A GENERAL METHOD FOR EVALUATING OUTAGE PROBABILITIES USING PADÉ APPROXIMATIONS

Transcription

1 A GENERAL METHOD FOR EVALUATING OUTAGE PROBABILITIES USING PADÉ APPROXIMATIONS Jack W. Stokes, Microsoft Corporation One Microsoft Way, Redmond, WA 9852, James A. Ritcey, University of Washington Dept. of Eectrica Engineering, Box 3525, Seatte, WA 98195, Abstract Many different methods have been derived for evauating the outage probabiity given fading, shadowing, or a combination of fading and shadowing. However, each of these methods imposes restrictions on the parameters or combination of fading and shadowing distributions. In this paper, we deveop a genera method to evauate outage probabiities without restrictions based on Padé approximations (PAs). Resuts show that imposing restrictions on the channe mode, such as the Ricean specuar parameter, can ead to arge errors in the outage probabiity. The PA method is appicabe to both the forward ink and the reverse ink and is extremey efficient due to the use of residues or sadde point integration. Padé approximations aow the use of any fading or shadowing modes given the moments of each distribution. In addition, this new method incudes the effects of compex AWGN. For picoceuar environments operating at ow vaues of carrier SNR, we show that the noise provides a significant contribution to the outage probabiity. 1 Introduction The evauation of outage probabiities (P o ) has been an important and active area of research. Previous works have modeed systems where the channe from the base station (BS) to the mobie station (MS) is distorted by either fading, shadowing, or a combination of fading and shadowing [1]. For the combined case of fading and shadowing, Linnartz [2] requires a channes to exhibit Rayeigh fading. Prasad and Kege [3] restricts the mode to a Ricean faded desired signa and Rayeigh faded interference signas. In Austin and Stüber [4], at east one of the desired or interference signas must exhibit Rayeigh fading. Tjhung, et.a. [5] is vaid assuming that the Ricean factors for each Ricean faded channes are identica. In addition, the mean powers of the shadowing must have identica ognorma statistics. This paper soves the genera outage probabiity probem using Padé approximations (PAs) [6] [7]. The PA method evauates the P o by integration of the moment generating function (MGF) in the compex pane using residues or sadde point integration. Any MGF can be represented by a Padé approximation to the truncated power series of exp(ux). This method does not impose any restrictions on the parameters or combination of Rayeigh and/or Ricean fading channes or the ognorma shadowing. Athough this paper discusses the forward ink, this method is appicabe to the reverse ink as we. In addition, the mode incudes the effects of compex AWGN. We present resuts for channes with mean square-enveope, ognorma shadowing and Rayeigh and Ricean fat fading, but the method is easiy extended to other fading and shadowing modes given the moments of the fading and shadowing distributions. Padé approximations have been used to compute the error probabiities resuting from Direct Sequence/Code Division Mutipe Access (DS/CDMA) communication systems [8]. The PA method is computationay efficient due to the evauation of a product of L+1 rationa functions where L is the number of interference base

2 stations. For the case of Rayeigh fading channes without shadowing, the order of the numerator poynomia is whie the order of the denominator poynomia is 1 for each of the rationa expressions. For the exampes presented in this paper, the maximum orders for the PA numerator and denominator are 3 and 5, respectivey. This paper is organized as foows. In section 2, we present the system mode and derive an expression for the P o based on the PA for each BS s MGF. We provide numerica resuts in section 3. 2 System Mode For the forward ink, the outage probabiity is defined as the probabiity that the ratio of the received power from the reference BS to the sum of the received powers from the interference BSs and compex AWGN is ess than the protection ratio, λ th P R, P o = P r[ L=1 P R, + N < λ th] (1) where N represents the power from compex AWGN. The power received from each BS is P R, = R 2 r β ˆP = R 2 P. (2) In the mode, fading and shadowing are represented by the parameters R and, respectivey. We choose to incude distance attenuation as a deterministic quantity r, which when combined with the transmitted base station power, ˆP, is represented by the power parameter P. The capture probabiity, P c is Thus, the outage probabiity is P c = 1 P o. (3) L P o = P r[s = P R, λ th P R, λ th N < ] = =1 p(s)ds (4) where p(s) is the probabiity density function (PDF) of the test statistic s. Instead of directy integrating (4), the PA method evauates the P o by the compex integration of the moment generating function (MGF) of s. Since P R,, P R,, and N in (4) are independent, the MGF of s is h(u) = E{exp( su)} = exp( su)p(s)ds = h (u)π L =1h ( λ th u)h N ( λ th u) (5) where h (u) and h N (u) represent the MGFs of the th source and the AWGN, respectivey. The outage and capture probabiities can be recovered by the inverse Lapace transform of the MGF P c,o = + u 1 h(u) du (6) C± 2πj { outage probabiity aong C+ 1 capture probabiity aong C where C+(-) is a vertica contour in the compex u-pane that crosses the rea-u axis in the right(eft) haf pane. The evauation of the contour integra aong C produces a negative vaue so P c 1. The MGF of the compex AWGN is h N (u) = E{exp( un)} = 1/(1 + 2σ 2 Nu). (7) For each BS, the MGF is h (u) = E{exp( ur 2 P )} = exp( ur 2P )p(r )p( )dr d (8) A cosed form expression for the integra in (8) does not exist uness the shadowing is negected and the channe exhibits Rayeigh fading. By substituting a Padé approximation (PA) of the truncated series representing the average MGF, we obtain a generaized method for computing the outage probabiity for any channe with fading or shadowing. A [M N /M D ] PA is a rationa approximation with numerator order M N and denominator order M D obtained by the moment matching approach [6]. The PA is produced by expanding the exponentia in (8) as a Tayor series and approximating the infinite summation as a rationa approximation. Substituting gives

3 h (u) = = k= ( up ) k k! k= R 2k p(r )dr ξ k p( )d ( up ) k µ (2k) R k! (9) k= where µ (2k) R are the even moments of fading distribution, and are the moments of shadowing distribution for the th source. Thus, ( up ) k µ (2k) R k! = P (u) + O(u M N +M D +1 ) (1) where the PA, [M N /M D ] P (u), is P (u) = g MN i=1 (u z,i) MD j=1 (u p,j). (11) In (11), we negect the terms of order M N +M D + 1 and higher. The infinite summation in (1) is mutipied by the denominator of (11). Noting that the numerator of (11) ony affects terms up to order M N of the infinite summation, terms of order M N + 1 to M N + M D determine a set of inear equations which yied the denominator coefficients. Once the denominator coefficients are determined, we can sove for the numerator coefficients. 2.1 Fading The evauation of the MGF in (8) requires the even moments of the fading distribution. For Rayeigh Fading, the PDF is p(r ) = R σ 2 R exp( R 2 /2σ 2 R ) (12) and the moments are R = (2σR 2 ) k k 2 Γ(1 + ). (13) 2 Likewise for Ricean fading with specuar parameter, s, the PDF is p(r ) = R σr 2 exp( (R2 + s2 ) 2σR 2 )I ( R s σr 2 ). (14) We compute the even moments for Ricean fading, µ (2k) R, using the efficient recursion in Hestrom [1, p.524]. 2.2 Shadowing For the mean square-enveope, ognorma shadowing, the PDF and moments are and p( ) = 1 2πσ 2 exp( (n( ) m ξ ) 2 /2σ 2 ) = E{exp( k)} = (15) exp( k)p( )d = exp(km ξ +.5k 2 σ 2 ). (16) Some authors mode og norma shadowing using the mean enveope instead of the mean squareenveope [1, p.86]. By simpy using the haf moments from (16) instead of the fu moments, we can mode mean enveope, og norma shadowing. 2.3 Residues The evauation of (8) requires numerica contour integration aong the Bromwich contour. The resuts presented in section 3 have been evauated using sadde point integration (SPI) [8]. However, the efficiency of the PA method can be further improved by evauating the contour integra using residue theory by cosing the contour found from SPI. By incuding the MGF of the noise (7), or by requiring the M N < M D for at east one of the PAs and M N M D for the remaining PAs, the anaytic function representing the overa MGF is guaranteed to have zero contribution at c ± j. Thus, we can cose the contour and evauate the outage probabiity using residues. The outage probabiity can be found from the Cauchy principe vaue 3 Resuts c+j P o = p.v. = im ρ c j c+jρ c jρ u 1 h(u) du 2πj u 1 h(u) du 2πj (17) In this section, we use the PA method to investigate the outage probabiities for modes with either Rayeigh or Ricean fading and with or with-

4 out ognorma shadowing. The carrier signa-tonoise ratio (SNR) is defined to be SNR db = 1 og 1 (E{R 2 }E{ξ }P /σ 2 n). (18) The signa-to-interference ratio (SIR), the ratio of the received power from the reference BS to an individua interference BS, is the reference BS has a specuar parameter of 6 db and the specuar parameter of the five interference channes range from 4 to 8 db in 1 db steps. The resuts from figure 1 show an error of amost an order of magnitude when forced to average the channe statistics of a independent channes. SIR db = 1 og 1 ( (E{R2 }E{ξ }P ) (E{R 2}E{ξ ). (19) }P ) 1 Independent Ricean Specuar Parameters Averaged Ricean Specuar Parameters In a our exampes, the SIR for each of the interference BSs is 1 db. We normaize the fading attenuation by E{R 2 } = 1 for a Rayeigh and Ricean channes. This corresponds to a Ricean factor of K = s 2 /(2σ2 R ) = db. For each of the modes incuding shadowing, the shadowing intensity is 6 db. For the modes which do not incude shadowing, we negect the E{ } terms in (18) and (19). The protection ratio is set to λ th = db for each exampe. The numerator and denominator orders for each of the PAs are given in tabe 1 for the various system modes. The orders are chosen to match Monte Caro simuations. Lognorma Compex PA Order Fading Shadowing AWGN [M N /M D ] Rayeigh Not Inc. Not Inc. [ / 1 ] Rayeigh Not Inc. Incuded [ / 1 ] Rayeigh Incuded Not Inc. [ 2 / 3 ] Rayeigh Incuded Incuded [ 3 / 5 ] Ricean Not Inc. Not Inc. [ 2 / 3 ] Ricean Not Inc. Incuded [ 3 / 4 ] Ricean Incuded Not Inc. [ 3 / 4 ] Ricean Incuded Incuded [ 3 / 5 ] Tabe 1: Padé approximation orders In figure 1, we demonstrate that a genera method for evauating outage probabiities is critica because imposing restrictions on the channe parameters can ead to arge errors. In this figure, we evauate the outage probabiities for L=5 interference base stations with Ricean fading channes and no shadowing. In the P o curve with averaged Ricean specuar parameters, a channes assume a Ricean specuar parameter of 6 db. In the other curve, the channe from Po, Outage Probabiity Carrier SNR (db) Figure 1: P o vs. carrier SNR for Ricean fading channes with and independent and dependent specuar parameters. In figure 2, we compare the outage probabiities for a singe interference BS. A mode combinations of fading, shadowing, and noise are considered. Ceuar radio systems operate at high SNR rates reative to the receiver noise. Thus, the interference from adjacent BSs masks the receiver noise. However, for pico-ceuar environments operating at much ower vaues of SNR, figure 2 iustrates that the noise cannot be negected. In addition, the figure shows that P o is increased by.2 by incuding shadowing with an intensity of 6 db. For the two cases of Rayeigh and Ricean fading without shadowing or noise, P o =.99 (Rayeigh) and P o =.727 (Ricean) which matches the cosed form resuts given in [1, p.13]. Next in figure 3, we evauate each of the modes in the previous exampe but increase the number of interference BSs to L=3. As in figure 2, we see that the compex AWGN has ess of an effect on the P o for three interference sources than for a singe interference source at ower vaues of SNR. Finay, we consider the capture probabiity reative to the number of BSs in figure 4. For this exampe, the carrier SNR is 1 db. As ob-

5 1.9 Ric fad Po, Outage Probabiity Ray fad Ric fad, shad Ray fad, shad Ric fad, noise Ray fad, noise Ric fad, shad, noise Ray fad, shad, noise Pc, Capture Probabiity Rayeigh fading, no shadowing, noise Ricean fading, no shadowing, noise Rayeigh fading, ognorma shadowing, noise Ricean fading, ognorma shadowing, noise Carrier SNR (db) L, Number of Interference Base Stations Figure 2: P o vs. carrier SNR for Rayeigh and Ricean fading channes with and without ognorma shadowing, L=1 interference source. Po, Outage Probabiity Ric fad Ray fad Ric fad, shad Ray fad, shad Ric fad, noise Ray fad, noise Ric fad, shad, noise Ray fad, shad, noise Carrier SNR (db) Figure 3: P o vs. carrier SNR for Rayeigh and Ricean fading channes with and without ognorma shadowing, L=3 interference sources. served in figures 2 and 3, we see that Ricean fading channe without shadowing offers the best performance foowed by Rayeigh fading, Ricean fading with shadowing, and Rayeigh fading with shadowing. However, near L=14 interference BSs, we see that the capture probabiity curves cross with the channe exhibiting Ricean fading ony performing worse than any of the other channes. 4 Concusion The outage probabiities for a cases of Rayeigh and Ricean fading with and without ognorma shadowing are evauated using the Padé approximation method. We approximate the truncated series representing each BS s MGF with a Padé approximation based on the moments of the fad- Figure 4: P c vs. number of interference BSs. ing and shadowing distributions. In addition to providing a new genera method for evauating the P o which does not impose restrictions on the parameters or combinations of the fading and shadowing distributions, the computation time is significanty reduced since the PA method uses residues or sadde point integration. The PA method is vaid for both the forward and reverse inks and does not introduce potentiay arge errors due to channe parameter restrictions. By incuding the effects of compex AWGN, we show that the noise cannot be negected for pico-ceuar environments operating at ow vaues of carrier SNR. References [1] G.L. Stüber, Principes of Mobie Communications, Norwe, Massachusetts:Kuwer Academic Pubishers, [2] J-P.M. Linnartz, Exact Anaysis of the Outage Probabiity in Mutipe-User Mobie Radio, IEEE Trans. Commun., COM-4, No. 1, pp. 2-23, January, [3] R. Prasad and A. Kege, Effects of Ricean Faded and Log-Norma Shadowed Signas on Spectrum Efficiency in Microceuar Radio, IEEE Trans. on Veh. Tech., VT-42, pp , August, [4] M.D. Austin and G.L. Stüber, Exact Cochanne Interference Anaysis for Log-Norma Shadowed Ricean Fading Channes, Eectron. Lett., vo. 3, pp , May, 1994.

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