Specular Coherent and Noncoherent Optimal ... - Semantic Scholar
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 4, APRIL 2002
Specular Coherent and Noncoherent Optimal Detection for Unresolved Multipath Ricean Fading Channels Florence A. Danilo-Lemoine, Member, IEEE, and Harry Leib, Senior Member, IEEE
Abstract—This paper considers specular coherent and noncoherent optimal detection for unresolved multipath Ricean fading channels with known delays. The focus is on receiver structures and performance. Specular coherent detection employs the carrier phase of the Ricean specular component, while noncoherent detection does not. Therefore, a specular coherent detector must be augmented with a carrier phase estimator for the specular component. The structures considered in this paper are generalization of the well-known Rake receiver to the unresolved multipath case. It is shown that both optimal structures perform a decorrelation operation before combining, which is essential to eliminating error floors under multipath unresolvability conditions. Furthermore, the noncoherent optimal receiver includes an inherent estimator for the specular component phasor. It is shown that the specular coherent and noncoherent structures converge at high SNR. This result is confirmed through analytical and numerical performance evaluation. Little performance gains can be obtained by the use of specular coherent detection for orthogonal frequency-shift keying and to a lesser extent for differential phase-shift keying over mixed mode Ricean/Rayleigh fading channels, making noncoherent demodulation attractive in these cases. Index Terms—Decorrelators, detection, estimation, multipath fading channels, noncoherent, quadratic receivers, specular coherent, wireless communications.
I. INTRODUCTION YSTEMS for personal communication services have to operate in indoor and outdoor radio environments that are characterized as multipath fading channels. The effect of multipath fading and consequently the choice (or validity) of a channel model depends on the transmission bandwidth. Narrow-band systems (compared to channel coherence bandwidth) yield a flat fading channel [1] while wide-band systems yield frequency-selective channels. Diversity gains can be achieved over frequency-selective fading channels by the use of Rake receivers [2]–[4] or their variations [5]–[13]. Rake receivers exploit path diversity by using wide-band signals that resolve multipath [14]. Such bandwidth requirement can
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Paper approved by A. Ahlen, the Editor for Modulation and Signal Design of the IEEE Communications Society. Manuscript received May 10, 2001; revised October 11, 2001. This work supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. F. A. Danilo-Lemoine is with the Department of Systems and Computer Engineering, Carleton University, Ottawa, ON K1S 5B6, Canada (e-mail: [email protected]). H. Leib is with the Department of Electrical and Computer Engineering, McGill University, Montréal, QC H3A 2A7, Canada (e-mail: [email protected]). Publisher Item Identifier S 0090-6778(02)03519-5.
be achieved outdoors by spread-spectrum signals, but it is not practical indoors or in dense urban environments where up to 50 MHz would be needed [15] to resolve the small inter-path delays [16], [17]. This paper considers detection techniques for unresolved multipath fading channels, thus not requiring bandwidth spreading. The best performance can be achieved when the channel realizations are available, implying that the channel impulse response or channel parameters have to be estimated. Carrier phase estimation in mobile environments is complex and may yield inaccurate estimates [18, p. 953]. An alternative is to use noncoherent detection at the expense of some performance degradation. Specular coherent detection, using the carrier phases of specular components only, yields little performance gains at high signal-to-noise ratio (SNR) for single-path Ricean channels [19], [20]. The noncoherent optimal receiver for unresolved multipath Rayleigh channels with Doppler and known delays was derived in [21]. Its performance was considered in [22] only for signals that are orthogonal to all time shifts, resolving the multipath. Optimal receivers over two-path unresolved Rayleigh channels were derived in [23] for known delays and different levels of channel knowledge, assuming that the autocorrelation magnitude of the transmitted signal complex envelope at inter-path delays is independent of the signal shape. Two-fold diversity-like effects were found in the performance of envelope orthogonal frequency-shift keying (FSK) and variants of chirp or linear frequency sweep modulation. The noncoherent optimal receiver over unresolved multipath Ricean fading channels was derived in [24]. However, its performance was not evaluated and the corresponding specular coherent receiver was not considered. This work considers unresolved multipath Ricean fading channels, focusing on similarities and differences between specular coherent and noncoherent optimal detection in terms of receiver structures and single pulse performance. For small inter-path delays, the effects of intersymbol interference (ISI) are negligible, making the single-pulse bound close to the achievable performance for sequential transmission. The multipath delays, assumed to be known, could have been estimated by using super-resolution techniques [25], [26] or by sounding the channel with a wide-band pulse. The effects of path-delay estimation errors, which may degrade performance if they are not kept small, is beyond the scope of this paper. This paper is organized as follows. Section II presents the system model and receiver structures for unresolved multipath Ricean fading channels. Section III considers the error rate
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DANILO-LEMOINE AND LEIB: SPECULAR COHERENT AND NONCOHERENT OPTIMAL DETECTION
calculation for mixed mode Ricean/Rayleigh channels. The Ricean component can model a line-of-sight path [18]. Section IV presents a performance analysis of specular coherent and noncoherent detection as well as numerical results for variations of binary FSK and differential phase-shift keying (DPSK). Section V presents the conclusions. II. SYSTEM MODEL AND RECEIVER STRUCTURES A. Channel Modeling possible bandpass signals Assume transmission of one of of finite energy over a multipath Ricean fading channel. Under , the received signal is given by hypothesis
(1) denotes the real part of the argument, and where are independent circularly complex Gaussian random and variance variables [27] with mean . The multipath comare either fixed and known, or unknown ponent phase shifts independent random variables uniformly distributed between and , and represents the complex envelope of the th possible transmitted signal. The multipath delays are assumed to be known and distinct. The channel noise is satisfying modeled by a zero mean white Gaussian process and statistically independent of the signal . For Ricean multipath channels, each path can be considered as the phasor sum of a Rayleigh component with uniformly distributed phase and a fixed (specular) component. To model absence of reference phase information (noncoherent detection), an additional random phase needs to be added to each , yielding the model (1). multipath component, can be expressed as In (1), where have the same joint probability since are density function (pdf) as zero mean circularly complex Gaussian. Therefore, whenever the ’s are known, (1) represents a multipath Ricean fading channel that implicitly assumes that the specular component phases and amplitudes are known at the receiver (specular coherent detection). , are conAssume that tained in the interval [0, ]. Under practical conditions, the delayed signals are linearly independent over [0, ] for distinct multipath delays [14]. In this work, the multipath resolvability condition is not assumed. Finally, , are continuous on assume that [0, ], which is not too limiting since any square-integrable function can be approximated arbitrarily close in the Euclidean norm by a continuous function [28, p. 71]. This ensures the , required for the existence of mean square continuity of its Karhunen–Loève expansion [29, pp. 379–380]. We use the following notation. Bold capital letters denote matrices and bold lowercase letters denote vectors, , and
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denote respectively the transposition, complex conjugation, th entry of a matrix and Hermitian conjugation. The is and the th entry of a vector is . We define . The baseband signal energy under is given by . The signal cross-corand , , is relation matrix between the hypotheses defined as . The th signal correlation matrix is is defined as channel covariance matrix , where and .
. The
B. Specular Coherent Optimal Decision Rule for an -Path Ricean Channel In this section, we assume that the specular components amplitudes and phases are known at the receiver (i.e., known ). Multipath Ricean channels when is fixed yield the classical problem of detecting a Gaussian signal in additive white Gaussian noise [29, pp. 419–421]. A minimum probability of error receiver forms the likelihood ratio between each and hypothesis . With equiprobable hya null hypothesis potheses, the decision is made in favor of the largest likelihood follows ratio [29, p. 11]. The discrete representation of from the Karhunen–Loève expansion that exists since is a second order mean-square continuous process [29, pp. given is 379–380]. The covariance function of
(2) where function of
and , given
, the covariance
is
(3) and . As shown in [24]1 has at most positive which are those of the matrix eigenvalues and has corresponding eigenfunctions given by
with
(4) where2 equations
,
is an
matrix that satisfies the (5a) (5b)
identity matrix and where is the diagonal matrix defined by
K
1
2The
(t; u) given by (3) is denoted as
denotes the . The signal
K (t; u) in [24]. superscript m is not an exponent and refers to the hypothesis H .
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Fig. 1. Specular coherent optimum receiver for an L-path Ricean channel (SPECCOH) assuming T
narrow-band assumption and (2) imply that the eigenvalues and eigenfunctions of are given by
associated with (6a)
associated with (6b)
for all l.
Since the term is independent of the hy, an equivalent decision variable is obpothesis can be tained by removing this term from (7). The variable , and [see (12)] can be obobtained from for tained by using a bank of matched filters [24]. When can be generated by sampling the output all , the variables for of the matched filter , at , as shown in Fig. 1, or by using a tapped-delay line after the matched filter. , , When the multipath is resolved (i.e., ), (7) reduces to [3], [30]
, it can be shown Using the Karhunen–Loève expansion of [14] that the likelihood ratio associated with the specular co, herent optimal scheme (SPECCOH) for short, is given by
(13)
(7) where
, (8)
The receiver implementing (13) is the specular coherent optimum receiver for resolved multipath fading channels and is denoted SPECCOHR in this paper. Since the linear transformation on the signals [see (4)] is invertible, the th hypothesis (1) can be equivalently expressed as
(9) (10) (14) where
(11) i.e.,
(12) and the integrals in (11) and (12) are Wiener integrals.
, , while are orthogonal signals of same energy as . When is fixed, for all the new random is Gaussian with mean vector and covariance [14]. Therefore, under each hypothesis, the received signal can be represented as a linear combination of orthogonal functions
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Fig. 2. Noncoherent optimum receiver for an L-path Ricean channel (OPT) assuming T
weighted by uncorrelated circularly complex Gaussian random variables, similar to the resolvable multipath case. Using (5) with several matrix manipulations, the specular coherent likelihood ratio when the multipath is unresolved (7) can be rewritten as
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for all l.
integrating (7) over all components of the vector between and resulting in
(16) where
and
are given by
(17) (15) is Gaussian Noting that for fixed given by (9) and covariance , it is seen from with mean (13) and (15) that the specular coherent optimal receiver for unresolved multipath Ricean channels consists of an orthogonalinto ization (or decorrelation) stage that transforms , and then implements a resolved multipath optimal de. Note that besides the orthogonalization of cision rule for , the matrix also perthe signals forms statistical decorrelation in the sense that the new vari, unlike , are uncorreables . lated as coefficients of the Karhunen–Loève expansion of The decorrelation stage of unresolved multipath fading channels optimal receivers was already identified for Rayleigh channels [24] and thus is generalized here to Ricean channels with specular coherent detection. C. Noncoherent Optimal Decision Rule for an -Path Ricean Channel The likelihood ratio associated with the SPECCOH scheme corresponds to the conditional likelihood ratio (given ) associated with the noncoherent optimal receiver (OPT) given by (7). is obtained by Thus, the noncoherent likelihood ratio
(18)
(19) (20) , and are given by (5), is where are respectively given by (8) and (9), given by (10), and and are respectively given by (11) and (12), denotes the diagonal matrix composed of the main diagonal entries denotes the lower triangular matrix composed of , and of the lower triangular elements of with zero main diagonal entries. It can be shown that the closed-form solution of the inis the sum of multidimensional infinite series of tegral of products of Bessel and trigonometric functions [14]. From (16) to (20) it is seen that the OPT scheme uses the as the SPECCOH scheme does. same decision variables for all , the receiver of Fig. 2 is obThus, when tained. Note that the OPT structure is identical to the one of
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[24] up to a different random vector . However, it can be of this paper [see (11)] and of [24] ( shown that ) are identically distributed. Therefore, identical receiver structures based on either version of are equivalent and yield the same probability of error.
low SNR. Using asymptotical properties of is very large, then
, when
D. Comparison of Specular-Coherent and Noncoherent Optimal Structures When
Using (5) and (10), it can be shown that
Thus, for a mixed mode Ricean/Rayleigh channel, i.e., , (7) reduces to
(21) and . For a mixed where mode Ricean/Rayleigh channel, (18) reduces to . Hence, (16) reduces to
is small, then
reducing the contribution of the specular phasor estimate. Assuming equally likely signals, the optimal receiver decision rule is obtained by selecting the hypothesis corresponding to the largest log-likelihood ratio. If the signals are further asis identical under all hypotheses sumed to have equal energy, we obtain an equivalent decision rule. The and dividing by behavior of the SPECCOH and OPT schemes at high SNR is illustrated in the following proposition proven in the appendix. Proposition 2.1: The SPECCOH and OPT scheme log-likelihood ratios “converge” almost surely (a.s.) to the same term as goes to infinity, i.e.,
At high SNR, the SPECCOH and OPT schemes use the same decision rule for equally likely equal energy signals, meaning that knowledge of the specular component phases is not necessary in this case. Section IV-A confirms this property in terms of performance for mixed mode Ricean/Rayleigh channels. (22) Comparing (21) with (22) shows that the mixed mode noncoherent likelihood ratio can be obtained from the mixed mode specular coherent likelihood ratio by substituting the specular in (21) with an estimate given by phasor
(23) From (23), it is seen that the specular phasor estimate is obtained by substituting the unknown phase by an estimate and then scaling the phasor by . The scaling factor acts as a soft limiter to reduce the phasor estimate effects at
III. PERFORMANCE EVALUATION The SPECCOH scheme is quadratic in a Gaussian statistic and the bit error probabilities over mixed Ricean/Rayleigh fading channels can be calculated as in [24]. The nonlinearities of the OPT scheme [see (16)] make its performance analysis very tedious, if not impossible. Therefore, upper and lower bounds for the bit error probability of the OPT scheme are employed. Upper bounds are obtained by evaluating the bit error probability of suboptimum quadratic receivers (such as the QDR (various ), QR and R OPT schemes derived in [24]) as a . For each value of function of the received SNR per bit , the lowest probability of error among all suboptimum receivers is retained to provide the tightest upper bound. An example for FSK with frequency deviation over a two-path mixed mode Ricean/Rayleigh fading channel is illustrated in Fig. 3. For a two-path mixed mode channel where , is the relative Rayleigh component strength is the Ricean between the first and second path, is the energy per bit of the real signal. All parameter and
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of
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It is known that (24) can be evaluated by integrating the pdf , obtained from its characteristic function, [30]
(25) ,
where
. Such integration can be performed by
and
0
=
Fig. 3. Bounds for OPT with FSK signaling with f f 1=2T over a two-path Ricean/Rayleigh channel ( = 0:1T , s = 1 and = 15 dB).
two-path mixed mode Ricean/Rayleigh channels are labeled by the values of their parameters and and the relative . For delay between the first and the second path convenience, is expressed as a percentage of the duration of the signaling waveform ( ) and is expressed in decibels. Lower bounds are obtained by evaluating the performance of the noncoherent optimum receiver over a Gaussian nonfading channel and the performance of the SPECCOH scheme from Section II-B over the considered mixed mode Ricean/Rayleigh multipath fading channel. For each received SNR per bit, the highest probability of error is retained to provide the tightest lower bound as seen by the example of Fig. 3. As seen in Section II-B, the SPECCOH scheme is also quadratic. Therefore, the technique of upper and lower bounding the bit error probabilities of the OPT receiver requires performance evaluation of quadratic receivers. We present now the method used for computing the bit-error probability of quadratic receivers in a Gaussian statistic. Let be the decision variable under hypothesis . The decision variables of a specular coherent receiver (SPECCOH, SPECCOHR) employ , while those of a noncoherent receiver (QDR, R OPT and QR) do not. In both cases, the decision variables may depend on through the received signal. With equiprobable equal energy binary signals, the bit error probability with held fixed is
,
using the residue method; however, for Ricean channels, rendering this method is not practical due to the complicated [24]. Also, the dependency of exponential factor in on via the mean 3 implies that the pdf of needs to be integrated with respect to and , which presents difficulties. However, for mixed mode Ricean/Rayleigh fading channels, no integration with respect to needs to be performed [24]. For example, the pairwise probability of error can be evaluated by integrating numerically the following improper integral:
(26)
(27) (28) where
,
. The matrix
is chosen
such that it diagonalizes while satisfying , . Since the function resulting in the diagonal matrix increases monotonically to , the integration is carried . It appears that the function over a finite range is quite suitable for numerical integration. The MATLAB program was designed such that the absolute error formed by the sum of truncation error and numerical integration error was less than 1.7 10 [14]. IV. PERFORMANCE OF BINARY MODULATION SCHEMES OVER MIXED MODE RICEAN/RAYLEIGH FADING CHANNELS WITH OPTIMAL DETECTION A. Convergence of the SPECCOH and OPT Performance at High SNR
(24)
where
is a Hermitian quadratic form
in jointly Gaussian random variables, and is a bias term such . The probability of error is then obthat tained by averaging (24) over . Assuming equal energy signals, , and for Table I presents the expressions of the quadratic receivers considered in this paper.
Following Section III, the performance of the OPT scheme at high SNR will be studied by considering the upper and lower bounds provided, respectively, by the bit error probabilities of R be the OPT and SPECCOH. Let for fixed cumulative distribution function (cdf) of under , where the subscript indicates that it depends generally on . Similarly, let us define the function as in (27). In 3Note that for all quadratic schemes considered here, including SPECCOH and SPECCOHR, R is independent of .
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TABLE I EXPRESSIONS OF Q , r , AND A FOR THE CONSIDERED QUADRATIC RECEIVERS
this section, we denote as to emphasize its dependency , for SPECCOH and R OPT, on . Let us first show that converge to the same value as tends to infinity. and do not depend Recall from (5) and (10) that . Hence, from Table I, as on and tends to infinity the vectors of the SPECCOH and R OPT schemes converge almost surely to the same vector, namely since the term
,
tinuous on [ ] [33, p. 454] and from [32, p. 9] the conas tends to infinity, is uniform. convergence of on [ , verges uniformly to the continuous function ], for all we have hence for both SPECCOH and R OPT we have since for both schemes
, 2 is deter-
. Thereministic. Furthermore, from (10), fore, as tends to infinity, for the SPECCOH and R OPT schemes converge almost surely to the same quadratic form . From [31, p. 20], converges also in probability. Let be the cdf of under for fixed . Then, at every continuity point of , [31, p. 23], which says that converges in distribution. Note that the convergence will be uniform [32, p. 9]. Considering in any closed interval of continuity of , similar to (26) it can be shown that
(29) and are functions equivalent to and [see (27) and (28)] as tends to infinity. is continuous on , is and has a finite continuous on is conlimit as tends to zero, hence . It can be shown [14] that tinuous on is uniformly convergent for , where is an upper bound for when given by where
Therefore, the bit error probabilities of the SPECCOH and R OPT schemes converge to the same value as tends to infinity. Since they are lower and upper bounds to the bit error probabilities of OPT, the bit error probabilities of SPECCOH and OPT also converge to the same value as tends to infinity. This performance result agrees with Proposition 2.1 concerning receiver structures. B. Performance of the SPECCOH and OPT Schemes (Numerical Results) Performance is assessed by calculating bit error probabilities . Commonly as functions of the received SNR per bit, used modulation schemes such as variations of FSK and DPSK , , 2 the complex envelope of are considered. Under a FSK binary signal is
where (carrier frequency). FSK modulation with freis denoted as FSK( ), where quency separation . For FSK we have , thus the energy per , , 2, i.e., bit of FSK signals is given by . Under and the complex envelopes for DPSK over a two-symbol interval are
(30) where
, , and . Therefore, from (29), the function
is con-
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Fig. 4. Performance of the SPECCOH and OPT schemes with FSK(1) 0:1T , s = 1 and signaling over two-path Ricean/Rayleigh channels ( = 5–20 dB).
=
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Fig. 5. Performance of the SPECCOH and OPT schemes with FSK(1/2) signaling over two-path Ricean/Rayleigh channels ( = 0:1T , s = 1 and = 5–20 dB).
The complex envelopes for symmetrical DPSK [34] over a twosymbol interval are
Since with DPSK and SDPSK the transition between the carrier phase of consecutive bits carries the information, we have , thus for (S)DPSK , , 2, i.e., . Note that the present definitions of FSK and DPSK waveforms are not continuous on the observation interval [0, ] since they present discontinuand . However, they can be approximated ities at arbitrarily close in terms of Euclidean distance by continuous complex waveforms on with compact support4 since they are square integrable functions [28, p. 71]. Furthermore, the performance analysis of quadratic receivers does not use the continuity assumption. Figs. 4–7 present the performance of the specular coherent (SPECCOH) and the noncoherent (OPT) optimum receivers over two-path mixed mode Ricean/Rayleigh fading channels ), and . with equal Rayleigh path strength ( Results similar to Figs. 4–7 are obtained when [14]. The SPECCOH bit error probabilities are represented by short dashed lines. Absence of those lines for some values of means that in that case the SPECCOH bit error probability is equal to the lower bound to the OPT bit error probability on the entire range of received SNR presented in this paper. The performances of the noncoherent and coherent optimum 4The support of a complex function f on a topological space X is the closure of the set fx : f (x) 6= 0g.
Fig. 6. Performance of the SPECCOH and OPT schemes with DPSK signaling over two-path Ricean/Rayleigh channels ( = 0:1T , s = 1 and K = 5–20 dB).
receivers over a single path nonfading Gaussian channel, as well as the performance of the noncoherent optimum receiver over a two-path equal strength Rayleigh channel with are added as references. Figs. 4–7 show that the lower and upper bounds for the OPT scheme are tight at high SNR for FSK(1) and DPSK, but are less tight for FSK(1/2) or SDPSK. For FSK this phenomenon can be explained for one-path fading channels as follows. On one-path Ricean channels, the probability of error for coherent detection depends only on the real part of the cross-correlation coefficient [1]. Hence FSK(1) and FSK(1/2) with coherent detection have the same probabilities of error in this case. At high SNR, the best modulus of the signal cross correlation for noncoherent detection over one-path Ricean channels is zero [19] and FSK(1/2) having nonzero cross correlation performs worse than FSK(1). Specular coherent detection with a sufficiently
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Fig. 7. Performance of the SPECCOH and OPT schemes with SDPSK 0:1T , s = 1 and signaling over two-path Ricean/Rayleigh channels ( K = 5–20 dB).
=
strong specular component has similar features to coherent detection. The gap between coherent and noncoherent detection over one-path Ricean channels is smaller for FSK(1) than for FSK(1/2). Figs. 4–7 illustrate that the performance of the SPECCOH and OPT schemes improves as the Ricean path dominates the Rayleigh path, tending to the performance for a Gaussian channel. It is also seen that for both detection techniques DPSK (and SDPSK) give the best performance. For example, at high SNR DPSK detected with the OPT scheme gives at least a 3.6-dB improvement compared to FSK(1/2) and at least 4.2 dB compared to FSK(1) in the error probability range 10 –10 , but 3 dB are gained because the observation interval used with DPSK is twice the one used with FSK. Figs. 4 and 5 dB at high SNR ( dB), show that for performance of the OPT scheme is better with FSK(1/2) than dB) FSK(1) with FSK(1), while for lower SNR ( dB, FSK(1) performs better performs better. For than FSK(1/2) for all SNR values, confirming our earlier observation that with a sufficiently strong specular component the zero cross-correlation feature of FSK(1) is an asset with the OPT receiver. Table II presents typical SNR gains (evaluated by the upper bounds) that can be obtained by using SPECCOH instead of the OPT scheme for three bit error probabilities, 10 (speech) and 10 , 10 (data). SNR gains greater than 2 dB are set in boldface indicating cases where specular component phase estimation may yield significant improvement. Table II and Figs. 4 and 6 show that little performance gains can be obtained by the knowledge of the specular term phase for FSK(1) and to a lesser extent for DPSK (ex. maximal gain of 0.2 –0.7 dB for ). Lower gains FSK(1) and 0.8–1.2 dB for DPSK at are obtained for lower values of and for higher SNR. These observations along with the difficulties inherent in phase estimation justify the use of noncoherent detection for FSK(1) or DPSK especially at high SNR. It may be argued here that the OPT scheme is more complex than the SPECCOH. However, as shown by Table II, the loss in performance of the simpler
noncoherent suboptimum scheme QDR ( ) compared to SPECCOH is also small. Table II and Figs. 5 and 7 show that significant gains can be obtained for FSK(1/2) and SDPSK by the knowledge of the specular component phase (ex. maximal with dB and 1.8 dB at gain of 3 dB at with dB for FSK(1/2), 1.5 dB at with dB and 1.8 dB at with dB for SDPSK). Gains larger than 3 dB are also obtained for FSK(1/2) dB favoring the use of coherent detection for with – dB), all modulation FSK(1/2). At very low SNR ( schemes yield significant losses (around and greater than 2 dB). These represent the losses of OPT with respect to SPECCOH, assuming perfect Ricean specular term phase estimation. However, at such low SNR, the specular term phase estimate is very likely to be imperfect and this will degrade the performance of the SPECCOH scheme, thus lowering the losses of OPT with respect to SPECCOH. Little performance gains with specular term phase estimation at high SNR and lower gains with lower values of have been already noticed over one-path Ricean channels for binary signaling with complex cross-correlation coefficient magnitude ) varying from 0 to 0.95 ( ) [20]. Note [19] and for binary orthogonal signaling ( that, in [19], the convergence between specular coherent and noncoherent detection over one-path Ricean channels is said to be better for large , which is true when comparing FSK(1/3) ) with FSK(1/2) ( ). However, it can be ( ) shown numerically that the convergence for FSK(1) ( ) which is better than is better than for FSK(2/3) ( ). Such observations agree with results for FSK(1/2) ( in our paper for mixed mode mode Ricean/Rayleigh channels, where the convergence for FSK (1) was shown to be much better than for FSK(1/2). The convergence of one-path Ricean specular coherent and noncoherent performance at high SNR was explained by the fact that the phase of the received signal has a contribution from the channel random component, so knowledge of the specular term phase provides only partial information [20]. At high SNR, the fading is causing the most degradation, making phase estimation less important in that case. C. Effect of the Decorrelation Operation (SPECCOH Versus SPECCOHR) Fig. 8 presents bit error probabilities of the unresolved multipath Ricean channels specular coherent optimum receiver (SPECCOH) and the resolved multipath Ricean channels specular coherent optimum receiver (SPECCOHR) over two-path unresolved mixed mode Ricean/Rayleigh fading channels. The main difference between these two schemes is that SPECCOH implements the decorrelation operation while SPECCOHR does not. From Fig. 8, it is seen that with FSK(1/2) and DPSK the SPECCOHR scheme yields error floors, which are eliminated by SPECCOH. This shows the importance of the decorrelation operation to handle path unresolvability for specular coherent detection. From Figs. 4–7, the OPT bit error probabilities have no error floor similar to the SPECCOH. This suggests that the decorrelation operation also eliminates error floors for noncoherent detection as well [14].
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TABLE II SNR GAINS (IN DB) OBTAINED BY SPECCOH COMPARED TO OPT AND QDR (
IS IN
dB)
tion. For FSK(1/2) and SDPSK, the performance degradation is larger (0.5–1.8 dB for FSK(1/2) and 1.2–1.5 dB for SDPSK at ) with losses greater than 2 dB at for FSK(1/2). Thus, noncoherent detection could be of interest for FSK(1) and DPSK, while specular phase estimation may be needed for FSK(1/2). Finally, we showed that the specular coherent optimal receiver over unresolved multipath Ricean channels includes a decorrelation stage and then implements a resolved specular coherent optimal decision rule. The decorrelation operation is also present in optimal noncoherent detection structures. The importance of the decorrelation operation in yielding diversity gains and eliminating the error floors was demonstrated for commonly used binary modulation schemes such as FSK and variants of DPSK over unresolved mixed mode Ricean/ Rayleigh fading channels. Fig. 8. Performance of the SPECCOH and SPECCOHR schemes with FSK(1/2) and DPSK signaling over two-path Ricean/Rayleigh channels ( 0:1T , s = 1, and = 10, 15 dB).
=
V. CONCLUSION This paper considered specular coherent and noncoherent optimal detection for unresolved multipath Ricean fading channels, emphasizing receiver structures and single pulse performance. Specular coherent detection needs estimation of the specular term phases, while noncoherent does not. It was shown that for mixed mode Ricean/Rayleigh fading channels the noncoherent likelihood ratio can be obtained by substituting the specular component phasor of the specular coherent likelihood ratio by an estimate. It was also shown that the specular coherent (SPECCOH) and noncoherent (OPT) optimal receivers converge to the same structure at high SNR for equally likely equal energy signals. The performance of the SPECCOH scheme was assessed by calculating its exact bit error probabilities. Due to the nonlinearities of the OPT scheme, its performance was assessed by using asymptotically tight lower and upper bounds to its bit error probabilities. For FSK(1) and, to a lesser extent for DPSK, at sufficiently high SNR, the performance loss due to the lack of specular term phase knowledge is quite small (0.2–0.7 dB loss with FSK(1) and 0.8–1.2 dB loss with DPSK, ), not justifying the use of carrier phase estimaat
APPENDIX A. Proof of Proposition 2.1 , , the term
are independent of . Hence, since is deterministic, converges a.s. to as tends to , and from (7) the infinity. From (10), SPECCOH log-likelihood ratio satisfies a.s. and from (17)
(31) in (16) satisfies a.s.
The function random vector be shown that
(32)
given by (18) depends on the Gaussian . Using the joint pdf of , from (18) it can
where
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(33)
and (33) is obtained where and (18)–(20). Hence, from (16) and using (32), the log-likelihood ratio for the OPT scheme satisfies
and
Hence,
and a.s.
(34)
Comparing (31) and (34) completes the proof. Furthermore, since and is measurable in
is measurable in (
) for fixed
( ) for fixed , can be defined as a Lebesgue integral of for almost all sample functions the sample functions of , [31, p. 45]. From (10), hence from (19)
ACKNOWLEDGMENT The authors would like to thank the three anonymous reviewers and the editor for their suggestions and time invested in handling this paper. REFERENCES
where
with defined here as from [18]
for any matrix
, and
with
Since
is independent of
and
, from the Lebesgue dominated convergence theorem a.s. a.s.
[1] J. Proakis, Digital Communications. New York: McGraw-Hill, 1989. [2] R. Price, “Optimum detection of random signals in noise, with application to scatter-multipath communication, I,” IRE Trans. Inform. Theory, vol. 2, pp. 125–135, Dec. 1956. [3] G. Turin, “Communication through noisy, random-multipath channels,” 1956 IRE National Convention Rec., vol. pt. 4, pp. 154–166, Mar. 1956. [4] R. Price and P. Green Jr, “A communication technique for multipath channels,” Proc. IRE, vol. 46, pp. 555–570, Mar. 1958. [5] H. Ochsner, “Direct-sequence spread-spectrum receiver for communication on frequency-selective fading channels,” IEEE J. Select. Areas Commun., vol. 5, pp. 188–193, Feb. 1987. [6] U. Fawer, “A coherent spread-spectrum diversity-receiver with AFC for multipath fading channels,” IEEE Trans. Commun., vol. 42, pp. 1300–1311, Feb./Mar./Apr. 1994. [7] F. Hagmanns and V. Hespelt, “On the detection of bandlimited direct-sequence spread-spectrum signals transmitted via fading multipath channels,” IEEE J. Select. Areas Commun., vol. 12, pp. 891–899, June 1994. [8] T. Eng and L. Milstein, “Partially coherent DS-SS performance in frequency selective multipath fading,” IEEE Trans. Commun., vol. 45, pp. 110–118, Jan. 1997. [9] H. Abdel-Ghaffar and S. Pasupathy, “Partially coherent, suboptimal detectors over Rayleigh fading diversity channels,” in Proc. GLOBECOM94, 3rd Comm. Theory Mini-Conf (CTMC), San Francisco, CA, Nov.-Dec. 1994, pp. 76–80. [10] K. Cheun, “Performance of direct-sequence spread-spectrum rake receivers with random spreading sequences,” IEEE Trans. Commun., vol. 45, pp. 1130–1143, Sept. 1997. [11] M. Win and Z. Kostic´, “Impact of spreading bandwidth on rake reception in dense multipath channels,” IEEE J. Select. Areas Commun., vol. 17, pp. 1794–1806, Oct. 1999. [12] K. Kim et al., “Effect of tap spacing on the performance of direct-sequence spread-spectrum RAKE receiver,” IEEE Trans. Commun., vol. 48, pp. 1029–1036, June 2000. [13] G. Bottomley et al., “A generalized RAKE receiver for interference suppression,” IEEE J. Select. Areas Commun., vol. 18, pp. 1536–1545, Aug. 2000. [14] F. Danilo, “Detection techniques and performance analysis for fading multipath channels with unresolved components,” Ph.D. dissertation, McGill University, Montreal, QC, Canada, 2000. [15] J. Shapira, “Channel characteristics for land cellular radio and their systems implications,” IEEE Antennas Propagat. Mag., vol. 34, pp. 7–16, Aug. 1992. [16] W. Lee, Mobile Communications Engineering, 2nd ed. New York: McGraw-Hill, 1998. [17] D. Devasirvatham, M. Krain, and D. Rappaport, “Radio propagation measurements at 850 MHz, 1.7 GHz and 4 GHz inside two dissimilar office buildings,” Electron. Lett., vol. 26, pp. 445–447, Mar. 1990.
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[18] H. Hashemi, “The indoor radio propagation channel,” Proc. IEEE, vol. 81, pp. 943–968, July 1993. [19] G. Turin, “Error probabilities for binary symmetric ideal reception through nonselective slow fading and noise,” Proc. IRE, vol. 46, pp. 1603–1619, Sept. 1958. [20] P. Kam, “On orthogonal signaling over the slow nonselective Ricean fading channel with unknown specular component,” IEEE Trans. Commun., vol. 41, pp. 817–819, June 1993. [21] R. Aiken, “Communication over the discrete-path fading channel,” IEEE Trans. Inform. Theory, vol. IT-13, pp. 346–347, Apr. 1967. , “Error probability for binary signaling through a multipath [22] channel,” Bell Syst. Tech. J., vol. 46, pp. 1601–1631, Sept. 1967. [23] M. Alles and S. Pasupathy, “Channel knowledge and optimal performance for two-wave Rayleigh fading channels,” IEEE Trans. Veh. Technol., vol. 43, pp. 8–20, Feb. 1994. [24] F. Danilo and H. Leib, “Detection techniques for fading multipath channels with unresolved components,” IEEE Trans. Inform. Theory, vol. 44, pp. 2848–2863, Nov. 1998. [25] R. Vaughan and N. Scott, “Super-resolution of pulsed multipath channels for delay spread characterization,” IEEE Trans. Commun., vol. 47, pp. 343–347, Mar. 1999. [26] Z. Kostic´ and G. Pavlovic´, “Resolving subchip-spaced multipath components in CDMA communication systems,” IEEE Trans. Veh. Technol., vol. 48, pp. 1803–1808, Nov. 1999. [27] W. McGee, “Complex Gaussian noise moments,” IEEE Trans. Inform. Theory, vol. IT-17, pp. 149–157, Mar. 1971. [28] W. Rudin, Real and Complex Analysis, 2nd ed. New York: McGrawHill, 1974. [29] H. Poor, An Introduction to Signal Detection and Estimation. New York: Springer-Verlag, 1988. [30] G. Turin, “Some computations of error rates for selectively fading multipath channels,” in Proc. National Electronics Conf., vol. 15, Mar. 1959, pp. 431–440. [31] E. Wong and B. Hajek, Stochastic Processes in Engineering Systems. New York: Springer-Verlag, 1985. [32] J. Doob, Stochastic Processes. New York: Wiley, 1953. [33] W. Kaplan, Advanced Calculus. Reading, MA: Addison-Wesley, 1984. [34] J. Winters, “Differential detection with intersymbol interference and frequency uncertainty,” IEEE Trans. Commun., vol. COM-32, pp. 25–33, Jan. 1984.
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Florence A. Danilo-Lemoine (S’94–M’00) was born in Troyes, France, in 1969. She received the Diplôme d’Ingénieur degree in electrical engineering from the National School of Electronics and Radio-Electricity (E.N.S.E.R.G.), Grenoble, France, in 1992 and the Ph.D. degree in electrical engineering from McGill University, Montreal, PQ, Canada, in 2000. From 1992 to 2000, she was a Teaching and Research Assistant and held several part-time lecturer positions at McGill University. In 1999, she was a faculty lecturer in the Department of Electrical Engineering at McGill University. Since November 2000, she has been an Assistant Professor in the Department of Systems and Computer Engineering at Carleton University, Ottawa, ON, Canada, where she conducts research and teaches undergraduate and graduate courses in telecommunications. Her research interests are in digital telecommunications, detection, and statistical signal processing with emphasis on wireless communication systems. Dr. Danilo-Lemoine was awarded the Canadian Advanced Technology alliance (CATA) telecommunications graduate scholarship in 1999. In 2001, she received a university faculty award from the Natural Sciences and Engineering Research Council of Canada (NSERC).
Harry Leib (M’83–SM’95) was born in 1953. He received the B.Sc. (cum laude) and M.Sc. degrees from the Technion—Israel Institute of Technology, Haifa, Israel, in 1977 and 1984, respectively, and the Ph.D. degree from the University of Toronto, Toronto, ON, Canada, in 1987, all in electrical engineering. From 1977 to 1984, he was with the Israel Ministry of Defense, working in the area of Communication Systems. During his Ph.D. studies at the University of Toronto (1984–1987), he was a Teaching and Research Assistant in the Department of Electrical Engineering, working in the areas of Digital Communications and Signal Processing. After completing his Ph.D. studies, he was with the University of Toronto as a Post-Doctoral Research Associate in Telecommunications (September 1987 to December 1988) and an Assistant Professor (January 1989–August 1989). Since September 1989, he has been with the Department of Electrical and Computer Engineering at McGill University in Montreal, initially as an Assistant Professor, then as an Associate Professor, and now as a Full Professor. He spent part of his sabbatical leave of absence at Bell Northern Research in Ottawa, ON, Canada (September 1995 to February. 1996) working on a CDMA-related project. During the rest of his sabbatical (March 1996 to August 1996), he was a Visiting Professor in the Communications Lab/Institute of Radio Communications at the Helsinki University of Technology, Espoo, Finland, where he taught a condensed course on channel coding and modulation and worked with graduate students in the communications area. At McGill, he teaches under graduate and graduate courses in Communications and directs the research of graduate students. His current research activities are in the areas of Digital Communications, Wireless Communication Systems and Code Division Multiple Access. Dr. Leib is an Editor for Communication and Information Theory for the IEEE TRANSACTIONS OF COMMUNICATIONS and an Associate Editor for the IEEE TRANSACTIONS OF VEHICULAR TECHNOLOGY.
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Specular Coherent and Noncoherent Optimal Detection for Unresolved Multipath Ricean Fading Channels Florence A. Danilo-Lemoine, Member, IEEE, and Harry Leib, Senior Member, IEEE
Abstract—This paper considers specular coherent and noncoherent optimal detection for unresolved multipath Ricean fading channels with known delays. The focus is on receiver structures and performance. Specular coherent detection employs the carrier phase of the Ricean specular component, while noncoherent detection does not. Therefore, a specular coherent detector must be augmented with a carrier phase estimator for the specular component. The structures considered in this paper are generalization of the well-known Rake receiver to the unresolved multipath case. It is shown that both optimal structures perform a decorrelation operation before combining, which is essential to eliminating error floors under multipath unresolvability conditions. Furthermore, the noncoherent optimal receiver includes an inherent estimator for the specular component phasor. It is shown that the specular coherent and noncoherent structures converge at high SNR. This result is confirmed through analytical and numerical performance evaluation. Little performance gains can be obtained by the use of specular coherent detection for orthogonal frequency-shift keying and to a lesser extent for differential phase-shift keying over mixed mode Ricean/Rayleigh fading channels, making noncoherent demodulation attractive in these cases. Index Terms—Decorrelators, detection, estimation, multipath fading channels, noncoherent, quadratic receivers, specular coherent, wireless communications.
I. INTRODUCTION YSTEMS for personal communication services have to operate in indoor and outdoor radio environments that are characterized as multipath fading channels. The effect of multipath fading and consequently the choice (or validity) of a channel model depends on the transmission bandwidth. Narrow-band systems (compared to channel coherence bandwidth) yield a flat fading channel [1] while wide-band systems yield frequency-selective channels. Diversity gains can be achieved over frequency-selective fading channels by the use of Rake receivers [2]–[4] or their variations [5]–[13]. Rake receivers exploit path diversity by using wide-band signals that resolve multipath [14]. Such bandwidth requirement can
S
Paper approved by A. Ahlen, the Editor for Modulation and Signal Design of the IEEE Communications Society. Manuscript received May 10, 2001; revised October 11, 2001. This work supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. F. A. Danilo-Lemoine is with the Department of Systems and Computer Engineering, Carleton University, Ottawa, ON K1S 5B6, Canada (e-mail: [email protected]). H. Leib is with the Department of Electrical and Computer Engineering, McGill University, Montréal, QC H3A 2A7, Canada (e-mail: [email protected]). Publisher Item Identifier S 0090-6778(02)03519-5.
be achieved outdoors by spread-spectrum signals, but it is not practical indoors or in dense urban environments where up to 50 MHz would be needed [15] to resolve the small inter-path delays [16], [17]. This paper considers detection techniques for unresolved multipath fading channels, thus not requiring bandwidth spreading. The best performance can be achieved when the channel realizations are available, implying that the channel impulse response or channel parameters have to be estimated. Carrier phase estimation in mobile environments is complex and may yield inaccurate estimates [18, p. 953]. An alternative is to use noncoherent detection at the expense of some performance degradation. Specular coherent detection, using the carrier phases of specular components only, yields little performance gains at high signal-to-noise ratio (SNR) for single-path Ricean channels [19], [20]. The noncoherent optimal receiver for unresolved multipath Rayleigh channels with Doppler and known delays was derived in [21]. Its performance was considered in [22] only for signals that are orthogonal to all time shifts, resolving the multipath. Optimal receivers over two-path unresolved Rayleigh channels were derived in [23] for known delays and different levels of channel knowledge, assuming that the autocorrelation magnitude of the transmitted signal complex envelope at inter-path delays is independent of the signal shape. Two-fold diversity-like effects were found in the performance of envelope orthogonal frequency-shift keying (FSK) and variants of chirp or linear frequency sweep modulation. The noncoherent optimal receiver over unresolved multipath Ricean fading channels was derived in [24]. However, its performance was not evaluated and the corresponding specular coherent receiver was not considered. This work considers unresolved multipath Ricean fading channels, focusing on similarities and differences between specular coherent and noncoherent optimal detection in terms of receiver structures and single pulse performance. For small inter-path delays, the effects of intersymbol interference (ISI) are negligible, making the single-pulse bound close to the achievable performance for sequential transmission. The multipath delays, assumed to be known, could have been estimated by using super-resolution techniques [25], [26] or by sounding the channel with a wide-band pulse. The effects of path-delay estimation errors, which may degrade performance if they are not kept small, is beyond the scope of this paper. This paper is organized as follows. Section II presents the system model and receiver structures for unresolved multipath Ricean fading channels. Section III considers the error rate
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calculation for mixed mode Ricean/Rayleigh channels. The Ricean component can model a line-of-sight path [18]. Section IV presents a performance analysis of specular coherent and noncoherent detection as well as numerical results for variations of binary FSK and differential phase-shift keying (DPSK). Section V presents the conclusions. II. SYSTEM MODEL AND RECEIVER STRUCTURES A. Channel Modeling possible bandpass signals Assume transmission of one of of finite energy over a multipath Ricean fading channel. Under , the received signal is given by hypothesis
(1) denotes the real part of the argument, and where are independent circularly complex Gaussian random and variance variables [27] with mean . The multipath comare either fixed and known, or unknown ponent phase shifts independent random variables uniformly distributed between and , and represents the complex envelope of the th possible transmitted signal. The multipath delays are assumed to be known and distinct. The channel noise is satisfying modeled by a zero mean white Gaussian process and statistically independent of the signal . For Ricean multipath channels, each path can be considered as the phasor sum of a Rayleigh component with uniformly distributed phase and a fixed (specular) component. To model absence of reference phase information (noncoherent detection), an additional random phase needs to be added to each , yielding the model (1). multipath component, can be expressed as In (1), where have the same joint probability since are density function (pdf) as zero mean circularly complex Gaussian. Therefore, whenever the ’s are known, (1) represents a multipath Ricean fading channel that implicitly assumes that the specular component phases and amplitudes are known at the receiver (specular coherent detection). , are conAssume that tained in the interval [0, ]. Under practical conditions, the delayed signals are linearly independent over [0, ] for distinct multipath delays [14]. In this work, the multipath resolvability condition is not assumed. Finally, , are continuous on assume that [0, ], which is not too limiting since any square-integrable function can be approximated arbitrarily close in the Euclidean norm by a continuous function [28, p. 71]. This ensures the , required for the existence of mean square continuity of its Karhunen–Loève expansion [29, pp. 379–380]. We use the following notation. Bold capital letters denote matrices and bold lowercase letters denote vectors, , and
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denote respectively the transposition, complex conjugation, th entry of a matrix and Hermitian conjugation. The is and the th entry of a vector is . We define . The baseband signal energy under is given by . The signal cross-corand , , is relation matrix between the hypotheses defined as . The th signal correlation matrix is is defined as channel covariance matrix , where and .
. The
B. Specular Coherent Optimal Decision Rule for an -Path Ricean Channel In this section, we assume that the specular components amplitudes and phases are known at the receiver (i.e., known ). Multipath Ricean channels when is fixed yield the classical problem of detecting a Gaussian signal in additive white Gaussian noise [29, pp. 419–421]. A minimum probability of error receiver forms the likelihood ratio between each and hypothesis . With equiprobable hya null hypothesis potheses, the decision is made in favor of the largest likelihood follows ratio [29, p. 11]. The discrete representation of from the Karhunen–Loève expansion that exists since is a second order mean-square continuous process [29, pp. given is 379–380]. The covariance function of
(2) where function of
and , given
, the covariance
is
(3) and . As shown in [24]1 has at most positive which are those of the matrix eigenvalues and has corresponding eigenfunctions given by
with
(4) where2 equations
,
is an
matrix that satisfies the (5a) (5b)
identity matrix and where is the diagonal matrix defined by
K
1
2The
(t; u) given by (3) is denoted as
denotes the . The signal
K (t; u) in [24]. superscript m is not an exponent and refers to the hypothesis H .
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Fig. 1. Specular coherent optimum receiver for an L-path Ricean channel (SPECCOH) assuming T
narrow-band assumption and (2) imply that the eigenvalues and eigenfunctions of are given by
associated with (6a)
associated with (6b)
for all l.
Since the term is independent of the hy, an equivalent decision variable is obpothesis can be tained by removing this term from (7). The variable , and [see (12)] can be obobtained from for tained by using a bank of matched filters [24]. When can be generated by sampling the output all , the variables for of the matched filter , at , as shown in Fig. 1, or by using a tapped-delay line after the matched filter. , , When the multipath is resolved (i.e., ), (7) reduces to [3], [30]
, it can be shown Using the Karhunen–Loève expansion of [14] that the likelihood ratio associated with the specular co, herent optimal scheme (SPECCOH) for short, is given by
(13)
(7) where
, (8)
The receiver implementing (13) is the specular coherent optimum receiver for resolved multipath fading channels and is denoted SPECCOHR in this paper. Since the linear transformation on the signals [see (4)] is invertible, the th hypothesis (1) can be equivalently expressed as
(9) (10) (14) where
(11) i.e.,
(12) and the integrals in (11) and (12) are Wiener integrals.
, , while are orthogonal signals of same energy as . When is fixed, for all the new random is Gaussian with mean vector and covariance [14]. Therefore, under each hypothesis, the received signal can be represented as a linear combination of orthogonal functions
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Fig. 2. Noncoherent optimum receiver for an L-path Ricean channel (OPT) assuming T
weighted by uncorrelated circularly complex Gaussian random variables, similar to the resolvable multipath case. Using (5) with several matrix manipulations, the specular coherent likelihood ratio when the multipath is unresolved (7) can be rewritten as
591
for all l.
integrating (7) over all components of the vector between and resulting in
(16) where
and
are given by
(17) (15) is Gaussian Noting that for fixed given by (9) and covariance , it is seen from with mean (13) and (15) that the specular coherent optimal receiver for unresolved multipath Ricean channels consists of an orthogonalinto ization (or decorrelation) stage that transforms , and then implements a resolved multipath optimal de. Note that besides the orthogonalization of cision rule for , the matrix also perthe signals forms statistical decorrelation in the sense that the new vari, unlike , are uncorreables . lated as coefficients of the Karhunen–Loève expansion of The decorrelation stage of unresolved multipath fading channels optimal receivers was already identified for Rayleigh channels [24] and thus is generalized here to Ricean channels with specular coherent detection. C. Noncoherent Optimal Decision Rule for an -Path Ricean Channel The likelihood ratio associated with the SPECCOH scheme corresponds to the conditional likelihood ratio (given ) associated with the noncoherent optimal receiver (OPT) given by (7). is obtained by Thus, the noncoherent likelihood ratio
(18)
(19) (20) , and are given by (5), is where are respectively given by (8) and (9), given by (10), and and are respectively given by (11) and (12), denotes the diagonal matrix composed of the main diagonal entries denotes the lower triangular matrix composed of , and of the lower triangular elements of with zero main diagonal entries. It can be shown that the closed-form solution of the inis the sum of multidimensional infinite series of tegral of products of Bessel and trigonometric functions [14]. From (16) to (20) it is seen that the OPT scheme uses the as the SPECCOH scheme does. same decision variables for all , the receiver of Fig. 2 is obThus, when tained. Note that the OPT structure is identical to the one of
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[24] up to a different random vector . However, it can be of this paper [see (11)] and of [24] ( shown that ) are identically distributed. Therefore, identical receiver structures based on either version of are equivalent and yield the same probability of error.
low SNR. Using asymptotical properties of is very large, then
, when
D. Comparison of Specular-Coherent and Noncoherent Optimal Structures When
Using (5) and (10), it can be shown that
Thus, for a mixed mode Ricean/Rayleigh channel, i.e., , (7) reduces to
(21) and . For a mixed where mode Ricean/Rayleigh channel, (18) reduces to . Hence, (16) reduces to
is small, then
reducing the contribution of the specular phasor estimate. Assuming equally likely signals, the optimal receiver decision rule is obtained by selecting the hypothesis corresponding to the largest log-likelihood ratio. If the signals are further asis identical under all hypotheses sumed to have equal energy, we obtain an equivalent decision rule. The and dividing by behavior of the SPECCOH and OPT schemes at high SNR is illustrated in the following proposition proven in the appendix. Proposition 2.1: The SPECCOH and OPT scheme log-likelihood ratios “converge” almost surely (a.s.) to the same term as goes to infinity, i.e.,
At high SNR, the SPECCOH and OPT schemes use the same decision rule for equally likely equal energy signals, meaning that knowledge of the specular component phases is not necessary in this case. Section IV-A confirms this property in terms of performance for mixed mode Ricean/Rayleigh channels. (22) Comparing (21) with (22) shows that the mixed mode noncoherent likelihood ratio can be obtained from the mixed mode specular coherent likelihood ratio by substituting the specular in (21) with an estimate given by phasor
(23) From (23), it is seen that the specular phasor estimate is obtained by substituting the unknown phase by an estimate and then scaling the phasor by . The scaling factor acts as a soft limiter to reduce the phasor estimate effects at
III. PERFORMANCE EVALUATION The SPECCOH scheme is quadratic in a Gaussian statistic and the bit error probabilities over mixed Ricean/Rayleigh fading channels can be calculated as in [24]. The nonlinearities of the OPT scheme [see (16)] make its performance analysis very tedious, if not impossible. Therefore, upper and lower bounds for the bit error probability of the OPT scheme are employed. Upper bounds are obtained by evaluating the bit error probability of suboptimum quadratic receivers (such as the QDR (various ), QR and R OPT schemes derived in [24]) as a . For each value of function of the received SNR per bit , the lowest probability of error among all suboptimum receivers is retained to provide the tightest upper bound. An example for FSK with frequency deviation over a two-path mixed mode Ricean/Rayleigh fading channel is illustrated in Fig. 3. For a two-path mixed mode channel where , is the relative Rayleigh component strength is the Ricean between the first and second path, is the energy per bit of the real signal. All parameter and
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of
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It is known that (24) can be evaluated by integrating the pdf , obtained from its characteristic function, [30]
(25) ,
where
. Such integration can be performed by
and
0
=
Fig. 3. Bounds for OPT with FSK signaling with f f 1=2T over a two-path Ricean/Rayleigh channel ( = 0:1T , s = 1 and = 15 dB).
two-path mixed mode Ricean/Rayleigh channels are labeled by the values of their parameters and and the relative . For delay between the first and the second path convenience, is expressed as a percentage of the duration of the signaling waveform ( ) and is expressed in decibels. Lower bounds are obtained by evaluating the performance of the noncoherent optimum receiver over a Gaussian nonfading channel and the performance of the SPECCOH scheme from Section II-B over the considered mixed mode Ricean/Rayleigh multipath fading channel. For each received SNR per bit, the highest probability of error is retained to provide the tightest lower bound as seen by the example of Fig. 3. As seen in Section II-B, the SPECCOH scheme is also quadratic. Therefore, the technique of upper and lower bounding the bit error probabilities of the OPT receiver requires performance evaluation of quadratic receivers. We present now the method used for computing the bit-error probability of quadratic receivers in a Gaussian statistic. Let be the decision variable under hypothesis . The decision variables of a specular coherent receiver (SPECCOH, SPECCOHR) employ , while those of a noncoherent receiver (QDR, R OPT and QR) do not. In both cases, the decision variables may depend on through the received signal. With equiprobable equal energy binary signals, the bit error probability with held fixed is
,
using the residue method; however, for Ricean channels, rendering this method is not practical due to the complicated [24]. Also, the dependency of exponential factor in on via the mean 3 implies that the pdf of needs to be integrated with respect to and , which presents difficulties. However, for mixed mode Ricean/Rayleigh fading channels, no integration with respect to needs to be performed [24]. For example, the pairwise probability of error can be evaluated by integrating numerically the following improper integral:
(26)
(27) (28) where
,
. The matrix
is chosen
such that it diagonalizes while satisfying , . Since the function resulting in the diagonal matrix increases monotonically to , the integration is carried . It appears that the function over a finite range is quite suitable for numerical integration. The MATLAB program was designed such that the absolute error formed by the sum of truncation error and numerical integration error was less than 1.7 10 [14]. IV. PERFORMANCE OF BINARY MODULATION SCHEMES OVER MIXED MODE RICEAN/RAYLEIGH FADING CHANNELS WITH OPTIMAL DETECTION A. Convergence of the SPECCOH and OPT Performance at High SNR
(24)
where
is a Hermitian quadratic form
in jointly Gaussian random variables, and is a bias term such . The probability of error is then obthat tained by averaging (24) over . Assuming equal energy signals, , and for Table I presents the expressions of the quadratic receivers considered in this paper.
Following Section III, the performance of the OPT scheme at high SNR will be studied by considering the upper and lower bounds provided, respectively, by the bit error probabilities of R be the OPT and SPECCOH. Let for fixed cumulative distribution function (cdf) of under , where the subscript indicates that it depends generally on . Similarly, let us define the function as in (27). In 3Note that for all quadratic schemes considered here, including SPECCOH and SPECCOHR, R is independent of .
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TABLE I EXPRESSIONS OF Q , r , AND A FOR THE CONSIDERED QUADRATIC RECEIVERS
this section, we denote as to emphasize its dependency , for SPECCOH and R OPT, on . Let us first show that converge to the same value as tends to infinity. and do not depend Recall from (5) and (10) that . Hence, from Table I, as on and tends to infinity the vectors of the SPECCOH and R OPT schemes converge almost surely to the same vector, namely since the term
,
tinuous on [ ] [33, p. 454] and from [32, p. 9] the conas tends to infinity, is uniform. convergence of on [ , verges uniformly to the continuous function ], for all we have hence for both SPECCOH and R OPT we have since for both schemes
, 2 is deter-
. Thereministic. Furthermore, from (10), fore, as tends to infinity, for the SPECCOH and R OPT schemes converge almost surely to the same quadratic form . From [31, p. 20], converges also in probability. Let be the cdf of under for fixed . Then, at every continuity point of , [31, p. 23], which says that converges in distribution. Note that the convergence will be uniform [32, p. 9]. Considering in any closed interval of continuity of , similar to (26) it can be shown that
(29) and are functions equivalent to and [see (27) and (28)] as tends to infinity. is continuous on , is and has a finite continuous on is conlimit as tends to zero, hence . It can be shown [14] that tinuous on is uniformly convergent for , where is an upper bound for when given by where
Therefore, the bit error probabilities of the SPECCOH and R OPT schemes converge to the same value as tends to infinity. Since they are lower and upper bounds to the bit error probabilities of OPT, the bit error probabilities of SPECCOH and OPT also converge to the same value as tends to infinity. This performance result agrees with Proposition 2.1 concerning receiver structures. B. Performance of the SPECCOH and OPT Schemes (Numerical Results) Performance is assessed by calculating bit error probabilities . Commonly as functions of the received SNR per bit, used modulation schemes such as variations of FSK and DPSK , , 2 the complex envelope of are considered. Under a FSK binary signal is
where (carrier frequency). FSK modulation with freis denoted as FSK( ), where quency separation . For FSK we have , thus the energy per , , 2, i.e., bit of FSK signals is given by . Under and the complex envelopes for DPSK over a two-symbol interval are
(30) where
, , and . Therefore, from (29), the function
is con-
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DANILO-LEMOINE AND LEIB: SPECULAR COHERENT AND NONCOHERENT OPTIMAL DETECTION
Fig. 4. Performance of the SPECCOH and OPT schemes with FSK(1) 0:1T , s = 1 and signaling over two-path Ricean/Rayleigh channels ( = 5–20 dB).
=
595
Fig. 5. Performance of the SPECCOH and OPT schemes with FSK(1/2) signaling over two-path Ricean/Rayleigh channels ( = 0:1T , s = 1 and = 5–20 dB).
The complex envelopes for symmetrical DPSK [34] over a twosymbol interval are
Since with DPSK and SDPSK the transition between the carrier phase of consecutive bits carries the information, we have , thus for (S)DPSK , , 2, i.e., . Note that the present definitions of FSK and DPSK waveforms are not continuous on the observation interval [0, ] since they present discontinuand . However, they can be approximated ities at arbitrarily close in terms of Euclidean distance by continuous complex waveforms on with compact support4 since they are square integrable functions [28, p. 71]. Furthermore, the performance analysis of quadratic receivers does not use the continuity assumption. Figs. 4–7 present the performance of the specular coherent (SPECCOH) and the noncoherent (OPT) optimum receivers over two-path mixed mode Ricean/Rayleigh fading channels ), and . with equal Rayleigh path strength ( Results similar to Figs. 4–7 are obtained when [14]. The SPECCOH bit error probabilities are represented by short dashed lines. Absence of those lines for some values of means that in that case the SPECCOH bit error probability is equal to the lower bound to the OPT bit error probability on the entire range of received SNR presented in this paper. The performances of the noncoherent and coherent optimum 4The support of a complex function f on a topological space X is the closure of the set fx : f (x) 6= 0g.
Fig. 6. Performance of the SPECCOH and OPT schemes with DPSK signaling over two-path Ricean/Rayleigh channels ( = 0:1T , s = 1 and K = 5–20 dB).
receivers over a single path nonfading Gaussian channel, as well as the performance of the noncoherent optimum receiver over a two-path equal strength Rayleigh channel with are added as references. Figs. 4–7 show that the lower and upper bounds for the OPT scheme are tight at high SNR for FSK(1) and DPSK, but are less tight for FSK(1/2) or SDPSK. For FSK this phenomenon can be explained for one-path fading channels as follows. On one-path Ricean channels, the probability of error for coherent detection depends only on the real part of the cross-correlation coefficient [1]. Hence FSK(1) and FSK(1/2) with coherent detection have the same probabilities of error in this case. At high SNR, the best modulus of the signal cross correlation for noncoherent detection over one-path Ricean channels is zero [19] and FSK(1/2) having nonzero cross correlation performs worse than FSK(1). Specular coherent detection with a sufficiently
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Fig. 7. Performance of the SPECCOH and OPT schemes with SDPSK 0:1T , s = 1 and signaling over two-path Ricean/Rayleigh channels ( K = 5–20 dB).
=
strong specular component has similar features to coherent detection. The gap between coherent and noncoherent detection over one-path Ricean channels is smaller for FSK(1) than for FSK(1/2). Figs. 4–7 illustrate that the performance of the SPECCOH and OPT schemes improves as the Ricean path dominates the Rayleigh path, tending to the performance for a Gaussian channel. It is also seen that for both detection techniques DPSK (and SDPSK) give the best performance. For example, at high SNR DPSK detected with the OPT scheme gives at least a 3.6-dB improvement compared to FSK(1/2) and at least 4.2 dB compared to FSK(1) in the error probability range 10 –10 , but 3 dB are gained because the observation interval used with DPSK is twice the one used with FSK. Figs. 4 and 5 dB at high SNR ( dB), show that for performance of the OPT scheme is better with FSK(1/2) than dB) FSK(1) with FSK(1), while for lower SNR ( dB, FSK(1) performs better performs better. For than FSK(1/2) for all SNR values, confirming our earlier observation that with a sufficiently strong specular component the zero cross-correlation feature of FSK(1) is an asset with the OPT receiver. Table II presents typical SNR gains (evaluated by the upper bounds) that can be obtained by using SPECCOH instead of the OPT scheme for three bit error probabilities, 10 (speech) and 10 , 10 (data). SNR gains greater than 2 dB are set in boldface indicating cases where specular component phase estimation may yield significant improvement. Table II and Figs. 4 and 6 show that little performance gains can be obtained by the knowledge of the specular term phase for FSK(1) and to a lesser extent for DPSK (ex. maximal gain of 0.2 –0.7 dB for ). Lower gains FSK(1) and 0.8–1.2 dB for DPSK at are obtained for lower values of and for higher SNR. These observations along with the difficulties inherent in phase estimation justify the use of noncoherent detection for FSK(1) or DPSK especially at high SNR. It may be argued here that the OPT scheme is more complex than the SPECCOH. However, as shown by Table II, the loss in performance of the simpler
noncoherent suboptimum scheme QDR ( ) compared to SPECCOH is also small. Table II and Figs. 5 and 7 show that significant gains can be obtained for FSK(1/2) and SDPSK by the knowledge of the specular component phase (ex. maximal with dB and 1.8 dB at gain of 3 dB at with dB for FSK(1/2), 1.5 dB at with dB and 1.8 dB at with dB for SDPSK). Gains larger than 3 dB are also obtained for FSK(1/2) dB favoring the use of coherent detection for with – dB), all modulation FSK(1/2). At very low SNR ( schemes yield significant losses (around and greater than 2 dB). These represent the losses of OPT with respect to SPECCOH, assuming perfect Ricean specular term phase estimation. However, at such low SNR, the specular term phase estimate is very likely to be imperfect and this will degrade the performance of the SPECCOH scheme, thus lowering the losses of OPT with respect to SPECCOH. Little performance gains with specular term phase estimation at high SNR and lower gains with lower values of have been already noticed over one-path Ricean channels for binary signaling with complex cross-correlation coefficient magnitude ) varying from 0 to 0.95 ( ) [20]. Note [19] and for binary orthogonal signaling ( that, in [19], the convergence between specular coherent and noncoherent detection over one-path Ricean channels is said to be better for large , which is true when comparing FSK(1/3) ) with FSK(1/2) ( ). However, it can be ( ) shown numerically that the convergence for FSK(1) ( ) which is better than is better than for FSK(2/3) ( ). Such observations agree with results for FSK(1/2) ( in our paper for mixed mode mode Ricean/Rayleigh channels, where the convergence for FSK (1) was shown to be much better than for FSK(1/2). The convergence of one-path Ricean specular coherent and noncoherent performance at high SNR was explained by the fact that the phase of the received signal has a contribution from the channel random component, so knowledge of the specular term phase provides only partial information [20]. At high SNR, the fading is causing the most degradation, making phase estimation less important in that case. C. Effect of the Decorrelation Operation (SPECCOH Versus SPECCOHR) Fig. 8 presents bit error probabilities of the unresolved multipath Ricean channels specular coherent optimum receiver (SPECCOH) and the resolved multipath Ricean channels specular coherent optimum receiver (SPECCOHR) over two-path unresolved mixed mode Ricean/Rayleigh fading channels. The main difference between these two schemes is that SPECCOH implements the decorrelation operation while SPECCOHR does not. From Fig. 8, it is seen that with FSK(1/2) and DPSK the SPECCOHR scheme yields error floors, which are eliminated by SPECCOH. This shows the importance of the decorrelation operation to handle path unresolvability for specular coherent detection. From Figs. 4–7, the OPT bit error probabilities have no error floor similar to the SPECCOH. This suggests that the decorrelation operation also eliminates error floors for noncoherent detection as well [14].
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DANILO-LEMOINE AND LEIB: SPECULAR COHERENT AND NONCOHERENT OPTIMAL DETECTION
597
TABLE II SNR GAINS (IN DB) OBTAINED BY SPECCOH COMPARED TO OPT AND QDR (
IS IN
dB)
tion. For FSK(1/2) and SDPSK, the performance degradation is larger (0.5–1.8 dB for FSK(1/2) and 1.2–1.5 dB for SDPSK at ) with losses greater than 2 dB at for FSK(1/2). Thus, noncoherent detection could be of interest for FSK(1) and DPSK, while specular phase estimation may be needed for FSK(1/2). Finally, we showed that the specular coherent optimal receiver over unresolved multipath Ricean channels includes a decorrelation stage and then implements a resolved specular coherent optimal decision rule. The decorrelation operation is also present in optimal noncoherent detection structures. The importance of the decorrelation operation in yielding diversity gains and eliminating the error floors was demonstrated for commonly used binary modulation schemes such as FSK and variants of DPSK over unresolved mixed mode Ricean/ Rayleigh fading channels. Fig. 8. Performance of the SPECCOH and SPECCOHR schemes with FSK(1/2) and DPSK signaling over two-path Ricean/Rayleigh channels ( 0:1T , s = 1, and = 10, 15 dB).
=
V. CONCLUSION This paper considered specular coherent and noncoherent optimal detection for unresolved multipath Ricean fading channels, emphasizing receiver structures and single pulse performance. Specular coherent detection needs estimation of the specular term phases, while noncoherent does not. It was shown that for mixed mode Ricean/Rayleigh fading channels the noncoherent likelihood ratio can be obtained by substituting the specular component phasor of the specular coherent likelihood ratio by an estimate. It was also shown that the specular coherent (SPECCOH) and noncoherent (OPT) optimal receivers converge to the same structure at high SNR for equally likely equal energy signals. The performance of the SPECCOH scheme was assessed by calculating its exact bit error probabilities. Due to the nonlinearities of the OPT scheme, its performance was assessed by using asymptotically tight lower and upper bounds to its bit error probabilities. For FSK(1) and, to a lesser extent for DPSK, at sufficiently high SNR, the performance loss due to the lack of specular term phase knowledge is quite small (0.2–0.7 dB loss with FSK(1) and 0.8–1.2 dB loss with DPSK, ), not justifying the use of carrier phase estimaat
APPENDIX A. Proof of Proposition 2.1 , , the term
are independent of . Hence, since is deterministic, converges a.s. to as tends to , and from (7) the infinity. From (10), SPECCOH log-likelihood ratio satisfies a.s. and from (17)
(31) in (16) satisfies a.s.
The function random vector be shown that
(32)
given by (18) depends on the Gaussian . Using the joint pdf of , from (18) it can
where
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 4, APRIL 2002
(33)
and (33) is obtained where and (18)–(20). Hence, from (16) and using (32), the log-likelihood ratio for the OPT scheme satisfies
and
Hence,
and a.s.
(34)
Comparing (31) and (34) completes the proof. Furthermore, since and is measurable in
is measurable in (
) for fixed
( ) for fixed , can be defined as a Lebesgue integral of for almost all sample functions the sample functions of , [31, p. 45]. From (10), hence from (19)
ACKNOWLEDGMENT The authors would like to thank the three anonymous reviewers and the editor for their suggestions and time invested in handling this paper. REFERENCES
where
with defined here as from [18]
for any matrix
, and
with
Since
is independent of
and
, from the Lebesgue dominated convergence theorem a.s. a.s.
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[18] H. Hashemi, “The indoor radio propagation channel,” Proc. IEEE, vol. 81, pp. 943–968, July 1993. [19] G. Turin, “Error probabilities for binary symmetric ideal reception through nonselective slow fading and noise,” Proc. IRE, vol. 46, pp. 1603–1619, Sept. 1958. [20] P. Kam, “On orthogonal signaling over the slow nonselective Ricean fading channel with unknown specular component,” IEEE Trans. Commun., vol. 41, pp. 817–819, June 1993. [21] R. Aiken, “Communication over the discrete-path fading channel,” IEEE Trans. Inform. Theory, vol. IT-13, pp. 346–347, Apr. 1967. , “Error probability for binary signaling through a multipath [22] channel,” Bell Syst. Tech. J., vol. 46, pp. 1601–1631, Sept. 1967. [23] M. Alles and S. Pasupathy, “Channel knowledge and optimal performance for two-wave Rayleigh fading channels,” IEEE Trans. Veh. Technol., vol. 43, pp. 8–20, Feb. 1994. [24] F. Danilo and H. Leib, “Detection techniques for fading multipath channels with unresolved components,” IEEE Trans. Inform. Theory, vol. 44, pp. 2848–2863, Nov. 1998. [25] R. Vaughan and N. Scott, “Super-resolution of pulsed multipath channels for delay spread characterization,” IEEE Trans. Commun., vol. 47, pp. 343–347, Mar. 1999. [26] Z. Kostic´ and G. Pavlovic´, “Resolving subchip-spaced multipath components in CDMA communication systems,” IEEE Trans. Veh. Technol., vol. 48, pp. 1803–1808, Nov. 1999. [27] W. McGee, “Complex Gaussian noise moments,” IEEE Trans. Inform. Theory, vol. IT-17, pp. 149–157, Mar. 1971. [28] W. Rudin, Real and Complex Analysis, 2nd ed. New York: McGrawHill, 1974. [29] H. Poor, An Introduction to Signal Detection and Estimation. New York: Springer-Verlag, 1988. [30] G. Turin, “Some computations of error rates for selectively fading multipath channels,” in Proc. National Electronics Conf., vol. 15, Mar. 1959, pp. 431–440. [31] E. Wong and B. Hajek, Stochastic Processes in Engineering Systems. New York: Springer-Verlag, 1985. [32] J. Doob, Stochastic Processes. New York: Wiley, 1953. [33] W. Kaplan, Advanced Calculus. Reading, MA: Addison-Wesley, 1984. [34] J. Winters, “Differential detection with intersymbol interference and frequency uncertainty,” IEEE Trans. Commun., vol. COM-32, pp. 25–33, Jan. 1984.
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Florence A. Danilo-Lemoine (S’94–M’00) was born in Troyes, France, in 1969. She received the Diplôme d’Ingénieur degree in electrical engineering from the National School of Electronics and Radio-Electricity (E.N.S.E.R.G.), Grenoble, France, in 1992 and the Ph.D. degree in electrical engineering from McGill University, Montreal, PQ, Canada, in 2000. From 1992 to 2000, she was a Teaching and Research Assistant and held several part-time lecturer positions at McGill University. In 1999, she was a faculty lecturer in the Department of Electrical Engineering at McGill University. Since November 2000, she has been an Assistant Professor in the Department of Systems and Computer Engineering at Carleton University, Ottawa, ON, Canada, where she conducts research and teaches undergraduate and graduate courses in telecommunications. Her research interests are in digital telecommunications, detection, and statistical signal processing with emphasis on wireless communication systems. Dr. Danilo-Lemoine was awarded the Canadian Advanced Technology alliance (CATA) telecommunications graduate scholarship in 1999. In 2001, she received a university faculty award from the Natural Sciences and Engineering Research Council of Canada (NSERC).
Harry Leib (M’83–SM’95) was born in 1953. He received the B.Sc. (cum laude) and M.Sc. degrees from the Technion—Israel Institute of Technology, Haifa, Israel, in 1977 and 1984, respectively, and the Ph.D. degree from the University of Toronto, Toronto, ON, Canada, in 1987, all in electrical engineering. From 1977 to 1984, he was with the Israel Ministry of Defense, working in the area of Communication Systems. During his Ph.D. studies at the University of Toronto (1984–1987), he was a Teaching and Research Assistant in the Department of Electrical Engineering, working in the areas of Digital Communications and Signal Processing. After completing his Ph.D. studies, he was with the University of Toronto as a Post-Doctoral Research Associate in Telecommunications (September 1987 to December 1988) and an Assistant Professor (January 1989–August 1989). Since September 1989, he has been with the Department of Electrical and Computer Engineering at McGill University in Montreal, initially as an Assistant Professor, then as an Associate Professor, and now as a Full Professor. He spent part of his sabbatical leave of absence at Bell Northern Research in Ottawa, ON, Canada (September 1995 to February. 1996) working on a CDMA-related project. During the rest of his sabbatical (March 1996 to August 1996), he was a Visiting Professor in the Communications Lab/Institute of Radio Communications at the Helsinki University of Technology, Espoo, Finland, where he taught a condensed course on channel coding and modulation and worked with graduate students in the communications area. At McGill, he teaches under graduate and graduate courses in Communications and directs the research of graduate students. His current research activities are in the areas of Digital Communications, Wireless Communication Systems and Code Division Multiple Access. Dr. Leib is an Editor for Communication and Information Theory for the IEEE TRANSACTIONS OF COMMUNICATIONS and an Associate Editor for the IEEE TRANSACTIONS OF VEHICULAR TECHNOLOGY.
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