated signals over generalized fading channels,” IEEE Trans. Commun., vol. 47, no. 9, pp. 1324–1334, Sep. 1998. [6] L.-L. Yang and L. Hanzo, “Performance of generalized multicarrier DS-CDMA over Nakagami- fading channels,” IEEE Trans. Commun., vol. 50, no. 6, pp. 956–966, Jun. 2002. [7] A. Scaglione, S. Barbarossa, and G. B. Giannakis, “Optimal adaptive precoding for frequency-selective Nakagami- fading channels,” in Proc. 52nd IEEE Veh. Technol. Conf., vol. 3, 2000, pp. 1291–1295. [8] A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. New York: McGraw-Hill, 1991. [9] Z. Kang, K. Yao, and F. Lorenzelli, “Nakagami- fading modeling in the frequency domain for OFDM system analysis,” IEEE Commun. Lett., vol. 7, no. 10, pp. 484–486, Oct. 2003. [10] T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun., vol. 45, no. 12, pp. 1613–1621, Dec. 1997. [11] J. Cheng, N. C. Beaulieu, and X. Zhang, “Precise BER analysis of dualchannel reception of QPSK in Nakagami fading and cochannel interference,” IEEE Commun. Lett., vol. 9, no. 4, pp. 316–318, Apr. 2005. [12] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic, 2000. [13] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. [14] A. Abdi, H. Hashemi, and S. Nader-Esfahani, “On the PDF of the sum of random vectors,” IEEE Trans. Commun., vol. 48, no. 1, pp. 7–12, Jan. 2000. [15] A. Erdélyi, Tables of Integral Transforms. New York: McGraw-Hill, 1954, vol. II. [16] M. K. Simon, “On the probability density function of the squared envelope of a sum of random phase vectors,” IEEE Trans. Commun., vol. COM-33, no. 9, pp. 993–996, Sep. 1985. [17] H. Exton, Multiple Hypergeometric Functions and Applications, G. M. Bell, Ed. Sussex, U.K.: Ellis Horwood, 1976. [18] Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun., vol. 51, no. 8, pp. 1389–1398, Aug. 2003. [19] A. Erdélyi, Higher Transcendental Functions. New York: McGrawHill, 1953, vol. I.
m
m
m
m
Julian Cheng (S’96–M’04) received the B. Eng. degree (First Class) in electrical engineering from the University of Victoria, Victoria, BC, Canada in 1995, the M.Sc. (Eng.) degree in mathematics and engineering from Queen’s University, Kingston, ON, Canada in 1997, and the Ph.D. degree in electrical engineering from the University of Alberta, Edmonton, AB, Canada, in 2003. He is currently an Assistant Professor with the Department of Electrical Engineering, Lakehead University, Thunder Bay, ON, Canada. Previously, he worked for Bell Northern Research (BNR) and Northern Telecom (now NORTEL Networks). His current research interests include digital communications over fading channels, orthogonal frequency division multiplexing, spread spectrum communications, and statistical signal processing for wireless applications. Dr. Cheng was the recipient of numerous scholarships during his undergraduate and graduate studies, which included a President Scholarship from the University of Victoria and a postgraduate scholarship from the Natural Sciences and Engineering Research Council of Canada (NSERC). He was also a winner of the 2002 NSERC Postdoctoral Fellowship competition.
Norman C. Beaulieu (S’82–M’86–SM’89–F’99) received the B.A.Sc. (honors), M.A.Sc., and Ph.D degrees in electrical engineering from the University of British Columbia, Vancouver, BC, Canada, in 1980, 1983, and 1986, respectively. He was a Queen’s National Scholar Assistant Professor with the Department of Electrical Engineering, Queen’s University, Kingston, ON, Canada, from September 1986 to June 1988, an Associate Professor from July 1988 to June 1993, and a Professor from July 1993 to August 2000. In September 2000, he became the iCORE Research Chair in Broadband Wireless Communications with the University of Alberta, Edmonton, AB, Canada, and, in January 2001, the Canada Research Chair in Broadband Wireless Communications. His current research interests include broad-band digital communications systems, ultrawide-bandwidth systems, fading channel modeling and simulation, diversity systems, interference prediction and cancellation, importance sampling and semianalytical methods, and space–time coding. Dr. Beaulieu is a member of the IEEE Communication Theory Committee and served as its Representative to the Technical Program Committee of the 1991 International Conference on Communications and as Co-Representative to the Technical Program Committee of the 1993 International Conference on Communications and the 1996 International Conference on Communications. He was General Chair of the Sixth Communication Theory Mini-Conference in association with GLOBECOM 97 and Co-Chair of the Canadian Workshop on Information Theory 1999. He has been an Editor for Wireless Communication Theory of the IEEE TRANSACTIONS ON COMMUNICATIONS since January 1992 and was Editor-in-Chief from January 2000 to December 2003. He served as an Associate Editor for Wireless Communication Theory of the IEEE COMMUNICATIONS LETTERS from November 1996 to August 2003. He has also served on the Editorial Board of the PROCEEDINGS OF THE IEEE since November 2000. He was the recipient of the Natural Science and Engineering Research Council of Canada (NSERC) E.W.R. Steacie Memorial Fellowship in 1999. He was elected a Fellow of the Engineering Institute of Canada in 2001 and was awarded the Médaille K.Y. Lo Medal of the Institute in 2004. He was elected Fellow of the Royal Society of Canada in 2002 and was awarded the Thomas W. Eadie Medal of the Society in 2005. He was awarded the University of British Columbia Special University Prize in Applied Science in 1980 as the highest standing graduate in the faculty of Applied Science.
