On the Detection of a Class of Fast Frequency-Hopped
Multiple Access Signals
Yu T. Su, Member, IEEE, Ye-Shun Shen, and Chu-Ya Hsiao
Abstract—The capacity of a fast frequency-hopped multiple access (FHMA) system is dictated by two major related design concerns: the hopping pattern and the receiver structure. This paper studies the impact of these two factors. We present a maximum likelihood (ML) diversity combiner for detecting asyn-chronous FHMA multilevel FSK (MFSK) signals in Rician fading channels and analyze the performance of a close approximation of the ML receiver. We compare systems using random hopping patterns and those using optimal hopping patterns of Einarsson [8]. Performance comparisons between chip-synchronous and chip-asynchronous systems are made as well. We propose and ex-amine the effectiveness of a two-stage multiuser detector, in which the first stage makes an initial decision while the second stage tries to reduce multiple access interference (MAI) and resolve the ambiguity left by the first stage detector. The MAI caused by undesired users is constituted by a cochannel interference (CCI) contribution and an interchannel interference (ICI) contribution. This detector is of modest complexity and is capable of removing most of the CCI and part of the ICI. Two methods for mitigating MAI are also examined.
Index Terms—Frequency hop communications, multiaccess communications, signal detection.
I. INTRODUCTION
T
HE DETECTION of a code division multiple access (CDMA) signal is often hampered by the interference generated by system users whose carrier frequencies are the same [i.e., cochannel interference (CCI)] or close to [inter-channel interference (ICI)] that used by the desired user. Therefore, in order to enhance the signal’s detectability and increase the system’s capacity, one needs to devise a detection scheme for reducing both the CCI and ICI. This paper suggests several detector structures for the fast frequency-hopped mul-tiple access systems studied in [1]–[10]. We also examine their efficiency in dealing with the above-mentioned interference and evaluate the related system capacity as well as spectral efficiency.Earlier analysis on noncoherent frequency-hopped multiple access (FHMA) systems [1]–[6] assume that: 1) a single-stage detector is employed by the receiver and 2) random memory-less FH patterns (users’ addresses) are used. Moreover, except
Manuscript received December 30, 1999; revised June 18, 2000. This work was supported in part by the National Council of Taiwan under Grant NSC 84-2612E009-003. This work was presented in part at the VTC’96, Atlanta, GA, USA.
Y. T. Su is with the Department of Communication Engineering, National Chiao Tung University, Hsinchu 30056, Taiwan R.O.C.
Y.-S. Shen is with the International Business Group, Chunghwa Telecom, Kaohsiung 80053, Taiwan R.O.C.
C.-Y. Hsiao is with Novatek Microelectronics, Hsinchu 30077, Taiwan R.O.C.
Publisher Item Identifier S 0733-8716(01)09081-3.
for [3], they also assume that a chip(hop)-synchronous mode is maintained, i.e., all users switch their carrier frequencies si-multaneously. Since multilevel FSK (MFSK) is used in these systems, 1) implies that the detector will simply choose the channel exhibiting the largest combined value as its output and 2) tells us that the performance obtained in this analysis is ac-tually the performance averaged over all possible address as-signment schemes. The third (chip-synchronous) assumption is a more practical assumption for the forward link than for the reverse link. This assumption also exempts the receiver from dealing with ICI, if an appropriate channel separation is used by the system. Removal of this assumption, as we shall see, will lead to serious ICI that may substantially impair a receiver’s de-tection capability.
Solomon [7] and Einarsson [8] had proposed FH pattern assignment schemes, which render minimum MA interfer-ence. Taking advantage of the inherent algebraic structure of these constructions, Timor [9] presented a two-stage chip-syn-chronous demodulator that is capable of eliminating most of the CCI if the thermal noise effect can be neglected. He later [10] suggested a multistage decoding algorithm to further improve the performance of the two-stage algorithm. Mabuchi et al. [11] proposed a class of multistage demodulation algorithms for FHMA/MFSK systems, which is similar to the multistage scheme developed in [10]. These algorithms, like that de-scribed in [10], are much more complicated than the two-stage decoding algorithm [9], since they require first-stage detection of all users. According to these studies [9]–[11], multiuser detection of fast FHMA/MFSK signals can be decomposed into two major successive stages. The first stage combines noncoherently detected outputs matched to various chips (hops) in an appropriate manner and makes a decision if possible, while the second stage attempts to resolve the ambiguity (if exists) left by the first stage of the detector. The first stage can reduce CCI and ICI but is incapable of eliminating them. The CCI and ICI cancellation is accomplished mostly by the second stage. Furthermore, the second stage detection can be divided into several substages, depending on whether outputs from other users’ detectors are available.
Our effort reported in this paper can be summarized as fol-lows. After a brief description of the system under investiga-tion in Secinvestiga-tion II, we derive a nearly maximum likelihood (ML) single stage detector for asynchronous FHMA/MFSK systems in Section III and analyze the performance of a receiver that is a close approximation of the ML one. In Section IV, we examine the performance of both the synchronous and asynchronous sys-tems using the optimal hopping pattern suggested by Einarsson. Means for containing the MA interference are suggested and
Fig. 1. A block diagram for the FHMA/MFSK system under investigation.
the associated statistical properties are derived in this section as well. Section V proposes a two-stage multiuser detection scheme while Section VI summarizes the major results and ob-servations obtained in this contribution. The Appendixes present derivations of some key equations and a modified interference test.
II. SYSTEMDESCRIPTION ANDDEFINITIONS
A block diagram of the FHMA/MFSK system proposed by Goodman et al. [1] is depicted in Fig. 1. The data source pro-duces a data bit stream of rate bits/s. The data stream is mapped into an MFSK signal sequence of rate symbols/s,
where and . The carrier
fre-quency of an MFSK symbol is then hopped times within s, i.e., the subsymbol (chip) duration is s and the chip rate is . Suppose (Hz) is the hopping bandwidth and the channel spacing is Hz, then the channel (tone) number available for hopping is given by
(1) where is the integer part of and is usually larger than . In this paper, however, we will only consider the case of , for such a system yields a spectral efficiency higher than those having other frequency structures of [6].
Thus the candidate values of ( ) for a given are re-lated by
(2) and the spectral efficiency , measured in bits/s/Hz, is given by [4]
(3) where is the system capacity, i.e., the maximum number of active users the system can support while guaranteeing a given performance. When the channel spacing is wider than the minimum required value, i.e., when , , then the spectral efficiency should be calculated by . It is clear that a system’s spectral efficiency can be improved by using a wider channel spacing, only if it brings a large enough capacity increase.
Frequency hopping and dehopping, as illustrated in Fig. 2, is performed in a deterministic manner. Each user is given an address (hopping pattern) which is an -tuple,
, where ,
, represents a candidate MFSK tone of duration s. The actual transmitted frequencies within a symbol
interval is , where
Fig. 2. Time-frequency matrices for (a) the transmitter and (b) the receiver.
bits message, denotes modulo- addition, and . An optimal assignment scheme suggested in [8] associates the th user an element and the address
(4) where is a fixed primitive element in .
Since the transmitted signal, , is required to hop within
the frequency band , where
is the lowest frequency in the time-frequency (T-F) matrix as shown in Fig. 2, it can be expressed as
(5)
where the random phases result from the noncoherent frequency hopper. Furthermore, ,
is a set of basis waveforms defined by (6) is the signal power, is the average signal energy per chip, and
otherwise. (7)
The received waveform is first dehopped and then detected by a bank of energy detectors as shown in Fig. 3. Typical re-ceived and dehopped spectrums are shown in the bottom half of Fig. 2, where circles in the T-F matrix are interfering tones
from other users. The received waveform, corresponding to the desired user, consists of three components
(8)
where
desired signal;
interference caused by the th interferer; number of interferers (hence there are users in the system);
additive white Gaussian noise (AWGN) whose one-sided power spectral density is W/Hz.
We model the channel effect as such that all candidate basis waveforms suffer from frequency nonselective fading obeying an independent and identically Rician distribution. More specifically, we assume that different MFSK tones at the same chip experience independent and identically distributed (i.i.d.) frequency nonselective Rician fading and statistics of a user signal at different chips are i.i.d. as well. The former (tone-independent) assumption is appropriate, since different user signals—even if the same carrier frequency are em-ployed in the same interval—are propagated over different physical paths. The second (chip-independent) assumption is approximately valid, because it is either impossible [7], [8] or unlikely (when a set of random hopping patterns with large is used) for neighboring chips of a user address to use the same frequency or even the adjacent frequencies. Ignoring the propagation delay, the desired signal in the first symbol period can therefore be written as
(9)
where are random variables ( s) uniformly distributed within the interval of and are i.i.d. Rician s. Note that represent the noncoherent receiver’s tainty about the phase of the received waveform. This uncer-tainty is the sum of the random phases associated with the dif-fused component (induced by the channel) and the specular component [i.e., of (5)] of the received waveform. Given the phase of the transmitter , the phase of the diffused com-ponent follows an unimodal distribution and is correlated with . However, after taking the average over , which is uni-formly distributed within , we can show that is also uniformly distributed within and is independent of . Let be the ratio of the power in the specular component to the power, , in the diffused component. Using the
normal-ization, , we obtain
and , respectively. When (no direct path) we have flat Rayleigh fading and (no diffused part) rep-resents the AWGN-only case.
The th interfering signal, being transmitted through a similar but independent Rician fading channel, has the form
Fig. 3. A nonlinear diversity-combining receiver for noncoherent FHMA/MFSK signals.
where and are independent random variables having the same distributions as and of the desired signal. For each pair, however, only one is nonzero, i.e.,
, where for all and ,
is the frequency tone used by the th interferer during the th chip interval. is the corresponding normalized timing mis-alignment with respect to the desired signal and is assumed to be uniformly distributed within (0, 1]. For chip-synchronous sys-tems, we have for all .
With the received waveform given by (8)–(10), the dehopped signal becomes
(11) where
(12)
(13)
and . Since the phase of the local
frequency synthesizer is not continuous (noncoherent) when it hops from one band to another, i.e., the “local phases” at different hops are independent, the dehopped random phases, , , , are i.i.d. s that are uniformly distributed
within .
The above equations, (11)–(13), imply that perfect power control has been assumed so that all the received signals have the same average strength. In the absence of channel perturbation, frequency dehopping followed by hard-decision detection is equivalent to a modulo- subtraction. Thus the
message chips will fill a complete row in the receiver’s T-F matrix whose index is equal to the desired message ; see Fig. 2(b). The presence of the channel impairment may not only prevent the receiver from forming a complete row but also produce false rows [1], [9].
When hard-limited detection is applied to each diversity branch, the majority vote method that selects the row with the largest number of detected chips as the desired message is a simple and reasonable detection rule. In fact, this de-tection scheme is equivalent to a decision rule that chooses the codeword having minimum Hamming distance from the hard-limited detected word. If there are two or more rows having equal minimum distance, one of them will be selected randomly in order to remove the ambiguity. But the probability that hard-limited combining results in more than one largest row is an increasing function of the number of users . These ambiguous events eventually become the dominant factor in limiting the system capacity. In general, there are two methods to resolve this kind of ambiguity. One is to use soft-decision schemes [6] while the other uses the information of the other users’ addresses to eliminate the faulty entries in the receiver’s T-F matrix. The first approach is studied in the following two sections while one alternative to the second approach will be presented in Section V.
III. RANDOMMEMORYLESSFH PATTERNSYSTEM
This section considers asynchronous random FH pattern sys-tems. The interfering signals in (13) are assumed to have used mutually independent random and memoryless FH patterns (ad-dresses). In other words, each component of a user’s address is randomly and independently selected from the set . Thus each element of the desired T-F matrix is hit by an interferer with probability if the chip-synchronous mode is as-sumed. Note that the multiple i.i.d. Rician channel assumption that leads to (8)–(10) and therefore (11)–(13) implies , , and are i.i.d. s. Furthermore, we emphasize that, as mentioned before, this assumption is only approximately
true. Hence the claims of maximum likelihood and optimal non-linearity in the following section are to be understood as such.
A. Maximum Likelihood Receiver
Let us first assume that the a priori probability of transmit-ting any bin is the same, i.e., . Denote the output of the th energy detector at the th diversity branch (hop) during a given symbol interval (see Fig. 3) as and the associated condi-tional probability density function (pdf) given that the th mes-sage bin was sent as . By invoking the approximation that are i.i.d. [3]–[6] and following a procedure similar to that presented in [3] and [6], we can show that the (nearly) optimal noncoherent Bayes’ decision rule is given by
Accept the hypothesis that the th tone was transmitted if
(14) It can be shown that is the squared magnitude of the com-plex random variable , which is given by
(15)
where
is given by (11);
complex conjugate of defined by (6);
, , and components contributed by the desired signal, the th interferer and AWGN, respectively.
In particular, is a zero-mean complex Gaussian r.v. whose real and imaginary parts have the same variance . Substituting (6) and (11)–(13) into the upper part of (15), we immediately have
(16)
where , and
(17)
where .
The magnitude of the product 1 represents the fre-quency separation between the dehopped th interferer at the th 1Note that our interference model (10) allows only one nonzeroc for each
(j; l) pair, hence only the corresponding g does not vanish.
hop and the th energy detector. The term
represents the contribution to the output of the th energy de-tector associated with the th MFSK tone, which is generated by the th interferer in the th chip interval. According to (17) we can see that the orthogonality among the MFSK tones can no longer be maintained, since or
is nonzero unless or 1 for all , and, simultaneously is an integer. When the latter conditions hold, (15) becomes
(18)
where is the Kronecker delta function. In short, unless per-fect chip synchronization can be achieved and the tone spacing equals an integer multiple of the chip rate, an MFSK tone trans-mitted by one user will impose ICI on all other FSK channels, hence on other active users’ detectors.
A closer look at (15)–(17) reveals that the magnitude of is a function of the basis waveform (6) because rep-resents the energy spectrum value of a rectangular-waveform-modulated carrier at a certain frequency. It is possible to select a proper set of basis waveforms that minimize ICI. But this can only be accomplished at the expense of greater CCI, larger tone spacing, or longer symbol duration. It is very difficult to quan-tify the impact of the baseband design on the system perfor-mance. We claim no optimal property of (6) though we have ex-amined several other nonrectangular baseband waveforms (like Barker-coded, square-root raised-cosine, Hamming-weighted, and Gaussian waveforms) and found no improvement in terms of spectral efficiency.
Although the channel separation can be as large as , the corresponding ICI is a decreasing function of this param-eter. Hence, we will only consider adjacent channels around the desired tone ( channels above and another channels below the desired one in the receiver T-F matrix) in the fol-lowing analysis when examining the effect of ICI. An appro-priate value for can be found via numerical evaluation of the associated performance (see Fig. 5).
For the multiple i.i.d. Rician fading channel, the relationship of implies that the conditional pdf and cumulative distribution function (cdf) of are given by (see Appendix A and [12] and [13])
(19) (20) In Appendix A, we show that the characteristic function of is given by
where
(22a)
(22b) If the tone spacing is greater than the minimum required, i.e., if , , then a procedure similar to that presented in Appendix A yields (23) where (24a) (24b) (24c) B. A Suboptimal Realization
The noncoherent Bayes receiver of (14) is depicted in Fig. 3 where the nonlinearity used in the receiver is given by
(25) with
(26) Note that is the same for all and the value of is in fact only dependent on and not on or additionally. Furthermore, is derived according to
, where the saturation point
is defined by .
The normalized form of , (25), is similar to of [2] with the approximation,
, and results in . Such a normalization is useful for comparing the optimal nonlinearity under various
Fig. 4. The normalized optimal nonlinearity for both chip-asynchronous and synchronous modes in dependence of the number of interferersJ, and the ratio
0 between the specular and the diffused channel power, for a fixed SNR = 20 dB. Three curves not grouped by circles are nonlinearities for synchronous
systems. The inclusions of CCI and ICI have led to higher saturation levels and slower saturation speeds (as a function ofR=d). When only CCI is considered, we setB = 0 in (21), (23), and (24a).
channel conditions. Because defined above is often difficult to locate, as in [6], we choose , where , as the reference point and redefine as
the value such that .
Define the average bit signal-to-noise ratio (SNR) as , where is a Rician or Rayleigh random variable characterizing the fading of the channel. The normalized non-linearity as a function of the normalized energy detector output for different number of interferers is depicted in Fig. 4. The nonlinearities of the chip-synchronous systems in Rayleigh [2] or Rician fading [6] have similar shapes but (25) renders different slopes and saturation levels; the difference is more evident when is large. The saturation levels of the non-linearity over Rayleigh fading ( ) channels are much lower than that over Rician fading channels but the corresponding con-vergence (to the saturation level) speeds are faster. The influence of CCI and ICI is also indicated in the same figure. As we take into account more interference [larger in (21), (23), and (24a)] the saturation level tends to increase and the corresponding con-vergence speed decreases; see also the caption of Fig. 4.
Similar to the optimal normalized nonlinearity for syn-chronous systems [6], can be well approximated by a soft-limiter. As mentioned before, the normalized nonlin-earity has a slope close to one in its “more linear” region (see Fig. 4). We can then further approximate the soft-lim-iter by a -level quantizer, , with a uniform step size of , where is the upper threshold of the quantizer. Such an approximation renders a larger quantization error for large normalized input . A properly designed nonuniform quantizer will have a much smaller quantization error. However, we have compared the performance of both types of quantizers and found no signif-icant difference in bit error probability performance. Hence, we will only consider uniform quantizers. The probability
mass function (pmf) [15] of a uniform quantizer’s outputs can be expressed as
(27) where the probabilities
are given by [see (20)]
(28)
assuming that the th bin has been transmitted. Designating as the probability mass
the th bin output when the number of diversity branches used for combining is
the th bin is transmitted (29) we have
the th bin is transmitted (30) where is the output of an -diversity com-biner. The conditional pmf of is given by
(31) The associated probability mass can be computed recursively by [6]
(32)
where , ,
.
With the aid of the above equations and the assumption that are approximately i.i.d. r.v.s [3]–[6], the symbol error probability can be expressed as [6]
correct symbol decision
(33)
IV. OPTIMALFH PATTERNSYSTEM
A set of optimal addresses that guarantee minimum CCI has been proposed by Solomon [7] and Einarsson [8]. In this section, two important properties of the optimal FH addresses are used in order to derive (34) and (35). These properties are: a) the
cross correlation (pair-wise agreement) between any two users’ addresses is at most one, i.e., any two different users’ carriers hop to the same frequency at most once within a symbol period ( hops) and b) no address uses the same frequency twice within one symbol period.
A. Performance Analysis
Let , , be the number of interfering tones that coincide with the th diversity of a given row in the desired receiver’s T-F matrix and define , . The cross-correlation property of the optimal FH pattern implies that , where , as defined before, is the number of interferers. Property a) indicates that once interfering tones (that come from different users) have coincided with the desired user’s pattern in the first hops, there can be at most interfering tones at the
th hop. Property b) implies that at the th hop (starting at the zeroth hop) a user can only choose one out of the remaining frequencies available. The assumptions that and information symbols are i.i.d. tell us that each of these remaining candidate frequencies are to be equally likely selected with probability . Hence the probability of the interference pattern is given by
(34)
where is the conditional probability
that there are interfering tones at the th diversity branch given that tones hit the th diversity branch, . In Appendix B, we show that (34) can be simplified to
(35) For a hard-limited combiner with a common threshold [3], the probabilities of having entries detected on the desired (in-formation-bearing) row and an arbitrary undesired (noise-only) row, denoted by and , are given by
(36a)
respectively, where , and
the th bin is the message bin and the th energy detector is hit by interfering tones
during the th hop (37)
The event is the set
: ,
, where or , depending on whether a miss or a detection occurs. As
and , we can easily compute the associated bit (or symbol) error rate by substituting , (36a) and (36b), into (33).
Note that the product terms in the right side of (36a) and (36b) represent the conditional probabilities that the combiner output at the message bin and a noise bin are given the inter-ference pattern . Therefore, to evaluate the performance of a synchronous soft-limited combiner we should replace these conditional probabilities by
and
where the pmf of the soft-limited (quantized) output , , is defined by (28) but the corresponding cdf is now given by
or (38)
B. Reference Clock and Guard Time
As has been mentioned, in a chip-asynchronous system, dif-ferent received MFSK tones are no longer orthogonal and the nonorthogonality results in ICI, as shown in (13) and (A2). This section examines two methods for alleviating ICI and, to a lesser extent, CCI. The first approach is designed to minimize the timing offset among system users. This can be accomplished if the base station broadcasts a reference clock signal with which all the mobile units’ local clocks will synchronize. Since all mobile units’ symbol timing are synchronous with the refer-ence, the timing misalignments among the received signals are only caused by the round-trip delays between mobile units and the base station. Such an arrangement is equivalent to the most primitive master–slave synchronization technique [16] that is usually used as part of the time distribution subsystem of a time-division multiple access (TDMA) network.
Given a data rate and the corresponding performance require-ment, one can always find an appropriate cell size and diver-sity order that guarantee a maximum timing offset less than one chip duration. Let be the maximum normalized timing offset and assume that s are i.i.d. random variables
uniformly distributed within . The corresponding char-acteristic function, , associated with the energy de-tector bank’s output becomes
(39) where
(40a)
(40b)
(40c) The above equations are similar to (21)–(22b), except that the timing offset is now uniformly distributed within the interval
instead of (0, 1].
The second approach achieves a similar result by inserting a guard time between two neighboring chips so that the chip duration is increased to . However, in order to maintain the same data rate the diversity order has to be reduced according to
(41) Suppose a reference clock is available and a guard time is in-serted such that equals , then, during any given chip interval, the th interferer with a random hopping pattern and a nonzero normalized timing offset can hit the desired tone for either or of the time. Hence, without loss of generality, the interference model (17) becomes
(42) where is assumed. Note that there exists a tradeoff be-tween the diversity order and the guard time length if a fixed data rate is to be maintained. Following an analogy procedure for deriving (21), we obtain
where
(44) The above equations can be used to derive the corresponding optimal or suboptimal nonlinearity.
V. TWO-STAGEDETECTION
A. Principle of Two-Stage Detector
Using the algebraic properties associated with the optimal FH pattern, Timor [9] proposed a multistage decoding method to cancel CCI in a noise-free environment. He noticed that the modulo arithmetic operation of two Reed–Solomon codewords is still a Reed–Solomon codeword, i.e., the interferer’s FH pat-tern after dehopping is also a legitimate hopping patpat-tern (code-word). Hence, after dehopping and hard-limiting, there are codewords and one complete signal row in the desired user’s T-F matrix, if thermal noise is negligible. An error occurs when the codewords constitute one or more complete rows. The al-gorithm suggested in [9] provides a systematic procedure to test whether a complete row is the result of (at least) interferers (since no two users will coincide in at more than one chip). In the absence of thermal noise, the first stage (majority vote) detector always results in at least one complete row. If there are more than one complete rows, the detector enters the second stage using Timor’s chip-by-chip test. However, the second stage cannot re-solve the ambiguity completely. The probability of the residual ambiguity can be further reduced by employing multistage de-coders having higher complexity [10].
B. A Novel Two-Stage Detector
The multistage detector suggested by Timor [9] assumed a hard-decision diversity combining rule and a zero thermal noise environment. When the effect of thermal noise is taken into con-sideration, the probability of both chip deletion (a message chip fails to pass the hard limiter threshold) and chip insertion (a non-message chip passes the hard limiter threshold) will not be zero. It is likely that the message chips do not form a complete row and the interference test is incapable of detecting a false com-plete row. To deal with such a practical situation we propose the following two-stage detector. The first stage locates the row (in the T-F matrix) with the largest hard-limited combining output. If there are rows having the same largest output value, the detector then enters the second stage where it:
1) performs an interference test on each chip in the th largest row and records the number of chips that pasess the test, say . A chip that passes the interference test is assumed to have been corrupted by an interferer. 2) repeats the above step for .
3) finds the row that yields the smallest number of passed chips, say , and decodes as the correct message. 4) determines if there are two or more smallest s and
chooses one at random as the message.
The so-called interference test mentioned in Step 1) is designed to test whether a particular chip has CCI from other users. In other words, the test checks if any interferer might use a partic-ular frequency in a given chip interval.
When soft-limited combining is used in the first stage, we select those rows with the largest combining output value say plus those whose combined output values are greater than but less than to enter the second stage. The threshold value can be optimized to yield a minimum error probability. We further divide the selected rows into two cat-egories—the first category, , are those that have the largest output while the rest, , belong to the second category. In
other words, , and
. Step 4) is then modified as follows.
4′) If there are two or more smallest s and the associated rows belong to both categories, choose one out of at random and decode it as the message.
The interference test proposed in [9] is designed for use with a special class of optimal hopping patterns. This class is not suitable for asynchronous operations. Appendix C shows that for the class of the optimal asynchronous hopping patterns [8] the interference test can still be used if the initial realignment procedure is properly modified.
Multistage detectors that consider all possible combinations of users’ addresses and message patterns were discussed in [10], [11]. Such a complete (multistage) detection scheme is very complicated: instead of cycle-shifting operations for resolving an ambiguous row, a complete decoding requires cycle-shifting operations. Since , the two-stage detector has a much smaller complexity than that of a complete detector.
VI. NUMERICALRESULTS ANDDISCUSSIONS
Unless otherwise specified, the following system parameters
are used: MHz, kHz, , ,
and .
To examine the effect of ICI we consider adja-cent channels around the desired channel since no noticeable ca-pacity degradation is found if is greater than 10; see Fig. 5. In other words, ICI caused by an interferer whose carrier is Hz away from the desired tone is negligible. System capacity degradation caused by ICI and CCI can be found in Fig. 6 where we have depicted the BER performance as a function of the number of active users. The two CCI curves are obtained by ig-noring ICI ( ). The system performance of both hard-lim-ited and soft-limhard-lim-ited combining are also demonstrated in Fig. 6 from which we find that at a BER below 10 , single-stage soft-limited receivers can accommodate 10 to 26 more users (or 1.6% to 4.3% higher spectral efficiency) than their hard-lim-ited counterparts. The threshold is used for hard-limited combining.
The influence of channel spacing is illustrated in Fig. 7. Using the definition of spectral efficiency (3), one can easily come to the conclusion that the system with is the best choice in terms of spectral efficiency. Fig. 8 demonstrates the
Fig. 5. Effect of ICI on the BER performance of the asynchronous FHMA/ MFSK system, where2B is the number of adjacent channels considered and
q = 8 quantization levels along with random frequency hopping (RFH) patterns
are assumed.
Fig. 6. Performance comparison: hard versus soft decisions and the influence of ICI and CCI; ICI is not included (B = 0) in obtaining those two curves labeled by CCI;q is the number of quantization levels of the soft-limiter.
effect of using a reference clock to minimize the maximum normalized timing offset among users to . Obviously, the smaller is (see those two curves with ), the better the BER performance becomes. If guard time is inserted, further improvement can be obtained, as also demon-strated in Fig. 8. For a system with and
or , (41) implies that the diversity order has to be reduced to 17 or 16. The effect of reference clock and guard time insertion in mitigating ICI is obvious but that in lessening CCI is not so obvious. The “net” effect of CCI is assessed by evaluating the system performance without taking ICI into ac-count ( ). Fig. 9 shows that CCI can indeed be reduced if proper design parameters are selected. It also indicates that systems using both mitigation schemes do necessarily yield
per-Fig. 7. The impact of normalized channel spacing on the BER performance of asynchronous FHMA/MFSK systems.
Fig. 8. BER performance improvements for chip-asynchronous systems when a reference clock is used or/and a guard time is inserted. The maximum chip-duration normalized timing offset is denoted by . The performance of the synchronous system (the right-most curve and dots) is provided for comparison purposes.
formance better than those using only a pilot clock, for, as just mentioned, inserting a guard-time is likely to force us to reduce the diversity order in order to maintain the same data rate.
Figs. 5–9 assume random hopping patterns are used by the system. The effectiveness of the proposed two-stage detector is shown in Fig. 10. Comparing Figs. 8 and 10, we find that the use of an optimal pattern brings about little improvement. This is due to the fact that the probability of more than one collision during a symbol period is very small for the system parameters under consideration ( , or ). It is noticed that the improvement over the single-stage detector is most impressive when the system is in the symbol-synchronous mode. Obviously, the algebraic properties of the hopping pattern that we have used in designing the interference test is not very useful for chip-asynchronous interference.
Fig. 9. Effects of reference clock or/and guard time on CCI. These curves were obtained by ignoring ICI (B = 0). The left-most curve and dots correspond to the performance of the chip-asynchronous system without these two remedies.
Fig. 10. Effect of the second stage detection: optimal frequency-hopping (OFH) pattern set is used and random frequency-hopping (RFH) patterns are used.
From the above error probability curves we can derive the ultimate system performance measure—system capacity and spectral efficiency . We summarize some of the selected ( ) values at BER along with the corresponding system parameters in Table I. The performance of chip-asyn-chronous systems is much inferior to that of synchip-asyn-chronous sys-tems, especially in high s. For example, Table I indicates that at BER the latter allow 64 more users when
dB, and 97 more users when dB, .
The values in parentheses are obtained when the system inserts a proper guard time; see (41)–(44). Similar to the conclusion we derived from Fig. 10, the advantage of using the optimal hop-ping patterns is most obvious in synchronous systems in which the two-stage receiver obtains a 10% to 20% increase in system capacity. However, when the system becomes asynchronous this benefit is drastically reduced and, as increases, it gradually disappears.
TABLE I
SYSTEMCAPACITYK , SPECTRALEFFICIENCY ,AND THEASSOCIATED
SYSTEMPARAMETERS
VII. CONCLUSION
This paper deals with noncoherent detection of fast FHMA/ MFSK signals in an environment characterized by frequency-nonselective Rician fading. We derived a nearly optimal non-linearity for diversity combining in the asynchronous mode. A suboptimal realization using quantizers (soft limiters) was pro-posed and analyzed. For other operation scenarios (limited time offsets, guard time insertion) considered we also derived the cor-responding characteristic functions that are needed to obtain the associated optimal nonlinearities.
We investigated the impact of the FH patterns used and found that, for single-stage detectors, the use of optimal FH patterns yields little BER reduction when compared with systems using random FH patterns. On the other hand, the algebraic proper-ties of the optimal pattern renders itself suitable for multistage detection. We suggested a two-stage detector which was shown to be effective in the symbol-synchronous mode for combating both ICI and CCI. Users’ clock misalignment destroys the signal orthogonality and makes the algebraic properties of the optimal hopping pattern less useful, which resulted in a significant per-formance degradation for both single- and two-stage detectors. Two methods for lessening the performance degradation and containing the interference were thus examined. Numerical re-sults proved that these methods provide a sufficient interfer-ence-reduction capability under the assumed channel condition.
APPENDIX A DERIVATIONS OF(19)–(22b)
The complex random variable , as can be seen from (15), is the sum of three independent complex random variables. For a fixed set of , each of these r.v.s has a spherical symmetric probability distribution [13], i.e., the corresponding pdf is independent of the complex r.v.s direc-tion (phase) and is a funcdirec-tion of its magnitude only. It follows immediately from the independence assumption that the
charac-teristic function of for given ,
which is expressed as , is the
product of the conditional characteristic functions of , and [see (15)]; i.e.,
(A1) where, on the right-hand side of the equation, for simplicity, we express the conditional characteristic functions of , and
as , and .
Assuming , substituting (16) and (17) into (15), and restricting ICI to adjacent channels, we have
(A2) Applying the theory of spherical symmetric random vectors [12], [13], we obtain
(A3)
(A4) (A5) where is the zeroth order Bessel function of the first kind. As mentioned in Section II, for a given pair, are not independent and only one of them is nonzero. However, it has been shown in [3] and [6] that when and the inde-pendent random hopping patterns are used, are approxi-mately i.i.d. with a common pdf
(A6)
where . Taking expectation on with respect to and substituting the resulting expression together with (A3) and (A5) into (A1), we obtain
(A7)
where
(A8)
Averaging (A7) over and and using (6.633.4) of [17] we obtain
(A9)
where
(A10)
but , , are i.i.d. r.v.s with a common uni-form distribution within (0, 1]. By taking the average over , we then obtain (21) and the ensuing two equations. As is the length of a two-dimensional random vector that has a spherical symmetric distribution, we can apply the Hankel-transform-like relations between and the pdf and
cdf of , , [12, (31)]
(A11) (A12) (A13)
and the change of variables to obtain (19) and (20). The above equations, (A11)–(A13), or their variations can also be found in [18] and [19]. The pdf (cdf) derivations for the other cases, e.g., , are similar hence will not be repeated.
APPENDIX B DERIVATION OF(35)
Expanding the right-hand side of (34), we obtain
(B1) By rearranging (B1), we obtain
(B2) It is easy to see that the above equation is equal to (35).
APPENDIX C ANINTERFERENCETEST
The interference test proposed by Timor [9] for use in a two-stage decoding algorithm is valid for the address sets gener-ated by (4). Timor did not deal with the other address assign-ment scheme presented in [8] that determines the transmitted frequency vector for the th user by
(C1)
where and
is a constant -tuple representing the address of the th user and is the message symbol (MFSK tone); both and are elements of . This address assignment is optimal for chip-asynchronous systems [8].
We want to demonstrate that a simple modification on the original interference test [9] is sufficient for use in a system that employs the FH patterns (C1). For such a system, the
fre-quency-dehopper has to perform the inverse operation of (C1). Chip-by-chip detection would result in a complete th row in the dehopped T-F matrix of user if no noise is present. Sup-pose CCI results in another complete row in . The address assignment scheme assures that every chip in must come from different users and therefore at least users
are needed to produce a complete row. The user that contributes to the th entry of is assumed to be the th user. If there are more than one user that cause this hit, let be any one of them. From this address assignment, the th entry of the th user in will be at row and column , where
(C2)
and . Note that for all , ,
and is the row in which the entry of the th user appears at the th column. Since this row is we obtain
(C3) Comparing the above equation with [9, (5)], we conclude that the rest of the interference test procedure can be carried out in exactly the same way as that presented in [9] if we replace by .
ACKNOWLEDGMENT
The authors would like to thank the anonymous reviewers for their careful reading and valuable suggestions.
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Yu T. Su (M’98) received the B.S.E.E. degree from
Tatung Institute of Technology, Taipei, Taiwan, and the Ph.D. degree from the University of Southern Cal-ifornia, in 1974 and 1983, respectively.
From 1983 to 1989 he was with LinCom Corpora-tion, Los Angeles, CA, where he was involved in the design of various measurement and digital satellite communication systems. Since September 1989 he has been with the Department of Communication En-gineering and Microelectronics and Information Sys-tems Research Center, National Chiao Tung Univer-sity, Hsinchu, Taiwan. His research interests include communication theory and statistical signal processing.
Ye-Shun Shen received the M.S. degree in
commu-nication engineering from the National Chiao Tung University, Hsinchu, Taiwan, in 1996. He is currently a Ph.D. candidate at the Institute of Electrical En-gineering, National Cheng Kung University, Tainan, Taiwan.
Since 1993, he has been with Chunghwa Telecom, where he is a Senior Engineer. His research interests include spread spectrum communications, multiple access, and antijam techniques.
Chu-Ya Hsiao was born in Hsinchu, Taiwan in 1970.
He received B.S. and M.S. degrees in communication engineering from the National Chiao Tung Univer-sity, Hsinchu, Taiwan, in 1993 and 1995, respectively. He is currently with Novatek Microelectronics, Hsinchu, Taiwan. His research interests include wireless access technologies, modulation, and coding theory.
