Coverage Analysis of Broadcast Networks with Users Having Heterogeneous Content/Advertisement Preferences
11 Coverage Analysis of Broadcast Networks withUsers Having HeterogeneousContent/Advertisement Preferences
Kanchan K. Chaurasia, Reena Sahu, Abhishek K. Gupta
Abstract
This work is focused on the system-level performance of a broadcast network. Since all transmittersin a broadcast network transmit the identical signal, received signals from multiple transmitters can becombined to improve system performance. We develop a stochastic geometry based analytical frameworkto derive the coverage of a typical receiver. We show that there may exist an optimal connectivity radiusthat maximizes the rate coverage. Our analysis includes the fact that users may have their individualcontent/advertisement preferences. We assume that there are multiple classes of users with each userclass prefers a particular type of content/advertisements and the users will pay the network only whenthen can see content aligned with their interest. The operator may choose to transmit multiple contentssimultaneously to cater more users’ interests to increase its revenue. We present revenue models to studythe impact of the number of contents on the operator revenue. We consider two scenarios for users’distribution- one where users’ interest depends on their geographical location and the one where itdoesn’t. With the help of numerical results and analysis, we show the impact of various parametersincluding content granularity, connectivity radius, and rate threshold and present important designinsights.
Index Terms
Stochastic geometry, Broadcast networks, Coverage.
I. I
NTRODUCTION
Broadcasting networks provide society with various services including TV communication,delivery of critical information and alerts, general entertainment, and educational services and
The authors are with the Department of Electrical Engineering, Indian Institute of Technology Kanpur, Kanpur, India 208016.Email: [email protected]. a r X i v : . [ c s . I T ] J a n thus have been a key wireless technology. With the recent advancement in wireless technologiesand handheld electronic devices including smartphone and tablets, the use of broadcastingservices has been extended to include many modern applications including delivery of traffic in-formation to vehicles in vehicle-to-infrastructure networks, advertisement industry and mobile TVservices. Many broadcasting standards have been recently proposed including the digital videobroadcast-terrestrial standard (DVB-T2), the advanced television systems committee standard(ATSC 3.0), and the DVB-next generation handheld standard (DVB-NGH) to assist deliveringof TV broadcasting services to mobile devices [1] [2] [3]. The advent of digital broadcastinghas lead to a significant increase in the demand for multimedia services for handheld devicesincluding mobile TV, live video streaming, and on-demand video in the last decade [1]. Fromthese applications’ perspective, broadcasting based multimedia services can provide better datarate and performance compared to the uni-cast cellular network based mobile TV. Note that in acellular network where the desired data is transmitted to each user via orthogonal resources, usersmay suffer from spectral congestion in regions with high density due to limit bandwidth resultingin the performance degradation. But, in a broadcasting network providing multimedia services,all transmitters transmit identical data to all users and hence, do not require orthogonal resource.In these networks, each transmitter can use the complete spectrum to serve their users, andhence, these are also called single frequency networks (SFN). Due to this, users may experiencea better quality of service. A. Related Work
Given increasing demand of broadcasting services, it is very interesting to analyze the broad-cast networks in terms of signal-to-noise-ratio (SINR) and achievable data rate to understandtheir limitations and potential to meet these demands. There have been some recent works in thesystem-level analysis of broadcast networks. In [4], the authors evaluated the blocking probabilityfor users accessing a network delivering mobile TV services over a hybrid broadcast unicast com-munication. In [5], the authors studied a cellular network with uni-cast and multicast-broadcastdeployments. However, these works didn’t include the effect of transmitters’ locations in theevaluations which is required for the system-level analysis of broadcast networks. Stochasticgeometry framework can be utilized to analyze wireless networks from system level perspective[6]–[8]. Stochastic geometry based models have been validated for various types of networksincluding cellular networks,and ad hoc networks [6], [9]–[11]. In [12], the authors describe the analytical approach to calculate the coverage probability of a hybrid broadcast and uni-castnetwork, however, the authors have only considered a single broadcast transmitter along withmany uni-cast transmitters. In our past work [13], we have considered a broadcast network withmultiple broadcasting transmitters to compute the coverage performance of users. However, thework assumed a static connectivity region around the user where transmitters need to be locatedto be able to serve the users. As shown in this paper, this connectivity region is of variable sizedepending on the location of the first closest transmitter. To the best of our knowledge, thereexists no other past work which analyzes the SINR and rate performance of a broadcast networkwith multiple broadcasting transmitters which is one of the main focuses of this paper.Another important metric to evaluate broadcast networks is the revenue earned by the networkoperator. In a broadcast network, the revenue is generated either from subscribers as networkaccess fees for viewing content of their choice or from advertisers to show their advertisementsto interested subscribers. With digital broadcast, subscriptions and user-targeted advertisementsadded a new dimension in the revenue. Due to advancements in technologies over the past fewdecades, the advertising has become more user targeted and location-adaptive which can beplanned according to the user demographics and their preferences to improve network revenue.In [14], the authors studied location-based mobile marketing and advertising to show the positiveinterest of mobile consumers in receiving relevant promotions. It is intuitive that a targeted andlocalized content will have a better engagement factor. It is interesting to analyze the networkrevenue earned from the users with their preference dependent on their choices and geographicallocation. As far we know, there does not exist any past work that analyses the network revenueof a broadcast network with subscribers having preferences for content and advertisement whichis another focus of this paper.
B. Contributions
In this paper, we derive an analytical framework to evaluate the performance of a broadcastnetwork with multiple broadcasting transmitters with users having content preference. We alsopresent a revenue model to quantify the network revenue earned by the network operator. Inparticular, the contributions of this paper are as follows:1) We consider a broadcast network with multiple transmitters. Since all transmitters ina broadcast network are transmitting the same signal, received signals from multipletransmitters from a certain connectivity region around the user can be combined to improve the coverage at this user. Using tools from stochastic geometry, we derive the expressionfor SINR coverage and rate coverage of a typical receiver located at the origin. Dueto the contribution in the desired signal power from multiple transmitters, the analysis issignificantly different and difficult than their cellular counterpart. Our main contribution liesin developing the framework and deriving techniques to evaluate the analytical expressionsof SINR and rate coverage. We show that this connectivity region depends on networkbandwidth.2) We present some numerical results to validate our analysis and present design insights.We show the impact of connectivity region size, path-loss exponents, and the networkdensity on the SINR and rate coverage. We also find that there exists an optimal size ofconnectivity region that maximizes the rate coverage.3) In this paper, we also include the fact that users may have their individual content oradvertisement preferences. We assume that there are multiple classes of users with eachclass of users prefers a particular type of content/advertisements and the users will paythe network only when then can see a particular content of their interest. We assumethat one unit of revenue comes to the network from a particular class of users if everyuser of this class can see the content as per the preference of this class. We study therevenue thus obtained by the network from users. The broadcast operator may chooseto transmit multiple contents simultaneously to cater more users’ interest to increase itsrevenue. However, given the limited resources, the network can cater only to few classesand this capability depends on how these user classes are distributed spatially. There aretwo scenarios considered for users’ distribution. In one scenario, users’ interest dependon their geographical position in the network and in the second scenario it does not. Wecalculate the analytical expression for SINR coverage and rate coverage at a typical userand evaluate the total revenue. We present many important design insights via numericalresults.
Notation:
Let B ( x , r ) denote the ball of radius r with center at x . (cid:107) x (cid:107) denotes the norm ofthe vector x and (cid:107) x i (cid:107) = r i denotes the random distance of BBS located at x i . Let o denote theorigin. B ( x, y ; z ) is the incomplete Beta function which is defined as B ( x, y ; z ) = (cid:90) z u x − (1 − u ) y − d u. Let c denote the speed of EM waves in the media. A (cid:123) denotes the complement of set A . II. S
YSTEM M ODEL
In this paper, we consider a broadcast network with multiple broadcasting base stations (BBSs),deployed in the 2D region R = R . The considered system model is as follows: R S X BBS
User
Fig. 1. Illustration of system model of a broadcast network. A typical user is considered at the origin. X is the distance ofthe nearest BS from the typical user. The 2D region B ( o , X + R s ) denotes the connectivity region of the user. A. Network Model
The location of BBSs can be modeled as a homogeneous Poisson point process
Φ = { X i ∈ R } with density λ in the region R (See Fig. 1). Let R e = 1 / √ λπ which represents the cellradius of an average cell. The subscribers (users) of the broadcasting service are assumed to forma stationary point process. We assume a typical user located at the origin o . Consider each BS isoperating in the same frequency band with the transmission bandwidth W . Let T s is the symboltime of the transmitted symbol which is inversely proportional to the bandwidth W . Assume thetransmit power of each BBS be p t and all devices are equipped with a single isotropic transmitantenna. The analysis can be extended for finite networks by taking R = B ( o , R ) with a finite R . B. Channel Model
We assume the standard path-loss model. Hence, the received signal power from the i th BBSat the typical user at origin is given as P i = p t aβ i (cid:107) X i (cid:107) − α , (1) where X i = (cid:107) X i (cid:107) denotes the random distance of this BBS from the typical user. Here, α isthe path-loss exponent and a is near-field gain which depends on the propagation environment. β i denotes the fading between the i th BBS and the user. We assume Rayleigh fading, i.e. , β i ∼ Exp(1) for tractability.
C. Serving Signal and Interference Model
In a broadcast system, multiple BBSs may transmit the same data at the one frequency band(as suggested by the name SFN). Therefore, at the receiver end, it can be seen as a singletransmission with multi-path propagation and signals transmitted from multiple BBSs can becombined at the user. However, since the signals from different BBSs are delayed accordingto time delays dependent on their distance, some of these signals may be delayed significantlyand may overlap with the next transmission slots. Therefore, only those signals that have delaywithin a certain limit can be combined to successfully decode the received symbol [15]. Therest BBSs contribute to the ISI (inter-symbol interference) which can be significant dependingon the BBSs density.Let X denotes the nearest serving BBS. The probability density function of the distance X = (cid:107) X (cid:107) to the nearest BS from the user is given as [7] f X ( u ) = 2 πλue − πλu ( u ≥ . (2)The time taken by the signal to reach from the i th BBS located at X i to the typical user at o be T i = X i /c . In particular, T denotes the time taken by signal to reach from X to a typicaluser at o . Let the propagation delay of transmitted signal from i th BBS compared to the nearestserving BBS is ∆ i = T i − T .We assume that the receiver design allows the maximum delay of δT s for the received signalsto be combined at the user where δ ∈ [0 1] is a design parameter. This means that the receivedsignal from the i th BBS may contribute in the serving signal power if ∆ i ≤ δT s . This conditionis equivalent to the condition (cid:107) X i (cid:107) − (cid:107) X (cid:107) ≤ R s ∆ = T s δc on the BBSs location X i . In otherwords, this means that all the BBSs that are located in the 2D region { X : (cid:107) X (cid:107) ≤ (cid:107) X (cid:107) + R s } = B ( o , X + R s ) can contribute to the serving signal at the typical receiver at origin o . We termthis region B ( o , X + R s ) as the connectivity region for the user and X + R s can be termed asthe connectivity radius. Let m = λπR denote the mean number of BBSs in this connectivityradius. On the other hand, all the BBSs located outside B ( o , X + R s ) i.e. all the BBSs with X i ≥ X + R s will contribute to the interference power even when they are transmitting the same dataas their signal will be delayed beyond the specified limit. D. Modeling Content Preferences of Users
In this paper, we also include the fact that users may have their individual content or ad-vertisement preferences. We assume that there are N c classes of users. Here, N c is termed content/advertisement granularity . Each class of users prefers a particular type of content/advertisements. We assume that the users will pay the network only when then can see a particularcontent of their interest. Each class consists of some quanta of users. For simplicity, we assumethat each class has the same number of users, however, the presented framework can be triviallyextended to include user classes with unequal sizes. We assume that one unit of revenue comesto the network from a particular class of users if every user of this class can see the content asper the preference of this class. Given the limited resources, the network can cater only to fewclasses and this capability depends on how these user classes are distributed spatially. We willconsider two types of users class distributions over the geographical space. We will also discussa revenue model to characterize the network’s revenue to help us understand optimal schedulingpolicies for the two scenarios.III. C OVERAGE A NALYSIS FOR C OMMON C ONTENT T RANSMISSION
We first start with the scenario that all users seek the same content, hence, all BBSs aretransmitting the same content to everyone. Examples include systems transmitting emergencyinformation, or traffic data which is common to every user. In this section, we will derive theSINR and rate coverage probability for a typical user at the origin o for such system. A. SINR
Since all BBSs are transmitting the same content, all BBSs located inside the connectivityregion B (0 , X + R s ) contribute to the signal power. Therefore, the desired received signal powerfor the typical user at origin is given as S (cid:48) = p t aβ (cid:107) X (cid:107) − α + (cid:88) X i ∈ Φ ∩B (0 ,X + R s ) \ X p t aβ i (cid:107) X i (cid:107) − α . (3) Similarly, the total interference can be given as I (cid:48) = (cid:88) X j ∈ Φ ∩B (0 , X + R s ) (cid:123) p t aβ j (cid:107) X j (cid:107) − α . (4)The signal-to-interference-plus-noise ratio (SINR) at the typical receiver is given as SINR = S (cid:48) I (cid:48) + N = β X − α + (cid:80) X i ∈ Φ ∩B (0 ,X + R s ) \ X β i X i − α (cid:80) X j ∈ Φ ∩B (0 ,X + R s ) (cid:123) β j X j − α + σ . (5)Here, σ is the normalized noise power given as σ = N/ ( p t a ) where N is the noise power.Similarly normalized desired received signal power and interference are denoted by S and I which are given as S = S (cid:48) / ( p t a ) and I = I (cid:48) / ( p t a ) . Hence, the SINR is equal to SINR = SI + σ . (6)Let K = σ /R − αe which represents the SNR at cell edge of an average cell. B. SINR Coverage Probability
The SINR coverage probability p c ( τ, λ ) of a user is defined as the probability that the SINRat the user is above the threshold τ i.e. p c ( τ, λ ) = P [ SINR > τ ] (7)Using the conditioning on the nearest serving BBS’s location X , the SINR coverage for typicaluser at o is given as p c ( τ, λ ) = E X [ P ( SINR > τ ) | X ]= E X (cid:20) P (cid:18) SI + σ > τ (cid:19) | X (cid:21) = E X (cid:2) P (cid:0) S > ( I + σ ) τ | X (cid:1)(cid:3) . (8)Using the distribution of (cid:107) X (cid:107) = X , the SINR coverage probability can be further written as p c ( τ, λ ) = E X [ P ( S > τ ( I + N ) | X )]= (cid:90) ∞ πλue − πλu P (cid:0) S > τ (cid:0) I + σ (cid:1) | (cid:107) X (cid:107) = u (cid:1) d u. (9)To solve the inner term further, we will use Gil Pelaez’s Lemma [16] which states that the CDFof a random variable Y can be written in term of its Laplace transform L Y ( t ) as F Y ( s ) = P [ Y ≤ s ] = 12 − π (cid:90) ∞ t Im (cid:2) e − jts L Y ( − jt ) (cid:3) d t. (10) Using this Lemma, we get, P (cid:0) S > ( I + σ ) τ | X (cid:1) = E I | X (cid:20)
12 + 1 π (cid:90) ∞ t Im (cid:104) e − jtτ ( I + σ ) L S ( − jt ) (cid:105) d t (cid:21) = 12 + 1 π (cid:90) ∞ t Im (cid:104) E I | X (cid:104) e − jtτ ( I + σ ) (cid:105) L S | X ( − jt ) (cid:105) d t = 12 + 1 π (cid:90) ∞ t Im (cid:104) L I | X ( jtτ ) e − jtτσ L S | X ( − jt ) (cid:105) d t. (11)Now, using in (9), the SINR coverage probability is p c ( τ, λ ) = 12 + 1 π (cid:90) ∞ (cid:90) ∞ πλue − πλu t Im (cid:104) L I | X ( jtτ ) e − jtτσ L S | X ( − jt ) (cid:105) d t d u. (12)Here, L I | X ( . ) and L S | X ( . ) are the Laplace transform of the sum interference I and of thedesired received signal power S respectively which are given in the following Lemma. Lemma 1.
The Laplace transforms of the desired signal power and the sum interference at thereceiver located at origin o are given as L S | X ( s ) = 11 + sX − α exp (cid:18) − πλ (cid:90) X + R s X sr − α sr − α r d r (cid:19) (13) L I ( s ) = exp (cid:18) − πλ (cid:90) ∞ X + R s sr − α sr − α r d r (cid:19) (14) Proof.
See Appendix A.Using Lemma 1 in (12), we can get the SINR coverage which is given in Theorem 1.
Theorem 1.
The probability of the SINR coverage for the user located at the origin in a broadcastnetwork with λ density of BBSs, is given as p c ( τ, λ ) = 12 + 1 π (cid:90) ∞ πλue − πλu (cid:90) ∞ t Im (cid:34) e − jtτσ − jtu − α × exp (cid:18) − πλ (cid:18)(cid:90) u + R s u − jtr − α − jtr − α r d r + (cid:90) ∞ u + R s jtτ r − α jtτ r − α r d r (cid:19)(cid:19)(cid:21) d t d u = 12 + 1 π (cid:90) ∞ (cid:90) ∞ v s (cid:20)
11 + s v − α (cid:21) e − v e − s α M d ( s,v ) × (cid:104) sv − α cos (cid:16) s α N d ( s, v ) + τ sK (cid:17) − sin (cid:16) s α N d ( s, v ) + τ sK (cid:17)(cid:105) d v d s (15) where M d ( t, u ) and N d ( t, u ) are given as M d ( s, v ) = 1 α (cid:20) Q (cid:18) α , s ( v + m ) − α , s v. − α (cid:19) + τ /α Q (cid:18) α , , τ s ( v + m ) − α (cid:19)(cid:21) (16) N d ( s, v ) = 1 α (cid:20) − Q (cid:18) α + 12 , s ( v + m ) − α , s v − α (cid:19) + τ α Q (cid:18) α + 12 , , τ s ( v + m ) − α (cid:19)(cid:21) (17) with Q ( z, a, b ) = B (cid:18) z, − z + 1; 11 + a (cid:19) − B (cid:18) z, − z + 1; 11 + b (cid:19) . (18) Proof.
See Appendix B.Theorem 1 provides the SINR coverage in terms of two parameters: K which denotes theinverse of SNR at the cell edge and m which denotes the mean number of BBSs in connectivityradius circle. Further we can derive the following remarks. Remark 1.
For interference limited scenario, K = 0 , which means the coverage probability isa function of m only. In case m is fixed, individual variation of λ and R s will not change thecoverage. Remark 2.
For a broadcast network, an increase in the BBS density λ improves both the desiredsignal power and the interference power. However, due to increase in number of serving BBSsdue to increase in λ which improve the overall SINR coverage (which can also be seen in thenumerical results). This behavior is different than conventional cellular case. Recall that withsingle serving BS density, the SINR in an interference-limited cellular network does not getaffected by any increase in the BS density which is known as SINR invariance [6]. This can beshown from (15) by performing a comparative study between λ and λ (1 + (cid:15) ) with (cid:15) < forsome λ . Remark 3.
It can been observed that the SINR coverage probability increases with an increasesconnectivity radius R s as it increases the serving power and decreases the interference.C. Rate Coverage Probability The rate coverage probability of a user is defined as the probability that the maximumachievable rate for the considered user is above some threshold ρ i.e. r c ( ρ ) = P [ Rate > ρ ] . Note that the maximum achievable rate for the typical user is given as
Rate = ξW log (1 + SINR ) (19)where ξ is some coefficient that denotes the spectrum utilization. W denotes the system band-width available to each BBS. Hence, the rate coverage for the typical user is r c ( ρ ) = P [ Rate > ρ ]= P [ ξW log (1 + SINR ) > ρ ]= P (cid:2) SINR > ρ/ ( ξW ) − (cid:3) = p c (2 ρ/ ( ξW ) − (20)where p c is the SINR coverage probability given in (15). Note that the available bandwidth W affects T s and hence, R s . If the BBSs use orthogonal frequency division multiplexing (OFDM)for transmission with FFT size N s , then, W is related to T s as W = N s T s . Hence, the connectivity radius is R s = T s δc = N s δcW . (21)Hence, an increase in the system bandwidth increases the pre-log factor in (19), however, it alsodecreases the connectivity radius resulting in the lower SINR coverage probability. Therefore,we can observe a trade-off on the rate coverage with increasing bandwidth. D. Numerical Results
We now validate our results for SINR and rate coverage probabilities through numerical sim-ulation. We will also explore the impact of different parameters on the coverage probabilities vianumerical evaluations of derived expressions to develop design insights. The default parametersare given in Table I which are according to [17], [18].
Validation of results:
Fig. 2 shows the SINR coverage probability vs SINR threshold ( τ ) fordifferent values of BBSs density ( λ ). Here, the solid lines represent the analytical expression andmarkers represent simulation values. It can be seen that the analysis matches with simulationresults which establishes the validity of the presented analytical results. From Fig. 2, it can beseen that SINR coverage increases with an increase in the BBS density consistent with Remark2. Similarly, Fig. 3 shows the rate coverage probability vs rate threshold ( ρ ) for different values TABLE ID
EFAULT PARAMETERS FOR NUMERICAL EVALUATIONS
Parameters Numerical value Parameters Numerical value R s λ N a, α . × − , W p t
20 dB N s δ
512 Coefficients a = 10 − , ξ = 1 N c
15 Simulation radius km -15 -10 -5 0 5 10 15 20 25SINR threshold(dB) 00.20.40.60.81 C o ve r a g e P r ob a b ili t y ( p c ) Sim =1 10 -3 Ana =1 10 -3 Sim =1.8 10 -3 Ana =1.8 10 -3 Sim =4.5 10 -3 Ana =4.5 10 -3 Sim =10 10 -3 Ana =10 10 -3 Fig. 2. SINR coverage vs. SINR threshold ( τ ) for various BBS density λ in a broadcast system with multiple BBSs. Here, thesolid lines represent the analytical expression and markers represent simulation values. The parameters are according to TableI. It can be seen that the analysis matches with simulation results. of λ . It can be seen that the rate coverage increases with an increase in λ which is expected dueto the SINR coverage behavior with λ . Impact of connectivity radius on SINR and rate coverage:
Fig. 4 shows the variation ofSINR coverage with the connectivity radius ( R s ) for different values of target SINR threshold.It is observed that the SINR coverage increases with R s . It can be justified as R s increases thenumber of contributing BBSs increases and the number of interfering BBSs decreases.Fig. 5 shows the variation of the rate coverage with R s . We can observe that with R s , the ratecoverage first increases up to a certain value of R s and afterwards starts decreasing again. From(21), increase in R s requires a decrease in the bandwidth W in order to allow larger symboltime. This causes a trade-off in the system performance. As bandwidth is a pre-log factor inthe rate expression, it has a negative and larger impact on the rate coverage. Hence, beyond a Rate threshold(Mbps) 00.20.40.60.81 R a t e C o ve r a g e ( r c ) Sim =1 10 -3 Ana =1 10 -3 Sim =1.8 10 -3 Ana =1.8 10 -3 Sim =4.5 10 -3 Ana =4.5 10 -3 Sim =10 10 -3 Ana =10 10 -3 Fig. 3. Rate coverage vs. rate threshold ( ρ ) for various BBS density λ in a broadcast system with multiple BBSs. Here, thesolid lines represent the analytical expression and markers represent simulation values. The parameters are according to TableI. It can be seen that the analysis matches with simulation results. s (km)00.20.40.60.81 C o ve r a g e P r ob a b ili t y ( p c ) = 4 dB = 8 dB = 12 dB Fig. 4. SINR coverage vs. R s for different values of SINR threshold ( τ ) in a broadcast network. Here, bandwidth varies with R s according to (21) with maximum value at 80 MHz. The rest of the parameters are according to Table I. It is observed thatthe SINR improves with R s . certain value of R s , the impact of decrease in W dominates the increase in SINR caused byincreased R s which results in the decrease in the rate. Due to the same reasons, there may existan optimal R s that maximizes the rate coverage. The knowledge of optimal R s can be helpfulin designing the broadcast network. Impact of network density on SINR coverage:
Fig. 6(a) and (b) show the variation of SINRand rate coverage with the network density λ for different value of target SINR threshold andconnectivity radius (which is achieved by changing bandwidth while keeping other parameters s (km)00.20.40.60.81 R a t e C o ve r a g e ( r c ) = 5 Mbps = 10 Mbps = 12 Mbps Fig. 5. Rate coverage vs. R s for different values of rate threshold ( ρ in Mbps) in a broadcast network. Here, bandwidth varieswith R s according to (21) with maximum value at 50 MHz. The rest of the parameters are according to Table I. A trade-off isseen in rate the with varying R s . the same as Table I). The coverage while ignoring the noise is also shown. From Fig. 5(a), it canbe seen that densification of the network helps in SINR coverage. When the BBS density is verysmall, network is noise limited. As λ increases, BBSs comes closer to the user improving SINRcoverage while SIR coverage remains constant. After a certain λ , the increase in λ reduces theinterference also. Hence, both SIR and SINR coverage improves. At high value of λ , coveragebecomes 1 as all dominant BBSs provide serving power. The behavior of SIR coverage is similarto as seen in networks with dual-slope pathloss [19]. -4 -3 -2 -1 BBS Density (/km )00.20.40.60.81 C o ve r a g e P r ob a b ili t y ( p c ) R s =25 km R s =5 km -4 -3 -2 -1 BBS Density (/km )00.20.40.60.81 R a t e C o ve r a g e ( r c ) R s =5 km R s =25 km Fig. 6. Coverage vs. BBS density λ for different values of R s in a broadcast network. Here, rest of the parameters are accordingto Table I. (a) SINR coverage. Dashed lines indicate SIR coverage while ignoring noise. (b) Rate coverage. Dashed lines indicaterate coverage ignoring the noise. IV. S
CENARIO
I: C
OVERAGE A NALYSIS FOR N ETWORKS WITH U SERS HAVING A H IGH L EVEL OF S PATIAL H ETEROGENEITY IN C ONTENT P REFERENCE
We now extend the system model to include networks with users having their individualcontent or advertisement preferences. In this section, we consider the first scenario where thereis high level of spatial heterogeneity in users. This means that all classes of users are present inany region. Given the limitation of resources, network selects n classes of users and shows n contents (one for each class) at any point of time. Here, n is a design parameter decided by thebroadcast network. Since the user classes are spatially inseparable, each BBSs should transmitto the same n contents to improve SINR coverage. We have assumed OFDM based transmissionwhere a BBS transmits the n number of contents on orthogonal resources. A. SINR and SINR Coverage
To improve coverage, the network can use the same bands for a particular content across allBBSs. Since all the BBSs are transmitting the same data in a band, the SINR for a typical useris the same as given in (5). Similarly, in this case, the SINR coverage probability of a typicaluser is the same as given in Theorem 1.
B. Rate Coverage
Now, the available resources are divided among n contents. If the total available bandwidthis W , the bandwidth available for each content is W/n . The instantaneous achievable rate for atypical user located at origin, for each content is
Rate = ξ Wn log (1 + SINR ) . (22)From (20), the rate coverage probability is given as: r c ( ρ ) = P (cid:2) SINR > nρ/ ( ξW ) − (cid:3) = p c (2 nρ/ ( ξW ) − (23)where p c is given in (15). C. Network Revenue
Let ρ denote the minimum rate required for a user to be able to view the content. Then, therate coverage r c at ρ denotes the fraction of users that are able to view this content. Therefore, r c unit of revenue will be earned by the network from a particular class, since only r c fractionof users can watch it. Therefore, the network’s total revenue R n can be given as: R n = n r c ( ρ ) . (24) N e t w o r k R eve nu e ( R n ) = 5 Mbps = 10 Mbps = 15 Mbps = 20 Mbps Fig. 7. Variation of total revenue with respect to allowed number of user classes for different rate threshold ρ (Mbps) (which isa proxy for content quality requested). Content granularity is N c = 15 . Other parameters are according to Table I. It is observedthat an optimal value of n can provide the maximum revenue to the network which depends on the content quality requested. D. Numerical Results: Impact of n on Total Revenue Fig. 7 shows the variation of the network revenue R n with n for a system with N c = 15 .We can observe that initially, the revenue increases with an increase in n up to a certain value,and thereafter, starts decreasing. This can be justified in the following way. If n increases, thefollowing two effects take place– (1) more user classes are served, causing a linear increase in R n , and (2) available bandwidth W/n for each content/advertisement decreases causing the ratecoverage to drop. As a combined effect dictated by (24), the revenue is optimal at a particularvalue of n . However, this behavior also depends on the target rate threshold. At a higher ratethreshold, R n decreases with n , showing n = 1 as the optimal choice. This implies that theoptimal number of user classes that can be served, depends on the quality of the content. Ifthe quality requested is high, then it may be better to serve less user classes, while more userclasses can be served when the quality requested is lower. V. S
CENARIO
II: C
OVERAGE A NALYSIS FOR N ETWORKS WITH U SERS HAVING S PATIALLY S EPARATED C LASSES FOR C ONTENT P REFERENCE
The second scenario we consider corresponds to the case where user classes are spatiallyseparated. For tractability, we assume that coverage area of each BBS comprises of users froma single class. Thus, users of different classes are spatially separated. Let S i denote the class ofthe i th BBS located at X i which is a uniform discrete random variable with PMF given as p S i ( k ) = 1 N c (1 ≤ k ≤ N c ) . (25)We assume that S i ’s are independent of each other.The network can cater to all user classes by letting each BBS to transmit content accordingto the class of users lying in its coverage area. Note that for a typical user, only those BBSsthat transmit the same content, can contribute to the serving signal power at this user. Therefore,this strategy will reduce the number of serving BSs and hence reduce the SINR. On the otherhand, network can decide to cater only one user class by forcing all BBSs to transmit only onecontent, will reduce the revenue as users of only one class will receive their preferred content. Itwill be interesting to find the optimal number of user-classes that can be catered by the network.As a general problem, we consider that the network decides to cater n user classes out of total N c classes. Let us denote the set of all selected index by M .Let us denote the content transmitted by i th BS by M i . If the BBS’s user class is one ofthe n selected classes ( i.e. S i ∈ M ), it will transmit the content corresponding to its user class i.e. M i = S i . If the BBS’s user class is not one of the selected classes, it will transmit the contentcorresponding to a randomly selected user class to help boost its signal strength.Let us consider a typical user at o . Without loss of generality, assume that its user class is 1.The probability that its class is one of the n selected classes is n/N c . Let us condition on thefact that it is one of the selected classes.Let the tagged BS of this typical user transmits the content M = 1 . Then, for the i th BBS, M i = M if1) S i = M which occurs with probability N c , or2) S i / ∈ M and S i = M i which occurs with probability N c − nN c · n . Therefore, the probability that the i th BBS is transmitting the content as per the preference ofthis typical user is p = P [ M i = M ] = 1 N c + N c − nN c · n = 1 n . (26) A. SINR
Now, note that the BBSs that are transmitting the same content as M and are located inside B ( o , X + R s ) will contribute to the desired signal power to the typical user at origin o . Therefore,the desired signal power for the typical user at origin is given as S = p t aβ X − α + (cid:88) X i ∈ Φ ∩B (0 ,X + R s ) \ X p t aβ i X i − α ( M i = M ) . (27)Similarly, the interference for the typical user is caused by the BBSs that are either locatedoutside B ( o , u + R s ) or located inside the B ( o , X + R s ) but transmitting a different content than M . Hence, the total interference is given as I = (cid:88) X i ∈ Φ ∩B (0 ,X + R s ) \ X p t aβ i X i − α ( M i (cid:54) = M ) + (cid:88) X j ∈ Φ ∩B (0 ,X + R s ) (cid:123) p t aβ j X j − α . (28)Now, the SINR for this user is given as SINR = SI + N = β (cid:107) X (cid:107) − α + (cid:80) X i ∈ Φ ∩B (0 ,u + R s ) \ X β i (cid:107) X i (cid:107) − α ( M i = M ) (cid:80) X i ∈ Φ ∩B (0 ,u + R s ) \ X β i (cid:107) X i (cid:107) − α ( M i (cid:54) = M ) + (cid:80) X j ∈ Φ ∩B (0 ,u + R s ) (cid:123) β j (cid:107) X j (cid:107) − α + σ . (29)Here, σ is the normalized noise power which is given as σ = N/ ( p t a ) where N is the noisepower. B. SINR Coverage Probability
We now calculate the SINR coverage for a typical user. Similar to Section III-B, the SINRcoverage probability is given as p c ( τ, λ ) = 12 + 1 π (cid:90) ∞ (cid:90) ∞ πλue − πλu t Im (cid:104) L I | X ( jtτ ) e − jtτσ L S | X ( − jt ) (cid:105) d t d u (30)where S and I are given in (27) and (28). Lemma 2.
Conditioned on the location of the closest serving BBS, the Laplace transforms ofthe desired signal power and the sum interference at the receiver are given as L S | X ( s ) = 11 + su − α exp (cid:18) − πλp (cid:90) X + R s X sr − α sr − α r d r (cid:19) (31) L I | X ( s ) = exp (cid:18) − πλ (1 − p ) (cid:90) X + R s X sr − α sr − α r d r − πλ (cid:90) ∞ X + R s sr − α sr − α r d r (cid:19) . (32) Proof.
See Appendix C.Using Lemma 2 and (30) we can calculate the SINR coverage which is given in Theorem 2.
Theorem 2.
The probability of SINR coverage for the user located at the origin in a broadcastnetwork with λ density of BBSs, is given as: p c ( τ, λ ) = 12 + 1 π (cid:90) ∞ πλue − πλu (cid:90) ∞ t Im (cid:34) e − jtτσ − jtu − α × exp (cid:18) − πλ (cid:18) p (cid:90) u + R s u − jtr − α − jtr − α r d r + (1 − p ) (cid:90) u + R s u jtτ r − α jtτ r − α r d r + (cid:90) ∞ u + R s jtτ r − α jtτ r − α r d r (cid:19)(cid:19)(cid:21) d t d u = 12 + 1 π (cid:90) ∞ (cid:90) ∞ v s (cid:20)
11 + s v − α (cid:21) e − v e − s α M (cid:48) d ( s,v ) × (cid:104) sv − α cos (cid:16) s α N (cid:48) d ( s, v ) + τ sK (cid:17) − sin (cid:16) s α N (cid:48) d ( s, v ) + τ sK (cid:17)(cid:105) d v d s (33) where M (cid:48) d ( t, u ) and N (cid:48) d ( t, u ) is given as M d ( s, v ) = 1 α (cid:20) pQ (cid:18) α , s ( v + m ) − α , s v − α (cid:19) + (1 − p ) τ /α Q (cid:18) α , τ s ( v + m ) − α , τ s v − α (cid:19) + τ /α Q (cid:18) α , , τ s ( v + m ) − α (cid:19)(cid:21) (34) N d ( s, v ) = 1 α (cid:20) − pQ (cid:18) α + 12 , s ( v + m ) − α , s v − α (cid:19) +(1 − p ) τ /α Q (cid:18) α + 12 , τ s ( v + m ) − α , τ s v − α (cid:19) + τ /α Q (cid:18) α + 12 , , τ s ( v + m ) − α (cid:19)(cid:21) (35) Proof.
See Appendix D. C. Rate Coverage
Since each BBS shows only one content, it can use the total available bandwidth W . Theinstantaneous achievable rate for a typical user located at origin, while receiving the content, is Rate = ξW log (1 + SINR ) . (36)From (20), the rate coverage probability is given as: r c ( ρ ) = P (cid:2) SINR > ρ/ ( ξW ) − (cid:3) = p c (2 ρ/ ( ξW ) − (37)where p c is given in Theorem 2. D. Network Revenue
Let ρ denote the minimum rate required for a user to be able to view the content. Then, therate coverage r c at ρ denotes the fraction of users that are able to view this content. Therefore, r c unit of revenue will be earned by the network from a particular class, since only r c fraction ofusers can watch it. The probability that the typical user receives the content as per its preferenceis n/N c which is also the probability that the network will receive revenue from this typicaluser. Similar to previous sections, the total revenue can be computed as R n = nN c · r c ( ρ ) , (38)where the rate coverage r c for the typical user is given by (37). E. Numerical Results
We now present numerical results for the considered scenario II. The parameters are stated inTable I.
Impact of n on SINR and rate coverage probability: Fig. 8(a) shows the variation of SINRcoverage with n for different values of connectivity radius R s . Here, SINR threshold τ = 10 dBand there are N c = 15 user classes. From Fig. 8, we observe that for a fix value of R s , SINRcoverage decreases with increase in n . This is due to the fact that more BBSs interfere as n increases. However, after a certain n , the coverage doesn’t changes much with n . This is becauseadditional fraction of BBSs that interfere when n increases by 1, is equal to nn +1 − n − n = n ( n +1) which decreases very fast with n . Therefore, after a certain n , there will not be a significantincrease in the interference, which makes the SINR constant with n . S I NR C o ve r a g e ( P c ) R s = 5 kmR s = 10 kmR s = 20 kmR s = 50 km R a t e C o ve r a g e ( R c ) R s = 5 kmR s = 10 kmR s = 20 kmR s = 50 km Fig. 8. Variation of SINR and rate coverage probability with respect to allowed number of user classes for different valuesof connectivity radius ( R s in km) at a typical user in a broadcast system with geographically separated user classes. Contentgranularity is N c = 15 , SINR threshold τ = 10 dB and rate threshold ρ = 15 Mbps. Network density is λ = . /km . Thebandwidth varies with R s according to (21) with maximum value at 80 MHz. Other parameters are according to Table I. Thecoverage decreases with n due to increased interference at the typical user. It is also observed that the decrease in the SINR coverage probability is faster when R s islarge. This can be justified as follows. First note that n only affects the BBSs that can either be ainterferer or a serving BBS, depending on the content they are showing. These BBSs lie in the ringof R s width denoting the region and their number approximately scales as λπR . Note that thisnumber is large when R s is large. When we allow BBSs to show more advertisements/contents,a large number of these BBSs means that there is a larger number of potential interferes. When R s is small, there are less number of these potential BBSs (or even 0), hence allowing moreadvertisements doesn’t affect the coverage significantly. Fig. 8(b) shows the variation of ratecoverage with respect to n for different size of connectivity region. The rate coverage alsofollows similar behavior as SINR coverage as described in (37). Impact of n on the total network revenue: Fig. 9 shows the impact of n on the total networkrevenue for different values of rate threshold. Increase in n means catering to more number ofuser classes. From Fig. 9, we can observe that the revenue initially decreases and then, increaseswith n . The initial decrease in the revenue seen from n = 1 to n = 2 for some configurationsis due to the decrease in rate coverage from n = 1 to n = 2 , as observed in Fig. 8(b) whichdominated the increase in the revenue generated due to catering to an additional user class.However, this behavior may depend on the target rate threshold and the value of R s . Impact of R s on the total network revenue Fig. 10 shows the total revenue with respect to N e t w o r k R eve nu e ( R n ) = 5 Mbps = 10 Mbps = 15 Mbps R s =5 km R s =50 km Fig. 9. Variation of the total network revenue with respect to allowed number of user classes n for different values of ratethreshold ρ (in Mbps) with R s = 150 km for a broadcast system with geographically separated user classes. Content granularityis N c = 15 . Here, bandwidth varies with R s according to (21). Other parameters are according to Table I.
20 40 60 80 100R s (km)00.10.20.30.40.50.60.7 N e t w o r k R eve nu e ( R n ) n = 1n = 2n = 5n = 10n = 15 Fig. 10. Variation of the total network revenue with respect to the connectivity radius R s (in km) for different values of n with ρ = 5 Mbps for a broadcast system with geographically separated user classes. Content granularity is N c = 15 . Here,bandwidth varies with R s according to (21) with maximum value at 50 MHz. Other parameters are according to Table I. size of connectivity region R s for different n with ρ = 10 Mbps and N c = 15 . We can observethat, the revenue decreases with increase in R s . Also the revenue increases with n for lowervalues of R s . For a higher values of R s , the behavior with n is not monotonic. The revenuefor n ≥ may fall below the revenue for n = 1 . As discussed previously, the rate coveragedecreases drastically from n = 1 to n = 2 for large R s which can dominate the increase in therevenue generated due to catering to an additional user class. VI. C
ONCLUSIONS
In this paper, we presented an analytical framework for the system performance of a broadcastnetwork using stochastic geometry. Since all BBSs in the broadcast network are transmitting thesame signal, signals from multiple BSs can be used to improve the coverage. We show that thereexists a region such that all BBSs lying in this region may contribute to the desired signal power.We computed the SINR and rate coverage probability for a typical user located at the origin. Wevalidated our results using numerical analysis. Using these results, we found that there existsan optimal region size which maximized the rate coverage. When users consists of many userclasses having heterogenous content preference, network can schedule content to maximize itsrevenue. We presented an analytical model of revenue thus obtained from users. The results arevalidated through numerical analysis. We also present the variation of total revenue with respectto various parameters including number of user classes to be catered, size of connectivity region,and rate threshold. We show how content quality also affects the network decision on variety ofcontent shown by the operator. A
PPENDIX AP ROOF FOR L EMMA I | X is given as L I | X ( s ) = E (cid:2) e − sI | X (cid:3) = E exp − s (cid:88) X j ∈ Φ ∩B (0 ,X + R s ) (cid:123) β j (cid:107) X j (cid:107) − α ( a ) = exp (cid:18) − λ (cid:90) Φ ∩B (0 ,X + R s ) (cid:123) (cid:16) − E β (cid:104) e − sβ (cid:107) X (cid:107) − α (cid:105)(cid:17) d x (cid:19) ( b ) = exp (cid:18) − πλ (cid:90) ∞ X + R s (cid:16) − E β (cid:104) e − sβr − α (cid:105)(cid:17) r d r (cid:19) (39)where (a) is due to the probability generating functional (PGFL) of homogeneous PPP [7] and(b) is due to conversion to polar coordinates. Now, since β ’s are exponetially distributed, usingtheir MGF, we get L I ( s ) = exp (cid:18) − πλ (cid:90) ∞ u + R s sr − α sr − α r d r (cid:19) . (40) Similarly, the Laplace transform of desired signal power conditioned on the nearest is BSlocated at X , is given as: L S | X ( s ) = E S | X (cid:2) e − sS | X (cid:3) = E β i , X i exp − s β (cid:107) X (cid:107) − α + (cid:88) X i ∈ Φ ∩B ( o ,X + R s ) \ X β i (cid:107) X i (cid:107) − α = E Φ | X (cid:104) E β | X (cid:104) e − sβ (cid:107) X (cid:107) − α (cid:105) × (cid:89) X i ∈ Φ ∩B ( o ,X + R s ) \ X E β i | X i , X (cid:104) e − sβ i (cid:107) X i (cid:107) − α ) (cid:105) (41)Now, from Slivnyak theorem [7], we know that conditioned on X , { X i : X i ∈ Φ } \ { X } is a PPP with the same density. Therefore, using the PGFL of a PPP and noting that β i ’s areexponential RVs, we get L S | X ( s ) = 11 + s (cid:107) X (cid:107) − α exp (cid:18) − πλ (cid:90) X + R s X (cid:18) −
11 + sr − α (cid:19) r d r (cid:19) = 11 + sX − α exp (cid:18) − πλ (cid:90) X + R s X sr − α sr − α r d r (cid:19) . (42)A PPENDIX BP ROOF OF T HEOREM L S | X = u ( − jt ) = 1 + jtu − α t u − α exp (cid:18) − πλ (cid:90) u + R s u − jtr − α − jtr − α r d r (cid:19) (43) L I | X = u ( jtτ ) = exp (cid:18) − πλ (cid:90) ∞ u + R s jtτ r − α jtτ r − α r d r (cid:19) . (44)Substituting the above values in (11), we get P (cid:0) S > ( I + σ ) τ | X = u (cid:1) = 12 + 1 π (cid:90) ∞ t Im (cid:20)(cid:18) jtu − α t u − α (cid:19) e − jtτσ × exp (cid:18) − πλ (cid:18)(cid:90) ∞ u + R s jtτ r − α jtτ r − α r d r + (cid:90) u + R s u − jtr − α − jtr − α r d r (cid:19)(cid:19)(cid:21) d t = 12 + 1 π (cid:90) ∞ t Im (cid:20)(cid:18) jtu − α t u − α (cid:19) e − jtτσ × exp (cid:18) − πλ (cid:18)(cid:90) ∞ u + R s t τ r − α +1 t τ r − α d r + (cid:90) u + R s u t r − α +1 t r − α d r + j (cid:90) ∞ u + R s tτ r − α +1 t τ r − α d r − j (cid:90) u + R s u tr − α +1 t r − α d r (cid:19)(cid:19)(cid:21) d t (45)where the last step is obtained using multiplication of conjugate terms. Now, if we define M ( t, u ) = 2 αt − /α (cid:20)(cid:90) u + R s u t r − α +1 t r − α d r + (cid:90) ∞ u + R s t τ r − α +1 t τ r − α d r (cid:21) (46) N ( t, u ) = 2 αt − /α (cid:20)(cid:90) ∞ u + R s tτ r − α +1 t τ r − α d r − (cid:90) u + R s u tr − α +1 t r − α d r (cid:21) (47)(45) can be written as P (cid:0) S > ( I + σ ) τ | X = u (cid:1) = 12 + 1 π (cid:90) ∞ t Im (cid:20)(cid:18) jtu − α t u − α (cid:19) × exp (cid:0) − πλt /α M ( t, u ) − j πλt /α N ( t, u ) − jtτ σ (cid:1)(cid:3) d t. (48)Now, with some trivial manipulations and substituting (48) in (9), we get p c ( τ, λ ) = 12 + 1 π (cid:90) ∞ (cid:90) ∞ πλue − πλu · t · (cid:20)
11 + t u − α (cid:21) e − πλt /α M ( t, u ) / α × (cid:104) tu − α cos (cid:16) πα λt /α N ( t, u ) + tτ σ (cid:17) − sin (cid:16) πα λt /α N ( t, u ) + tτ σ (cid:17)(cid:105) d t d u. (49)Further, the forms of M and N can be simplified by trivial manipulations and definition ofincomplete Beta function in (46) and (47) to get M ( t, u ) = Q (cid:18) α , t ( u + R s ) − α , t u − α (cid:19) + τ /α Q (cid:18) α , , t τ ( u + R s ) − α (cid:19) , and (50) N ( t, u ) = − Q (cid:18) α + 12 , t ( u + R s ) − α , t u − α (cid:19) + τ /α Q (cid:18) α + 12 , , t τ ( u + R s ) − α (cid:19) (51)Now, we can substitute t → s/ ( λπ ) α/ u → v/ √ λπ (52)in (49), (50) and (51) to get the desired result. A PPENDIX CP ROOF OF L EMMA S conditioned that the nearest BSlocated at X is given as: L S | X ( s ) = E S | X (cid:2) e − sS (cid:3) = E { β i , X i }| X exp − sβ X − α − (cid:88) X i ∈ Φ ∩B ( o ,X + R s ) \ X sβ i X i − α ( M i = M ) which is similar to (41) except the fact that the summation in the last term is over only thosepoints that satisfy an additional condition M i = M . From the independent thinning theorem,these points also form a PPP with density λ P [ M i = M ] = λp . Now using the PGFL of thisPPP, we get L S | X ( s ) = 11 + sX − α exp (cid:18) − πλp (cid:90) X + R s X sr − α sr − α r d r (cid:19) . (53)Now, from (24), the Laplace transform of sum interference is L I ( s ) = E I (cid:2) e − sI (cid:3) = E I exp − (cid:88) X i ∈ Φ ∩B (0 ,X + R s ) \ X sβ i X i − α ( M i (cid:54) = M ) − (cid:88) X j ∈ Φ ∩B (0 ,X + R s ) (cid:123) sβ j X j − α ( a ) = exp (cid:18) − πλ (1 − p ) (cid:90) X + R s X (cid:16) − E β (cid:104) e − sβr − α (cid:105)(cid:17) r d r − πλ (cid:90) ∞ X + R s (cid:16) − E β (cid:104) e − sβr − α (cid:105)(cid:17) r d r (cid:19) ( b ) = exp (cid:18) − πλ (1 − p ) (cid:90) X + R s u sr − α sr − α r d r − πλ (cid:90) ∞ X + R s sr − α sr − α r d r (cid:19) , (54)where (a) is due to the probability generating functional of homogeneous PPP and independentthinning theorem and (b) is due to MGF of exponentially distributed β i ’s.A PPENDIX DP ROOF OF T HEOREM L S | X = u ( − jt ) = 1 + jtu − α t u − α exp (cid:18) − πλp (cid:90) u + R s u − jtr − α − jtr − α rdr (cid:19) (55) L I | X = u ( jtτ ) = exp (cid:18) − πλ (cid:20) (1 − p ) (cid:90) u + R s u jtτ r − α jtτ r − α r d r + (cid:90) ∞ u + R s jtτ r − α jtτ r − α r d r (cid:21)(cid:19) . (56) Substituting the above values in (11), we get P (cid:0) S > ( I + σ ) τ | X = u (cid:1) = 12 + 1 π (cid:90) ∞ t Im (cid:20)(cid:18) jtu − α t u − α (cid:19) e − jtτσ · exp (cid:18) − πλ (cid:18)(cid:90) ∞ u + R s jtτ r − α jtτ r − α r d r + (1 − p ) (cid:90) u + R s u jtτ r − α jtτ r − α r d r + p (cid:90) u + R s u − jtr − α − jtr − α r d r (cid:19)(cid:19)(cid:21) d t = 12 + 1 π (cid:90) ∞ t Im (cid:20)(cid:18) jtu − α t u − α (cid:19) e − jtτσ · exp (cid:18) − πλ (cid:18)(cid:90) u + R s u p t r − α +1 t r − α d r (cid:90) ∞ u + R s t τ r − α +1 t τ r − α d r + (cid:90) u + R s u (1 − p ) t τ r − α +1 t τ r − α d r + j (cid:20)(cid:90) ∞ u + R s tτ r − α +1 t τ r − α d r + (cid:90) u + R s u (1 − p ) tτ r − α +1 t τ r − α d r − (cid:90) u + R s u p tr − α +1 t r − α d r (cid:21)(cid:19)(cid:19)(cid:21) d t (57)where the last step is obtained using multiplication of conjugate terms and rearranging into thereal and imaginary parts. Now, if we define M (cid:48) ( t, u ) =2 αt − α/ (cid:20)(cid:90) u + R s u p t r − α +1 t r − α d r + (cid:90) Ru + R s t τ r − α +1 t τ r − α d r + (cid:90) u + R s u (1 − p ) t τ r − α +1 t τ r − α d r (cid:21) , (58) N (cid:48) ( t, u ) =2 αt − α/ (cid:20)(cid:90) Ru + R s tτ r − α +1 t τ r − α d r + (cid:90) u + R s u (1 − p ) tτ r − α +1 t τ r − α d r − (cid:90) u + R s u p tr − α +1 t r − α d r (cid:21) , (59)(57) can be written as P (cid:0) S > ( I + σ ) τ | X = u (cid:1) = 12 + 1 π (cid:90) ∞ t Im (cid:20)(cid:18) jtu − α t u − α (cid:19) × exp (cid:0) − πλt /α M (cid:48) ( t, u ) − j πλt /α N (cid:48) ( t, u ) − jtτ σ (cid:1)(cid:3) d t. (60)Now, with some trivial manipulations and substituting (60) in (9), we get p c ( τ, λ ) = 12 + 1 π (cid:90) ∞ (cid:90) ∞ πλue − πλu · t · (cid:20)
11 + t u − α (cid:21) · e − πα λt /α M (cid:48) ( t, u ) × (cid:104) tu − α cos (cid:16) πα λt /α N (cid:48) ( t, u ) + tτ σ (cid:17) − sin (cid:16) πα λt /α N (cid:48) ( t, u ) + tτ σ (cid:17)(cid:105) d t d u. (61) Further, the forms of M and N (cid:48)(cid:48) can be simplified by trivial manipulations and definition ofincomplete Beta function in (58) and (59) to get M (cid:48) ( t, u ) = pQ (cid:18) α , t ( u + R s ) − α , t u − α (cid:19) + τ /α (1 − p ) Q (cid:18) α , ( tτ ) ( u + R s ) − α , ( tτ ) u − α (cid:19) + τ /α Q (cid:18) α , , t τ ( u + R s ) − α (cid:19) . (62)and, N (cid:48) ( t, u ) = − pQ (cid:18) α + 12 , t ( u + R s ) − α , t u − α (cid:19) + τ /α (1 − p ) Q (cid:18) α + 12 , ( tτ ) ( u + R s ) − α , ( tτ ) u − α (cid:19) + τ /α Q (cid:18) α + 12 , , t τ ( u + R s ) − α (cid:19) . (63)Now, we can substitute t → s/ ( λπ ) α/ u → v/ √ λπ (64)in (61), (62) and (63) to get the desired result.R EFERENCES [1] M. El-Hajjar and L. Hanzo., “A survey of digital television broadcast transmission techniques,”
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