RICIAN FADING DISTRIBUTION FOR 40GHZ CHANNELS

Transcription

1 Jan 006 RICIAN FADING DISTRIBUTION FOR 40GHZ CHANNELS.0 Background and Theory Amlitude fading in a general multiath environment may follow different distributions deending recisely on the area covered by measurements that is the resence or absence of a dominating strong comonent, and some other conditions. The model behind Rician fading is similar to that for Rayleigh fading excet that in Rician fading includes a strong dominant comonent, which for instance can be the lineof-sight wave. Rayleigh fading used for aths with no dominant signal is derived from Clarke [] studies. In 968 Clarke has resented theoretical model that statistically describes the behavior of small scale fading in urban multiath environments. In this model the arriving multiath comonents are confined to the horizontal lane. The hases of these multiath comonents are assumed to be uniformly distributed over [0-π]. The Rice distribution roosed by Rice [] in 944 is a fading model is suitable for reresenting the small scale fluctuations of the signal enveloe in a narrowband channel where a LOS exists between the transmitter and the receiver terminals such as in microcellular or a satellite channels. In other words, the effect of a dominant signal arriving with many weaker scatter signals (Rayleigh fading) gives rise to the Rician distribution. As the dominant signal becomes weaker, the comosite signal resembles a noise signal, which has an enveloe that is described by Rayleigh distribution. Thus, the Rician distribution degenerates to a Rayleigh distribution when the dominant comonent fades away. This is shown in the next formulation section. The acceted hidden assumtion behind the Rician model [] is that the numerous individual constituent roagation aths of both the in-hase comonent and the quadrature hase comonent of the received signal are seen as jointly Gaussian random variables, i.e. the distribution of the sum of a large number of random variables is a normal distribution that can comletely described by its mean and variance. This known as the Central-Limit-Theorem and it justifies the use random variables for many engineering alications. In the case of a Rayleigh fading channel, the in-hase and quadrature hase comonents will both have zero-means, however in the case of a Rician fading, the mean value of (at least) one comonent is non-zero due to the strong constant comonent in the received signal.

2 Jan PDF Formulation Let R X+jY be comlex Gaussian with real and imaginary comonents that are indeendent, in the olar coordinate reresentation thats X R cosθ and Y Rsinθ As mentioned formally, now assume that X and Y have equal variances σ. While the imaginary comonent Y has a zero mean m q the real comonent X which will reresent the dominant signal, has a ositive mean m i. The df of the Rician distribution is widely given by [3]: r > 0, s > 0, σ > 0 P r + s r σ RICE ( r) e I 0 () σ rs σ Where r is the amlitude of the received enveloe, and the arameter s is the so called non-centrality arameter comuted from the first moment and is given by: s n ( m i ) i () I 0 is the zero th modified Bessel function of first kind. On the contrary, as 0 s, ( 0) I yields the Rayleigh distribution given by: 0 P r r σ RAYLEIGH ( r) e (3) σ 3.0 Rician K Factor According to [] [3] [4] and elsewhere in the literature, s / dominates the coherent (LOS ath) and σ dominates the incoherent ower (indirect aths). The Rician K-factor is hysically defined as the ratio of signal ower in dominant comonent over the (localmean) scattered ower. Thus it is given in db by: s 0 log0 σ K (4)

3 Jan 006 Knowledge of the Rician K Factor can be useful in determining the signal strengths i.e. SIR, bit error rate, link budget calculation and caacity of a channel among other useful metrics. It also is imortant in describing the distribution of the density functions in a wireless communication channel. Given () we can now write the PDF of the Rician distribution as function of K as follows: r r r K PRICE ( r) e e I 0 K σ σ σ (5) Therefore it can be clearly concluded if there is no dominant signal in the T-R K roagation ath, K0, I ( 0) and e roducing a Rayleigh PDF CDF Formulation Scanning the literature, a number of attemts have been given for aroximating the cumulative distribution function for the Rician distribution. In [3] the Rician CDF takes the shae of: C RICE ( r) e r n K + σ σ K r I n n 0 r K σ (6) Clearly this formula is more difficult to evaluate than the PDF of (3) due to the summation of an infinite number of terms, requiring double or quadrule recision, which referably is avoided in numerical evaluations. Nonetheless [3] has shown a range of CDF evaluated from the above equation. On the other hand, in ractical terms it can be sufficient to increase n to a certain value such as 50 terms that could be sufficient enough to reduce the remaining terms contribution to a negligible level [8]. Other alternatives forms have also been addressed in the literature. In [5] the CDF is exressed as: Where ( a b) r C RICE ( r) Q K, (7) σ Q, is the Marcum s Q function and a & b are non-negative real numbers. Derivation of (7) is given in [6] & [7]. Recently another aroach has also been resented in [9], based on non-linear tabulated coefficients, however it is similar in a way to the revious ones in the ublished literature. 3

4 Jan Extracting the Rician K Factor from Measurements Recall the K factor reresents the ratio of the received LOS comonent to the received scattered comonents. This concet of K factor, first ut forth in [9], is very difficult to measure in a hysically meaningful manner; i.e., by isolating the direct signal from the scattered comonents. From the available literature, a number of aroaches are available for extracting the Rician K factor from a measured time-series data; these are summarized in the following sections. 5. Method of Moments Traditional methods of estimating the Rician K factor from the measured received ower versus time such as the Maximum Likelihood method are relatively imractical and time consuming as we will see in the following sections. Given the Rician distribution it is ossible to obtain its moments from the measured data, and as a result it can then be ossible to estimate a locally constant signal s from magnitude data oints r. The well-known authors in [0] have resented a very significant moment method of estimating the Rician K factor. The method is simle and raid aroach where the K factor is an exact function of the moments estimated from the time series data. The method seems to be reliable and widely used in the technical literature by a number of rofessional eole such has Prof. Theodore Raaort in his work given in [4] as art of the IEEE 80.6 Broadband Wireless Access Working Grou roject. The method simly alies the definition of K factor given in (4) which requires the estimation of the Rician arameters s (mean) and σ (variance). These can be calculated from the mean and standard deviation of the measured data, conversely it is set by finding the first two moments of the measured data. Using the same notation in [4], the first moment is the time averaged amlitude of the measured data, i.e. its theoretical mean value µ. The second moment of interest is σ the local RMS fluctuation of the received signal about µ. It is worth nothing that the standard method for using the local RMS value for normalization is roosed by Clarke and given in [3] and is defined as: RMS r( x ) (8) n i n The definitions of the first and second moments [4] are given by: 4

5 Jan 006 µ s + σ (9) σ 4 σ + s σ (0) Note that σ here the standard deviation differs from σ the nd momentum of the measured data. By using the above definitions and subsequently by simly solving for Rician arameters s and σ, the Rician K factor (in db) is now given by: s K 0 log0 σ 0 log0 µ µ σ µ σ () This leads to the same derivation given in [0], which also contains the validation of the roosed moment method. 5. Grahical Moment Method Another fine method for extracting the K factor is given in []. This method is very much similar to the moment method described formally, and is given by the following exression: E [ r] [ ] E r π K ex 4K + K K ( K + ) I 0 + KI () Where E[ r] is the average amlitude defined by µ ( st moment) earlier in this reort, E is the average of the squared amlitude defined by σ ( nd moment) earlier in this reort. The following figure shows equation (0) grahically. Therefore from the ratio of the st moment to the nd, the K factor can be estimated from the grah in Fig.. and [ r ] 5

6 Jan 006 Equation () st/nd moment K Factor Equation () st/nd moment K Factor (db) Fig. (In the bottom figure K is in db) In [] the K values generated from the above exression are checked otimally by lotting the CDF obtained emirically from the measured data and the theoretical CDF. They also have used the Kolmogorov-Smirnov test [3] in order to best fit the data. As a matter of interest, the Kolmogorov-Smirnov test is a distribution fitting algorithm that rovides robust, but nevertheless, comutationally comlex rocedure which is not easy to imlement online. For curiosity, a K factor mean value of 0.5dB and 9.3dB was 6

7 Jan 006 obtained for there indoor measurements in a university common room and a worksho, resectively. 5.3 Second and Forth Moment Method Another moment-based method very similar to the grahical method is roosed by [], which aims on roviding a closed-form exression for extracting K factor. The method develos from the grahical method by the use of the second and forth moments of the measured data. The motivation behind this method corresonds to finding the exact value for K to solve equation (), which involves a comlex inverting numerical rocedure. ( K ) ( K + ) f,4 (3) K + 4K + Where f, 4 is a function of K and refers to the ratio of the second moment to the forth of the measured data. Clearly, calculating the inverse of the above equation involves finding the roots of a second-order olynomial which can be done in a closed form, and is straightforwardly given by: µ + µ + µ µ µ 4 4 K,4 (4) µ µ 4 Note that notation of the second moment here µ is the same as σ defined formerly. 5.4 Maximum Likelihood Estimation The authors in [3] have constructed an otimum maximum likelihood estimator for the Rician distributed data. The maximum estimator of s is defined as the estimator maximizing L or logl as a function of s. Where L is the joined PDF of a samle of n indeendent observations r i. This is called the likelihood function of the measured samle and is given by: L n i P RICE ( r s), (5) Hence using equation () in the above equation gives: log L n i sri log I o ς n i s σ (6) The maximum likelihood is therefore the global maximum of logl: 7

8 Jan 006 ML ^ arg max ( log L) s s (7) From the above, finding the otimum value for s is not direct and requires finding the maximum of logl which in general will be an exhaustive iterative numerical rocess. It will be only necessary at certain instances [3] to utilize this method in which is out of our scoe. Moreover the authors of [4] have studied the statistical erformances of the momentbased method as a less comlex alternative to the maximum likelihood method. The asymtotic analysis given reveals that both methods are equally efficient, but, however the maximum likelihood is unsuitable for real-world alications. 6.0 True Values for the Rician K Factor From measurements our research grou has conducted and elsewhere in the literature, antennas with narrow beamwidths seem to be more beneficial to obtain and model a high value of K as ossible. This directly imlies isolating the direct ath by limiting the Rayleigh fading that occurs from multiath owers. Now since the K factor can serve as a metric for differentiating Rician from Rayleigh distributions, the question becomes, how large must K be to truly have a Rician distribution? The ossible answer to this question is given from measurements ublished literature, below are a coule of them. Measurements done by Theodore Raaort [4] and his colleagues where conducted at 38GHz utilizing a transmitter horn antenna of 9dBi and a receiver arabolic antenna of 39dBi gain. The T-R searation is 65m and is a artly obstructed (mostly by tress) LOS link. The urose of the measurements was to evaluate the K factor at different rain conditions as art of the IEEE 80.6 Broadband Wireless Access Working Grou roject. The K factor was extracted using the method of moments. According to the authors, the value of the K factor was surrisingly as high as 8dB at the rain rate of 0mm/hr and 7dB at dry conditions. Another set of measurements where conducted at 900MHz by John Davies [5] in 994 at the University of California. One tremendous urose of the measurements was to determine the Rician arameters in vehicle to vehicle RF roagation. The values of K factors obtained were between 6-8dB. Its worth noting out here that in this articular interesting measurements, desite the high values of K, the transmitter and receivers where of same heights. However at some rare instants they exerienced low values of K factors (~.38dB) which does is not of any concern as the author had exlained. As long as a Rician channel actually exists with a dominant comonent of at least two aths; a 8

9 Jan 006 LOS and a reflection of the roadway; these two comonents may interact with each other in such a way to have a sinusoidal tye of frequency that might affect the comutations. Furthermore data resented in [6] and again from the IEEE 80.6 Broadband Wireless Access Working Grou, it was suggested that the median value for the Rician K factor is in the range from 5-0dB According to [7] values of the Rician K factor in indoor and outdoor systems can in general be as high as 5dB. Finally, all the K values excavated from the above resented measurements agree totally with the general rule of thumb given by [8] which quotes that a near LOS condition is associated with a Rician K Factor of 0dB or more. 7.0 Imlementation and Evaluation Now that the arameter extraction methods have been resented, this section imlements the Moment Method and the Grahical Method. The data used in this examle are the industrial estate link time-series measurements conducted on Se 05. Fig. shows the results obtained from the moment method. 5 Industrial Link Tx K Factor (db) Rx Antenna Gain Fig. Moment Method 9

10 Jan 006 Moreover, the Grahical Method roduces the following values for the ratio of st moment over the nd moment given in eq.; , , 0.965, and which from Fig. refer to K factors of 5.44, 6.53, 7.78 and.46db for antenna gains of 0, 5, 0 and 9dBi resectively. In comarison, the values generated from this method and the ones obtained directly from the Moment Method are relatively close, desite the slight differences which can be due to exact lotting errors. Furthermore, to evaluate the accuracy of the K Factors values obtained, a Rician distribution can be fitted to mach the emirical measured PDF. Fig.3 shows two examles for the K Factors of 5.96 and 4.46dB obtained from the Moment Method at a receiver antenna gain of 0 and 9dBi resectively Industrial G R 0dBi Emerical PDF Rician PDF with K5.96dB Industrial G R 9dBi Emrical PDF Rician PDF with K4.6dB P D F 0. P D F Received Power dbm Received Power (dbm) As the wavelength at 40GHz is only 7.5mm, the received signal will exerience great fluctuation over small sace variations of the receiver s terminals; therefore the values of the K factor will strongly deend on the receiver s recise location. This comes to an extreme level when the T-R searation is small and when there is a large number of multiaths at the receiver s vicinity which reresents a tyical microcell environment. The measurement conducted in our camus is one very good examle. It came to my attention, at some articular instant, a very low value for the K Factor has been observed, at around -.75dB, this occurred while moving the 0dBi receiver antenna in a direction erendicular to the link. In this extreme case the emirical PDF distribution of the measured data reasonably follows a Rayleigh distribution and has been fairly fitted with a -.75dB K Factor Rician distribution. On the other side, very high K Factors values occur at some instances, such as 5dB. Exclusively for our measurements, where the receiver is intentionally moved once in the direction of the link and another in erendicular direction; it would seem more aroriate to take the mean K value between the both. 0

11 Jan 006 Hence, for the industrial link, these values become 7.63, 8.4, 8.57 and.3 for antenna gains of 0, 5, 0 and 9dBi resectively. 8.0 Summary From the literature it seems that the Moment Method has been widely used and is the most aroriate technique for extracting the Rician arameters. The justification of the Grahical Moment Method and The Second and Fourth Moment Method use the same basis for there derivations and refers to a number of ublished aers not available to me at the moment. However it has been shown that the Grahical method yields relatively similar values to the Moment method. Additionally, it has been shown that one, very exlicit method for accomlishing the evaluation of these methods is to match various Rician distribution functions to the tabulated results from the measurements. By varying the Rician arameters, the distribution that most closely matches is otimum. 9.0 References [] R. H. Clarke, A statistical theory of mobile-radio recetion, B.S.T.J., vol. 47, , 968. [] Rice, S. O., Mathematical Analysis of Random Noise, Bell Syst. Tech. J. 3, 944 [3] Mobile Radio Communications, Raymond Steel (ed.), Pentech Press, New York, 99. [4] H. Xu, T. S. Raaort, R. J. Boyle, J. H. Schaffner, "38 GHz Wideband Point-to- Multioint Measurements under Different Weather Conditions," IEEE Communications Letters, Vol. 4, No., January 000, [5] George Karagiannidis, Stavros Kotsooulos, and Chris Georgooulos, Outage Probability Analysis for a Rician Signal in L Nakagami Interferers with Arbitrary Parameters, Journal of Communications and Networks, Vol., No., March 999 [6] George K. Karagiannidis and Stavros A. Kotsooulos, On the Distribution of the Weighted Sum of L Indeendent Rician and Nakagami Enveloes in the Presence of AWGN, 00 KICS [7] Jeremiah Hu, Norman C. Beaulieu, Accurate Closed-Form Aroximations to Ricean Sum Distributions and Densities, IEEE Communications Letters, Vol. 9, No., February [8] Raaort, Theodore. Wireless Communications Princiles and Practice, Prentice Hall, New Jersey, , 39-45, [9] Jeremiah Hu, Norman C. Beaulieu, Accurate Closed-Form Aroximations to Ricean Sum Distributions and Densities, IEEE Communications Letters, Vol. 9, No., February [0] L. J. Greenstein, D. G. Michelson, and V. Erceg, Moment-Method Estimation of the Ricean K-Factor, IEEE Communications Letters, Vol. 3, No. 6, June 999

12 Jan 006 [] Jeetinder Purwaha, Arthur Mank, Dušan Matic, Klaus Witrisal and Ramjee Prasad, Wide-Band Channel Measurements at 60GHz in Indoor Environments, Center for Wireless Personal Communications (CEWPC), IRCTR, Delft University of Technology, Mekelweg 4, 68 CD Delft, The Netherlands [] Cihan Teedelenlioglu, Ali Abdi, Georgios B. Giannakis and Mostafa Kaveh, Performance Analysis of Moment-Based Estimators for the K Parameters of the Rice Fading Distribution, De of Elec. And Com. Eng. Unv of Minnesota, 00 IEEE [3] Jan Sijbers, Arnold J. den Dekker, Paul Scheunders and Dirk Van Dyck, Maximum Likelihood estimation of Rician distribution arameters, IEEE Transactions on Medical Imaging. [4] Ali Abdi, Cihan Teedelenlioglu, Mostafa Kaveh, and Georgios Giannakis,, On the Estimation of the K Parameter for the Rice Fading Distribution, IEEE Communications Letters, Vol. 5, No. 3, March 00 [5] John S. Davis, Vehicle to Vehicle RF Proagation Measurements, II 8th Annual Asilomar Conference November, 994 [6] Erceg, V., Michelson, D. G., Ghassemzadeh, S.S., Greenstein, L.J., Rustako, A.J., Guerlain, P.B., Dennison, M.K., Roman, R.S., Barnickel, D.J., Wang, S.C., and R. Miller, A model for the multiath delay rofile of fixed wireless channels,' IEEE JSAC, March 999, [7] George K. Karagiannidis and Stavros A. Kotsooulos, On the Distribution of the Weighted Sum of L Indeendent Rician and Nakagami Enveloes in the Presence of AWGN, 00 KICS [8] D. Gesbert, L. Haumont_E, H. B Olcskei, R. Krishnamoorthy, and A. J. Paulraj, Technologies and Performance For Non-Line-Of-Sight Fixed Broadband Wireless Access Networks, IEEE Comunications Magazine Submitted (Broadband Access Series March 00 [9] K.A. Norton, L.E. Vogler, W.V. Mansfieldand P.J. Short, The robability distribution of the amlitude of a constant vector lus a Rayleigh-distributed vector, Proc. of the IRE, Oct. 955,