User-Cognizant Scalable Video Transmission over Heterogeneous Cellular Networks

Wu, Liang; Zhang, Wenyi

doi:10.1007/978-981-10-1394-2_36

Liang Wu^2,3 &
Wenyi Zhang^4,5

1037 Accesses

Abstract

With the increase of mobile video applications in people’s daily life as well as industrial manufacture, such as video streaming, surveillance, and so on, video has been the main service in cellular networks. Operators and service providers are struggling to enhance the mobile video service, while user requirements for abundant, high-definition, and low-delay video have nearly drained the transmission capacity of current networks. Moreover, the large population of user equipments (UEs) exhibit differentiated video demands and various network transmission environments. Traditional networking, which is static and base station (BS) concentric, can hardly deal with these challenges. Thus, adaptive video transmission schemes are needed by jointly considering the interplay among user demand, video source characteristics, and networking. This work focuses on user-cognizant scalable video transmission over heterogeneous cellular networks. The video source is encoded using scalable video coding, which enables dynamic adaption of source information to the requirements of UEs and is suitable for cellular networks in which the transmission link quality varies substantially over space and time. Three novel transmission schemes are proposed, layered digital transmission, layered hybrid digital-analog transmission, and cooperative digital transmission. Leveraging tools from stochastic geometry, a comprehensive analysis is conducted focusing on three key performance metrics: outage probability, high-definition probability, and average distortion. The associated spectrum allocation and video transmission are chosen based on the user-cognizant information, such as the requirements for video service, wireless channel status, and the connections with the BSs. The results show that the proposed user-cognizant transmission schemes can provide a scalable video experience for UEs.

Access provided by Autonomous University of Puebla. Download reference work entry PDF

User-Cognizant Scalable Video Transmission over Heterogeneous Cellular Networks

Reliable rate-adaptive video transmission over cognitive cellular networks using multiple description scalable coding

Article 07 August 2018

Adaptive Resource Allocation for Scalable Video Transmission over OFDMA Systems

Article 27 June 2016

Keywords

Introduction

Emerging wireless communication technologies as well as powerful and versatile mobile terminal devices have changed people’s daily life, and the data traffic grows explosively, among which a substantial portion is attributed to multimedia such as mobile video. According to the Cisco Visual Networking Index, mobile video is expected to grow at an average growth rate of 62% until 2020, and within the 30.6 exabytes of data per month crossing mobile networks by 2020, 23.0 exabytes will be video related, such as video on demand, real-time streaming video, video conferencing, and so on. The ever-increasing demand for abundant, high-definition, and low-latency mobile video brings great challenges to the mobile network with time-varying wireless channel. Moreover, with the release of different types of UEs, the requirements on data rate of video transmission vary in a wide range.

Compared to the IP transmission network and cellular core network, the bottleneck of the end-to-end transmission degrading the Quality of Experience (QoE) of video lies in the radio access network due to user traffic congestion and packet loss. Current LTE/LTE-A networks are not inheritably built for QoE-aware video delivery. The application-specific information exists at Packet Data Network Gateway (P-GW), while the wireless channel quality and connection status are restrictively known by eNodeB.

State-of-art design of cellular networks is base station (BS) concentric, which means that the resource allocation and transmission schedule are completed at the BSs and are not on-demand for UEs. The information bits are treated equally and the transmission strategy is not UE specific. As for the video transmission of a typical UE, the UE would require the video content with different video qualities based on its terminal capacity. Meanwhile, the UE can choose different service mechanisms provided by the cellular network based on the connection status.

Taking into account of time-varying wireless channel conditions and congestion, video streams are adapted to reduce the transmission bitrates. Traditionally, rate adaptation of video streams is realized by packet/frame dropping or transcoding with some serious drawbacks, since packet/frame dropping significantly degrades the video quality and transcoding is computationally complex. Advanced source coding techniques provide a new dimension of dynamically provisioning wireless resources for the varying requirements and the varying link conditions of UEs, thus creating the possibility of extracting video scaled in multiple dimensions, e.g., spatial, temporal, and quality. Scalable Video Coding (SVC) is an extension of the H.264/MPEG-4 AVC video compression standard [1] and has been evolved to Scalable High-Efficiency Video Coding (SHVC) [2], in which the bitstream is encoded into multiple layers, namely, a base layer (BL) and at least one enhancement layer (EL). The quality of reconstructed video depends on the number of layers decoded and stays the same until a higher enhancement layer is successfully decoded. The number of layers and their code rates may be determined by the requirement and the link condition of the subscribing UE.

On the other hand, cellular networks are evolving from a homogenous architecture to a composition of heterogeneous networks, comprised of various types of base stations (BSs) [3, 4]. Each type of BSs has its characteristic transmit power and deployment intensity: for example, macro BSs (MBSs) have larger transmit power, aiming at providing global coverage; femto access points (FAPs) are small BSs targeted for home or small business usages. As the distance between a UE and its serving FAP is small, the UE enjoys a high-quality link and achieves power savings. Meanwhile, the reduced transmission range also enhances spatial reuse and alleviates multiuser interference. In addition, different types of BSs can transmit cooperatively the same video content to the UE to enhance the quality of experience. The user-cell association approach for heterogenous networks should be addressed to exploit context information as well as channel-related information extracted from UEs. Generally speaking, there are two different service modes, separate mode and cooperative mode. In separate mode, the macro cells and the small cells (e.g., femto cells) transmit different video streams to the UEs in a manner of dual connectivity. In cooperative mode, both the macro cells and small cells transmit the same video content to the UE in a manner of coordinated multipoint transmission. In this paper, we study the problem of scalable transmission over heterogeneous networks and demonstrate the performances of several user-cognizant transmission schemes to exploit the combination of multi-layer video transmission and multitier cellular networks.

Related Work

The prior works that consider video transmission over wireless networks mainly focus on scalable coding of video source or adaptive networking techniques separately. Furthermore, UE is regarded as a dummy terminal, and thus their differentiated demand and status are neglected. The analysis usually focuses on homogeneous networks, and the common feature of the layered structure of SVC and HCNs is not exploited. In [5], an overview of SVC and its relationship to mobile content delivery are discussed focusing on the challenges due to the time-varying characteristics of wireless channels. In [6], a per-subcarrier transmit antenna selection scheme is employed to support multiple scalable video sequences over a downlink cognitive network, and the outage probability is reduced because of video scalability. In [7], real-time use cases of mobile video streaming are presented, for which a variety of parameters like throughput, packet loss ratio, and delay are compared with H.264 single-layer video under different degrees of scalability. In [8], the proposed scheme employs WiFi: the BL is always transmitted over a reliable network such as cellular, whereas the EL is opportunistically transmitted through WiFi. Technical issues associated with the simultaneous use of multiple networks are discussed. In [9], HCNs with storage-capable small-cell BSs are studied: versions and layers of video have different impacts on the delay-servicing cost tradeoff, depending on the user demand diversity and the network load.

Some works related to QoE-aware or adaptive strategies have also been studied previously. Chen et al. [10] proposed an admission control strategy that was designed to maximize the number of video users satisfying the QoE constraints on their second-order statistics. Although the admission control strategy damages the QoE of the blocked users, the overall percentage of users satisfying the QoE constraints among both admitted and blocked users can be significantly improved. Thakolsri et al. [11] proposed a content-aware scheduling and resource allocation, taking into account the content characteristics of the video streams, and performs video rate adaptation at the BS. Fu et al. [12] proposed a QoE-aware video delivery by considering the hierarchical architecture of LTE/LTE-A network. Different video flows are marked at the core network to transform the video content information into QoE-aware priority classes. A packet dropping strategy addressing the transmission capacity at the eNodeB is also proposed. But the priority marking process at the core network is unaware of channel status.

Considering the wireless video transmission techniques, there are several approaches to enhance the QoE of video users. The above literature is based on digital transmission, consisting of digital source coding (e.g., quantization and entropy coding), digital modulation (e.g., QPSK, 64QAM), and digital channel coding (e.g., turbo or LDPC). Unfortunately, digital transmission results in cliff effect. The cliff effect refers to the drastic degradation in video quality when the signal strength fades below the decoding threshold (as opposed to a graceful degradation). There exist certain SINR thresholds at which the video quality changes drastically; in between these thresholds, the quality stays approximately constant. The recently revitalized analog transmission has shown promising potential in handling channel variations and user heterogeneities for wireless video communication. The analog scheme consists of analog source coding and analog modulation that directly maps a source signal into a linearly transformed channel signal without channel coding. SoftCast [13] is an analog video broadcast scheme that transmits a linear transform of the video signal without quantization, entropy coding, or channel coding. It is claimed to realize continuous quality scalability. However, information-theoretic studies (such as [14, 15]) show that analog schemes with linear mapping (from source signals to channel signals) are relatively inefficient for video transmission, while hybrid digital-analog transmission is asymptotically optimal under matched channel conditions for optimally chosen power allocations between the analog and digital parts. The hybrid digital-analog scheme combines digital with analog schemes, transmitting digital and analog signals simultaneously using TDMA, FDMA, or superposition transmission. The authors in [16] propose a hybrid digital-analog scheme for broadcasting, showing a substantial performance gain. However, these works did not consider the impacts of HCNs and the spatial distribution of wireless networks, let alone the design of scalable transmission algorithms utilizing the structure of HCNs.

Moreover, coordinated multipoint (CoMP) transmission is intensively studied to enhance the system performance of LTE-A. By coordinating multiple BSs, the interference at the UE can be alleviated, or multiple received signals can be merged. The studies in [17, 18] evaluate the potential system gain of CoMP and discuss the appropriate deployment scenarios. The authors also review the necessary techniques of signal processing, backhaul link design, and supported protocols. There exist two types of cooperative transmissions, namely, coherent and noncoherent joint transmission. Many previous studies considered noncoherent joint transmission because it requires less channel status information. The authors in [19, 20] analyze the performance of noncoherent joint transmission in heterogeneous cellular networks and give the distribution of SINR for a user in a random position and cell edge, respectively. In addition, the impact of channel status information is also studied. Most previous works neglect the spectrum sub-band allocation, but the same sub-band is required when two BSs transmit cooperatively. Bang et al. [21] combines frequency fraction reuse and CoMP, and proposes an allocation to minimize the system power. Zhang et al. [22] and Kosmanos et al. [23] propose a joint sub-band allocation and power optimization scheme to improve the spectrum efficiency for LTE-A when BS and relay transmit cooperatively.

In order to give a theoretic analysis of the system performance, stochastic geometry has been utilized as an effective tool for modeling and analyzing cellular networks; see, e.g., [24,25,26] and references therein. Generally, the spatial distribution of BSs is modeled as a spatial point process, such as the homogeneous Poisson point process (PPP) for single-tier networks, for which the coverage probability is derived in [27]. For HCNs, the spatial distribution of heterogeneous BSs is often modeled as multiple independent tiers of PPPs, and several key statistics are analyzed in [20, 28]. A comprehensive treatment of the application of stochastic geometry in wireless communication and content can be found in [29, 30].

A User-Cognizant Solution

Most existing works on heterogenous networks have focused on admission control, resource allocation, and transmission coordination. The serving BS associated to a particular UE is assigned based on indicators of the wireless link quality at the UEs, such as the received signal strength indicators (RSSIs) or the SINRs. The same priority is allocated to each information bit for different video data flows in the scheduling stage. All these networking designs again verify that the current network is inefficient for video transmission. To enhance the QoE, one promising approach for efficient networking is by making the network better informed of its environment and user requirements.

Thus, considering scalable video transmission over HCNs, we have previously proposed two user-cognizant transmission schemes. In [31], the common layer structures of both video source data and network topology are employed to enhance the video transmission, based on the user’s video requirement and association status. In [32], analog transmission of enhancement layer of video stream is proposed in order to make the video reception quality changing continuously with channel quality, thus degrading the staircase effect of digital transmission. Here the cooperative transmission is taken into consideration, where the macro BS and small BS work in a manner of coordinated multipoint transmission.

In all, three user-cognizant transmission schemes are proposed, which are layered digital transmission, layered hybrid digital-analog transmission, and cooperative digital transmission, respectively. The user is cognizant of its video service requirement, mobility, and connection status.

An analytical performance assessment of user-cognizant SVC transmission over two-tier HCNs utilizing tools from stochastic geometry is studied. The contributions of the are:

1.
Three user-cognizant transmission schemes are proposed to enhance the QoE of video users exploiting the interplay among user demand, video source characteristic, and networking.
2.
An analytical framework is proposed for scalable video transmission exploiting the common feature of a layered structure of SVC and HCNs. A digital and a hybrid digital-analog transmission scheme are proposed and studied. The impact of UE load, i.e., the number of UEs served in a cell, is also considered.
3.
The power allocation between the digital BL signal and the analog EL signal is also analyzed to minimize the average distortion. The hybrid digital-analog scheme can further improve the system performance by avoiding the cliff effect and realizing continuous quality scalability when the proportion of frequency resource allocated to the femto tier exceeds a certain threshold.
4.
A noncoherent joint transmission cooperative scheme is proposed, and moreover, the impact of sub-band allocation is also studied.

The remaining part of this paper is as follows: section “System Model” describes the system model, including the transmission schemes and spectrum allocation methods. Section “UE Load and Sub-band Occupancy” derives the distributions of the number of UEs per cell and sub-band occupancy probabilities. Section “SINR Distribution and Data Rate” derives the SINR and data rate distributions. Section “System Performances” evaluates the performance metrics, namely, outage probability, HD probability, and average distortion. Section “Simulation and Discussion” presents simulation results and related discussions. Section “Conclusion and Future Directions” concludes this paper.

System Model

The downlink performance of SVC over a two-tier HCN is considered; see Fig. 1. When a video user needs to request for a particular video, it would first collect the information about video quality requirement due to possessing ability of the UE and connection status due to its mobility and the network topology. The user sends the user-cognizant information to the serving BS (or BSs), and the BS or BSs choose an appropriate transmission strategy. The video data stream is traversing through the video server, IP transmission network, cellular core network, radio access network, and finally reaching at the user.

Layered Video Model

The SVC video content is split into two layers, BL and EL. The BL is always modulated into a digital signal, and the data rate is R _B, while the EL is modulated into a digital signal or an analog signal in these transmission schemes accordingly. If the EL is modulated into a digital signal, then the data rate is R _E. Here we focus on the streaming video service; the video can be decoded successfully when the data rate requirements of the BL and the EL are met.

Actually, the proposed analytical framework can be extended to video signals that are encoded to J layers using a fine granularity, and the BS chooses the first J ₁ layers for the BL and the following J ₂ layers for the EL based on the channel quality for each UE, where J ₁ + J ₂ ≤ J.

It should be aware that SVC allows three types of scalable encoding (spatial, temporal, SNR quality) to be combined and create a single layer [1, 5]. The proposed layered video model is generic and is not restricted by the specifications of the layered encoding and the optimal selection of scalability combinations. Each layer is generated by some combinations of video scalabilities, and the required data rates are the main parameters from the view of networking.

Network Model

The two-tier HCN consists of two types of BSs, namely, MBSs and FAPs. These two types of BSs are modeled by two independent tiers of homogeneous PPPs, Φ _mb and Φ _fb, whose intensities are λ _mb and λ _fb, respectively. FAPs aim at providing network access to UEs in their vicinity within a coverage radius R _f. Suppose that there exist N sub-bands each of bandwidth W. The transmit powers of an MBS and an FAP over each sub-band are set as P _m and P _f, respectively. The path loss model is r^−α, and here for simplicity, it is assumed that the path loss exponent is the same for MBS and FAP, and the effect of shadowing is ignored. The small-scale fading distribution is exponential with mean unity in squared magnitude, i.e., Rayleigh fading. The fading is assumed to be frequency flat within each sub-band and independent among different sub-bands. The noise variance at each UE is denoted by σ².

There are two types of UEs, macro UEs and femto UEs. The locations of macro UEs form a homogeneous PPP Φ _mu with intensity λ _mu, and each macro UE connects to the nearest MBS. The locations of the femto UEs form a Matern cluster process Φ _fu [30] with parent process Φ _fb (the FAPs), i.e., the UEs in each cluster form a finite PPP of intensity λ _fu on the disk of radius R _f centered at each FAP, implying that the mean number of users per cluster is $\bar {U}_{\mathrm {f}}=\lambda _{\mathrm {fu}} \pi R_{\mathrm {f}}^2$. Each femto UE connects to the FAP located at the parent point of the corresponding cluster, called the parent FAP. The access mechanism is as follows: a femto UE always connects to its parent FAP when accessing a femto BS and connects to the MBS closest to its parent FAP when accessing a macro BS; a macro UE can only connect to the nearest MBS, even if it is situated within the coverage of an FAP. This corresponds to a closed-access femto network, in which only subscribers are allowed to be served by an FAP.

Transmission Schemes

Considering the connection status of UEs and their differentiated demand, three transmission schemes are proposed, i.e., layered digital (LD) [31], layered hybrid digital-analog (LHDA) [32], and cooperative digital (CD). For the macro UEs, they only connect to the MBS. Since the MBS aims at providing the coverage service, macro UEs attempt to obtain their BL contents from their serving MBSs and forego the EL. For the femto UEs, since they are covered by the MBS and the FAP, they have two choices: one is that they attempt to obtain their EL contents from their serving FAPs, and they attempt to obtain their BL contents either from their serving MBSs with probability p or from their serving FAPs with probability 1 − p. The other one is that they receive the video contents which are transmitted cooperatively by the MBS and the FAP in the manner of CoMP.

1.
LD transmission: See Fig. 2. Both the BL and the EL are modulated into digital signals. For a macro UE, the data stream of encoded BL signals is transmitted from the serving MBS. For a femto UE, the data stream of encoded BL signals for small SINR or jointly encoded signals of both the BL and the EL for large SINR is transmitted from the serving FAP when p = 0; the digital BL data stream is transmitted from its serving MBS, while the digital EL data stream is transmitted from its serving FAP when p = 1; a mixed transmission is adopted when 0 < p < 1.
Fig. 2
Qualitative illustration of the performances of LD and LHDA transmissions. LD transmission shows a staircase effect, while LHDA transmission shows continuous quality reception with respect to the channel quality
Full size image
2.
LHDA transmission: See Fig. 2. The BL is modulated into a digital signal, while the EL is modulated into an analog signal. For a macro UE, the data stream of encoded BL signals is transmitted from the serving MBS. For a femto UE, the superposition of the digital BL signal and the analog EL signal is transmitted from the serving FAP when p = 0; the digital BL data stream is transmitted from its serving MBS, while the analog EL data stream is transmitted from its serving FAP when p = 1; a mixed transmission is adopted when 0 < p < 1.
3.
CD transmission: See Fig. 3. Both the BL and the EL are modulated into digital signals. For a macro UE, the data stream of encoded BL signals is transmitted from the serving MBS. For a femto UE, if it can claim the same sub-band from both the MBS and the FAP, then the data stream of jointly encoded signals of both the BL and the EL is transmitted from the serving MBS and FAP cooperatively in the manner of noncoherent joint transmission, otherwise, the data stream of jointly encoded signals of both the BL and the EL is transmitted from the serving FAP.
Fig. 3
Illustration of the CD transmission model
Full size image

Since the video source is encoded into multiple layers, different layers are transmitted to the UE based on the channel quality, thus providing scalable video quality. Specifically, for those UEs in less favorable conditions, only the BL with relatively low data rate is received in order to ensure basic video experience. When the channel quality improves, the EL is also received for enhanced video experience. Thus, the LD and CD transmissions can provide two-level scalable video for the UEs, and LHDA can provide a continuous quality scalability.

UE Load and Sub-band Occupancy

UE Load

Since the distribution of femto UEs in an FAP coverage disk is a PPP with intensity λ _fu, the number of femto UEs connected to an FAP is a Poisson random variable (r.v.) with mean $\bar {U}_{\mathrm {f}}$,

$$\displaystyle \begin{aligned} \mathbb{P}\{U_{\mathrm{f}}=i\}=\frac{(\bar{U}_{\mathrm{f}})^i}{i!}e^{-\bar{U}_{\mathrm{f}}}, \quad i=0,1,\cdots. \end{aligned} $$

(1)

For LD and LHDA transmissions, an MBS not only serves the macro UEs situated in its Voronoi cell but also the femto UEs that belong to the FAPs in this Voronoi cell and connect to the MBS to receive the BL contents. We denote the number of macro UEs in the Voronoi cell as U _MBS and the total number of femto UEs served by the MBS as U _FAP, which is given by $U_{\mathrm {FAP}}=\sum _{i=1}^{N_{\mathrm {c}}}N_{\mathrm {f}, i}$, where N _c denotes the number of the FAPs in the Voronoi cell and N_f,i denotes the number of femto UEs which belong to the ith FAP but connect to the MBS to receive the BL contents. The total number of UEs served by an MBS is thus

$$\displaystyle \begin{aligned} U_{\mathrm{m}}=U_{\mathrm{MBS}}+U_{\mathrm{FAP}}. \end{aligned} $$

(2)

U_MBS is conditionally independent of U_FAP given the area of the Voronoi cell. Denote the area of a Voronoi cell by S; the probability generating function (pgf) of U_m conditioned on S, denoted by G_m(z∣S), is

$$\displaystyle \begin{aligned} G_{\mathrm{m}}(z\mid S)=G_{\mathrm{MBS}}(z\mid S)G_{\mathrm{FAP}}(z\mid S), \end{aligned} $$

(3)

where G_MBS(z∣S) and G_FAP(z∣S) are the pgfs of U_MBS and U_FAP conditioned on S, respectively.

U_MBS is a Poisson r.v. with mean λ_mu S, and the conditional pgf of U_MBS is

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} G_{\mathrm{MBS}}(z\mid S)=e^{\lambda_{\mathrm{mu}}S(z-1)}. \end{array} \end{aligned} $$

(4)

Since a femto UE attempts to connect to its serving MBS with probability p in LD and LHDA transmissions, a thinning occurs, i.e., N_f,i is a Poisson random variable with mean $p\bar {U}_{\mathrm {f}}$. Meanwhile, N_c is also a Poisson r.v. with mean λ_fb S because of the PPP distribution of the FAP locations. U_FAP is a compound Poisson r.v. with conditional pgf

$$\displaystyle \begin{aligned} G_{\mathrm{FAP}}(z\mid S)= e^{\lambda_{\mathrm{fb}}S(e^{p\bar{U}_{\mathrm{f}}(z-1)}-1)}. \end{aligned} $$

(5)

There is no known closed form expression of the probability density function (pdf) of the area S of the typical Poisson Voronoi cell, but the following approximation [33]

$$\displaystyle \begin{aligned} f_S(x)\approx\frac{(\lambda_{\mathrm{mb}}c)^c}{\varGamma(c)}x^{c-1}e^{-c\lambda_{\mathrm{mb}}x}, \end{aligned} $$

(6)

where $c=\frac {7}{2}$ and $\varGamma (c)=\int _0^\infty t^{c-1}e^{-t}\mathrm {d}t$, has been known to be handy and sufficiently accurate (see, e.g., [34]). Aided by this approximation, with some manipulations, the pgf of U_m is

$$\displaystyle \begin{aligned} G_{\mathrm{m}}(z)=c^c\Bigg(c-\frac{\lambda_{\mathrm{mu}}}{\lambda_{\mathrm{mb}}}(z-1)+\frac{\lambda_{\mathrm{fb}}}{\lambda_{\mathrm{mb}}}\left(1-e^{p\bar{U}_{\mathrm{f}}(z-1)}\right)\Bigg)^{-c}, \end{aligned} $$

(7)

and the distribution of U_m follows as

$$\displaystyle \begin{aligned} \mathbb{P}\{U_{\mathrm{m}}=i\}=\frac{G_{\mathrm{m}}^{(i)}(0)}{i!}, \quad i=0,1,\cdots, \end{aligned} $$

(8)

where $G_{\mathrm {m}}^{(i)}(0)$ is the i-th derivative of G_m(z) evaluated at z = 0.

For CD transmission, all the femto UEs attempt to connect to the MBS to obtain cooperative gain; thus distribution of U_m is similar to that in LD and LHDA transmissions with p = 1.

Sub-band Occupancy

Since the number of served UEs for each BS is random, the sub-band frequency resource will be underutilized in some BSs and overutilized in some other BSs. As the UE loads in the MBS and the FAP are different under different transmission schemes, the sub-band occupancy is calculated accordingly.

Spectrum Allocation for LD and LHDA

Of the N sub-bands, let N_m sub-bands be allocated to the macro tier and N_f sub-bands to the femto tier. Each UE requires one sub-band for each transmission. We consider the following two spectrum allocation methods [35]:

1.
Orthogonal spectrum allocation: The N sub-bands are split as N = N_m + N_f, where the N_m sub-bands used by all the MBSs of the macro tier are orthogonal to those N_f sub-bands used by all the FAPs of the femto tier. So there is no inter-tier interference.

It is assumed that the available sub-bands are uniformly and independently allocated to the UEs by the BS. There are N_m available sub-bands for the MBS, and each sub-band is equally likely to be chosen. If the number of UEs is smaller than that of sub-bands, the MBS randomly chooses U_m out of the total N_m sub-bands. Otherwise, all the sub-bands are chosen. The probability that a sub-band is used by an MBS is
$$\displaystyle \begin{aligned} P_{\mathrm{busy}}^{\mathrm{m},\perp}=\frac{1}{N_{\mathrm{m}}} \sum_{i=0}^{\infty}\min\{i,N_{\mathrm{m}}\}\mathbb{P}\{U_{\mathrm{m}}=i\}, \end{aligned} $$
(9)
and similarly the probability that a sub-band is used by an FAP is
$$\displaystyle \begin{aligned} P_{\mathrm{busy}}^{\mathrm{f},\perp}=\frac{1}{N_{\mathrm{f}}} \sum_{i=0}^{\infty}\min\{i,N_{\mathrm{f}}\}\mathbb{P}\{U_{\mathrm{f}}=i\}. \end{aligned} $$
(10)
2.
Non-orthogonal spectrum allocation: Compared with the orthogonal case, here the two sets of sub-bands may overlap: each MBS (resp. FAP) independently randomly selects N_m (resp. N_f) sub-bands from the N sub-bands. The values of both N_m and N_f can be chosen from 1 to N flexibly and need not add to N. So there is inter-tier interference, while the available spectrum will be abundant as N_m and N_f grow large.

For the non-orthogonal case, both the MBS and the FAP choose a sub-band randomly from N sub-bands, so the probability that a sub-band is used by an MBS is
$$\displaystyle \begin{aligned} P_{\mathrm{busy}}^{\mathrm{m},\not\perp}=\frac{1}{N} \sum_{i=0}^{\infty}\min\{i,N_{\mathrm{m}}\}\mathbb{P}\{U_{\mathrm{m}}=i\}, \end{aligned} $$
(11)
and similarly the probability that a sub-band is used by an FAP is
$$\displaystyle \begin{aligned} P_{\mathrm{busy}}^{\mathrm{f},\not\perp}=\frac{1}{N} \sum_{i=0}^{\infty}\min\{i,N_{\mathrm{f}}\}\mathbb{P}\{U_{\mathrm{f}}=i\}. \end{aligned} $$
(12)

The spatial point process of BSs that use a given sub-band is an approximately independent thinning of the original point process Φ_mb (resp. Φ_fb) by the probability $P_{\mathrm {busy}}^{\mathrm {m,s}}$ (resp. $P_{\mathrm {busy}}^{\mathrm {f,s}}$), denoted by $\tilde {\varPhi }_{\mathrm {mb}}$ (resp. $\tilde {\varPhi }_{\mathrm {fb}}$) with the intensity $\tilde {\lambda }_{\mathrm {mb}}=\lambda _{\mathrm {mb}}P_{\mathrm {busy}}^{\mathrm {m,s}}$ (resp. $\tilde {\lambda }_{\mathrm {fb}}=\lambda _{\mathrm {mb}}P_{\mathrm {busy}}^{\mathrm {m,s}}$) [34], where the superscript s ∈{⊥, ⊥̸} indicates whether the orthogonal or the non-orthogonal spectrum allocation method is used.

Spectrum Allocation for CD

Since both the macro UEs and femto UEs connect to the MBS, if the number of UEs connected to MBS U_m ≤ N, each UE is allocated one sub-band, otherwise, if U_m ≥ N, all the UEs share the sub-bands in a round-robin mechanism. Then the FAPs allocate sub-bands to femto UEs in a similar way by comparing U_f and N. For a femto UE, if it is chosen by both the MBS and the FAP, the MBS and the FAP allocate the same sub-band to the UE, thus making it working in a cooperative mode. Otherwise it is only served by the FAP and works in a noncooperative mode.

Since both the macro and femto tiers employ the total N sub-bands, similar to that of the LD case, the probability that a sub-band is used by an MBS is

$$\displaystyle \begin{aligned} P_{\mathrm{busy}}^{\mathrm{m},\mathrm{CoMP}}=\frac{1}{N} \sum_{i=0}^{\infty}\min\{i,N_{\mathrm{m}}\}\mathbb{P}\{U_{\mathrm{m}}=i\}, \end{aligned} $$

(13)

and similarly the probability that a sub-band is used by an FAP is

$$\displaystyle \begin{aligned} P_{\mathrm{busy}}^{\mathrm{f},\mathrm{CoMP}}=\frac{1}{N} \sum_{i=0}^{\infty}\min\{i,N_{\mathrm{f}}\}\mathbb{P}\{U_{\mathrm{f}}=i\}. \end{aligned} $$

(14)

SINR Distribution and Data Rate

SINR Distribution

The complementary cumulative distribution function (ccdf) of the SINR is defined as $\mathcal {P}(\theta )=\mathbb {P}\{\mathrm {SINR}>\theta \}$, where θ is the SINR threshold. The SINR distributions of a UE connected to the MBS and the FAP are derived under three transmission schemes.

LD Transmission

For analytical tractability, we assume that both the BL and the EL are modulated into digital signals according to a Gaussian codebook.

For the typical UE which is assumed to be located at the origin and connected to its MBS, the received signal denoted by Y can be written as

$$\displaystyle \begin{aligned} Y&=P_{\mathrm{m}}^{1/2}\|x_0\|{}^{-\alpha/2}h_{x_0}X_{x_0}+\sum_{x\in\tilde{\varPhi}_{\mathrm{mb}}\setminus\{x_0\}}P_{\mathrm{m}}^{1/2}\|x\|{}^{-\alpha/2}h_xX_x\notag\\ &\quad +\kappa\sum_{y\in\tilde{\varPhi}_{\mathrm{fb}}}P_{\mathrm{f}}^{1/2}\|y\|{}^{-\alpha/2}h_yX_y+Z, \end{aligned} $$

(15)

where the first item of right side of the equation denotes the received signal symbol, the second and the third items denote the interference symbols from the macro and the femto tier, respectively, and Z denotes the Gaussian noise with zero mean and variance σ². We use x₀ to denote the location of the serving MBS. $X_{x_0}$ is the signal symbol, while X_x is the interference symbol transmitted by the interfering MBS x. $X_{x_0}, X_x \sim \mathbb {C}\mathbb {N}(0, 1)$. X_y is the interference symbol transmitted by the interfering FAP y, and $X_y\sim \mathbb {C}\mathbb {N}(0, 1)$. The indicator κ ∈{0, 1} indicates the orthogonal and non-orthogonal spectrum allocation methods, respectively.

Thus, the received SINR is

$$\displaystyle \begin{aligned} \gamma_{\mathrm{LD}}^{\mathrm{m}}=\frac{P_{\mathrm{m}}\|x_0\|{}^{-\alpha}|h_{x_0}|{}^2}{I_{\mathrm{m}}+\kappa I_{\mathrm{f}}+\sigma^2}, \end{aligned} $$

(16)

where $I_{\mathrm {m}}=\sum _{x\in \tilde {\varPhi }_{\mathrm {mb}}\setminus \{x_0\}}P_{\mathrm {m}}\|x\|{ }^{-\alpha }|h_{x}|{ }^2$ is the interference from the macro tier, and $I_{\mathrm {f}}=\sum _{y\in \tilde {\varPhi }_{\mathrm {fb}}}P_{\mathrm {f}}\|y\|{ }^{-\alpha }|h_y|{ }^2$ is the interference from the femto tier.

For the typical femto UE which is assumed to be located at the origin and connected to its FAP, the received signal can be written as

$$\displaystyle \begin{aligned} Y&=P_{\mathrm{f}}^{1/2}\|y_0\|{}^{-\alpha/2}h_{y_0}X_{y_0}+\sum_{y\in\tilde{\varPhi}_{\mathrm{fb}}\setminus\{y_0\}}P_{\mathrm{f}}^{1/2}\|y\|{}^{-\alpha/2}h_yX_y\notag\\ &\quad +\kappa\sum_{x\in\tilde{\varPhi}_{\mathrm{mb}}}P_{\mathrm{m}}^{1/2}\|x\|{}^{-\alpha/2}h_xX_x+Z, \end{aligned} $$

(17)

where y₀ denotes the location of the serving FAP. Note that the FAP transmits the encoded EL signals only or the jointly encoded signals of both the BL and the EL to the typical UE based on user request. $X_{y_0}$ is the signal symbol transmitted by the serving FAP, and X_y is the interference symbol transmitted by the interfering FAP y.

Thus, the received SINR is

$$\displaystyle \begin{aligned} \gamma_{\mathrm{LD}}^{\mathrm{f}}=\frac{P_{\mathrm{f}}\|y_0\|{}^{-\alpha}|h_{y_0}|{}^2}{I_{\mathrm{f}}+\kappa I_{\mathrm{m}}+\sigma^2}, \end{aligned} $$

(18)

where $I_{\mathrm {f}}=\sum _{y\in \tilde {\varPhi }_{\mathrm {fb}}\setminus \{y_0\}}P_{\mathrm {f}}\|y\|{ }^{-\alpha }|h_y|{ }^2$ denotes the interference from the femto tier and $I_{\mathrm {m}}=\sum _{x\in \tilde {\varPhi }_{\mathrm {mb}}}P_{\mathrm {m}}\|x\|{ }^{-\alpha }|h_{x}|{ }^2$ denotes the interference from the macro tier.

The following theorem gives the ccdf of the SINR for the typical UE,

Theorem 1

For LD transmission, the ccdf of the SINR for the typical UE connected to its serving MBS is

$$\displaystyle \begin{aligned} \mathcal{P}_{\mathrm{LD}}^{\mathrm{m}}(\theta)&= \mathbb{P}\{\gamma_{\mathrm{LD}}^{\mathrm{m}}>\theta\}\notag\\ &= \int_{0}^{\infty}\pi\lambda_{\mathrm{mb}}\exp\Bigg(-\pi v(\lambda_{\mathrm{mb}}+\tilde{\lambda}_{\mathrm{mb}}\rho(\theta ,\alpha))\notag\\ &-\frac{ \theta v^{1/\delta}\sigma^2}{P_{\mathrm{m}}}-\kappa\left(\frac{P_{\mathrm{f}}\theta }{P_{\mathrm{m}}}\right)^{\delta} v\tilde{\lambda}_{\mathrm{fb}}\delta\pi^2\mathrm{csc}(\delta\pi)\Bigg)\mathrm{d}v, \end{aligned} $$

(19)

and the ccdf of the SINR for the typical femto UE connected to its serving FAP is

(20)

where δ = 2∕α, $\tilde {\lambda }_{\mathrm {mb}}=\lambda _{\mathrm {mb}}P_{\mathrm {busy}}^{\mathrm {m},\mathrm {s}}$ , $\tilde {\lambda }_{\mathrm {fb}}=\lambda _{\mathrm {fb}}P_{\mathrm {busy}}^{\mathrm {f},\mathrm {s}}$ , and $\rho (\theta ,\alpha )=\theta ^{\delta }\int _{\theta ^{-\delta }}^{\infty }\frac {1}{1+x^{1/\delta }}\mathrm {d}x$ . In orthogonal spectrum allocation, κ = 0, while in non-orthogonal spectrum allocation, κ = 1.

Proof

Let ∥x₀∥ be the distance from the typical UE to its serving MBS, which is the nearest MBS, so the pdf of ∥x₀∥ is $f_{\|x_0\|}(r) = e^{-\lambda _{\mathrm {mb}} \pi r^2}2\pi \lambda _{\mathrm {mb}} r.$

The SINR experienced by the typical UE connected to its serving MBS is given by $\gamma _{\mathrm {LD}}^{\mathrm {m}}=\frac {P_{\mathrm {m}}\|x_0\|{ }^{-\alpha }|h_{x_0}|{ }^2}{I_{\mathrm {m}}+\kappa I_{\mathrm {f}}+\sigma ^2}$, where $I_{\mathrm {m}}=\sum _{x\in \tilde {\varPhi }_{\mathrm {mb}}\setminus \{x_0\}}P_{\mathrm {m}}\|x\|{ }^{-\alpha }|h_{x_0}|{ }^2$ is the interference from the macro tier, and $I_{\mathrm {f}}=\sum _{y\in \tilde {\varPhi }_{\mathrm {fb}}}P_{\mathrm {f}}\|y\|{ }^{-\alpha }|h_y|{ }^2$ is the interference from the femto tier. κ ∈{0, 1} is the indicator that whether the orthogonal or the non-orthogonal spectrum allocation is used. Due to the independent thinning approximation, the set of interfering MBSs is a PPP $\tilde {\varPhi }_{\mathrm {mb}}$ with intensity $\tilde {\lambda }_{\mathrm {mb}}$, and the set of interfering FAPs is a PPP $\tilde {\varPhi }_{\mathrm {fb}}$ with intensity $\tilde {\lambda }_{\mathrm {fb}}$.

The ccdf of the SINR experienced by the typical UE connected to its serving MBS

$$\displaystyle \begin{aligned}&\mathcal{P}_{\mathrm{LD}}^{\mathrm{m}}(\theta )= \mathbb{P}\{\gamma_{\mathrm{LD}}^{\mathrm{m}}>\theta\}\notag\\ &=\int_0^\infty2\pi\lambda_{\mathrm{mb}}re^{-\pi\lambda_{\mathrm{mb}}r^2}\mathbb{P} \left\{\frac{P_{\mathrm{m}}|h_{x_0}|{}^2r^{-\alpha}}{I_{\mathrm{m}}+\kappa I_{\mathrm{f}}+\sigma^2}>\theta \right\}\mathrm{d}r \notag \\ &\overset{(a)}=\int_0^\infty2\pi\lambda_{\mathrm{mb}} re^{-\pi\lambda_{\mathrm{mb}}r^2-\frac{ \theta r^\alpha\sigma^2}{P_{\mathrm{m}}}}\mathcal{L}_{I_{\mathrm{m}}+\kappa I_{\mathrm{f}}}\left(\frac{ \theta r^\alpha}{P_{\mathrm{m}}}\right)\mathrm{d}r. \end{aligned} $$

(21)

where (a) follows from $|h_{x_0}|{ }^2\sim \mathrm {Exp}(1)$.

After excluding the serving BS x₀, $\tilde {\varPhi }_{\mathrm {mb}}\setminus \{x_0\}$ is still a PPP, so we apply the pgfl of PPP to obtain the Laplace transform of I_m

$$\displaystyle \begin{aligned} \mathcal{L}_{I_{\mathrm{m}}}(s)&=\exp{\left(-2\pi\tilde{\lambda}_{\mathrm{mb}}\int_{r}^{\infty}\left(1-\frac{1}{1+sP_{\mathrm{m}}x^{-\alpha}}\right)x\mathrm{d}x\right)}\notag\\ &=e^{-\pi\tilde{\lambda}_{\mathrm{mb}}r^2\rho(\frac{sP_{\mathrm{m}}}{r^{\alpha}}, \alpha)}.\end{aligned} $$

(22)

Since $\tilde {\varPhi }_{\mathrm {fb}}$ is a PPP, the Laplace transform of I_f is

$$\displaystyle \begin{aligned} \mathcal{L}_{I_{{\mathrm{f}}}}(s)&=\exp{\left(-2\pi\tilde{\lambda}_{\mathrm{fb}}\int_{0}^{\infty}\left(1-\frac{1}{1+sP_{\mathrm{f}}x^{-\alpha}}\right)x\mathrm{d}x\right)}\notag\\ &=e^{-\delta\pi^2\mathrm{csc}(\delta\pi)\tilde{\lambda}_{\mathrm{fb}}(sP_{{\mathrm{f}}})^{\delta}}. \end{aligned} $$

(23)

Substituting (22) and (23) into $\mathcal {P}_{\mathrm {LD}}^{\mathrm {m}}(\theta )$, we can obtain (19).

Let y₀ be the distance between the typical femto UE and its serving FAP. Since femto UEs are uniformly distributed in the circular coverage area of radius R_f of each FAP, the pdf of y₀ is given by $f_{y_0}(r)=\frac {2r}{R_{\mathrm {f}}^2}$.

The received SINR for the typical femto UE connected to its serving FAP follows as $\gamma _{\mathrm {LD}}^{\mathrm {f}}=\frac {P_{\mathrm {f}}\|y_0\|{ }^{-\alpha }|h_{y_0}|{ }^2}{I_{\mathrm {f}}+\kappa I_{\mathrm {m}}+\sigma ^2},$ where $I_{\mathrm {f}}=\sum _{y\in \tilde {\varPhi }_{\mathrm {fb}}}P_{\mathrm {f}}\|y\|{ }^{-\alpha }|h_y|{ }^2$ is the interference from the femto tier, and $I_{\mathrm {m}}=\sum _{x\in \tilde {\varPhi }_{\mathrm {mb}}}P_{\mathrm {m}}\|x\|{ }^{-\alpha }|h_{x_0}|{ }^2$ is the interference from the macro tier.

The ccdf of the SINR experienced by the typical femto UE connected to its serving FAP is

$$\displaystyle \begin{aligned} \mathcal{P}_{\mathrm{LD}}^{\mathrm{f}}(\theta )&= \mathbb{P}\{\gamma_{\mathrm{LD}}^{\mathrm{f}}>\theta\}\notag\\ &=\int_0^{R_{\mathrm{f}}}\frac{2r}{R_{\mathrm{f}}^2}\mathbb{P} \left\{\frac{P_{\mathrm{f}}|h_{y_0}|{}^2r^{-\alpha}}{I_{\mathrm{f}}+\kappa I_{\mathrm{m}}+\sigma^2}>\theta \right\}\mathrm{d}r \notag \\ &=\int_0^{R_{\mathrm{f}}}\frac{2r}{R_{\mathrm{f}}^2}e^{\frac{- \theta r^{\alpha}\delta^2}{P_{\mathrm{f}}}}\mathcal{L}_{I_{\mathrm{f}}+\kappa I_{\mathrm{m}}}\left(\frac{\theta r^{\alpha}}{P_{\mathrm{f}}}\right)\mathrm{d}r, \end{aligned} $$

(24)

which, after expanding the Laplace transform of I_m, I_f, and further manipulations, leads to (20).

LHDA Transmission

The BL is modulated to a digital signal, while the EL is modulated to an analog signal. The digital modulation is based on a Gaussian codebook, and the EL signal after analog modulation is also modeled as a Gaussian source with zero mean and unit variance [36, 37]. For analog modulation, it is assumed that the source bandwidth is equal to the channel bandwidth [14, 16].

For the typical UE which is assumed to be located at the origin and connected to its serving MBS, the received signal can be written as

$$\displaystyle \begin{aligned} Y&=P_{\mathrm{m}}^{1/2}\|x_0\|{}^{-\alpha/2}h_{x_0}X_{x_0}+\sum_{x\in\tilde{\varPhi}_{\mathrm{mb}}\setminus\{x_0\}}P_{\mathrm{m}}^{1/2}\|x\|{}^{-\alpha/2}h_xX_x\notag\\ &\quad +\kappa\sum_{y\in\tilde{\varPhi}_{\mathrm{fb}}}P_{\mathrm{f}}^{1/2}\|y\|{}^{-\alpha/2}h_yX_y+Z, \end{aligned} $$

(25)

which is nearly the same as (15) in LD transmission, the difference lies in that X_y is the analog EL interference symbol or the superposition of digital BL and analog EL interference symbol transmitted by the interfering FAP y based on the transmission scheme of y, and $X_y\sim \mathbb {C}\mathbb {N}(0, 1)$.

Thus the received SINR is

$$\displaystyle \begin{aligned} \gamma_{\mathrm{LHDA}}^{\mathrm{m}}=\frac{P_{\mathrm{m}}\|x_0\|{}^{-\alpha}|h_{x_0}|{}^2}{I_{\mathrm{m}}+\kappa I_{\mathrm{f}}+\sigma^2}, \end{aligned} $$

(26)

where $I_{\mathrm {m}}=\sum _{x\in \tilde {\varPhi }_{\mathrm {mb}}\setminus \{x_0\}}P_{\mathrm {m}}\|x\|{ }^{-\alpha }|h_{x}|{ }^2$ is the interference from the macro tier and $I_{\mathrm {f}}=\sum _{y\in \tilde {\varPhi }_{\mathrm {fb}}}P_{\mathrm {f}}\|y\|{ }^{-\alpha }|h_y|{ }^2$ is the interference from the femto tier.

For the typical femto UE which is assumed to be located at the origin and connected to its FAP, according to the transmission scheme, it receives only the EL, or it receives the superposition of the digital BL signal and the analog EL signal.

Case 1: The typical femto UE connected to its FAP receives only the EL. The received signal for the typical femto UE is
$$\displaystyle \begin{aligned} Y&=P_{\mathrm{f}}^{1/2}\|y_0\|{}^{-\frac{\alpha}{2}}h_{y_0}X_{y_0}^{\mathrm{E}}+\sum_{y\in\tilde{\varPhi}_{\mathrm{fb}}\setminus\{y_0\}}P_{\mathrm{f}}^{1/2}\|y\|{}^{-\frac{\alpha}{2}}h_yX_y\notag\\ &\quad +\kappa\sum_{x\in\tilde{\varPhi}_{\mathrm{mb}}}P_{\mathrm{m}}^{1/2}\|x\|{}^{-\frac{\alpha}{2}}h_xX_x+Z, \end{aligned} $$
(27)
where $X_{y_0}^{\mathrm {E}}$ is the EL signal symbol transmitted by the serving FAP and X_y is the interference symbol transmitted by the interfering FAP y.

Thus, the received SINR for the femto UE connected to its FAP to receive the EL is
$$\displaystyle \begin{aligned} \gamma_{\mathrm{LHDA}}^{\mathrm{f}}=\frac{P_{\mathrm{f}}\|y_0\|{}^{-\alpha}|h_{y_0}|{}^2}{I_{\mathrm{f}}+\kappa I_{\mathrm{m}}+\sigma^2}, \end{aligned} $$
(28)
where $I_{\mathrm {f}}=\sum _{y\in \tilde {\varPhi }_{\mathrm {fb}}\setminus \{y_0\}}P_{\mathrm {f}}\|y\|{ }^{-\alpha }|h_y|{ }^2$ is the interference from the femto tier and $I_{\mathrm {m}}=\sum _{x\in \tilde {\varPhi }_{\mathrm {mb}}}P_{\mathrm {m}}\|x\|{ }^{-\alpha }|h_{x}|{ }^2$ is the interference from the macro tier.
Case 2: The typical femto UE connected to its FAP receives the superposition of the digital BL signal and the analog EL signal. The received signal for the typical femto UE is
$$\displaystyle \begin{aligned} Y&=\|y_0\|{}^{-\alpha/2}h_{y_0}\bigg(\sqrt{P_{\mathrm{f}}^{\mathrm{B}}}X_{y_0}^{\mathrm{B}}+\sqrt{P_{\mathrm{f}}^{\mathrm{E}}}X_{y_0}^{\mathrm{E}}\bigg)+\sum_{y\in\tilde{\varPhi}_{\mathrm{fb}}\setminus\{y_0\}}P_{\mathrm{f}}^{1/2}\|y\|{}^{-\alpha/2}h_yX_y\notag\\ &\quad +\kappa\sum_{x\in\tilde{\varPhi}_{\mathrm{mb}}}P_{\mathrm{m}}^{1/2}\|x\|{}^{-\alpha/2}h_xX_x+Z, \end{aligned} $$
(29)
where $X_{y_0}^{\mathrm {B}}$ is the BL signal symbol transmitted by the serving FAP, and $X_{y_0}^{\mathrm {E}}$ is the EL signal symbol transmitted by the serving FAP.

Thus, the received SINR for the typical femto UE connected to its FAP to receive the BL, denoted by $\gamma _{\mathrm {LHDA}}^{\mathrm {f,B}}$, is
$$\displaystyle \begin{aligned} \gamma_{\mathrm{LHDA}}^{\mathrm{f,B}}=\frac{P_{\mathrm{f}}^{\mathrm{B}}\|y_0\|{}^{-\alpha}|h_{y_0}|{}^2}{P_{\mathrm{f}}^{\mathrm{E}}\|y_0\|{}^{-\alpha}|h_{y_0}|{}^2+I_{\mathrm{f}}+\kappa I_{\mathrm{m}}+\sigma^2}. \end{aligned} $$
(30)

Successive interference cancellation (SIC) [38] is adopted to demodulate the EL signal. Conditioned on the successful reception of the BL, the received SINR for the typical femto UE connected to the FAP to receive the EL signal, denoted by $\gamma _{\mathrm {LHDA}}^{\mathrm {f,E}}$, is
$$\displaystyle \begin{aligned} \gamma_{\mathrm{LHDA}}^{\mathrm{f,E}}=\frac{P_{\mathrm{f}}^{\mathrm{E}}\|y_0\|{}^{-\alpha}|h_{y_0}|{}^2}{I_{\mathrm{f}}+\kappa I_{\mathrm{m}}+\sigma^2}. \end{aligned} $$
(31)

The following theorem gives the ccdf of the SINR for the typical UE:

Theorem 2

For LHDA transmission, the ccdf of the SINR for the typical UE connected to its serving MBS is

$$\displaystyle \begin{aligned} \mathcal{P}_{\mathrm{LHDA}}^{\mathrm{m}}(\theta_{\mathrm{B}})=\mathbb{P}\{\gamma_{\mathrm{LHDA}}^{\mathrm{m}}>\theta_{\mathrm{B}}\}=\mathcal{P}_{\mathrm{LD}}^{\mathrm{m}}(\theta_{\mathrm{B}}); \end{aligned} $$

(32)

the ccdf of the SINR for the typical femto UE connected to its serving FAP to receive the EL is given by:

$$\displaystyle \begin{aligned} \mathcal{P}_{\mathrm{LHDA}}^{\mathrm{f}}(\theta_{\mathrm{E}})=\mathbb{P}\{\gamma_{\mathrm{LHDA}}^{\mathrm{f}}>\theta_{\mathrm{E}}\}=\mathcal{P}_{\mathrm{LD}}^{\mathrm{f}}(\theta_{\mathrm{E}}), \end{aligned} $$

(33)

and the joint ccdf of the SINR for the typical femto UE connected to its serving FAP to receive the superposition of the digital BL and the analog EL is given by (40) .

Proof

Similar to the derivation of $\mathcal {P}_{\mathrm {LD}}^{\mathrm {m}}(\theta )$, the ccdf of $\gamma _{\mathrm {LHDA}}^{\mathrm {m}}$ follows as:

$$\displaystyle \begin{aligned} \mathcal{P}_{\mathrm{LHDA}}^{\mathrm{m}}(\theta)=\mathbb{P}\{\gamma_{\mathrm{LHDA}}^{\mathrm{m}}>\theta\}=\mathcal{P}_{\mathrm{LD}}^{\mathrm{m}}(\theta). \end{aligned} $$

(34)

According to the transmission scheme, the FAP transmits the analog EL signal with probability p or the superposition of the digital BL signal and the analog EL signal with probability 1 − p.

Case 1::

The received SINR for the typical femto UE connected to the FAP receives the EL follows as $\gamma _{\mathrm {LHDA}}^{\mathrm {f}}=\frac {P_{\mathrm {f}}\|y_0\|{ }^{-\alpha }|h_{y_0}|{ }^2}{I_{\mathrm {f}}+\kappa I_{\mathrm {m}}+\sigma ^2}.$ Similar to the derivation of $\mathcal {P}_{\mathrm {LD}}^{\mathrm {f}}(\theta )$, the ccdf of $\gamma _{\mathrm {LHDA}}^{\mathrm {f}}$ follows as

$$\displaystyle \begin{aligned} \mathcal{P}_{\mathrm{LHDA}}^{\mathrm{f}}(\theta)=\mathbb{P}\{\gamma_{\mathrm{LHDA}}^{\mathrm{f}}>\theta\}=\mathcal{P}_{\mathrm{LD}}^{\mathrm{f}}(\theta). \end{aligned} $$

(35)

Case 2::

The received SINR for the typical femto UE connected to the FAP receives the superposition of the digital BL signal and the analog EL signal is

$$\displaystyle \begin{aligned} \gamma_{\mathrm{LHDA}}^{\mathrm{f,B}}=\frac{P_{\mathrm{f}}^{\mathrm{B}}\|y_0\|{}^{-\alpha}|h_{y_0}|{}^2}{P_{\mathrm{f}}^{\mathrm{E}}\|y_0\|{}^{-\alpha}|h_{y_0}|{}^2+I_{\mathrm{f}}+\kappa I_{\mathrm{m}}+\sigma^2}, \end{aligned} $$

(36)

where $P_{\mathrm {f}}^{\mathrm {E}}\|y_0\|{ }^{-\alpha }|h_{y_0}|{ }^2$ is the interference of the superposed EL.

$$\displaystyle \begin{aligned} \mathbb{P}\{\gamma_{\mathrm{LHDA}}^{\mathrm{f}, \mathrm{B}}>\theta\}&=\int_0^\infty\frac{2r}{R_{\mathrm{f}}^2}e^{-\frac{ \theta r^\alpha\sigma^2}{P_{\mathrm{f}}^{\mathrm{B}}-\theta P_{\mathrm{f}}^{\mathrm{E}}}}\mathcal{L}_{I_{\mathrm{f}}+\kappa I_{\mathrm{m}}}\left(\frac{ \theta r^\alpha}{P_{\mathrm{f}}^{\mathrm{B}}-\theta P_{\mathrm{f}}^{\mathrm{E}}}\right)\mathrm{d}r\notag\\ &=\int_{0}^{R_{\mathrm{f}}^2}\frac{1}{R_{\mathrm{f}}^2}\exp\bigg(-\frac{ \theta v^{1/\delta}\sigma^2}{P_{\mathrm{f}}^{\mathrm{B}}-\theta P_{\mathrm{f}}^{\mathrm{E}}}-\delta\pi^2\mathrm{csc}(\delta\pi)\theta ^{\delta}v\notag\\ &\quad \times\bigg(\tilde{\lambda}_{\mathrm{fb}}\bigg(\frac{P_{\mathrm{f}}}{P_{\mathrm{f}}^{\mathrm{B}}-\theta P_{\mathrm{f}}^{\mathrm{E}}}\bigg)^{\delta}+\kappa\tilde{\lambda}_{\mathrm{mb}}\bigg(\frac{P_{\mathrm{m}}}{P_{\mathrm{f}}^{\mathrm{B}}-\theta P_{\mathrm{f}}^{\mathrm{E}}}\bigg)^{\delta}\bigg)\bigg)\mathrm{d}v. \end{aligned} $$

(37)

SIC is adopted to decode the EL signal. After successful reception of the BL, the received SINR for the EL signal is $\gamma _{\mathrm {LHDA}}^{\mathrm {f,E}}=\frac {P_{\mathrm {f}}^{\mathrm {E}}\|y_0\|{ }^{-\alpha }|h_{y_0}|{ }^2}{I_{\mathrm {f}}+\kappa I_{\mathrm {m}}+\sigma ^2}$.

The ccdf of $\gamma _{\mathrm {LHDA}}^{\mathrm {f,E}}$ follows as:

$$\displaystyle \begin{aligned} \begin{aligned} \mathbb{P}\{\gamma_{\mathrm{LHDA}}^{\mathrm{f}, \mathrm{E}}>\theta\}&=\int_0^\infty\frac{2r}{R_{\mathrm{f}}^2}e^{-\frac{ \theta r^\alpha\sigma^2}{P_{\mathrm{f}}^{\mathrm{E}}}}\mathcal{L}_{I_{\mathrm{f}}+\kappa I_{\mathrm{m}}}\left(\frac{ \theta r^\alpha}{P_{\mathrm{f}}^{\mathrm{E}}}\right)\mathrm{d}r\\ &=\int_{0}^{R_{\mathrm{f}}^2}\frac{1}{R_{\mathrm{f}}^2}e^{-\frac{ \theta v^{1/\delta}\sigma^2}{P_{\mathrm{f}}^{\mathrm{E}}}-\delta\pi^2\mathrm{csc}(\delta\pi)\theta ^{\delta} v\bigg(\tilde{\lambda}_{\mathrm{fb}}\bigg(\frac{P_{\mathrm{f}}}{P_{\mathrm{f}}^{\mathrm{E}}}\bigg)^{\delta}+\ \kappa\tilde{\lambda}_{\mathrm{mb}}\bigg(\frac{P_{\mathrm{m}}}{P_{\mathrm{f}}^{\mathrm{E}}}\bigg)^{\delta}\bigg)}\mathrm{d}v. \end{aligned} \end{aligned} $$

(38)

The joint ccdf of $\gamma _{\mathrm {LHDA}}^{\mathrm {f,B}}$ and $\gamma _{\mathrm {LHDA}}^{\mathrm {f,E}}$ is

(39)

$$\displaystyle \begin{aligned} \mathcal{P}_{\mathrm{LHDA}}(\theta_{\mathrm{B}},\theta_{\mathrm{E}})&=\mathbb{P}\{\gamma_{\mathrm{LHDA}}^{\mathrm{f,B}}>\theta_{\mathrm{B}}, \gamma_{\mathrm{LHDA}}^{\mathrm{f,E}}>\theta_{\mathrm{E}}\}\notag\\ &=\mathbb{P}\left\{|h_{x0}|{}^2>\frac{\theta_{\mathrm{B}}I_{\mathrm{total}}}{(P_{\mathrm{B}}-\theta_{\mathrm{B}} P_{\mathrm{E}})\|x_0\|{}^{-\alpha}},|h_{x0}|{}^2>\frac{\theta_{\mathrm{E}}I_{\mathrm{total}}}{P_{\mathrm{E}}\|x\|{}^{-\alpha}}\right\}\notag\\ &=\mathbb{P}\left\{|h_{x0}|{}^2>\max\Bigg(\frac{\theta_{\mathrm{B}}I_{\mathrm{total}}}{(P_{\mathrm{B}}-\theta_{\mathrm{B}} P_{\mathrm{E}})\|x_0\|{}^{-\alpha}},\frac{\theta_{\mathrm{E}}I_{\mathrm{total}}}{P_{\mathrm{E}}\|x\|{}^{-\alpha}}\Bigg)\right\}\notag\\ &=\mathbb{P}\{\gamma_{\mathrm{LHDA}}^{\mathrm{f, B}}>\theta_{\mathrm{B}}\}\mathbf{1}\Bigg(\theta_{\mathrm{B}}>\frac{\theta_{\mathrm{E}}P_{\mathrm{f}}^{\mathrm{B}}}{(1+\theta_{\mathrm{E}})P_{\mathrm{f}}^{\mathrm{E}}}\Bigg)\notag\\ &\quad +\mathbb{P}\{\gamma_{\mathrm{LHDA}}^{\mathrm{f ,E}}>\theta_{\mathrm{E}}\}\mathbf{1}\Bigg(\theta_{\mathrm{B}}\leq\frac{\theta_{\mathrm{E}}P_{\mathrm{f}}^{\mathrm{B}}}{(1+\theta_{\mathrm{E}})P_{\mathrm{f}}^{\mathrm{E}}}\Bigg), \end{aligned} $$

(40)

where I_total = I_f + κI_m + σ².

CD Transmission

For the macro UE, the video transmission from the serving MBS is the same as that of LD; thus the ccdf of the SINR denoted by $\mathcal {P}_{\mathrm {CD}}^{\mathrm {m}}(\theta )$ is equal to $\mathcal {P}_{\mathrm {LD}}^{\mathrm {m}}(\theta )$ with p = 1 and κ = 1.

For the femto UE, based on the sub-band allocation from the MBS and the FAP, it can work in a cooperative or noncooperative modes.

In the noncooperative case, since the femto UE can only connect to the FAP, the received signal can be written as
$$\displaystyle \begin{aligned} Y&=P_{\mathrm{f}}^{1/2}\|y_0\|{}^{-\frac{\alpha}{2}}h_{y_0}X_{y_0}+\sum_{y\in\tilde{\varPhi}_{\mathrm{fb}}\setminus\{y_0\}}P_{\mathrm{f}}^{1/2}\|y\|{}^{-\frac{\alpha}{2}}h_yX_y\\&\quad +\sum_{x\in\tilde{\varPhi}_{\mathrm{mb}}}P_{\mathrm{m}}^{1/2}\|x\|{}^{-\frac{\alpha}{2}}h_xX_x+Z, \end{aligned} $$
(41)

Thus, the received SINR is
$$\displaystyle \begin{aligned} \gamma_{\mathrm{CD}}^{\mathrm{f,non}}=\frac{P_{\mathrm{f}}\|y_0\|{}^{-\alpha}|h_{y_0}|{}^2}{I_{\mathrm{f}}+I_{\mathrm{m}}+\sigma^2}. \end{aligned} $$
(42)

The ccdf of the $\gamma _{\mathrm {CD}}^{\mathrm {f,non}}$, denoted as $\mathcal {P}_{\mathrm {CD}}^{\mathrm {f,non}}(\theta )$, is equal to $\mathcal {P}_{\mathrm {LD}}^{\mathrm {f}}(\theta )$ with p = 1 and κ = 1.
In the cooperative case, since the femto UE is served jointly by the MBS and the FAP, the received signal can be written as
$$\displaystyle \begin{aligned} Y&=P_{\mathrm{m}}^{1/2}\|{{x}_{0}}{{\|}^{-\alpha /2}}{{h}_{{{x}_{0}}}}{{X}_{0}}+P_{\mathrm{f}}^{1/2}\|{{y}_{0}}{{\|}^{-\alpha /2}}{{h}_{{{y}_{0}}}}{{X}_{0}}+\sum_{x\in {{{\tilde{\varPhi }}}_{\mathrm{mb}}}\setminus \{{{x}_{0}}\}}{P_{\mathrm{m}}^{1/2}}\|x{{\|}^{-\alpha /2}}{{h}_{x}}{{X}_{x}} \notag\\ &\quad+\sum_{y\in {{{\tilde{\varPhi }}}_{\mathrm{fb}}}\setminus \{{{y}_{0}}\}}{P_{\mathrm{f}}^{1/2}}\|y{{\|}^{-\alpha /2}}{{h}_{y}}{{X}_{y}}+Z, \end{aligned} $$
(43)
where the fist and second items of right side of the equation denote the received signal symbols from the serving MBS and FAP, respectively, and the following two items denote the interference symbols from the macro and femto tiers, respectively

The noncoherent joint transmission is adopted, and the SINR of the received signal is
$$\displaystyle \begin{aligned} \gamma_{\mathrm{CD}}^{\mathrm{f,CoMP}}=\frac{{{\left| P_{\mathrm{m}}^{1/2}\|{{x}_{0}}{{\|}^{-\alpha /2}}{{h}_{{{x}_{0}}}}+P_{\mathrm{f}}^{1/2}\|{{y}_{0}}{{\|}^{-\alpha /2}}{{h}_{{{y}_{0}}}} \right|}^{2}}}{{{I}_{\mathrm{m}}}+{{I}_{\mathrm{f}}}+{{\sigma }^{2}}}, \end{aligned} $$
(44)
where the interference from the macro tier is $I_{\mathrm {m}}=\sum _{x\in {\tilde {\varPhi }_{\mathrm {mb}}\setminus {x_0}}}P_{\mathrm {m}}\|x\|{ }^{-\alpha }h_x$, and the interference from the femto tier is $I_{\mathrm {f}}=\sum _{y\in {\tilde {\varPhi }_{\mathrm {fb}}\setminus {y_0}}}P_{\mathrm {f}}\|y\|{ }^{-\alpha }h_y$.

Theorem 3 For CD transmission, when the femto UE works in a cooperative mode, the ccdf of the SINR is
$$\displaystyle \begin{aligned} &\mathcal{P}_{\mathrm{CD}}^{\mathrm{f,CoMP}}(\theta)\\&=\int_0^\infty\int_0^{R_{\mathrm{f}}} e^{-\frac{\mu\theta\sigma^2}{P_{\mathrm{m}}r_{\mathrm{m}}^{-\alpha}+P_{\mathrm{f}}r_{\mathrm{f}}^{-\alpha}}-\pi\tilde{\lambda}_{\mathrm{mb}}r_{\mathrm{m}}^2\rho\left(\frac{\theta}{1+\frac{P_{\mathrm{f}}r_{\mathrm{f}}^{-\alpha}}{P_{\mathrm{m}}r_{\mathrm{m}}^{-\alpha}}},\alpha\right)-\pi\tilde{\lambda}_{\mathrm{fb}}r_{\mathrm{f}}^2\rho\left(\frac{\theta}{1+\frac{P_{\mathrm{m}}r_{\mathrm{m}}^{-\alpha}}{P_{\mathrm{f}}r_{\mathrm{f}}^{-\alpha}}},\alpha\right)-\lambda_{\mathrm{mb}}\pi r_{\mathrm{m}}^2}\notag\\ &\qquad \qquad \qquad \times2\pi\lambda_{\mathrm{mb}}r_{\mathrm{m}}\frac{2r_{\mathrm{f}}}{R_{\mathrm{f}}^2}\mathrm{d}r_{\mathrm{m}}\mathrm{d}r_{\mathrm{f}}. \end{aligned} $$
(45)

Proof Let ∥x₀∥ be the distance from the typical femto UE to its serving MBS, and the pdf of ∥x₀∥ is $f_{\|x_0\|}(r_{\mathrm {m}}) = e^{-\lambda _{\mathrm {mb}} \pi r_{\mathrm {m}}^2}2\pi \lambda _{\mathrm {mb}} r_{\mathrm {m}}$.

Let ∥y₀∥ be the distance from the typical femto UE to its serving FAP, and the pdf of r_f is $f_{\|y_0\|}(r_{\mathrm {f}}) = \frac {2r_{\mathrm {f}}}{R_{\mathrm {f}}^2}$.

The ccdf of SINR $\gamma _{\mathrm {CD}}^{\mathrm {f,CoMP}}$ is
(46)
where (a) follows that h_m and h_f are independent Gaussian variable $\mathcal {N}(0,1)$.

The Laplace transform of I_m is
$$\displaystyle \begin{aligned} \mathcal{L}_{I_{\mathrm{m}}}(s)&=\exp{\left(-2\pi\tilde{\lambda}_{\mathrm{mb}}\int_{r_{\mathrm{m}}}^{\infty}(1-\mathcal{L}_h(sP_{\mathrm{m}}x^{-\alpha}))x\mathrm{d}x\right)}\notag\\ &=e^{-\pi\lambda^\prime_mr_{\mathrm{m}}^2\rho(\frac{sP_{\mathrm{m}}}{\mu r_{\mathrm{m}}^{\alpha}}, \alpha)}. \end{aligned} $$
(47)

The Laplace transform of I_f is
$$\displaystyle \begin{aligned} \mathcal{L}_{I_{\mathrm{f}}}(s)&=\exp{\left(-2\pi\tilde{\lambda}_{\mathrm{fb}}\int_{r_{\mathrm{f}}}^{\infty}(1-\mathcal{L}_h(sP_{\mathrm{f}}x^{-\alpha}))x\mathrm{d}x\right)}\notag\\ &=e^{-\pi\lambda^\prime_fr_{\mathrm{f}}^2\rho(\frac{sP_{\mathrm{f}}}{\mu r_{\mathrm{f}}^{\alpha}}, \alpha)}. \end{aligned} $$
(48)

Data Rate

The instantaneous data rate that a sub-band channel of bandwidth W can accommodate is R = Wlog₂(1 + SINR). For LD transmission, since both the MBS and the FAP transmit digital signals; the channel from the typical UE to its serving MBS can accommodate the data rate $R_{\mathrm {m}}=W\log _2(1+\gamma _{\mathrm {LD}}^{\mathrm {m}})$, and the channel from the typical UE to its serving FAP can accommodate the data rate $R_{\mathrm {f}}=W\log _2(1+\gamma _{\mathrm {LD}}^{\mathrm {f}})$. For LHDA transmission, only the BL is modulated to a digital signal, so the data rate is defined only for the BL, the channel from the typical UE to its serving MBS can accommodate the data rate $R_{\mathrm {m}}=W\log _2(1+\gamma _{\mathrm {LHDA}}^{\mathrm {m}})$, and the channel from the typical UE to its serving FAP can accommodate data rate $R_{\mathrm {f}}=W\log _2(1+\gamma _{\mathrm {LHDA}}^{\mathrm {f,B}})$.

The actually achieved UE data rates, after taking into consideration the UE load and sub-band occupancy, are given below. Without loss of generality, we take an MBS as an example. When the number of UEs in a macro cell does not exceed the total number of sub-bands (i.e., U_m ≤ N_m), each UE can exclusively occupy a sub-band, and its achieved data rate is R_m; when U_m > N_m, the U_m UEs share the N_m sub-bands, and the data rate is thus discounted into $\frac {N_{\mathrm {m}}}{U_{\mathrm {m}}}R_{\mathrm {m}}$, assuming a round-robin sharing mechanism. So the average achieved data rate of a UE served by an MBS is given by:

$$\displaystyle \begin{aligned} R_{\mathrm{mu}}=\xi_{\mathrm{m}}R_{\mathrm{m}}, \end{aligned} $$

(49)

where ξ_m is the scheduling index denoting the probability that a UE is scheduled by the MBS,

$$\displaystyle \begin{aligned} \xi_{\mathrm{m}}=\frac{\sum_{i=1}^{N_{\mathrm{m}}}\mathbb{P}\{U_{\mathrm{m}}=i\}+\sum_{i=N_{\mathrm{m}}+1}^{\infty}\mathbb{P}\{U_{\mathrm{m}}=i\}\frac{N_{\mathrm{m}}}{i}}{1-\mathbb{P}\{U_{\mathrm{m}}=0\}}. \end{aligned} $$

(50)

Similarly, the average achieved data rate of a UE served by an FAP is given by

$$\displaystyle \begin{aligned} R_{\mathrm{fu}}=\xi_{\mathrm{f}}R_{\mathrm{f}}, \end{aligned} $$

(51)

where ξ_f is the scheduling index denoting the probability that a UE is scheduled by the FAP,

$$\displaystyle \begin{aligned} \xi_{\mathrm{f}}=\frac{\sum_{i=1}^{N_{\mathrm{f}}}\mathbb{P}\{U_{\mathrm{f}}=i\}+\sum_{i=N_{\mathrm{f}}+1}^{\infty}\mathbb{P}\{U_{\mathrm{f}}=i\}\frac{N_{\mathrm{f}}}{i}}{1-\mathbb{P}\{U_{\mathrm{f}}=0\}}. \end{aligned} $$

(52)

System Performances

In this section we evaluate several important performance metrics, namely, the outage probability, the HD probability, and the average distortion. The outage probability is the probability that a UE cannot receive the BL, namely, the UE data rate is less than R_B. The HD probability is the probability that a UE can receive high-definition content, i.e., both the BL and the EL, namely, the UE data rate is greater than R_B + R_E. The average distortion evaluates the difference between the received video and source video, which is measured using the distortion-rate function. Note that, for LHDA transmission, the HD probability for the femto UE is not defined since the EL is transmitted as an analog signal and the data rate for an analog signal is undefined.