VIEWS: 7 PAGES: 6 CATEGORY: Education POSTED ON: 1/5/2010
Two-Dimensional Signaling in Ricean Fading with Imperfect Channel Estimation Xiaodai Dong Member, IEEE, and Lei Xiao Student Member, IEEE Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2V4 Email: {xdong, lxiao}@ece.ualberta.ca Tel: 780 492-6989 Fax: 780 492-1811 Abstract— In this paper, the analytical framework reported In this paper, we present a general analysis of the SER of in [1] for calculating the symbol error rate (SER) of two- arbitrary 2-D signaling in Ricean fading with channel estima- dimensional (2-D) signaling in Rayleigh fading with channel tion errors, as an extension to [1]. Our analysis is applicable to estimation errors is further developed to address the more general case of frequency-ﬂat Ricean fading. We show that in the any channel estimation scheme where the estimated fading is presence of channel estimation errors, the SER of arbitrary 2-D jointly Gaussian with the actual fading. The Doppler frequency signaling with polygonal decision regions in Ricean fading can shift in the line-of-sight component is included in the Ricean be expressed as a two-fold proper integral with ﬁnite integration channel modeling and its effect on the error performance of limits, which is suitable for numerical evaluation. Moreover, arbitrary 2-D signaling is easily studied with our general SER this new analysis is general in the sense that it is applicable to any channel estimation scheme where the estimated and the expression. The generality and ﬂexibility of our work make it actual channel gains are jointly complex-Gaussian. The effect of a useful tool in the design and analysis of practical coherent static channel estimation errors and dynamic channel estimation receivers in microcellular and indoor wireless environments. errors introduced by pilot symbol assisted modulation (PSAM) The remainder of this paper is organized as follows. We and minimum mean square error (MMSE) channel estimations ﬁrst discuss the system and channel models in Section II. are studied using the newly derived SER formula. The effect of Doppler frequency shift in the line-of-sight (LOS) component Section III presents the derivation for the PDF’s of the decision of the channel on the error performance is investigated in our statistics and the equivalent noise of the system. The PDF’s analysis. The analytical and numerical results presented in this are then used in Section IV to further develop the symbol error paper provide a useful tool on choosing suitable signaling formats probability of coherent 2-D signaling with channel estimation and optimizing parameters in the communication system design. errors. In Section V, the SER formula is applied to three types of channel estimation errors: static estimation errors, pilot symbol assisted modulation and MMSE estimation errors. I. I NTRODUCTION Finally, in Section VI and Section VII, we discuss some Extensive research has been conducted on the performance numerical results and present our conclusions. analysis of coherent signaling in fading channels. Ready-to- evaluate probability of error expressions can be found in [2] II. S YSTEM M ODEL for arbitrary 2-D signaling over various fading channels. These Consider transmission of a carrier modulated signal s(t) results, however, are acquired with the assumption of perfect over a slowly time-varying frequency ﬂat Ricean fading chan- knowledge of channel fading gains, which is not possible in nel. The channel gain is assumed to be almost constant during practical communication systems. In [1], a new method of one symbol interval T , but changes from symbol to symbol. analysis was presented for determining the SER of arbitrary 2- Further assuming perfect carrier frequency synchronization D signaling in Rayleigh fading with channel estimation errors. and symbol timing in the receiver, as well as intersymbol- Performance studies on Ricean fading channels, however, have interference free transmission, the sampled output of the been less reported in the literature and are largely conﬁned receiver matched ﬁlter at t = i T can be expressed as to perfect coherent and non-coherent detection [2], [3]. In [3], the SER’s of MDPSK, coherent MPSK and noncoherent 2Es KR j2πfp (iT ) i zi = e + αi si + n i MFSK were presented for Ricean fading channels. In [4], 1 + KR performance analysis of linearly modulated signals was carried g = (¯i + αi )si + ni out in Ricean fading with minimum mean square estimation = g i si + n i (1) errors. And in [5], the probability density functions (PDF’s) to describe the behavior of data demodulated with imperfect where the transmitted signal si takes value from a set of M channel estimation were derived and the SER and BER of symbols in a constellation normalized to yield unit average 16 rectangular-QAM were presented. Their study, however, energy, and gi is the multiplicative complex fading random is only concerned with the decision statistics and cannot be variable with mean gi which represents the line-of-sight (LOS) ¯ applied to arbitrary 2-D signaling easily. To the best of the component or the specular component. The zero-mean com- authors’ knowledge, there is no general result on the SER plex Gaussian random variable (R.V.) αi denotes the diffused performance of arbitrary 2-D signaling over Ricean fading or multipath component of the channel fading with variance channel with channel estimation errors. 2 E[|αi | ] = Es /(1 + KR ), where Es = 2 E[|gi | ] is the 1 2 1 2 average energy of the received signal since the transmitted Y respectively. Matrix L is the Hermitian covariance matrix signal si is normalized to have unit average energy, and KR = given by |¯i |2 g E[|αi |2 ] denotes the Ricean K factor which by deﬁnition is mxx mxy L= (7) the ratio of the LOS power to the power in the multipath m∗xy myy component of the fading. The additive white Gaussian noise n i ¯ ¯ where mxx = E[|X − X|2 ], myy = E[|Y − Y |2 ] and mxy = has zero-mean and variance N0 in both its real and imaginary ¯ ¯ E[(X − X)(Y − Y )∗ ] are the second central moments of the parts, and is assumed to be independent of the channel fading nonzero-mean complex Gaussian R.V.s X and Y . Eqn. (6) can gi . Parameter fp is the Doppler frequency shift of the LOS i be expressed in an alternative form as component at the i-th symbol. It is reasonable to assume the LOS Doppler shift fp approximately constant over several data i ¯ fX,Y (X, Y ) = π −2 |L|−1 exp{−|L|−1 (myy |X − X|2 frames, and therefore we drop the superscript notation i to +mxx |Y − Y ¯ ¯ ¯ |2 − 2 [m∗ (X − X)(Y − Y )∗ ])}. (8) xy arrive at fp . Furthermore, the LOS component and the diffused component have the following temporal property And in polar coordinates X = rx ejθx , Y = ry ejθy , the PDF (8) can be further expressed as gi /¯k ¯ g = ej2πfp (i−k)T (2a) 1 Es |L|−1 E[αi α∗ ] = J0 (2πfD |i − k|T ) (2b) f (rx , θx , ry , θy ) = rx ry exp(−P ) exp − |L|−1 2 k 1 + KR π2 × myy rx − m∗ rx ry ej(θx −θy ) − mxy rx ry e−j(θx −θy ) 2 where fD is the maximum Doppler frequency and J0 (·) is xy the zero-th order Bessel function of the ﬁrst kind. The LOS ¯ ¯ +mxx ry − 2 (myy X ∗ rx ejθx − m∗ Y ∗ rx ejθx 2 xy Doppler shift fp is related to the maximum Doppler frequency ¯ ¯ −mxy X ∗ ry ejθy + mxx Y ∗ ry ejθy ) (9a) fD by fp = fD cos θ0 where θ0 is the angle of arrival of the LOS component. where In coherent detection, the receiver generates a channel gain ¯ ¯ P = |L|−1 myy |X|2 + mxx |Y |2 estimation gi , and the decision variable D is given by ˆ ∗ ¯ ¯∗ ¯ ¯ zi gi si + n −mxy X Y − mxy X ∗ Y . (9b) D= = si + ( − si ) = si + t, (3) ˆ gi ˆ gi To obtain the PDF of the decision variable D = rd ejθd , we and then deﬁne another variable F = rf ejθf = Y . With the relation gi si + n rd = rx , θd = θx − θy , rf = ry , θf = θy , it is easy to verify t= − si = D − s i (4) ry ˆ gi that the Jacobian is |J| = r1 = r1 . Therefore, the joint PDF y f where t is the equivalent noise term that includes the effects of of D and F is given by both the channel estimation error and the AWGN. In most of the practical channel estimation schemes such as the prevailing |L|−1 −P 3 fD,F (rd , θd , rf , θf ) =e rd rf PSAM and MMSE, the channel estimate gi and the actual ˆ π2 2 channel gain gi are jointly complex Gaussian. × exp − Brf + 2rA rf cos(θf + θA ) (10a) We can see from (3) that it is the noise term t that causes where a decision error. The SER can be obtained by integrating the PDF of t outside si ’s decision region. In Section III, we present A = rA ejθA = ard ejθd + b (10b) our method to ﬁnd the PDF of t. ¯ ¯ a = ra ejθa = |L|−1 (myy X ∗ − m∗ Y ∗ ) (10c) xy III. T HE PDF OF THE E QUIVALENT N OISE T ERM b = rb e = |L| (mxx Y − mxy X ∗ ) jθb −1 ¯ ∗ ¯ (10d) 2 2 2 2 Recognizing the fact that the decision variable D in (3) is rA = ra rd + rb + 2ra rb rd cos(θa − θb + θd ) (10e) the ratio of two complex-valued nonzero-mean Gaussian R.V.s, B = |L|−1 (myy rd − 2 {m∗ ejθd }rd + mxx ). 2 xy (10f) we rewrite variable D in a general form as Then, we can integrate (10a) with respect to rf and θf to X obtain the PDF of the complex R.V. D. That is, D= (5) Y where X and Y are correlated nonzero-mean complex Gaus- e−P rd rA + B rA 2 2 fD (rd , θd ) = eB. (11) sian variables. The joint PDF of X and Y can be obtained via π|L| B3 slightly modifying the joint PDF of two zero-mean Gaussian Via a change of variables, it can be shown that (11) is R.V.s given by Wooding [6] as equivalent to [5, eqn. (15)], which was derived using a different fX,Y (X, Y ) = π −2 |L|−1 exp (−VH L−1 V) (6) method. Note that it is difﬁcult to apply the fading estimate model given by [5, eqn. (1)] to PSAM in Ricean fading. ¯ ¯ where V = [(X − X)∗ (Y − Y )∗ ]H with ∗ representing Our expression for the decision variable D is further devel- complex conjugation and A the Hermitian transpose of H oped to obtain the PDF of the combined noise term t, which ¯ ¯ matrix A. X and Y are the complex mean of R.V.s X and is required for arbitrary 2-D signaling. Recall that t and D have the relation t = rejθ = D − si given by (4). After some with well-behaved integrand that can be evaluated efﬁciently mathematical manipulations, we have ¯ ¯ with numerical methods. With X = 0 and Y = 0 as a special case representing the Rayleigh fading scenario, (14) can be e−P r G + H G ft (r, θ) = eH (12a) further simpliﬁed to yield the same single integral expression π|L| H 3 as given in [1]. where The symbol error probability of an arbitrary coherent two dimensional constellation with channel estimation errors is the G(r) 2 = ra r2 + 2f r + g (12b) 2 weighted sum of the probability of error in each erroneous H(r) = hr + 2cr + d (12c) subregion [2], i.e., 2 ra ¯ ¯ = |L|−2 [m2 |X|2 + |mxy |2 |Y |2 yy N −2myy {mxy X ¯ ¯ Y ∗ }] (12d) Pe = wj Pe,j (15) f = ra {ejθ s∗ } + ra rb cos(θa − θb + θ) 2 i (12e) j=1 g = ra |si |2 + rb + 2ra rb {si ej(θa −θb ) } 2 2 (12f) where N is the total number of erroneous subregions and wj h = myy |L|−1 (12g) is the weighting coefﬁcient related to the a priori probability of each symbol [2]. Eqns. (14) and (15) are a new general c = |L|−1 (myy {s∗ ejθ } − {m∗ ejθ }) i xy (12h) SER expression for arbitrary 2-D signaling in Ricean fading d = |L|−1 (myy |si |2 − 2 {si m∗ } + mxx ). xy (12i) with channel estimation errors. The PDF of the equivalent noise term t given by (12) is new. V. A PPLICATIONS IV. S YMBOL E RROR P ROBABILITY OF 2-D S IGNALING To apply our general analysis to various channel estimation methods, the only effort needed is to specify second central To provide a general analysis applicable to arbitrary polyg- moments and the means of R.V.s X = gi si + ni and Y = gi , ˆ onal 2-D constellations, we adopt the idea of Craig’s method which describe the statistical characteristics of the received (see [7] for details in AWGN, and [2] for analysis in fading signal and the channel estimation. Now, we analyze three scenarios). Instead of dealing with the corrupted received different kinds of channel estimation errors employing the signal, we work directly with the noise term t superimposed derived formula in Section IV. on the transmitted signal. Therefore, the probability of error can be calculated exactly by integrating the PDF of t over A. Static Estimation Error the erroneous decision region. Furthermore, the erroneous By the term static estimation error, we mean the estimated decision regions can be partitioned into subregions so that each channel gain gi is related to actual channel gain gi by a ˆ subregion can be conveniently expressed in polar coordinates constant factor. Thus, the estimated fading gi = qejφ is ˆ gi [2]. With the PDF of t available, the probability of error for also Gaussian with nonzero mean, where q and φ denote the the j-th erroneous subregion of symbol si is given by estimation error in amplitude and phase, respectively. In this θ2 ,j ∞ case, the central moments and means are given by Pe,j = ft (r, θ)drdθ, θ1 ,j R(θ) 2Es |si |2 −P θ2,j ∞ mxx = + 2N0 (16a) e G 1 G 1 + KR = r + 2 e H drdθ (13a) 2Es π|L| θ1,j R(θ) H3 H myy = (16b) (1 + KR )q 2 where G, H and P are given in (12) and (9b) and R(θ) has 2Es si the form mxy = ejφ (16c) xj sin ψj (1 + KR )q R(θ) = . (13b) sin(θ − θ1,j + ψj ) 2Es KR ¯ X = si (16d) Parameters xj ’s, θ1,j ’s, θ2,j ’s and ψj ’s depend on the geometry 1 + KR of the j-th subregion and they are deﬁned in [2]. ¯ e−jφ 2Es KR Y = . (16e) Further simpliﬁcation of (13a) is not easy. However, we can q 1 + KR use a simple change of variable u = 1 to transform the semi- r When the static error is small enough so that every signal inﬁnite integral to an integral with ﬁnite integration limits. point is still inside its own decision boundary, the analysis in That is, [2] with a rotated and/or scaled decision boundary yields a 1 1 −P θ2,j R(θ) 2 ra + 2f u + gu2 solution in the form of a sum of one-fold integral. However, Pe,j = e u Craig’s method is no longer applicable for situations where π|L| θ1,j 0 (h + 2cu + du2 )3 ra +2f u+gu2 2 the signal point is outside its decision boundary, which can be 1 + e h+2cu+du2 dudθ. (14) the case for severe static errors. The approach in this paper, (h + 2cu + du2 )2 given by (12)-(15) and (16), is more general since we do not where ra , f , g, h, c and d have been deﬁned in (12). The put any constraint on the severity of the static error in the probability of error (14) is now a two-fold proper integral channel estimation. B. Pilot Symbol Assisted Modulation And the temporal property in (2) should be modiﬁed as In the transmitter of a PSAM system, known symbols gl = g i ¯ ¯ (21a) termed pilot symbols are inserted periodically into the data 1 Es stream such that one frame of L symbols begins with a E[αl α∗ ] = i J0 (2πfD |l − i|T ) 2 1 + KR pilot symbol followed by L − 1 data symbols. The period of pilot symbols should satisfy the Nyquist sampling theorem to × e−j2πfp (l−i)T . (21b) ensure channel state information at non-pilot positions can be Accordingly, some quantities in (19) need to be modiﬁed as accurately estimated. That is L ≤ 2fD T . The receiver, with the 1 K2 priori knowledge of the position and value of pilot symbols, 2Es hk,l sl J0 (2πfD |kL − l|T ) mxy = can obtain a channel estimation as 1 + KR k=−K1 zk,0 nkp ˆk gp = k = gp + (17) × ej2πfp (kL−l)T (22a) pk pk K2 ¯ 2Es KR where gp and gp are the channel fading and fading estimate k ˆk Y = hk,l (22b) 1 + KR at the pilot position in the k-th frame, respectively. The ﬁrst k=−K1 received symbol of the k-th frame is denoted by zk,0 and pk ∗ 2Es J0 (2πfD |k − n|LT ) Ckn = E[gpk gpn ] = is the pilot symbol transmitted. The channel gain of the l-th 1 + KR data symbol in a frame is estimated by linear interpolation of × e −j2πfp (k−n)LT . (22c) channel gains of K nearby pilot symbols K2 It will be shown in Section VI through numerical evaluation ˆ gl = ˆk hk,l gp (18) that the LOS tracking method leads to signiﬁcantly degraded k=−K1 error performance compared to perfect carrier synchroniza- tion, unless some Doppler shift compensation mechanism is where K1 = (K − 1)/2 , K2 = K/2 and hk,l ’s are the implemented in the interpolation ﬁlter. Note that the moments coefﬁcients of the interpolator. Since gl is a linear combination ˆ and means in (19) or (22) as well as the SER’s are different of jointly Gaussian R.V.s gp 1 · · · gp 2 , it is also a Gaussian ˆ−K ˆK from symbol to symbol in a frame. The overall SER, therefore, R.V.. Several interpolators, including Cavers’ optimum Weiner needs to be averaged over l, where l = 1, . . . , L − 1. interpolator [8] and sinc interpolator [9], have been proposed in the literature. Our analysis applies to any linear interpolating C. Minimum Mean Square Error Estimator schemes. For an MMSE estimator, the error e = gi − gi and the ˆ Assume perfect carrier frequency synchronization, the sec- estimation gi are orthogonal (and hence independent, since ˆ ond central moments and means needed for calculating the they are Gaussian) and the estimation is unbiased [4]. That is, SER of the l-th symbol in a frame of size L are given by E[gi ] = E[ˆi ] and E[eˆi ] = 0. Therefore, g g∗ 2Es mxx = |sl |2 + 2N0 (19a) 2Es |si |2 1 + KR mxx = + 2N0 (23a) 2N0 1 + KR myy = Hl CHT + l Hl 2 (19b) 2Es |pk |2 myy = 2 − 2σe (23b) K2 1 + KR 2Es hk,l sl J0 (2πfD |kL − l|T ) mxy = myy si (23c) mxy = (19c) 1 + KR 2Es KR k=−K1 ¯ X = si (23d) 2Es KR 1 + KR ¯ X = sl (19d) 1 + KR ¯ 2Es KR Y = (23e) 2Es KR K2 1 + KR ¯ Y = hk,l ej2πfp (kL−l)T (19e) 1 + KR where σe = E[|gi − gi |2 ] denotes the energy of the estimation 2 ˆ k=−K1 error, which is a parameter indicating the accuracy of the esti- Hl = [h−K1 ,l , . . . , hK2 ,l ] (19f) mation. The orthogonality between e and gi can be exploited ˆ ∗ 2Es J0 (2πfD |k − n|LT ) to further simplify eqn. (12) as Ckn = E[gpk gpn ] = (19g) 1 + KR 2 G(r) = rb (24a) where Ckn is the element with index (n, k) in covariance H(r) = hr2 + d, (24b) matrix C. In practice, when an automatic frequency control (AFC) which make it possible to solve the inner integral in eqn. (14) loop is used to track the line-of-sight component, the received analytically as signal sample is written as ¯ |Y |2 ¯ |Y |2 myy V e − myy θ2,j V exp m3 R2 (θ)+V 2Es KR M Pe,j M SE = yy dθ (25) zl = ( + αl e−j2πfp lT )sl + nl . (20) 2π myy R2 (θ) + V 1 + KR θ1,j where V = mxx − myy |si |2 . The SER of arbitrary 2-D Fig. 4 shows how the SER of PSAM 16 rectangular-QAM signaling with MMSE channel estimation can be calculated depends on K in Ricean fading. We ﬁnd that K = 25 is using (25). Our result reconciles with the previous analysis sufﬁcient when the SNR is less than 30dB. For the case where reported in [4]. SNR is 40dB or 50dB, however, a larger K value of 30 or 32, respectively, is required. VI. N UMERICAL R ESULTS Fig. 5 depicts the effect of the LOS Doppler frequency In this section, we present some numerical results for shift on the error performance of pilot symbol assisted 2- the symbol error probabilities of 16 star-QAM [2] and 16 D modulation, using 16 rectangular-QAM as an example. rectangular-QAM in Ricean fading with channel estimation One interesting observation is that the LOS Doppler shift errors. We adopt KR = 7dB in our numerical calculation, has minimal effect on the symbol error probability in the which is a typical value of KR for practical microcellular case of perfect carrier frequency synchronization for low to channels. medium range of SNR values. When the synchronization is Fig. 1 indicates that the performance of 16 rectangular- not ideal, i.e. the receiver is locked at the frequency of the QAM is degraded more severely than that of 16 star-QAM LOS component, a substantial performance degradation is in the presence of static estimation errors. Furthermore, the observed. This degradation is caused by the spectral shift of rectangular-QAM is no longer usable when the actual channel the multipath component. In the ideal synchronization case, the gain is 2dB smaller in amplitude than the estimated channel power spectral density (PSD) of the fading process is band- gain and differs 10◦ in phase. This is because the amount limited to [−fD , fD ]. With an AFC loop locked to the LOS of estimation error has displaced some signal points of 16 component with a Doppler shift of fp , the non-zero region rectangular-QAM outside their decision boundaries. Although of the fading process PSD is now [−fD − fp , fD − fp ] and not shown here, our study ﬁnds that in the low to medium SNR −fD ≤ fp ≤ fD , which makes |f |max = fD + |fp |. This shift range, both constellations are more sensitive to static channel of PSD makes our choice of L according to L ≤ 2fD T no 1 estimation errors in Ricean fading than in Rayleigh fading. longer sufﬁcient in providing zero mean square error recovery The effect of dynamic estimation errors on the performance from the sinc interpolation. If an accurate estimate of fp is of 2-D signaling is investigated using the pilot symbol assisted available to the receiver, we can compensate this undesirable modulation scheme. Although our analysis is valid for any lin- shift by either adjusting the pass-band of the interpolation ﬁlter ear interpolator, we choose the sinc interpolator [9] because of to [−fD − fp , fD − fp ] or shifting the estimation from pilots its simple implementation and close-to-optimum performance. back to [−fD , fD ]. If the value of fp is not available to the For a sinc interpolator, we have receiver, we observe that |f |max = fD + |fp | ≤ 2fD . So the design criteria for the frame size L should be L ≤ 4fD T in 1 l this case, which requires double the number of pilots inserted hk,l = sinc − k). (26) L and thus leads to a reduced channel throughput. To smooth the abrupt truncation in the time domain, a window VII. C ONCLUSION should be applied. Our numerical study shows that Hamming window gives better performance than rectangular window, In this paper, we have presented a new, general analysis Blackman window and Hanning window. All the numerical of the symbol error probability of arbitrary 2-D signaling in results for PSAM in this paper, therefore, are evaluated using Ricean fading with imperfect channel estimation. The SER the sinc interpolator with Hamming window. of arbitrary 2-D signaling in Ricean fading with channel In Fig. 2, we plot the SER’s of PSAM 16 rectangular-QAM estimation errors has been expressed as a two-fold proper and 16 star-QAM in Ricean fading with KR = 7dB. The frame integral with ﬁnite integration limits, which can be evaluated size L is ﬁxed as 15 for a fading rate of fD T = 0.03. Our efﬁciently. Applications of the newly derived formula to numerical results agree with the simulations. In the high SNR three types of channel estimation errors have been studied. region, the error ﬂoor of 16 star-QAM is less signiﬁcant than Numerical examples of 16 rectangular-QAM and 16 star-QAM that of 16 rectangular-QAM. Another advantage of 16 star- with static and dynamic estimation errors in Ricean fading QAM is that the ring ratio of the constellation is adjustable have been presented. The effect of Doppler frequency shift on for systems operating in different SNR’s. the error performance has also been studied and it has been Fig. 3 plots the optimum ring ratios of various 16-ary shown that we need to double the number of pilot symbols circular constellations in terms of minimizing the SER of to achieve acceptable performance when a receiver is locked PSAM in Ricean fading. The signal space diagrams of these to the LOS component and the Doppler frequency shift is constellations were presented in [2]. We ﬁnd that the optimum unknown. It has been shown that key parameters in PSAM ring ratios for different constellations roughly have the same and the optimum ring ratios for circular 2-D signaling can be tendency as SNR increases and the more signaling points on easily and accurately determined by our analytical results. the outer ring, the larger the optimum ring ratio. R EFERENCES An important parameter in PSAM is the interpolation order [1] X. Dong and N.C. Beaulieu, “SER of two-dimensional signalings in K. While a small K value worsens performance, increasing Rayleigh fading with channel estimation error”, Proc. ICC’03, pp. 2763- K blindly implies huge buffer size and long detection delay. 2767, May 2003. 0 10 3.8 16 Star−QAM −2 3.6 10 Rotated (8,8) (5,11) Symbol Error Rate 3.4 (1,5,10) Optimum Ring Ratio 3.2 (4,12) −4 10 3 Perfect, 16 rect.−QAM q=2dB, φ=10° 16 rect.−QAM 2.8 −6 10 q=−2dB, φ=10° 16 rect.−QAM 2.6 Perfect, 16 star−QAM q=2dB, φ=10° 16 star−QAM 2.4 q=−2dB, φ=10° 16 star−QAM −8 10 2.2 0 10 20 30 40 50 Average E /N (dB) b 0 2 0 10 20 30 40 50 60 Average E /N (dB) b 0 Fig. 1. SER of 16 rectangular-QAM and 16 star-QAM in a Ricean fading channel (KR = 7dB) with static estimation error q = ±2dB in amplitude and φ = 10◦ in phase. Fig. 3. Optimum ring ratios of PSAM 16-ary constellations in Ricean fading with KR = 7dB, fD T = 0.03, fp T = 0, L = 15, K = 36 and Hamming window. 0 10 16 rect.−QAM, PSAM 16 star−QAM, PSAM −2 16 star−QAM, perfect 10 16 rect.−QAM, perfect 0 10 Symbol Error Rate simulation, PSAM −4 10 −2 10 Symbol Error Rate −6 10 −4 10 E /N = 10dB b 0 0 10 20 30 40 50 60 70 Eb/N0= 20dB Average Eb/N0 (dB) Eb/N0= 30dB −6 10 E /N = 40dB b 0 Fig. 2. SER of PSAM 16 rectangular-QAM and 16 star-QAM in Ricean E /N = 50dB b 0 fading with KR = 7dB, fD T = 0.03, L = 15 and K = 30. The ring ratio 5 10 15 20 25 30 35 40 for 16 star-QAM is 2.40. Interpolation order, K Fig. 4. The effect of K on the SER of 16 rectangular-QAM in Ricean fading [2] X. Dong, N.C. Beaulieu and P.H. Wittke, “Signal constellations for with KR = 7dB, fD T = 0.03 and L = 15. fading channels”, IEEE Trans. Commun., pp. 703-714, May 1999. [3] J. Sun and I. S. 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(25.5.1-25.5.5), 1991. p fpT=0.005, LOS Tracking [8] J.K. Cavers, “An Analysis of Pilot Symbol Assisted Modulation for fpT=0.01, LOS Tracking Rayleigh Fading Channels”, IEEE Trans. VT., Vol. 40, pp. 686-693, −5 Nov. 1991. 10 0 5 10 15 20 25 30 35 40 [9] Y.S. Kim, C.J. Kim, G.Y. Jeong, Y.J. Bang, H.K. Park and S.S. Choi, Average Eb/No (dB) “New Rayleigh Fading Channel Estimator Based on PSAM Channel Sounding Technique”, Proc. ICC’97, pp. 1518-1520, June 1997. Fig. 5. The effect of Doppler shift on the SER of PSAM 16 rectangular- QAM in Ricean fading with KR = 7dB, fD T = 0.03, L = 15, K = 36 and Hamming window.