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Performance of Direct-Sequence Spread-Spectrum RAKE Receivers with Random Spreading Sequences Kyungwhoon Cheun, Member, IEEE Abstract—In this paper, we derive error probability expressions for binary phase-shift keying (BPSK) and quaternary phaseshift keying (QPSK) spread direct-sequence spread-spectrum (DSSS) systems employing random spreading sequences with RAKE receivers. The derived expressions accurately take into account the effect of interpath interference which usually has been neglected in previous analyses. Selection, equal gain, and maximal ratio techniques are considered for diversity combining. Two possible finger assignment strategies, one based on the instantaneous amplitudes and another based on the average powers of the multipath components, are considered for the assignment of multipath components to the available demodulating fingers in the RAKE receiver. Also, various simple, and in many cases, closed-form approximations for the error probabilities are derived and their accuracies are assessed. Index Terms— Diversity methods, pseudonoise coded communication.
I. INTRODUCTION
T
HE use of a rake receiver in direct-sequence spreadspectrum (DSSS) systems provides a unique and valuable means of combating the adverse effects of short-term multipath fading in mobile radio propagation environments [1]–[13]. The processing gain employed in DSSS systems allows the receiver to distinguish between the separable multipath components with delays separated by more than one chip duration, in which case the channel becomes frequency selective [1]. With traditional narrow-band modulation techniques, the frequency selectivity of the channel causes ISI (intersymbol interference), and adaptive channel equalizers are required to cancel the undesirable effects of ISI. The design and implementation of channel equalizers with satisfactory performance for narrowband modulation techniques over time-varying frequencyselective mobile radio channels have been one of the most difficult issues in narrow-band receiver design. With DSSS modulation, the receiver may acquire synchronization to one of the separable multipath components and perform demodulation based on the information contained in that particular multipath component. The other multipath components with relative delays larger than one chip duration are not despread by the receiver, and hence act only as wide-
Paper approved by R. Kohno, the Editor for Spread Spectrum Theory and Applications of the IEEE Communications Society. Manuscript received October 21, 1996; revised February 21, 1997. This work was supported by the Korea Institute of Information Technology Assessment (IITA) U96-59. The author is with the Department of Electronic and Electrical Engineering, Pohang University of Science and Technology (POSTECH), and the POSTECH Information Research Laboratories (PIRL), Pohang, 790-784, Korea (e-mail:
[email protected]). Publisher Item Identifier S 0090-6778(97)06627-0.
band background noise and do not cause ISI.1 Although the problem of ISI can be avoided via DSSS modulation, the short-term fading problem still exists. This is due to the fact that each of the separable multipath components actually consists of many closely spaced (delay difference less than one chip duration) multipath components causing the amplitudes of the separable multipath components, in many cases, to follow the Rayleigh distribution [1], [2]. In this case, the link performance is normally not satisfactory, and some form of diversity is required. Space (antenna) and time diversity are the most popular forms of diversity employed to combat short-term fading. The problem with space diversity is that it requires multiple antennas separated sufficiently far apart in order to guarantee approximately uncorrelated fading among the received signals at the antennas. For this reason, space diversity cannot normally be used at the mobile terminals in the present cellular and PCS bands, and is usually employed only at the base stations. Time diversity is almost universally employed in digital cellular and PCS systems supporting high user mobility. The problem with time diversity is that it is only applicable to cases where the fade duration (channel coherence time) is much smaller than the interleaver depth. Hence, when the mobile velocity is small and thus the fade rate is low, time diversity cannot be employed without excessive delay and hardware complexity due to large interleaver depths required. When multiple (say ) demodulators are available at a DSSS receiver, a new dimension of diversity reception is possible which is usually referred to as path or delay diversity. The receiver operates a search engine which constantly searches the channel for separable multipath components of the desired signal. The search engine, having found the separable multipath components and ranked them in terms of the signal strength, assigns the top components to the available demodulators (usually referred to as fingers) that independently track and demodulate the assigned multipath components. The demodulated data, after being time aligned (deskewed) to compensate for the different, possibly time-varying time delays of the multipath components, are combined for a final decision or fed into a channel decoder. Demodulators of this type are called RAKE receivers. Unlike narrow-band systems where the frequency selectivity of the channel causes ISI, for DSSS receivers employing the RAKE receiver structure, the frequency selectivity of the channel actually improves performance by offering a new dimension of diversity due to the fact that the fade statistics of the separable multipath components are, in many cases, 1 Of course, this is assuming a spreading sequence with period sufficiently larger than the delay-spread of the channel.
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approximately uncorrelated. The RAKE receiver structure also opens up an entirely new concept of deliberate multipath injection by the system in order to improve network performance under multipath fading, especially when the fading is frequency nonselective. Examples of application of this concept are the cell-site diversity (soft-handover) and distributed antenna systems employed in the IS-95 based cellular and PCS networks [8], [17]. In this paper, we investigate the performance of the RAKE receiver with various diversity combining schemes when the employed spreading sequence is very long and sufficiently random so that it may be safely modeled as being purely random [9], [10]. Selection, equal gain, and maximal ratio diversity combining techniques are considered and compared. We also compare two different types of path assignment strategies for the fingers. The first strategy, which is possible when the fade rate is sufficiently small compared to the search engine search rate, is to assign the paths with the largest instantaneous amplitudes to the available fingers for demodulation of each data symbol. The second strategy, which is applicable when the fade rate is comparable to or larger than the search engine search rate, assigns the paths with the largest average powers to the fingers. The performance analysis of the RAKE receiver is carried out via a Gaussian approximation of the matched filter demodulator output similar to one used in [9] and [10]. The Gaussian approximation is known to yield accurate results within the practical region of interest, i.e., uncoded symbol error probabilities larger than 10 [18]. The analysis for equal gain and maximal ratio combining takes into account of the effects of interpath interference (IPI) among the combined paths [6] which usually have been neglected under the assumption that they are negligible. The results show when the IPI can be safely neglected, and when it should be taken into account for a more accurate estimate of the error probability. Also, various simple, and in many cases, closed-form approximations for the error probabilities are derived and their accuracies are assessed. The analyses may also easily incorporate the effects of multiple-access interference in CDMA networks employing random spreading sequences since, for the system model employed, the effect of a separable multipath component from an interfering user on the RAKE receiver performance is identical to that of a separable multipath component of the desired user that has not been assigned to a finger for demodulation. The organization of the paper is as follows. First, a description of the employed channel and system models are given in Section II, and performance analysis is carried out in Section III. Numerical results are presented for various cases in Section IV, and, finally, conclusions are drawn in Section V. II. SYSTEM
AND
CHANNEL MODELS
The transmitted signals under consideration are BPSK and QPSK spread DSSS signals with BPSK data modulation given below with powers normalized to 1/2. BPSK spreading
(1)
QPSK spreading
(2)
where and . Here, the data sequence and the spreading sequences , and are modeled as mutually independent i.i.d. (independent and identically distributed) random sequences taking on the values of +1 and with equal probability, for and zero otherwise, and and are the chip and the data symbol durations, respectively, with the ratio defined as the processing gain of the system. The carrier frequency is assumed to be sufficiently large for the double carrier frequency terms to be safely neglected. The channel is modeled as having a baseband equivalent impulse response given as follows: (3) where ’s represent the amplitudes of slowly varying independent stationary Rayleigh random processes with pdf (probability density function) given by (4) and
’s are i.i.d. random variables uniformly distributed on . The ’s are the relative delays of the separable multipath components with and are assumed to satisfy . The received signal can then be written as
(5) for BPSK spreading and
(6) for QPSK spreading where the background thermal noise is modeled as an AWGN process with two-sided power and spectral density equal to . The available fingers are assigned to the of the separable paths according to one of the following two strategies when (when , the unassigned fingers remain idle, and their outputs are not used for diversity combining). When the fade rate is sufficiently small, i.e., the bandwidth of the ’s is small enough so that it remains nearly constant while the search engine searches out the paths with the largest ’s, the fingers may be assigned to the paths with the largest instantaneous amplitudes for
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demodulation of each data symbol. The variation of the ’s should also be slow enough to allow time for the PN code and carrier phase-tracking loops in the fingers to pull in and start demodulation well before the values of ’s change significantly. This finger assignment scheme should be applicable to networks supporting relatively low-mobility users. When this is not practically possible, we consider assigning to the fingers the paths possessing the largest average powers. In this case, the searcher engine will search out the paths with the largest average powers, and assign them to the demodulating fingers. These schemes for finger assignment are referred to as the IAFA (instantaneous amplitude-based finger assignment) and the APFA (average power-based finger assignment) schemes, respectively. Clearly, IAFA will offer performance superior to that of APFA. Exactly when it is possible to employ IAFA depends on the search rate of the search engine relative to the fade rate. For any given fade rate sufficiently smaller than the data symbol rate, IAFA may be employed with a sufficient number of correlators in the search engine. The constraint is the hardware complexity required to implement the correlators and the settling time required for the PN code and carrier phase-tracking loops in the fingers after being assigned to a path. Let denote the deskewed (time-aligned) output of the th finger. Then, the final output of the RAKE receiver after diversity combining, denoted by for a given data symbol, is given as (7) where the selection of the combining weights ’s determine the specific diversity combining technique employed.
where the paths assigned to a finger are reindexed with and the paths that are not assigned to a finger are reindexed with (we will follow this convention in the following derivations). Here, is the desired symbol to be demodulated, is the instantaneous received symbol energy and , the contribution of the background thermal noise, is a zeromean Gaussian random variable with variance . Also, is the average SNR (signalto-noise ratio) of the th multipath component. We assume without loss of generality that the paths are ordered such that with IAFA and with APFA. The term denotes the interference contribution of the th multipath component to the output of the th finger given by (9) are i.i.d. random variables where uniformly distributed on and the random data modulation was subsumed into the random spreading sequence without loss of generality. The variance of denoted by can be computed via straightforward computations to be [9], [10] (10) where
and is the absolute normalized relative delay between paths and assumed to be independent and uniformly distributed on . With these results and noting that has zero mean, we find that has mean and variance equal to
III. PERFORMANCE ANALYSIS
(11)
To analyze the performance of the diversity combining schemes considered, we adopt a Gaussian approximation approach based on the central limit theorem. The Gaussian approximation is known not only to give accurate estimates of the error probability in the region of practical interest, but also to offer insights into the effects of various system parameters and interference sources on the performance of the RAKE receiver. We model the output of the th finger as being conditionally Gaussian given and The conditional mean and variance of for BPSK and QPSK spreading cases given , and are derived in the following two subsections. A. Conditional Mean and Variance of For BPSK spreading,
B. Conditional Mean and Variance of
: QPSK Spreading
For the QPSK spreading case, we take
to be (12)
where the in-phase and the quadrature components are given as
and
(13)
: BPSK Spreading
is given as (14) (8)
and and are independent zero-mean Gaussian random . The terms and are variables with variance
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the contribution of the th multipath component on the inphase and quadrature components of the th finger output respectively, given as (15) (16) where
pdf for and zero otherwise [1]. The average error probability may be computed by averaging (23) over the associated random variables. The averaging may most efficiently be carried out via the Monte Carlo technique [14]. At this point, let us consider approximating in (23) by its mean value of 1/3 for the special case when . In this case, the average error probability may be approximated as follows (Appendix A):
(17) (18)
(24)
(19) (20) (25) which are clearly uncorrelated with zero mean with identical variance equal to . Hence, are zero-mean, uncorrelated random variables with identical variance equal to
It is interesting to note that (25) may also be obtained by approximating the summation term in the denominator of (23) by its mean value as follows using (A2) of Appendix A:
(21) for the With these results and (12)–(14), we find that QPSK spreading case has mean and variance given as in (11), but with
(26)
(22) Hence, once we have an error probability expression for the BPSK spreading case, we may simply replace by its mean value of 1/2 to obtain the error probability expressions for the QPSK spreading case. Also, for the case with independent data modulation for the in-phase and quadrature channels, it is easy to see that the error probability will be identical to that of the BPSK data modulation/BPSK spreading case for a given bandwidth and data rate. With these results at hand, we now compute the performance of the diversity combining schemes with BPSK spreading starting with the single-finger receiver case. C. Single-Finger Receiver: SFR For the case when there is only one demodulating finger and the finger is locked onto an arbitrary path, say, the th multipath component, the above Gaussian approximation for results in the following expression for the conditional symbol error probability given and :
(23)
and are i.i.d. central chiwhere squared random variables with two degrees of freedom with
(27)
where case of
, which for the , reduces to
(28) (29)
which is identical to with (25). Note that (26) and (28) become exact as approaches infinity. We observe that the average error probability decreases linearly in as expected, and linearly increases and decreases with the number of paths and the processing gain , respectively. It is also clear that we obtain exactly the same results if we interpret the unassigned multipath components as multipath components of interfering multiple-access users. D. Selection Diversity Combining: SDC With selection, equal gain, and maximal ratio diversity combining, we consider the IAFA and APFA schemes for finger assignment. The conditional error probability for SDC with the IAFA scheme is given as follows (recall that
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with
for IAFA):
. Since (33)
(30)
The expression for the conditional error probability of the APFA scheme is identical to (30), but with the paths ordered according to the per path average SNR’s, . Clearly, approximations (25)–(29) for the SFR case may also be applied to the SDC–APFA case by setting equal to 1. For the IAFA case, we consider the approximation
we have the expressions (34) and (35), shown at the bottom of the page, for the pdf of and thus . After some straightforward manipulations and using (A2) of Appendix A, we find (36), shown at the bottom of the page, which is easily seen to reduce to the SFR case for . For the special case when , we have (Appendix B) (37) where
with (38)
Furthermore, (36) and (37) become exact as infinity.
approaches
E. Equal Gain Combining: EGC
(31) Under this approximation, we have (32)
Unlike SDC, with equal gain and maximal ratio combining, the effect of IPI comes into play. Consider Fig. 1, which depicts the case where two available fingers are assigned to the first two separable multipath components with delays and respectively. In order to demodulate the data bit , finger 1 will generate a decision variable by despreading and integrating in and finger 2 will generate in a similar fashion. The final decision variable is then where and are nonzero for the
(34)
(35)
(36)
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of
is given as (41)
When , the two integrals overlap, and are thus in this case as follows: correlated. We rewrite Fig. 1. Illustration of the case with three multipath components with the two earliest arriving paths assigned to the two available fingers.
cases of EGC and MRC. The effect of IPI, which in this case is the effect of path 1 (path 2) on ( ) and the possible correlation between and , has not been accurately taken into account in many of the previous analyses either under the assumption that it can be neglected or that the finger outputs are uncorrelated. Here, we accurately model the effect of IPI and analyze its effect on the performance of the RAKE receiver. The final decision variable after diversity combining is given by (7). This combined with (8) results in the following expression for :
(42)
where it was assumed, without loss of generality, that . Clearly, the three integrals are nonoverlapping, and are thus uncorrelated. Using the fact that the variance of for is , in this case is given as the variance of
(43)
(39) where, again, the paths that are not assigned to a finger are denotes the indexed by and sum of the IPI components between fingers and . We note that are uncorrelated, where and may be correlated. The last two summation terms in (39) have zero mean and variances and , respectively. We need to compute the variance of in order to compute the variance of . First, as we write
(40) It is clear that when , the two integrals do not overlap, and are thus uncorrelated. In this case, the variance
with being a positive where . integer and Combining (41) and (43), we obtain the expression for the as variance of
(44) . We note where that since , paths cause larger IPI than with relative delays smaller than . Therefore, when those with relative delays larger than the exact distribution of the delays of the separable multipath components is unknown, we may obtain an upper bound on the error probability by assuming that the relative delays between all of the multipath components assigned to a finger are well of each other by setting . Using within
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these results, we obtain the following expressions for the conditional error probability with EGC:
(45)
F. Maximal Ratio Combining: MRC The diversity combining weight for the th finger output that should be used in (7) for optimum performance is where is the variance of the th finger output [1]. We assume that the noise powers as seen by the fingers are approximately the same, and we set . Then, using the results derived in the previous subsections, we have the following expression for the conditional error probability with MRC:
(50)
,) if we neglect the For APFA (and for IAFA when IPI term, i.e., set , and approximate the first double summation term in the denominator of (45) by its mean value, we arrive at the following approximation for the conditional error probability:
(46)
with when
As with the EGC case, we may obtain the following approximation for the conditional error probability with APFA (and for IAFA when ): (51)
. Furthermore, (46) reduces to with
. Since the pdf of in this case is given as [1]
(47) (52) with . For IAFA, a similar procedure gives the following approximation for the conditional error probability for the case when :
with (53) we arrive at the following approximation for the average error probability using the techniques employed in [1]:
(48) (54) with . Using the results from the theory of order statistics [16], we have the following expression for the mean of for IAFA:
Furthermore, when
,
, (50) reduces to (55)
with
(49) approaches infinity. UnAlso, (46)–(48) become exact as fortunately, closed-form expressions for these approximations to the average error probability could not be found.
. Since follows the chi-squared distribution with degrees of freedom whose pdf is given as [1] !
(56)
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Fig. 2. Average error probability versus
T ,
SFR,
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L
= 3; 7; 15; 30;
N
we arrive at the following approximation for the average error probability for this case [12, Appendix A]:
(57)
This is easily seen to reduce to the SFR case for For IAFA and , we have
.
(58)
where with are given by (49). Equations (54), (57), and (58) become exact as approaches infinity. IV. NUMERICAL RESULTS For the numerical results presented in this section, we assume a processing gain of and that the per-path average SNR’s are identical, i.e., . The average error probabilities are plotted against the total received average and for the SDC case, we also show plots of SNR, the average error probabilities versus the number of multipath components . Fig. 2 shows the average error probability plots of the SFR and . First, we observe that approximating for in (23) by its mean value of 1/3 gives results almost
= 64.
indistinguishable from the exact2 results obtained by numerically averaging (23) over the associated random variables. This significantly simplifies the computation of the average error probabilities since we need only numerically average over the and need not perform the averaging over the random delays ( ’s) and phases ( ’s). We also note that QPSK spreading offers performance almost identical to the BPSK spreading case. Recall that our performance measure is the average error probability, and for the worst case scenario where all terms in are 1 for BPSK spreading, QPSK spreading, which is equivalent to setting in will offer better performance. Looking at the accuracies of the various approximations to the average error probability of the SFR, we note that the asymptotic approximations for the large SNR and processing gain (relative to ) case of (25) and those corresponding to the case of infinite processing gains are only accurate in the corresponding regions, and are quite inaccurate when the asymptotic conditions are not met. For example, the asymptotic approximation (25) predicts error probabilities larger than one for small SNR’s, and also gives inaccurate results when is not sufficiently larger than . On the other hand, the simple approximations (24) and (28) give extremely accurate estimates of the exact error probabilities for all values of SNR’s and the number of paths considered. Approximation (28) [(26)], which is given in a simple closed form, is especially useful in estimating the error performance of SFR’s. The error probability curves for the SDC scheme are shown in Figs. 3–6 for and with IAFA and APFA. 2 The terminology exact is used with the understanding that it only refers to the exactness of the averaging operation over ; , and when computing the average error probability, and not to imply that the resulting average error probability is exact, which it is not, due to the underlying Gaussian approximation.
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Fig. 3. Average error probability versus T , SDC, L = 3; N = 64.
Fig. 4. Average error probability versus T , SDC, L = 7; N = 64.
SDC with APFA is identical to the SFR with the same under our assumption that the per-path SNR’s value of are identical. We note that for IAFA, approximation (37) is but rather inaccurate for large SNR’s for small values of as increases, the accuracy improves. Since the asymptotic case of infinite processing gain quite accurately predicts the average error probability for small values of , we may use the infinite processing gain approximation for small values and approximation (37) for large values of to get a of first-hand approximation of the average error probability for
SDC with IAFA. Approximating by its mean value in (30) for IAFA gives quite accurate results for the of range of error probabilities considered with some error in the saturation region (large SNR’s) where the approximation gives slightly optimistic results. The QPSK spreading case with IAFA is also observed to offer performance close to the BPSK spreading case, except in the saturation region where it offers performance slightly superior to that of the BPSK spreading case, and is almost identical to the BPSK spreading case with approximation. The large gain of the IAFA the
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Fig. 5. Average error probability versus T , SDC, L = 15; N = 64.
Fig. 6. Average error probability versus T , SDC, L = 30; N = 64.
scheme compared to the APFA scheme is clearly seen from these figures, which is rather obvious for the SDC case since SDC with APFA for equal per-path SNR’s is identical to the SFR. One last interesting observation from these figures is that that minimizes the average there is an optimum value of for the IAFA scheme. To error probability for a given make this more clear, in Fig. 7, we plot the average error and probability for SDC (IAFA) versus when dB. The existence of an optimal value of for SDC with IAFA for a given SNR is due to the fact that for small values
of , the diversity gain resulting from multiple independent observations of the channel dominates, and a performance improvement is observed as increases, but for larger values of , the decrease in the per-path SNR for a fixed total received SNR dominates and performance degrades with . In Figs. 8 and 9, we plot the average error probabilities with and , for the case of EGC for respectively, for IAFA and APFA. The difference between ] and the upper bounds the lower bounds [ ) to the error probabilities, although it (
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Fig. 7. Average error probability versus L, SDC exact (30), T = 15; 20 dB, N = 64.
Fig. 8. Average error probability versus T , EGC, L = 13; J = 3; N = 64.
increases with , stays within 1/2 dB of one another. This implies that the exact arrival-time distribution of the multipath components as observed by the receiver has only a small effect on the performance of a RAKE receiver employing EGC whether the finger assignment scheme is IAFA or APFA. The stays within 1 dB results assuming zero IPI, i.e., of the upper bound. Hence, the effect of IPI may be neglected with errors less than 1 dB in the worst case for the system parameters considered. Also, note that approximations (47) for APFA and (48) for IAFA (which are based on the zero IPI assumption) accurately predict the exact error probabilities
under the zero IPI assumption for the parameters considered, resulting in a slight optimistic prediction. approximation As with the SFR case, the and the QPSK spreading case result in error probabilities almost identical to the exact error probability with BSPK spreading. The large gains achievable by using IAFA as opposed to APFA is evident from these figures, especially when the number of demodulating fingers is small. The difference in the required SNR needed to achieve an error probability of 10 for the two finger assignment strategies is and reduces to approximately approximately 12 dB for
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Fig. 9. Average error probability versus T , EGC, L = 13; J = 10; N = 64.
1.8 dB for . Also, with IAFA, only three fingers are required to achieve performance close to the infinite processing gain case, whereas many more are required with the APFA scheme. From these figures, we also note that the required SNR to achieve 10 error probability is approximately same for IAFA with and APFA with . Hence, with APFA, seven additional demodulating fingers are required to achieve performance close to a receiver using the IAFA scheme. On the other hand, a receiver employing the IAFA scheme requires a faster searcher engine. The decision on which finger assignment scheme to employ should be based on a tradeoff between the hardware complexity required to implement additional fingers versus implementing a faster searcher engine. Figs. 10 and 11 show the average error probability results for the MRC case, and we note that observations similar to the EGC case can be made. Gains of approximately 2.2 and 1.2 dB are observed with MRC compared to EGC in the SNR required to achieve an average error probability of 10 with APFA for and , respectively. The gain achieved with MRC compared to EGC with IAFA is much smaller and is well within 1/2 dB. Hence, with IAFA, we may use the simpler EGC which does not require normalization of the finger outputs before combining as does the MRC with negligible degradation in performance.
tions were derived for the average error probabilities which should be useful as a first-hand estimate of the average error probabilities. Two finger assignment strategies based on the instantaneous amplitudes (IAFA) and the average powers (APFA) of the multipath components were considered for assignment of fingers to the available multipath components, and their respective performances were compared. APPENDIX A DERIVATION OF (24) AND (25) We wish to compute
(A1)
where are i.i.d. central chi-squared random variables with two degrees of freedom with pdf for and zero otherwise. Let ; then follows the central chi-squared pdf with degrees of freedom with pdf for and zero, otherwise [1]. Since [19]
V. CONCLUSIONS In this paper, expressions for the error probabilities were derived for DSSS receivers employing RAKE receiver architectures under frequency-selective multipath fading based on a Gaussian approximation of the demodulator outputs. Selection, equal gain, and maximal ratio diversity combining schemes were considered. Along with the exact expressions, various simple, and in many cases closed-form approxima-
(A2) we have (A3)
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Fig. 10.
Average error probability versus T , MRC, L = 13; J = 3; N = 64.
Fig. 11.
Average error probability versus T , MRC, L = 13; J = 10; N = 64.
which gives (24) and (A4) (A6)
(A5)
(A7) which gives (25).
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APPENDIX B DERIVATION OF (37) For the special case when
N
reduces to (B1)
Hence [19],
(B2) Therefore, (B3)
(B4) where
. Making
a change of variables as
[3] U. Grob et al., “Microcellular direct-sequence spread-spectrum radio system using -path RAKE receiver,” IEEE J. Select. Areas Commun., vol. 5, pp. 772–780, June 1990. [4] S. Allpress et al., “An investigation of RAKE receiver operation in an urban environment for various spreading bandwidth allocations,” in Proc. 1992 IEEE Veh. Technol. Conf., Denver, CO, May 1992, pp. 506–510. [5] G. Bottomley, “Optimizing the RAKE receiver for the CDMA downlink,” in Proc. 1993 IEEE Veh. Technol. Conf., Secaucus, NJ, May 1993, pp. 742–745. [6] N. Chan, “Multipath propagation effects on a CDMA cellular system,” IEEE Trans. Veh. Technol., vol. 43, pp. 848–855, Nov. 1994. [7] P. Van Rooyen et al., “Performance of coded SSMA system and RAKE reception on a Nakagami fading environment,” in Proc. 1994 Int. Symp. Inform. Theory, Sydney, Australia, Nov. 1994, pp. 121–125. [8] The CDMA Network Engineering Handbook, Vol. 1: Concepts in CDMA. San Diego, CA: Qualcomm, Inc., 1992. [9] D. Torrieri, “Performance of direct-sequence systems with long pseudonoise sequences,” IEEE J. Select. Areas Commun., vol. 10, pp. 770–781, May 1992. [10] K. Cheun, Spread-Spectrum Communications. Pohang, Korea: POSTECH Press, 1995. [11] K. Wu and S. Tsaur, “Selection diversity for DS-SSMA communications on Nakagami fading channels,” IEEE Trans. Veh. Technol., vol. 43, pp. 428–438, Aug. 1994. [12] T. Eng and L. Milstein, “Coherent DS-CDMA performance in Nakagami multipath fading,” IEEE Trans. Commun., vol. 43, pp. 1134–1143, Feb./Mar./Apr. 1995. [13] D. Goeckel and W. Stark, “Performance of coded direct-sequence systems with RAKE reception,” European Trans. Telecommun., vol. 6, pp. 41–51, Jan./Feb. 1995. [14] W. Press et al., Numerical Recipes. Cambridge, U.K.: Cambridge Univ. Press, 1986. [15] M. Abramowits and I. Stegun, Handbook of Mathematical Functions. New York: Dover, 1970. [16] A. Papoulis, Probability, Random Variables and Stochastic Processes. Tokyo, Japan: McGraw-Hill, 1984. [17] Telecommunications Industry Association (TIA), TIA/EIA Interium Standard: Mobile Station—Base Station Compatibility Standard for Dual-Mode Wideband Spread Spectrum Cellular System, TIA/EIA/IS-95, July 1993. [18] E. Geraniotis and M. Pursley, “Performance of coherent direct-sequence spread-spectrum communications over specular multipath fading channels,” IEEE Trans. Commun., vol. COM-33, pp. 502–508, June 1985. [19] I. S. Gradshteyn and L. M. Ryzhik, Table of Integrals, Series and Products. New York: Academic, 1980.
, we have
(B5) Setting and
and and evaluating the integral by parts results in
(B6)
(B7) (B8) giving (37). REFERENCES [1] J. Proakis, Digital Communications, 2nd ed. New York: McGraw-Hill, 1989. [2] M. Kavehrad and B. Ramamurthi, “Direct-sequence spread spectrum with DPSK modulation and diversity for indoor wireless communications,” IEEE Trans. Commun., vol. COM-35, pp. 224–236, Feb. 1987.
Kyungwhoon Cheun (S’88–M’89) was born in Seoul, Korea, on December 16, 1962. He received the B.A. degree in electronic engineering from Seoul National University in 1985 and the M.S. and Ph.D. degrees from the University of Michigan, Ann Arbor, in 1987 and 1989, respectively, both in electrical engineering. From 1987 to 1989, he was a Research Assistant at the EECS Department, University of Michigan, and from 1989 to 1991, he was with the Department of Electrical Engineering, University of Delaware, as an Assistant Professor. In 1991, he joined the Department of Electronic and Electrical Engineering, Pohang University of Science and Technology (POSTECH), where he is currently an Associate Professor. His current research interests include cellular and packet radio networks, algorithm and VLSI design for U.S. and European HDTV modems, military communication networks, synchronization/equalization for radio systems, free space laser optic communication systems, and communication networks for public transportation systems. He has also served as an engineering consultant to various industries in the areas of modem, VLSI design, and cellular networks.
