On the Capacity of Joint Fading and Two-path Shadowing Channels
11 On the Capacity of Joint Fading and Two-pathShadowing Channels
I. Dey,
Student Member, IEEE , G. G. Messier,
Member, IEEE , and S. Magierowski,
Member, IEEE
Abstract —The ergodic and outage channel capacity of differentoptimal and suboptimal combinations of transmit power andmodulation rate adaptation strategies over a Joint fading andTwo-path Shadowing (JFTS) fading/shadowing channel is studiedin this paper. Analytically tractable expressions for channelcapacity are obtained assuming perfect channel side informationat the receiver and / or the transmitter with negligible feedbackdelay. Further, the impacts of the JFTS parameters on the chan-nel capacity achieved by these adaptive transmission techniquesare determined.
Index Terms —Fading, Shadowing, Channel Capacity, AdaptiveTransmission
I. I
NTRODUCTION
The high density and considerable individual data raterequirements of modern indoor wireless users has made highcapacity wireless communications a priority in indoor environ-ments. While the use of indoor pico-cells is expected to grow,this demand is primarily being served today by indoor wirelessaccess points. Therefore, it is essential to have an accuratepicture of what high throughput wireless communicationssystems can achieve when implemented on densely deployedindoor access points.This picture is provided by Shannon channel capacity. Withthe introduction of capacity achieving coding schemes [3],Shannon capacity is now of both theoretical and practicalinterest. In case of wireless links, Shannon channel capacitycharacterizes the long-term achievable information rate andtherefore is termed as the ergodic capacity [1].In a fading environment, the Shannon bound can beachieved by adapting a variety of parameters relative to thechannel quality, if perfect channel side information (CSI) isavailable at the receiver and/or the transmitter [5], [6], [7].Examples include Optimal Rate Adaptation (ORA) [8], whichadapts modulation constellation size, and Optimal Power andRate Adaptation (OPRA) [7], which adapts a combination ofmodulation rate and transmit power. The Shannon capacity canalso be achieved only through optimal power control by usingfading inversion to maintain a constant carrier signal-to-noiseratio (CSNR). This technique is known as Channel Inversion
Copyright (c) 2013 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to [email protected]. Dey and G. G. Messier are with Department of Electrical and ComputerEngineering, University of Calgary, 2500 University Drive N.W. Calgary,Alberta, Canada T2N 1N4 (E-mails: deyi and [email protected]).S. Magierowski is with the Department of Electrical Engineering and Com-puter Science, Lassonde School of Engineering, York University, LassondeBldg. 1012B, 4700 Keele St Toronto, Ontario, Canada M3J 1P3 (E-mail:[email protected]). with Fixed Rate (CIFR) [7], [8]. Another adaptive transmissiontechnique referred to as Truncated Channel Inversion withFixed Rate (TIFR) is introduced in [4], where the channelfading is compensated only when the received CSNR is abovea certain cut-off fade depth. The constant information rate thatcan be achieved using TIFR with an outage probability undera certain threshold is referred to as outage capacity [9].It is important to point out that ergodic Shannon capacityestimates are only as good as the channel model upon whichthey are based. It is well known that indoor wireless links areaffected by both small scale fading and shadowing effects. AShannon capacity estimate is meant to characterize throughputexperienced on time scales beyond a few seconds must bebased on channel models that take both large and small scaleeffects into account.To date, composite channel models that combine large andsmall scale effects have been developed primarily for outdoorchannels. In the bulk of these models, like the Suzuki [10]and Nakagami - log-normal [11] composite channel models,the log-normal distribution is used to model shadowing. Amore practical closed-form composite fading model is the K -distribution [12], [13], where log-normal shadowing is ap-proximated by Gamma shadowing. This is done because LMSand macro-cellular communication users are highly mobile inan outdoor environment transiting through several scatteringclusters. As a result, a range of main waves arrive at themobile, the strength of each of which can be drawn from thelog-normal or the Gamma distribution.These outdoor composite models do not accurately charac-terize the indoor wireless LAN link, primarily because thepath between the access point and users are too short forshadowing to be characterized by a log-normal distribution.A new composite channel model, called the Joint Fadingand Two Path Shadowing (JFTS) model, is proposed in [14].Based on an extensive channel measurement campaign, theJFTS model is shown to be a more accurate model forthe indoor wireless LAN channel than any other compositechannel model proposed to date.The primary contribution of this paper is to derive analyt-ically tractable expressions for JFTS ergodic capacity underdifferent adaptive transmission schemes. These expressionswill provide new insight into the behavior of ergodic capacityfor indoor WLAN systems due to the nature of the JFTSmodel. The JFTS distribution is a convolution of the Ricianfading distribution and the two-wave with diffused power(TWDP) shadowing model. The Rician distribution can beexpressed in terms of circular bivariate Gaussian randomvariable with potentially non-zero mean, while the TWDP [15] a r X i v : . [ c s . I T ] M a y distribution is the sum of two half-Ricians. Hence, the JFTScapacity expressions for adaptive modulation techniques donot approach a non-fading channel for high values of carriersignal-to-noise ratio (CSNR). This is a fundamentally differentbehavior from capacity expressions based on conventionalchannel models.The JFTS capacity expressions also have added valuescompared to similar capacity work based on other channelmodels. Unlike the K -fading model, the JFTS model has beenverified using a practical measurement campaign. The JFTSchannel also has a closed form PDF expression as opposedto the Suzuki or Nakagami-log-normal channel models. Theparameters of the JFTS distribution can also be varied torepresent a wide variety of channel conditions like no-fading(infinitely high fading parameter), no-shadowing (infinitelyhigh shadowing parameter), heavy fading (low fading param-eter) or heavy shadowing (low shadowing parameter). Hence,the capacity expressions evaluated over the JFTS channelmodel will provide us with the achievable ergodic capacitymeasures over a large variety of practical channel conditions,without assuming that the propagation environment is complexGaussian distributed.The second contribution of this paper is to explore the re-lationship between the optimal cut-off CSNR and the averagereceived CSNR for JFTS faded/shadowed links when adaptivetransmission techniques are applied. Our numerical resultsshow that in presence of heavy fading and shadowing, thecut-off CSNR remains significantly lower than 1 even at highreceived CSNR, as opposed to traditional fading models [1].A lower cut-off CSNR will result in lower achievable channelcapacity over a JFTS channel in comparison to other compos-ite channel models imparting same severity in fading and/orshadowing. These results will be used to analyze how the JFTSchannel capacity behaves in a fundamentally different waythan the other composite fading/shadowing models prevalentin literature. Our results will also demonstrate the effect ofJFTS parameters on the optimal achievable rate (capacity)assuming perfect CSI to be available at the transmitter and/orthe receiver.The rest of the paper is organized as follows. In Section II,we present the probability density function (PDF) of thereceived instantaneous CSNR over a JFTS communicationchannel. In Section III, expressions for the channel capacityunder different adaptive transmission policies are derived.Numerical results are presented in Section IV followed bysome concluding remarks in Section V.II. J OINT F ADING AND T WO - PATH S HADOWING M ODEL
In an indoor wireless LAN (WLAN) communication sce-nario representing an open concept office or laboratory layout,the PDF f A ( α ) of the received signal envelope, α ( t ) , can begiven by [16], f A ( α ) = (cid:88) i =1 b i α P P m (cid:88) h =1 R e − K − S h − α P r h · [ e S h ∆ M i I (cid:0) α (cid:112) KS h (1 − ∆ M i ) /P P (cid:1) + e − S h ∆ M i I (cid:0) α (cid:112) KS h (1 + ∆ M i ) /P P (cid:1) (1) where M i = cos(( i − π/ , I is the zeroth order modifiedBessel function of the first kind, m is the quadrature order (de-termining approximation accuracy) and R = w h | r h | e r h (2 P − P .The parameter K is the small scale fading parameter, S h is the shadowing parameter, ∆ is the shape parameter ofthe shadowing distribution, P and P are the mean-squaredvoltages of the diffused and the shadowed components respec-tively. An order of the shadowing distribution, i , of 4 is used.This is done because, the 4th order TWDP distribution [17]was found to offer the best fit of the extracted shadowingdistribution of the measurement campaign in [1]. In (1), b i = a i I (1) , where a = , a = , a = and a = . The multiplier w h denotes the Gauss-Hermitequadrature weight factors which is tabulated in [18] andis given by, w h = (2 m − m ! √ π ) / ( m [ H m − ( r h )] ) , where H m − ( . ) is the Gauss-Hermite polynomial with roots r h for h = 1 , , . . . , m . For our analysis, we have chosen m = 20 ,as is done for parameter estimation of composite gamma - log-normal fading channels in [19]. In this case, the mean-squaredvalue of the joint faded and two-path shadowed envelope, A ,can then be calculated using (1) and the integral solution from[20, eq. 6.643.2, p. 709] as, Ω A = E { A } = (cid:88) i =1 20 (cid:88) h =1 b i P R r h P e − K − S h · (cid:20) e S h ∆ M i − KS h (1 − ∆ M i ) r h P (cid:0) P + KS h r h (1 − ∆ M i ) (cid:1) + (cid:0) P + KS h r h (1 + ∆ M i ) (cid:1) e − S h ∆ M i − KS h (1+∆ M i ) r h P (cid:21) . (2)Let us denote the instantaneous received CSNR as γ andthe average received CSNR as γ . The expression for thePDF of the instantaneous CSNR per symbol over a JFTSfaded/shadowed channel has been derived in [16] in termsof the JFTS parameters, K , S h and ∆ . Putting (2) back inthat expression for the PDF of γ , the final expression can beobtained in terms of γ and Ω A as, f γ ( γ ) = (cid:88) h =1 Ω A γP r h (cid:20) − e − Ω Aγ γP r h (cid:21) . (3)In the next section, we will be using (3) to obtain expressionsfor the achievable ergodic and outage capacities of a JFTS fad-ing/shadowing communication channel with different adaptivetransmission techniques.III. A NALYSIS OF C HANNEL C APACITY
Given an average transmit power constraint, the optimal cut-off CSNR level ( γ ) for any adaptive transmission techniquemust satisfy the relationship [6], (cid:82) + ∞ γ (cid:16) γ − γ (cid:17) f γ ( γ )d γ = 1 .If the received instantaneous CSNR level γ falls below γ ,data transmission will be suspended. In order to find therelationship between γ and γ for adaptive transmission overa JFTS faded / shadowed channel, we need to solve two integrals, [20, eq. 3.351.6, p. 340] I = (cid:90) + ∞ γ f γ ( γ ) d γ = (cid:90) + ∞ γ B γ d γ − (cid:90) + ∞ γ B γ e − B γγ d γ = B Ei (cid:18) − B γ γ (cid:19) − B log ( γ ) (4)and I = (cid:90) + ∞ γ γ f γ ( γ )d γ = (cid:90) + ∞ γ B γ d γ − (cid:90) + ∞ γ B γ e − B γγ d γ = − B γ e − B γ γ − B γ Ei (cid:18) − B γ γ (cid:19) − B γ (5)where Ei ( · ) is the exponential integral given by [21] and B = (cid:80) h =1 Ω A P r h . Now, putting the integral solutions ob-tained in (4) and (5), back in the above mentioned relationship,we can find the equation which the optimal cut-off CSNRshould satisfy for adaptive transmission. Therefore, in case of aJFTS faded/shadowed channel, γ should satisfy the followingrelationship, (cid:18) B γ + B γ (cid:19) Ei (cid:18) − B γ γ (cid:19) + B γ (cid:18) − log ( γ ) + e − B γ γ (cid:19) = 1 . (6) A. Ergodic Capacity1) Optimal Power and Rate Adaptation (OPRA):
Assum-ing perfect CSI at the transmitter and the receiver, the er-godic channel capacity (cid:104) C (cid:105) OPRA in bits/sec under an av-erage transmit power constraint is given by, (cid:104) C (cid:105) OPRA = B (cid:82) + ∞ γ log (cid:16) γγ (cid:17) f γ ( γ )d γ , where B (Hz) is the channel band-width and γ is the optimal cut-off CSNR. A water-fillingalgorithm is used for optimal power adaptation given by S ( γ ) = γ − γ for all γ ≥ γ . The optimal rate adaptationsends a rate of log ( γ/γ ) bits/sec for a fade level of γ . Inorder to find the final expression for channel capacity per unitbandwidth over a JFTS faded / shadowed channel ( (cid:104) C/B (cid:105)
JFTSOPRA [bits/sec/Hz]), we need to solve four sets of integrals in, (cid:28) CB (cid:29) JFTSOPRA = 1 log (2) (cid:20) (cid:90) + ∞ γ log ( γ ) B γ d γ (cid:124) (cid:123)(cid:122) (cid:125) I − (cid:90) + ∞ γ log ( γ ) B γ d γ (cid:124) (cid:123)(cid:122) (cid:125) I − (cid:90) + ∞ γ log ( γ ) B γ e − B γγ d γ (cid:124) (cid:123)(cid:122) (cid:125) I + (cid:90) + ∞ γ log ( γ ) B γ e − B γγ d γ (cid:124) (cid:123)(cid:122) (cid:125) I (cid:21) . (7)The expression in (7) can be obtained in a tractable formthrough the following steps of integral solutions and math-ematical manipulations. Firstly we can express, I − I = B log ( γ ) . (8) Using the identities, Ei ( − x ) = − Γ(0 , x ) − log ( x ) + ( log ( − x ) − log ( − x )) and log ( − x ) = log ( x ) + ıπ , validfor x > [21] and [20, eq. 4.452.1, p. 573], and assumingthat, ( B γ /γ ) > and ( γ/ B γ ) > and after some algebraicmanipulations, we can express, I − I = B log ( γ ) log (cid:18) B γ γ (cid:19) + B E log ( γ ) − B log ( γ ) − B γ γ F (cid:18) , ,
1; 2 , , − B γ γ (cid:19) . (9)where E is the Euler-Mascheroni constant with a numericalvalue of E ≈ . . Finally, using (8) and (9), theexpression in (7) can be obtained as, (cid:28) CB (cid:29) JFTSOPRA = B log ( γ ) log (2) (cid:20) log (cid:18) B γ γ (cid:19) + E (cid:21) − B γ γ log (2) F (cid:18) , ,
1; 2 , , − B γ γ (cid:19) (10)where p F q ( · ) is the generalized confluent hyper-geometricfunction [21] and p, q are integers.
2) Optimal Rate Adaptation (ORA):
Assuming perfect CSIat the receiver only, the ergodic channel capacity (cid:104) C (cid:105) ORA inbits/sec with constant power over any composite fading andshadowing channel is given by, (cid:104) C (cid:105) ORA = B (cid:82) + ∞ log (1 + γ ) f γ ( γ )d γ . It is shown in [2] that (cid:104) C (cid:105) OPRA becomes equal to (cid:104) C (cid:105) ORA when the transmit power is kept constant for OPRA.Using the identity log (1 + y ) = log ( y ) − (cid:80) + ∞ n =1 ( − n ny n for | y | > , we can solve the integral in the above definition[20, eq. 3.351.2, p. 340]. Using (3), the final expression forchannel capacity per unit bandwidth with ORA transmission( (cid:104) C/B (cid:105)
JFTSORA ) over a JFTS faded / shadowed communicationlink can be written as, (cid:28) CB (cid:29) JFTSORA = 1 log (2) (cid:20) (cid:90) ∞ ∞ (cid:88) n =1 ( − n n B γ n +1 e − B γγ d γ (cid:21) = + ∞ (cid:88) n =1 B Γ( − n ) n log (2) (cid:18) − B γ (cid:19) n . (11)It is evident from (10) and (11), that ergodic capacity overa JFTS distributed link depends on the mean-squared valueof the joint faded and two-path shadowed envelope, Ω A . Nowfrom (2), we observe that Ω A decreases exponentially withthe increase either in K or S h or both. In (10), the capacityterm is directly proportional to (cid:2) log (cid:0) B γ γ (cid:1) + E (cid:3) . Hence, as Ω A decreases, (cid:12)(cid:12) log (cid:0) B γ γ (cid:1)(cid:12)(cid:12) increases, since Ω A < . As aresult, the term (cid:2) log (cid:0) B γ γ (cid:1) + E (cid:3) increases with the increasein the fading and/or the shadowing parameters resulting inthe overall increase in the ergodic capacity. Similar intuitiveconclusions can also be made from (11), where capacityincreases with the decrease in Ω A , since (cid:10) CB (cid:11) ∝ (cid:0) − B γ (cid:1) n for n > .
3) Channel Inversion with Fixed Rate (CIFR):
Assumingperfect CSI at the transmitter and the receiver, the channelcapacity of this technique for any fading/shadowing communi-cation link is given by, (cid:104) C (cid:105) CIFR = B log (cid:16) (cid:82) + ∞ γ f γ ( γ )d γ (cid:17) . Using the integral solution from (5), it can be shown that CIFRchannel capacity is equal to zero for the JFTS channel. In thatcase, a large amount of transmitted power will be requiredto compensate for the deep channel fades if this technique isused for adaptive transmission. A better approach will be touse truncated channel inversion with fixed rate, the channelcapacity for which has been derived in the next subsection.
B. Outage Capacity1) Truncated Channel Inversion with Fixed Rate (TIFR):
In case of TIFR, channel fading is inverted only if the receivedinstantaneous CSNR level is above the cut-off fade depth ( γ ).The channel capacity with TIFR over any fading channel isobtained by maximizing the outage capacity ( C out ) over allpossible γ and can be expressed as, C TIFR = max γ C out ,where C out is the outage capacity. The outage channel ca-pacity for a fading/shadowing channel can be calculated as, (cid:104) C out (cid:105) TIFR = B log (cid:16) (cid:82) + ∞ γ γ f γ ( γ )d γ (cid:17) (1 − P out ) , where P out is the outage probability. For a JFTS fading/shadowingchannel, P out can be calculated as, P out = (cid:90) γ f γ ( γ ) d γ = B log ( γ ) − B Ei (cid:18) − B γ γ (cid:19) (12)using the integral solution provided in (4). Using (5), we canevaluate the channel capacity with TIFR in a JFTS faded /shadowed communication link which can be expressed as, (cid:28) C out B (cid:29) JFTSTIFR = (cid:18) B Ei (cid:18) − B γ γ (cid:19) − B log ( γ ) (cid:19) · log (cid:32) − γ γ B γ e − B γ γ + B γ Ei (cid:0) − B γ γ (cid:1) + B γ (cid:33) . (13)IV. N UMERICAL R ESULTS AND D ISCUSSION
It has been claimed in [1] that for any fading channel, theoptimal cut-off CSNR or optimal threshold satisfies ≤ γ ≤ if both the transmit power and the modulation rate are variedfor optimal adaptation. Results from [2], [3] also indicate thatfor Rayleigh and Nakagami- m fading channels, γ convergesto 1 as γ increases. For a JFTS fading/shadowing channel, therelationship between γ and γ is demonstrated in Fig. 1(a).For a communication link with high K and S h (i.e. low fadingand shadowing severity) γ converges to 1 with the increasein γ , as observed in [1]. However, as the channel conditiondeteriorates with lower K and S h , γ remains significantlylower than 1 even at high γ . In such a scenario, perfectknowledge of both the transmit side and the receive side CSIshould provide an edge over the perfect knowledge of only thereceive side CSI, as claimed in [1]. As a result, regulating boththe transmit power and the modulation rate (OPRA) will resultin a considerable increase in ergodic capacity over adaptingonly the modulation rate (ORA). This will be verified below.The next set of curves in Fig. 1(b) are generated by compar-ing optimal achievable rate using OPRA over a JFTS channelwith that achievable over conventional joint fading/shadowing Fig. 1. (a) Calculated cut-off CSNR ( γ ) for different values of averagereceived CSNR ( γ ). (b) Ergodic Capacity per unit bandwidth achievable withOPRA over different channel models contributing the same AF of 3.45. channels like Nakagami-log-normal and K -fading models. Foreach channel model, the distribution parameters are chosensuch that the same amount of fading (AF) is contributed byeach channel model. The curves are plotted for Nakagami-log-normal ( m = 1 , σ = 3 . ) [22], K -fading ( k = 0 . )[12], and JFTS ( K = 5 dB, S h = − . dB, ∆ = 0 . ) [23]channel models, each contributing an AF of 3.45. An optimalcut-off CSNR, γ , which is significantly lower than 1 even athigh γ , results in lower achievable rate over a JFTS channelin comparison to other channel models. The reason can beattributed to the fact that the JFTS distribution has a verydifferent PDF from common composite fading/shadowing dis-tributions like Nakagami-log-normal or K -distribution. Both ofthese distributions can be described using Gamma distributionand therefore the received envelope can be expressed interms of zero mean complex Gaussian random variables withdifferent shape factors. Hence in all of these cases, at higherreceived CSNR, the channel approaches the no-fading and no-shadowing condition and the received signal envelope becomeszero mean complex Gaussian distributed with a shape factorof 1. As a result, the achievable ergodic channel capacitystarts approaching the Shannon bound as the received CSNRincreases. While JFTS distribution can only be expressed interms of bi-variate non-centralized chi-squared distributionand therefore can never be described using Gaussian randomvariables.It is claimed in [2] that the difference in channel capacitybetween OPRA and ORA is bounded by C OPRA − C ORA ≥ B log (cid:0) (cid:82) γ ( γ − γ ) f γ ( γ )d γ (cid:1) . As a result, the channelcapacity obtained using ORA starts approaching that achiev-able using OPRA with the increase in γ for JFTS channels,as is evident in Fig. 2(a) and Fig. 2(b). Hence it can beconcluded by summarizing the results from Fig. 2(a) andFig. 2(b) that OPRA offers improvement in ergodic capacityover ORA only when γ remains significantly lower than 1.These observations are similar to that made in [2] and [1] for Fig. 2. Ergodic capacity per unit bandwidth of JFTS communication linkwith OPRA and ORA, where the curves are generated by (a) varying the K -factor ( S h = − dB and ∆ = 0 . ) and (b) varying the S h -factor ( K = 5 dB and ∆ = 0 . ). Rayleigh and Nakagami- m fading channels. The gap between C OPRA and C ORA increases at lower γ with the increase inseverity of fading (decrease in K ) and shadowing (decreasein S h ) both of which degrade the channel quality. These resultsare in accordance with the general behavior of a wirelesscommunication system over a JFTS faded/shadowed channel.As noted in [16], performance of any communication systemover a JFTS channel deteriorates with the decrease in K and S h -factors.The degradation in ergodic capacity due to the decrease in K -factor from 8 dB to 2 dB (refer to Fig. 2(a)) is much lesscompared to the decrease in optimal achievable rate due to thelowering of S h -factor from 5 dB to − dB (refer to Fig. 2(b)).These results do not agree with the observations made in [16],where bit error rate performance of BPSK is found degradeequally either due to the decrease in the K -factor or the S h -factor. The reason for this can be attributed to the ∆ -valuechosen for each plot. For Fig. 2(a) a low ∆ of 0.4 is chosen.In this case shadowing severity is reduced by the fact that onlyone scattering cluster dominates instead of two clusters. ForFig. 2(b) a high ∆ of 0.9 is chosen, where the magnitudesof the shadowing values contributed by each scattering clusterare almost equal. As a result, even for a high γ of 12 dB apenalty of 3 bits/sec/Hz of achievable rate is observed onlyfor decreasing the S h factor.On the other hand, the outage capacity with TIFR degradesequally with the lowering of either the small scale fading ( K )factor or the shadowing ( S h ) factor, as is evident in Fig. 3.Hence it can be concluded that the outage capacity of a JFTScommunication channel is more sensitive than ergodic capacityto the changes in small scale fading and shadowing. Thisobservation agrees with that made in case of Rician channel inpresence of shadow fading in [24]. It has also been observedin [24], increase in the severity of shadow fading improvesergodic capacity and degrades outage capacity of a shadowedRician channel. However, for a JFTS faded/shadowed channelboth ergodic and outage capacities are degraded significantly Fig. 3. Outage capacity per unit bandwidth of JFTS communication link withTIFR, where the curves are generated by (a) varying the K -factor ( S h = − dB and ∆ = 0 . ) and (b) varying the S h -factor ( K = 5 dB and ∆ = 0 . ). due to the increase in shadowing severity, as is evident fromFig. 2 and Fig. 3. V. C ONCLUSION
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