Introduction Traditionally, studying outage or cellular systes has been based on the signal-to-intererence ratio (SIR) dropping below a required thres

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1 Miniu Duration Outages in Rayleigh Fading Channels Jie Lai and Narayan B. Mandaya WINLAB, Rutgers University 73 Brett Rd., Piscataway, NJ Eail: Abstract Miniu duration outages have been introduced in [, ] or lognoral shadow ading where the durations o signal ades were considered in evaluating outages. In this paper we develop and analyze such iniu duration outages or channels susceptible to Rayleigh ading. The results show that under typical Doppler requencies the outages due to Rayleigh ading are ore likely to cause rae or packet errors rather than call dropping due to the short tie scales in eect. Keywords: Miniu Duration Outages, Fading Channels, Level Crossings, Perorance Analysis This paper was presented in part at the 3st Annual Conerence on Inoration Sciences and Systes (CISS'97), Johns Hopkins University, Baltiore, MD, March, 997. This work is unded in part by DARPA under grant N66-96-C-853.

2 Introduction Traditionally, studying outage or cellular systes has been based on the signal-to-intererence ratio (SIR) dropping below a required threshold. In real cellular environents, it is not the instantaneous drop o SIR below the threshold that deterines outage. It is, in act, the duration o tie that the SIR stays below a threshold that deterines outage or cellular systes. Moreover, the static analysis o outage precludes the tie correlation in the signals which is iportant in real systes owing to obility, ading and power control. The notion o \iniu duration outage" was introduced in [, ] as the SIR staying below a threshold longer than a certain \iniu duration". The above notion is otivated by the act that a very short interval o signal dropping below a threshold results in little or no ipairent, while a longer duration ay cause quality degradation, reduced intelligibility, or even call dropping. Siilarly, in data counications, burst error correcting codes ay deterine these iniu durations which ay result in packet errors. In [, ], the ocus o tie-dependent perorance was or lognoral shadow ading situations. Perorance easures such as the average duration o outage, the requency o outage, and the probability o outage and the eect o on these quantities was analyzed. In [3], the sae ethodology was used or analysis o slow shadow ading with reception diversity. In this paper, we extend the iniu duration outage concept to ast Rayleigh ading. Rayleigh ading generally characterizes the rapid signal level uctuations in a cellular syste. Usually, the study o outage due to Rayleigh ading is based on the threshold crossing rate, the average duration o ade, and the probability o ade. Instead o directly assuing constant intererence as in [{3] we rst evaluate the approxiation o using a nonading noise type signal to represent the joint intererence coposed o ultiple utually independent Rayleigh signals under the condition o large nuber o intererers and high ade argins. Using the concept o iniu duration outage, we extend asyptotic results on level crossings ro the work by S.O. Rice [4] and analyze the average duration o outage, the requency o outage, and the probability o outage as unctions o the iniu duration, the ade argin F and Doppler requency. The paper is organized as ollows. In section, we present the general orulation o tie-

3 dependent perorance easures or Rayleigh ading channels and justiy the siplication o using a nonading signal to represent the joint intererence o ultiple utually independent Rayleigh signals with large intererer nuber and high ade argins. The analysis based on asyptotic level crossings is presented in 3. The nuerical results are presented in section 4, with discussions and conclusions in section 5. Proble Forulation The iniu duration outage and its perorance easures are dened in general as ollows [, ]. Denition. An outage event is dened as SIR(t) going below a required threshold z and staying below at least seconds. In atheatical or, SIR(t) < z; 8 t [ a ; b ] and ( b? a ) : () Denition. Frequency o outage is dened as the average rate o outage events out = li t! T otal nuber o outages in [; t] : () t Denition 3. Probability o outage is the raction o tie that the syste is in outage P out = li t! T otal outage tie in [; t] : (3) t Now we exaine the signal and intererence in Rayleigh ading channels. In a typical cellular syste, the received signal shows rapid uctuations around relatively slow variation o the ean level. This received signal can be well characterized by the ast Rayleigh signal around the lognorally distributed ean level [5,6]. Moreover, each interering signal also coprises a Rayleigh and lognoral superiposed ading signal. In this paper, we assue that the slowly varying shadow ading and distance loss are perectly copensated through power control whereas the ast uctuations o signal and intererence due to ultipath ading reain uncopensated. Thus we assue that at the receiver, the desired signal or the user o interest is a Rayleigh signal with constant ean power and the intererence signal is the su o ultiple utually independent Rayleigh signals each with constant ean power. 3

4 I the intererence is coposed o n utually independent Rayleigh signals with identical ean power p, the probability density unction (pd) o the intererence power p t can be expressed as a gaa unction [6] pt (p t jn) = p n? t (n? )!p exp(?p t ): (4) n p The pd o the desired signal power in Rayleigh ading is exponential ps (p s ) = p s exp(? p s p s ): (5) The cuulative distribution unction (CDF) o SIR can be calculated as P (SIR p s p t z) = Z ( Z zpt exp(? p s )dp s ) p s p s p n? t (n? )!p exp(?p t n p )dp t =? ( + z p p s )?n : (6) As the ean SIR being constant, p s p t = ps = const =, then np n! li P (SIR z) =? exp(? z ): (7) This result is the sae as we consider the intererence as a nonading noise type signal with ean power p t = np P (SIR p s p t z) = P (p s < z p t ) =? exp(? z ): (8) This iplies that we could use a nonading noise type signal to approxiate the joint intererence o ultiple independent Rayleigh signals when the nuber o intererers is large (n >> ). As known ro the central liit theore, the intererence tends toward band-liited Gaussian noise when the nuber o intererers increases. For the general case o intererence coposed o n Rayleigh signals with arbitrary dierent ean powers p k ; k = ; ; :::; n, the pd o the intererence power can be approxiated by the gaa distribution [6] pt (p t j) = p? t?()p exp(?p t p ) (9) 4

5 with p = P n k= p k P n k= p k () and = [P n k= p k ] P n k= p : () k Thus the intererence power p t approxiated by the gaa pd has the sae ean and standard deviation as the exact intererence power p t. Note that now is a real nuber with n. As in the case o equal ean power or each intererer, when the nuber o intererers gets larger (n; >> ), the joint intererence power approaches a constant. Furtherore, when we look at a real cellular syste, it should be designed to operate at a low outage probability. This iplies that the ade argin is high. The intererence power is uch lower than the desired signal power ( p t << p s ). The instantaneous SIR will drop below the required threshold only when the desired signal experiences a deep ade. Fluctuations o the intererence has a less eect on the SIR than uctuations in the desired signal. Hence, in the case o relatively high ade argin and large nuber o intererers (as or a heavily loaded syste), the joint intererence power can be well approxiated by a nonading noise type signal with ean power p t np. Siilar conclusions can also be ound in [6, 7]. Thus we can sipliy the tie dependent SIR outage proble as the level crossing proble o a single desired Rayleigh signal. The threshold level or the desired Rayleigh signal about which the crossings are considered is the value which is F db (F is the ade argin) down ro the desired Rayleigh signal ean power. Here the ade argin (in db) is dened as the dierence between the ean SIR and the SIR threshold [6]. In the siulation results, we will show the inuence o n (the nuber o intererers) on the accuracy o the analysis. 3 Analysis Based on Asyptotic Level Crossings As entioned in last section, we are interested in the situations where the SIR drops below the required threshold only when the desired signal has a deep ade, that is, the high ade argin case. S.O. Rice [4] gave an asyptotic result or the ade duration distribution unction o deep ades by a Rayleigh signal r(t). It is shown that when r(t) crosses downward through the level 5

6 R (R! ), the probability that it will reain below R or a duration is given as: P ( ) = u I ( u )exp(? u ) () where u is the noralized ade duration with respect to ean ade duration, dened as u =, and I () is the odied Bessel unction o the rst kind. Starting ro Rice's asyptotic result, we derive the ollowing analytical expressions or the tie-dependent perorance easures dened in section. The details o the derivations are given in the Appendix. Firstly, the average duration o a iniu duration outage is given as out = + I ( ( ) ) exp(? ( ) ) X k= (k)!? [( ( ) ) k + (k)( ( k!(k + )! k+ ) ) k? + ::: + (k)!] exp(? ( ) )g: (3) Further, the requency o a iniu duration outage can be shown to be out = p exp(? )( )I ( ( ) ) exp(? ( where is the axiu Doppler requency and = is expressed as ) ) (4) R R rs. Finally, the probability o outage P out = out out : (5) 4 Nuerical Results We use Clarke's odel [8] to siulate the Rayleigh ading signal. In the siulation two independent Gaussian rando signals are ltered by Doppler spectru shaped lters to generate a coplex Gaussian signal. The envelop o this coplex Gaussian signal ors the Rayleigh ading signal at baseband. The lter is ipleented in requency doain using the spectru shaping ethod described in [9]. Note that we are approxiating the probability o outage by ultiplying the requency and duration o outage. Such an approxiation ay not necessarily hold or rando processes, however, our analysis will be validated by siulations 6

7 Firstly, we evaluate the validation o using a nonading noise type signal to approxiate the joint intererence coposed o ultiple utually independent Rayleigh signals. We are especially interested in the inuence o such an approxiation on the ade duration statistics. Figure (a) shows the copleentary CDF o ade durations in the cases o 3, 6, and 9 intererers with identical ean power. The result using constant nonading intererence power and the asyptotic theoretical result (equation ()) are also shown or coparison. In all the cases, the required SIR threshold z is 5 db. According to the joint intererence power, the desired signal power is adjusted to ake the ean SIR to be 5 db. Hence the ade argin F is db. The Doppler requency o the desired signal is 38 Hz (or obile speed v = 3ph and carrier requency c = 85MHz). The Doppler requencies o intererers are uniorly distributed ro.7hz (v = ph) to 75.98Hz (v = 6ph), which iplies that the intererers can have dierent speeds ro that o the desired user. Notice that when the intererer nuber is large (e.g. n = 6; 9) the results o copleentary CDF o have a very good atch with the result using constant nonading intererence power and the asyptotic theoretical result. The atch is better or the larger intererer nuber. For a ore realistic situation, Figure (b) shows the copleentary CDF o ade durations in the cases o 3, 6, and 9 intererers with dierent ean powers. The intererer ean powers are assued to be lognoral distributed. The standard deviation o the intererer ean power is assued to be 6 db. All the other conditions are the sae as in Figure (a). The results are ound siilar to the cases o identical intererer ean power. For the case o dierent intererer ean powers, it is the paraeter in gaa distribution (equation (9)) which deterines the approxiation. Since is always saller than the intererer nuber n (and is always larger than ), the results are less atched than the cases o identical intererer ean power or the sae intererer nuber. However as n increases will also increase. Thus the atch will be better, as shown in Figure (b). Fro these siulation results, we can see that the joint intererence power o ultiple utually independent Rayleigh signals can be well approxiated by a nonading noise type signal when the nuber o intererers is large and the ade argin is high. Thus our iniu duration outage proble can be siplied as the level crossing proble o the desired Rayleigh signal with the threshold level being the value which is F db (F is the ade argin) down ro the desired Rayleigh signal ean power. 7

8 The siulation results are copared with analytical results derived in equations (3-5). Figure (a) shows the probability o outage as a unction o the iniu duration or dierent ade argins. The coparison o the requency o outage under the sae conditions is shown in Figure (b). The results show that there is a good atch between the analysis and siulations. Further, notice that the atch is better or deeper ades as expected. The eects o dierent Doppler requencies on the probability and requency o the outage are shown in Figure 3(a) and 3(b), respectively, with a xed ade argin o F = db. In Figure 3(a), the probabilities o outage or dierent Doppler requencies originate ro the sae static probability (when = ) since the probability o a ade is independent o Doppler requency in the static case. As increases, the probability o outage or higher Doppler requency decreases aster. In Figure 3(b), the requency o outage decreases with the increasing o the iniu duration, and the aily o curves have intersections with each other. The requency o outage or a higher Doppler requency starts ro higher value and eventually drops below that or lower Doppler requency. As the result, when the iniu duration is sall, the requency o outages is higher or higher Doppler requency. In contrast, when the iniu durations are longer, the requency o outages is higher or lower Doppler requency. Also notice that the outages due to Rayleigh ading are in the order o illiseconds. Thus these outages will have ore ipact on packet data counication in ters o packet error rate rather than the capacity or voice counication. Regarding the sensitivity o perorance easures to iniu durations, note that the above results can also directly be used to copare the traditional static analysis with the iniu duration analysis. Specically, when equals zero, the expressions or out, out and P out will becoe the ean duration o ade, the ean level crossing rate N R, and the probability o ade P (r R req ), respectively. Our results indicate that even a relatively sall value o, such as 5 s, ay cause the perorance easures out and P out to change draatically ro static values. 8

9 5 Discussion and Conclusion In obile radio counications, not only the probability o a signal ade but also the duration o a ade deterines the quality o counication. In this paper, we extended the \iniu duration" outage concept to cellular systes with Rayleigh ading. We justiy the validation o using a nonading noise type signal to approxiate joint intererence coposed o ultiple utually independent Rayleigh signals under the situations o large nuber intererers and high ade argins. The statistics o outages were derived based on Rice's asyptotic results on the probability distribution unction o the ade durations o Rayleigh ading signals. These analytical results were validated via siulations using Clarke's odel which is coonly used or land obile radio channels. It was shown that the asyptotic results were accurate and could be used to ake ore realistic evaluations o syste perorance than existing static ethods. With typical Doppler requencies, Rayleigh ading causes ast uctuations o the signal to intererence ratio and the ade durations are in the order o illiseconds. Thus it is ore likely that the outages due to Rayleigh ading will cause rae or packet errors rather than call dropping. In digital transission systes, with protection against burst errors (such as errorcorrection and interleaving), the iniu duration or a rae or packet is deterined based on the burst error-correcting capacity o the rae or packet. The study o ading statistics or Rayleigh ading will help the analysis o orward error-correcting coding and or wireless data counication []. Appendix We will derive the analytical expressions or out, out, and P out as presented in section 3. The well-known results or the level crossing rate N R and the ean ade duration or Rayleigh signal, which are needed in the derivation, are given as ollows [5] N R = p e? ; (6) 9

10 and? = p e : (7) The pd o the ade duration is derived ro equation () () =? dp ( ) du du d =? d du [ u I ( u ) exp(? )]: (8) u Using the above equation and Denition, the pd o the outage duration out can be obtained as out ( out ) = 8 >< >: The average duration o outage is now given as out = = = Z out out ( out )d out P ( ) Z P ( ) (?u) u I ( = + P ( ) ( out) P ( ) i out otherwise. u ] (?u)d[ = u I ( )e? u Z u )e? u j = = u I ( )e? u + Z = u I ( )e? u u dug (9) u du: () Consider the integral on the RHS o the above equation. It can be siplied as ollows: Z = u I ( )e? u u du = Z = u Now, substituting =(u) = x in the above equation, we obtain = = = Z = Z ( ) X k= X k= u I ( k= Z ( ) )e? u X X k= u du k!(k + )! ( u )k+ e? u du: () k!(k + )! (x )k e?x dx k!(k + )! k xk e?x dx k!(k + )! k+ (k)!? [( ( ) ) k +(k)( ( ) ) k? + ::: + (k)!]e? ( ) g; ()

11 which results in an innite series expression or easy calculation. Cobining equation () and (), the average duration o outage can be obtained as in equation (3) in section 3. The requency o outage is now given as out = crossing rateg Probthe crossing is an outageg = N R P ( ) = p exp(? ) I ( ( ) ) exp(? ( ) ): (3) Finally, using the above expressions, the probability o outage is given as P out = out out : (4) Reerences [] N. B. Mandaya, P.-C. Chen, and J. M. Holtzan, \Miniu duration outage or cellular systes: A level crossing analysis," in Proceedings o IEEE VTC'96, vol., pp. 879{883, April 996. [] N. B. Mandaya, P.-C. Chen, and J. M. Holtzan, \Miniu duration outages or CDMA cellular systes : A level crossing analysis," Wireless Personal Counications, vol. 7, no. /3, pp. 35{46, 998. [3] P.-C. Chen, N. B. Mandaya, and J. M. Holtzan, \Miniu duration outage or cellular systes with reception diversity," in Proceedings o IEEE ICUPC, Sept. 3-Oct [4] S. O. Rice, \Distribution o the duration o ades in radio transissions: Gaussian noise odel," Bell Syste Technology Journal, vol. 37, pp. 58{635, May 958. [5] W. C. Jakes, Jr., Microwave Mobile Counications. New York: John Wiley and Sons, 974. [6] J.-P. Linnartz, Narrowband Land-Mobile Radio Networks. Artech House, Inc., 993. [7] J.-P. Linnartz and R. Prasad, \Threshold crossing rate and average non-ade duration in a rayleigh-ading channel with ultiple intererers," AEU-Archiv ur Elektronik und Ubertragungstechnik-Electronics & Counication, vol. 43, pp. 345{349, Nov-Dec 989.

12 [8] R. H. Clarke, \A statistical theory o obile-radio reception," Bell Syste Technology Journal, July-Aug [9] J. I. Sith, \A coputer generated ultipath ading siulation or obile radio," IEEE Trans. on Veh. Technol., vol. VT-4, pp. 39{4, August 975. [] J. Lai and N. Mandaya, \Packet error rate or burst-error-correcting codes in rayleigh ading channels," in Proceeding VTC'98, May 998. (Also subitted to IEEE Trans. on Co.). P τ (τ > τ) coplientary CDF o τ, F=, di. v and identical ean power int. 6 int. 9 int. const. int. theoretical P τ (τ > τ) coplientary CDF o τ, F=, di. v and di. ean power 3 int. 6 int. 9 int. const. int. theoretical τ (s) τ (s) (a) (b) Figure : Analytical and siulation results o coplientary CDF o ade durations: (a) intererers with identical ean power; (b) intererers with dierent ean powers.

13 .5. =37.99 Hz (v=3ph, c=85mhz) analytical siulation =37.99 Hz (v=3ph, c=85mhz) analytical siulation Pout.5 out 5. F=6 db 5.5 F= db τ (s) 5 F= db F=6 db τ (s) (a) (b) Figure : Analytical and siulation results with dierent ade argins (F): (a) probability o outage; (b) requency o outage. Pout F= db =.7 Hz =6.33 Hz =37.99 Hz =75.98 Hz out F= db =.7 Hz =6.33 Hz =37.99 Hz =75.98 Hz τ (s) τ (s) (a) (b) Figure 3: Analytical and siulation results with dierent Doppler requencies ( ): (a) probability o outage; (b) requency o outage. 3