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Capacity gain and design tradeoffs for partialduplex OFDM wireless communications
EURASIP Journal on Wireless Communications and Networking volume 2018, Article number: 288 (2018)
Abstract
A hybrid transmission method between classical halfduplex and fullduplex is here considered for pointtopoint orthogonal frequency division multiplexing (OFDM) links experiencing frequency selective fading. This partialduplex solution uses only a portion of the available bandwidth for fullduplex transmission. It aims to increase the overall bidirectional system rate w.r.t. an equivalent halfduplex system, relaxing at the same time the high selfinterference cancelation requirements that practical fullduplex systems have to provide. In this paper, we analyze the regions of selfinterference cancelation values where partialduplex systems outperform halfduplex ones in terms of achievable spectral efficiency. We investigate the potential of the proposed hybrid method by deriving the analytical distributions of the spectral efficiency gain regions in the presence of frequency selective Rayleigh fading. For different strategies in the selection of the subcarriers that operate in fullduplex, we highlight the role of the different parameters involved and the peculiarity of this system in terms of design flexibility.
Introduction
The interest for wireless fullduplex (FD) communication in the future 5G and beyond networks is increasing due to its potential of contributing to some of the ambitious goals in the evolution of the next generation of wireless communication: increased distance and capacity of the wireless links, spectrum virtualization, and enhanced interference coordination [1, 2]. However, one of the most important deterrents in implementing an FD wireless system is the impact of the selfinterference signal that is generated locally by the transmission device. The amplitude of this echo, in fact, is tens of decibels above the received signal of interest, which comes from the other transmitter and is highly attenuated by the channel propagation path loss and fading. For this reason, the selfinterference cancelation (SIC) capability is the crucial technology that can make possible the FD promising throughput improvements [3, 4].
One of the earliest reports on experimental demonstrations of FD communication for narrowband wireless communication systems was presented in 1998 [5]. Since that time, many researchers have investigated this topic and proposed different methods to extend the implementation to larger bandwidths and have studied solutions for the transmission of a signal in FD, based on several types of SIC schemes, from the analog radio frequency domain to the digital baseband domain. One of the simplest SIC methods consists of exploiting transmit and receive directional antennas to decouple the transmitted from the received signals [6]; other solutions are also based on antenna cancelation methods [7], relying on proper antenna positions, beamforming tuning, or multipleinput multipleoutput (MIMO).
In [8–11], the innovative approach is in the antenna design, supporting multiple antennas twoway transmission with multiple levels of cancelation of the selfinterference signal. On the other hand, [12] presents the design and implementation of a single antenna with high values of SIC till 110 dB measured for WiFi signals in dense indoor office environments.
In [13], an auxiliary transmit path at radio frequency is used to feed a cancelation signal, which is a preprocessed version of the transmit signal, in order to match exactly the selfinterference signal and cancel it.
In [14], a prototype of the FD design is realized on a softwaredefined radio platform; this design combines a dualpolarization antennabased analog part with a digital selfinterference canceler that operates in real time. The results from tests on this prototype confirm that the proposed FD system achieves about 1.9 times higher throughput than the corresponding halfduplex (HD) system, even if for short distances.
In [15], the maximum achievable FD link range is reported as 30 m. The shortrange capability of FD systems, reported in many studies, is another way of expressing the strong impact of the residual selfinterference on the system performance.
In [16], we proposed the partialduplex (PD) system, a hybrid solution where HD and FD coexist on the same link and at the same time, differently from other studies.
In [17], the hybrid solution named Xduplex switches between HD and FD transmission, depending on the level of the selfinterference. The authors in [18–20] evaluate the performance of FD wireless communication networks consisting of base stations with FD capability and legacy HD users.
In [21–23], the hybrid solution named αduplex is designed for bidirectional singlecarrier transmission where the bands for the two directions are contiguous. The proposed solution is to overlap partially the two bands (the parameter α is the percentage of overlapped bandwidth), enlarging the bandwidth in each direction and increasing the capacity of each directional link. This objective is achieved at the expense of losing the signal orthogonality in the spectral domain. The authors report a performance gain on the bidirectional link through a proper selection of the pulse shaping and the related matched filtering in order to optimize the signal separation at the receiver.
On the contrary, our PD hybrid solution is a bidirectional communication link which operates in FD mode only in a portion of the bandwidth, partitioning the overall bandwidth into one FD segment serving simultaneously both directions and two HD segments, one for each direction.
The PD system investigated here is based on multicarrier transmission on frequency selective channels as in [16], where different criteria can be adopted to partition the bandwidth in FD and HD segments. Two different strategies in the assignment of the FD subcarriers are revised in Section 4.1, one based on a block allocation, named block partialduplex, and the other one based on selective allocation, named selective partialduplex. The PD scheme has obviously two extreme cases, the full HD and the full FD, and can be applied either to pointtopoint highcapacity links or to pointtomultipoint links in cellular scenarios. W.r.t. FD, our approach balances a capacity loss, coming from the partial use of the bandwidth simultaneously in the two transmission directions, with a decrease of the SIC requirements. Here, we limit our study on symmetrical pointtopoint links where the transceiver equipments at both ends have equivalent characteristics, e.g., noise figure and SIC capability, and the propagation introduces path loss and Rayleigh frequency selective fading. The numerical results are then extended to a standardized International Telecommunication Union  vehicular A (ITUVehA) frequency selective fading model.
The novel contribution of the paper is the definition and the derivation of the theoretical limits of a PD approach in terms of the achievable spectral efficiency gains w.r.t. a classical HD system and the investigation of the role of the main system parameters involved in the PD design.
We remark that a PD approach is intended to be complementary to the implementation of an efficient selfinterference canceler; the main motivation for the adoption of a PD scheme is the search of encoding solutions that take advantage from the bandwidth portion with higher SNR, possibly exploited in FD mode, for relaxing the high performance targets of the SIC scheme. In order to reveal the potential advantage of the method, the analytical results assume an ideal receiver model, neglecting possible nonidealities as imperfect synchronization or amplifier nonlinearities or possible leakages from FD to HD subcarriers due to impairments present in a real radio frequency receiver. On the other hand, these impairments have been taken into account in the simulation results of the channel model in order to show their impact on the SIC margin w.r.t. that predicted by the analysis.
In the sequel, Section 2 revises the main concepts of the PD wireless solution. The system model and the strategies adopted for PD are introduced in Section 3. Then, in Section 4, we present the analysis of the capacity gains and of the related SIC design tradeoffs in AWGN and frequency selective channels. Finally, in Section 5, the numerical results, obtained analytically and by the simulations, are reported and discussed.
Partialduplex
PD is a hybrid solution between classical HD and FD. Its rationale is the increase of the overall bidirectional system data rate w.r.t. HD operation, with a contemporary relaxation of the high SIC requirements. In the proposed PD approach, the PD parameter (PDP) is the fundamental factor that returns the fraction of the overall spectrum that will operate in FD mode. We will show that, in the presence of frequency selective fading, a smart choice of the portion of the bandwidth to be used in FD can maximize the spectral efficiency while operating with a selfinterference canceler of lower performance and complexity w.r.t. that required in an FD system.
Our reference application is a multicarrier transmission where, given a value of PDP, the N subcarriers are partitioned into two groups Π_{HD} and Π_{FD}, consisting of N_{HD} and N_{FD} subcarriers, that operate on one and two directions, respectively (considering two generic nodes A and B, the two directions are A to B and B to A), i.e.:
We assume that N_{FD}=PDP·N is approximated to the closest integer value and N_{HD} to the closest even number, so providing the same capacity on the two link directions.
In order to analyze the performance of PD systems, we derive the spectral efficiency gain region as the region of SIC capability in which PD system outperforms HD in terms of spectral efficiency. Our approach is similar to the approach adopted in [24, 25]. In [24], the authors define the rate gain region on the received signalofinterest strength, while here we define it on the capability of SIC. The spectral efficiency region is obtained by the condition η_{PD}>η_{HD}, where η_{PD} and η_{HD} are the PD and HD spectral efficiencies, respectively.
In a multicarrier system with N subcarriers, a sum spectral efficiency region will be investigated according to the spectral efficiency η [bits/subcarrier] averaged on the overall bandwidth as:
where η_{i} are the spectral efficiencies provided on each subcarrier (i=1,2,···,N). Considering that each subcarrier can work in HD, i.e., serving only one direction of the link, or in FD, serving both directions, the spectral efficiency formula for η_{i} on the ith subcarrier varies depending whether it is operated in HD or FD mode, respectively as η_{HD,i} and η_{FD,i}. Such spectral efficiency values depend on the signaltonoise ratio or signaltointerferenceplusnoise values, namely γ_{HD,i} and γ_{FD,i} on each subcarrier, as:
where the derivation computation of γ_{HD,i} and γ_{FD,i} will be discussed in Section 4. Therefore, the Shannon spectral efficiency (1) in the PD case is specified as:
and the PD spectral efficiency gain η_{G} [%] is defined w.r.t. the HD spectral efficiency \(\eta _{\text {HD}} = \frac {1}{N}\sum \nolimits _{i=1}^{N}\eta _{\text {HD},i}\) as:
On the contrary, in the presence of random frequency selective fading, for each jth fading realization, the generic ith subcarrier experiences the spectral efficiencies η_{FD,i,j} or η_{HD,i,j} respectively in FD and HD, and the jth sum spectral efficiencies η_{PD,j} and η_{HD,j} are evaluated by:
Therefore, given the statistical fluctuations of η_{FD,i,j} and η_{HD,i,j} per subcarrier and per fading realization, two definitions of the spectral efficiency gain regions for the PD scheme are considered:

The former considers the average values of the spectral efficiencies E_{j}[η_{PD,j}] and E_{j}[η_{HD,j}]. The efficiency gain region (4) is defined by:
$$\begin{array}{*{20}l} &\sum_{i\in\Pi_{\text{FD}}}E_{j}[\eta_{\text{FD},i,j}]+ \sum_{i\in\Pi_{\text{HD}}}E_{j}[\eta_{\text{HD},i,j}] \geq \\ &\sum_{i\in\{\Pi_{\text{FD}} \bigcup \Pi_{\text{HD}}\}}E_{j}[\eta_{\text{HD},i,j}], \end{array} $$which is simplified to:
$$\sum_{i\in\Pi_{\text{FD}}}E_{j}[\eta_{\text{FD},i,j}]\geq \sum_{i\in\Pi_{\text{FD}}} E_{j}[\eta_{\text{HD},i,j}], $$and it is satisfied if:
$$ E_{j}[\eta_{\text{FD},i,j}]\geq E_{j}[\eta_{\text{HD},i,j}], \forall i\in\Pi_{\text{FD}}. $$(8) 
The latter imposes a limit on the occurrence of the event η_{PD,j}<η_{HD,j}. Given the condition:
$$ P_{\text{out}}=\text{Prob}_{j}[\eta_{\text{PD},j} < \eta_{\text{HD},j}] = 0.01, $$or, equivalently, Prob_{j}[η_{PD,j}≥η_{HD,j}]=0.99, from (6) and (7), the gain region is defined as:
$$ \text{Prob}_{j}(\eta_{\text{FD},i,j}>\eta_{\text{HD},i,j})\geq 0.99, \forall i\in\Pi_{\text{FD}}. $$(9)This definition corresponds to a more restrictive efficiency gain region than the previous one.
In multicarrier systems with frequency selective fading, different strategies of allocation of HD and FD portions can be considered for taking advantage of the system flexibility. Two schemes have been already proposed in [16]: a selective strategy (selective partialduplex (SPD)) and a block strategy (block partialduplex (BPD)). In BPD, the subcarriers selected for FD mode are adjacent, as in a single block. Without any selection strategy on the block, we can assume to choose it centered in the available spectrum; this is the allocation strategy that reproduces the method proposed in [21–23] in order to increase the throughput in singlecarrier systems, by partially overlapping of the spectrum of the two adjacent bandwidths assigned to the two directions of the link. On the other hand, the SPD strategy exploits the maximum available selectivity of multicarrier transmission, allowing the selection of the FD subcarriers as those ones with the higher signaltointerferenceplusnoise ratio γ_{FD,i,j} thus providing potentially the best spectral efficiency gain. In [16], the numerical results have already highlighted the advantage from a flexible design of PD systems in the combination of SIC and PDP parameters, with the aim of increasing spectral efficiency performance for a fixed link distance or, alternatively, improving link distance performance for a target spectral efficiency. Depending on the characteristics of the random fading and on the PD strategy, numerical results in [16] showed indeed how longer distances can be reached for reasonable SIC capabilities by decreasing PDP w.r.t. the high demanding SIC capability in a full FD system.
System model
Let us consider a bidirectional link between the two nodes, A and B, and we assume that the link characteristics are reciprocal, from A to B and B to A. Two OFDM signals, one for each direction, occupy a total bandwidth equal to B [Hz]. As previously described, a portion of the subcarriers operates in FD mode while the rest in HD. For transmitting in the FD portion of the spectrum, both devices are equipped with a SIC module needed for contrasting efficiently the selfinterference w.r.t. the signal of interest, which is received heavily attenuated by the propagation path loss and the random fading. In Fig. 1, a highlevel block diagram of an OFDM FD transceiver with a SIC module is sketched.
The wireless propagation channel model between A and B, assumed reciprocal in the two directions, is modeled by a flat frequency path loss (PL) contribution and a random fading frequency selective transfer function H_{C}(f). The analysis has been conducted for two channel models: (a) an AWGN channel model that introduces additive white Gaussian noise, assuming all over the bandwidth a normalized channel transfer function H_{C}(f)=1, and (b) a frequency selective fading channel model where the received signal on the ith subcarrier with center frequency f_{i} is affected by the channel transfer function H_{C}(f_{i})=H_{i}. The frequency selective channel is simulated either by a Rayleigh channel model with different coherence bandwidths or by using the multipath intensity profile of the ITUVehA channel model [26].
In case of Rayleigh fading model, in order to reproduce the general impact of a frequency selective channel according to the coherence bandwidth definition B_{C}, i.e., the range of frequencies over which the channel response is highly correlated, we consider a simplified channel, as proposed in [16] where:

The bandwidth of the signal is divided into equal subbands of width B_{C} [Hz], where the channel is assumed flat;

In each subband, the complex channel amplitude is a zeromean Gaussian random process, with Rayleigh distributed envelope, and the channel coefficients in different subbands are independent and identically distributed;

As representative of the channel frequency selectivity, the single parameter λ_{C}=B_{C}/B allows to reproduce the frequency selectivity effect of different channel models.
Concerning the path loss (PL), we have adopted the path loss (PL) formula proposed in 3GPP Recommendation [26] for an Urban LoS (Line of Sight) environment:
where the link distance d is in meters.
We summarize the assumptions of our system as:

The channel is reciprocal in the two directions (i.e., A to B and B to A);

The overall propagation channel transfer function is characterized by the path loss PL, dependent on the link distance d, and the frequency selective Rayleigh fading H_{C}(f);

The effect of transceiver nonidealities is modeled at the receiver as an error vector magnitude (EVM) noise with average power proportional to the received power P_{R}, as P_{R}/γ_{E} [25];

An echo cancelation circuit, with capability defined by the parameter SIC, is able to mitigate the selfinterference generated by bidirectional transmission in FD subcarriers. The residual selfinterference signal after the canceler is assumed flat with power P_{T}/SIC, where P_{T} is the transmit power on each subcarrier. In practice, some frequency dependence could be present in the residual selfinterference power, but we assume that its effect on the final signaltointerferenceplusnoise ratio can be included in the frequency selectivity of the channel;

The effect of the leakage that the FD subcarriers generate into the HD ones is modeled as an EVM noise with average power proportional to the residual selfinterference generated by the FD subcarriers PDP·P_{T}/SIC, as PDP·(P_{T}/SIC)/γ_{E}.
According to these assumptions, the power of the received signal of interest P_{R,i} on the ith subcarrier at frequency f_{i}, assuming all the subcarriers at the same transmit power P_{T} transmit and receive antenna gains G_{T}·G_{R}=G_{ant} and channel power gain g_{i}=H_{i}^{2}, is given by:
and the value of the signaltonoise ratio on HD subcarriers γ_{HD,i} (2) is simply given by:
with the thermal noise power P_{N}, the EVM noise P_{R,i}/γ_{E}, and the leakage from FD subcarriers PDP·P_{T}/(SIC·γ_{E}).
Analogously, considering that on FD subcarriers the residual selfinterference power is given by P_{T}/SIC, the signaltointerferenceplusnoise ratio γ_{FD,i}, in (3), is given by:
Notice that, in the presence of random fading, the notation includes the ^{′}j^{′} subscript for denoting the jth fading realization, i.e.:
Methods for the analysis of rate gain regions
The analysis in AWGN channel is a preliminary step for the Rayleigh frequency selective model, separated in Section 4.1. In AWGN, all the subcarriers experience the same received power:
for i=1,2,⋯,N, and consequently, they have the same γ_{HD,i}=γ_{HD} and γ_{FD,i}=γ_{FD} and the same η_{HD,i}=η_{HD} and η_{FD,i}=η_{FD} for HD and FD modes, respectively. In order to analyze the spectral efficiency gain region w.r.t. the value of PDP, we make explicit in (4) the dependence of the spectral efficiency expression at varying PDP as:
and we study the sign of its derivative w.r.t. PDP, for 0<PDP<1,
It is straightforward to see two simple cases emerging:
highlighting that the optimal design scheme coincides with a full FD scheme (PDP_{OPT}=1) when η_{FD}>η_{HD} and with a full HD transmission (PDP_{OPT}=0) when η_{FD}<η_{HD}. The definition of the spectral efficiency gain region is:
that turns out to be:
Hence, from (12) and (13), for g_{i} = 1, the spectral efficiency gain condition becomes SIC>SIC_{0,AWGN}, with SIC_{0,AWGN} computed as in (17).
In particular, assuming in (17) γ_{E}=∞ and \(P_{N} \ll \frac {P_{T} \cdot G_{\text {ant}}}{PL}\), i.e., γ_{HD,AWGN}≫1, SIC_{0,AWGN} is reduced to:
Some observations can be made looking at the formulae:

For SIC≤SIC_{0,AWGN}, where η_{HD}≥η_{FD}, PD schemes are not able to provide any efficiency gain, and they suffer from an efficiency loss, which increases as SIC decreases. In this case, the best choice is a pure HD system (PDP=0) with spectral efficiency η_{min}=η_{HD} and efficiency gain η_{G,min}=0;

For SIC>SIC_{0,AWGN}, the efficiency gain of the PD scheme increases linearly with PDP with slope η_{FD}−η_{HD} as in (15), and the best design is a full FD system (PDP=1);

For values of SIC well above the threshold, the residual selfinterference becomes negligible w.r.t. the thermal noise, i.e., γ_{FD}=γ_{HD}, and the optimal design is with PDP=1, providing the maximum performance η_{max}=2η_{HD} and η_{G,max}=100%;

For SIC values slightly above the threshold, i.e., when γ_{FD} is dominated by the selfinterference, spectral efficiency and efficiency gain follow approximately a logarithmic function.
Frequency selective Rayleigh channels
In multicarrier systems with frequency selective fading, the PD strategies for partitioning the spectrum in HD and FD portions can take advantage from the flexibility of the system. For a Rayleigh fading channel, the channel power gain g_{i}≥0 that appears in the received power on the ith subcarrier (11) is exponentially distributed with probability density function f(g_{i}), cumulative density function F(g_{i}), and mean value \(\overline {g}_{i}=E\left [g_{i}\right ]\) as:
Differently from the frequency flat AWGN channel, in which the threshold SIC_{0,AWGN} discriminates clearly the convenience between FD and HD transmissions, the presence of frequency selective fading requires a more elaborated response. Following the approach in (14)–(18), for each ith subcarrier, we define a SIC_{0,i} threshold obtained substituting P_{R} with \(P_{R,i}=\frac {P_{T}\cdot G_{\text {ant}}\cdot g_{i}}{PL}\) in (17).
Now, the performance analysis of the allocation strategies SPD and BPD requires a probabilistic approach, through the determination of the statistics of SIC_{0,i} thresholds. Therefore, for these two allocation strategies, we define two SIC thresholds, SIC_{0} and SIC_{99}, according to the efficiency gain regions defined in (8) and (9), respectively:

When SIC=SIC_{0},
$$ E_{j}[\eta_{\text{FD},i,j}] = E_{j}[\eta_{\text{HD},i,j}], \forall i\in\Pi_{\text{FD}}; $$(20) 
When SIC=SIC_{99},
$$ \text{Prob}_{j}(\eta_{\text{FD},i,j} = \eta_{\text{HD},i,j})\geq 0.99, \forall i\in\Pi_{\text{FD}}. $$(21)
In the case of γ_{E}=∞ and \(P_{N} \ll \frac {P_{T} \cdot G_{\text {ant}}}{\text {PL}}\) (as in (18), i.e., for dominant selfinterference w.r.t. AWGN and neglecting the effect of transceiver nonidealities), SIC_{0,i} thresholds turn out to be inversely proportional to the corresponding amplitude channel gain g_{i}, i.e.:
and we are able to derive the theoretical expressions of SIC_{0} and SIC_{99} thresholds for the two allocation strategies from the distribution of the amplitude channel gain. This derivation constitutes, as it will be shown in the numerical results, a reference for the system potential gain.
Selective partialduplex
In order to maximize the spectral efficiency in a PD scheme, as the thresholds SIC_{0,i} in (22) are inversely proportional to the channel power gains g_{i}, the optimal allocation assigns the FD portion to the subcarriers with the highest gains. Therefore, ordering the power channel gains with decreasing values \({[g^{\prime }_{1}, g^{\prime }_{2},\cdots, g^{\prime }_{N}]}\), the FD subcarriers will be the ones that experience the highest N_{FD} gains \([g^{\prime }_{1}, g^{\prime }_{2},\cdots, g^{\prime }_{\phantom {\dot {i}\!}N_{\text {FD}}}]\) corresponding to the SIC thresholds \([\text {SIC}^{\prime }_{0,1}, \text {SIC}^{\prime }_{0,2},\cdots, \text {SIC}^{\prime }_{\phantom {\dot {i}\!}0,N_{\text {FD}}}]\) in increasing order. Denoted ρ the SIC level of the system, if less than N_{FD} subcarriers have thresholds SIC_{0,i}>ρ, the proposed allocation is the best choice; if more than N_{FD} subcarriers have SIC_{0,i}>ρ, the selection of the highest N_{FD} gains \({[g^{\prime }_{1}, g^{\prime }_{2},\cdots, g^{\prime }_{\phantom {\dot {i}\!}N_{\text {FD}}}]}\) is again the best assignment. In fact, from (22), the SIC value ρ corresponds to a gain value \(\tilde {g}\) and, consequently, to a signaltonoise ratio \(\tilde {\gamma }_{\text {HD}}\) and a signaltonoiseplusinterference ratio \(\tilde {\gamma }_{\text {FD}}\) that satisfy (16):
Now, for a generic gain \(g>\tilde {g}\), the FD capacity gain over HD is:
and, from (23):
Equation (24) shows the increasing trend of the capacity gain with g, justifying the best selection of the N_{FD} subcarriers with highest gains.
The derivation of the thresholds SIC_{0} and SIC_{99} starts from the knowledge of the cumulative distribution function of the power gain F(g_{i}) (19); given a vector of N power gains in ascending order, the cumulative distribution function \(F_{\phantom {\dot {i}\!}NN_{\text {FD}}+1}(g)\), i.e., the probability that the first N_{HD}=N−N_{FD} power gains are less than a defined g value and the remaining N_{FD} power gains are greater than g, is:
In order to derive the expression of SIC_{0}, we rewrite the condition (20), including γ_{FD}≫1 and γ_{HD}≫1, as
and then
with deriving \(f_{NN_{\text {FD}}+1}(g)\) from:
Hence, comparing (25) and (26), the threshold SIC_{0} can be expressed as the threshold SIC_{0,AWGN} (18) multiplied by a margin Δ_{0,SPD} as:
where the margin Δ_{0,SPD} is equal to:
and depends on the fading distribution and on the PDP value (i.e., N and N_{FD}) according to (27).
Now, in order to derive the expression of SIC_{99} (21), we consider the cumulative distribution of
Λ=1/g
that for the SPD scheme is:
From the definition of the threshold SIC_{99,SPD}, we define Λ_{99,SPD} as the value that satisfies the condition Prob(Λ<Λ_{99,SPD})=0.99, i.e., F_{Λ,SPD}(Λ_{99,SPD})=0.99, and we call Δ_{99,SPD} its square root:
Hence, analogously to the final expression derived for Δ_{0,SPD} (28), from (22), we can express also the threshold SIC_{99,SPD} as the threshold SIC_{0,AWGN} (18) multiplied by the margin Δ_{99,SPD}:
Again, analogously to the margin Δ_{0,SPD}, also the margin Δ_{99,SPD} depends on the fading distribution and on the values of N and N_{FD} as in (30).
Notice that (??) assumes that the N power gains are independent, i.e., the frequency selectivity factor is λ_{C}≤1/N. For higher values of λ_{C}, the derivations can be rewritten replacing, in the formulae, N and N_{FD} with \(\widetilde {N}=\lceil 1/ \lambda _{C} \rceil \) and \(\widetilde {N}_{\text {FD}}=\lceil N_{\text {FD}}/(N\lambda _{C}) \rceil \) respectively, introducing the impact of the frequency selectivity in the thresholds. Obviously, we expect a loss in the accuracy of the analytical results depending on the error introduced in the above approximations.
We remark that both expressions of SIC_{0,SPD} and SIC_{99,SPD} in (28) and (32) are composed of two factors: SIC_{0,AWGN}, which depends on the radio link parameters (transmitted power P_{T}, antenna gains G_{ant}, propagation path loss PL and consequently link distance d, noise power P_{N}), and a margin, Δ_{0,SPD} and Δ_{99,SPD}, respectively, which depends on the fast fading channel (distribution, frequency selectivity λ_{C}) and on the PDP parameter of the PD scheme.
Block partialduplex
In the block partial case, the gain vector is not ordered anymore, and the FD subcarriers are contiguous in a single block at the center of the bandwidth.
The derivation of the analytical expression of the thresholds SIC_{0,BPD} and SIC_{99,BPD}, as for the SPD strategy, requires the knowledge of the distributions of the channel gain g_{i} for the subcarriers operating in FD mode. Here, the probability that randomly chosen N_{FD} power gains are less than a defined g value, without caring about the values that the other N−N_{FD}+1 channel gains assume, is expressed simply by \(F_{\phantom {\dot {i}\!}N_{\text {FD}}}(g)\) as:
Then, following the same steps of the thresholds derivation for the SPD strategy, also the thresholds SIC_{0,BPD} and SIC_{99,BPD} turn out to be the product of the threshold SIC_{0,AWGN} by a margin, Δ_{0,BPD} and Δ_{99,BPD}, respectively, which returns the impact of the channel, of the frequency selectivity λ_{C} and of the PDP parameter:
As noted for SPD, when the N_{FD} power gains are not independent, i.e., for frequency selectivity factors λ_{C}>1/N, (33) has to be rewritten replacing N_{FD} with \(\widetilde {N}_{\text {FD}}=\lceil N_{\text {FD}}/(N\lambda _{C}) \rceil \).
It can be noticed that, differently from the SPD case, only N_{FD}, and not N_{FD} and N, appears in the cumulative distribution of the channel gains g_{i} operating in FD mode, i.e., \(F_{\phantom {\dot {i}\!}N_{\text {FD}}}(g)\) (33), and, consequently, in the margin expressions.
Numerical results
In this section, we report the results of the numerical analysis and the simulations of the PD system performance, with the aim of highlighting design tradeoffs and spectral efficiency gain regions for practical application in OFDM pointtopoint links. The main parameters of the simulated system are listed in Table 1. In a fixed bandwidth B=10 MHz with N=1024 subcarriers, for different link distances d, different values of the system parameters PDP and for the two PD partition strategies (BPD and SPD), the requirements on the SIC capability have been derived. First, PD transmission is analyzed in the absence of fading, i.e., on a AWGN channel. Then, the numerical results are extended to a Rayleigh frequency selective fading channel model; for each value of the frequency selectivity parameter λ_{C} = [ 0.001,0.008,0.016,0.031,0.062,0.125,0.25,0.5,1], corresponding to coherence bandwidths B_{C} equivalent to [1,8,16,32,64,128,256,512,1024] subcarriers, 1000 realizations of the transfer function H_{C}(f) are generated. Finally, the simulations are repeated for the ITUVehA channel model.
AWGN channel
Figure 2 shows the behavior of SIC_{0,AWGN} as a function of the distance d for two different values of G_{ant}, G_{ant}=40 dB (a) and G_{ant}=0 dB (b). Moreover, each subfigure shows the curves for three different values of γ_{E}=[∞,10^{5},10^{3}]. First, we can notice that the gain of the antennas G_{ant} has a significant impact. For γ_{E}=∞, as expected from the deterministic expression (18), decreasing the antennas gain reduces the received power and, consequently, selfinterference becomes dominant with corresponding higher SIC requirements. Moreover, the antennas gain affects also the EVM noise power P_{R}/γ_{E}, for γ_{E}<∞. Therefore, the greater the value of G_{ant}, the higher the impact of EVM noise, i.e., the γ_{E} parameter, on the SIC_{0,AWGN} value. On the other hand, the leakage contribution from FD to HD subcarriers, which is related to the PDP parameter, has no effect on the SIC_{0,AWGN} values.
Frequency selective channel
Figure 3 reports PD performance in terms of E[η_{G}], the spectral efficiency gain averaged on all the subcarriers and all the fading realizations. The results refer to a frequency selective Rayleigh channel model with frequency selectivity parameter λ_{C}=1/N, γ_{E}=∞, SPD strategy and link distance d=200 m. The curves show that the SIC value that guarantees an efficiency gain depends on PDP; this suggests that a spectral efficiency advantage might be tuned by implementing a partial form of the FD scheme. Same results are reported in Fig. 4 vs. PDP for different values of SIC and are compared with the AWGN case: while in AWGN the gain increases linearly with PDP for any value of SIC, in a channel with a high frequency selectivity, the SPD scheme can take advantage of a frequency diversity gain when PDP is low. In fact, the curve for SIC=80 dB shows a higher gain at PDP=0.5 than at PDP=1.
In order to better investigate the role of the parameters PDP and SIC on the spectral efficiency gain in PD schemes, we derived analytically and simulated the thresholds SIC_{0}, SIC_{99} for the SPD and BPD strategies. Then, we plotted their margins w.r.t. SIC_{0,AWGN}, i.e., Δ_{0} and Δ_{99}, as defined in (28) and (32), and (34) and (35), respectively. Figures 5 and 6 show the resulting Δ_{0,SPD}, Δ_{99,SPD} as a function of PDP for different values of the frequency selectivity factor λ_{C}, antenna gain G_{ant}=40 dB and two different values of γ_{E}: γ_{E}=∞ (a) and γ_{E}=10^{3} (b). In the subfigures (a), the analytical results (continuous lines) and the simulated ones (dotted lines) agree quite well and validate the theoretical analysis. The results in subfigures (b), with the modeling of nonidealities and leakage noise, show increased values w.r.t. subfigures (a) but maintain a similar behavior w.r.t. PDP. The margins for the SPD strategy show variations from about − 2 dB to 17 dB with PDP for low values of λ_{C}. The selective strategy, exploiting the diversity gain offered by the frequency selectivity, for low PDP and low λ_{C} shows negative margins. Interestingly, the curves show that, with a limited SIC capability, it may be an advantage to use PDP values lower than 1.
Figures 7 and 8 refer to the BPD strategy; they report the margins Δ_{0,BPD} and Δ_{99,BPD} for G_{ant}=40 dB and γ_{E}=∞. Also in this case, the simulation results (dotted lines) and the theoretical ones (continuous lines) agree except for some slight differences. We observe that the two strategies provide the same performance for the extreme cases PDP=1 and λ_{C}=1, as expected. But the margin for the BPD strategy is not so sensitive to PDP variations, differently from the SPD strategy, as BPD is unable to select the best subcarriers for FD. So, BPD shows the highest Δ_{99} at the lowest λ_{C}, as the probability to find heavily faded subcarriers increases.
ITUVehA channel
Figure 9 shows the ITUVehA frequency autocorrelation vs. λ_{C}, here interpreted as Δf/B. As expected, the autocorrelation differs from our simplified model, which is reported on the same figure assuming λ_{C}=0.5. Then, Fig. 10 reports the values of the four margins, Δ_{0} and Δ_{99} for both strategies, as a function of PDP. Due to the difference in the frequency correlation function of the two models, the correspondence between the two models looking at the Δ margin curves is not evident; anyway, ITUVehA curves are inside the range of the λ_{C} model. The curves for ITUVehA model confirm the advantage of the SPD strategy w.r.t. BPD and the flexibility of the design of a PD scheme, in terms of tradeoff between SIC capability and PDP. Particularly for the SPD strategy, Δ_{99} passes from 5 dB (PDP=0.1) to 26 dB (PDP=1), allowing to implement partial versions of the FD modality and providing spectral efficiency gains also with limited SIC capabilities.
Conclusions
The selfinterference canceler is a key component of wireless FD communication transceivers, limited in their application and performance by the extremely low levels of the received signal when compared to the residual selfinterference signal. In this paper, we analyze the theoretical limits of a PD approach, characterized by FD operation in a portion of the available spectrum. The motivation behind this approach is the relaxation of the constraints on the selfinterference canceler and the investigation of new tradeoffs between the achievable spectral efficiency on the bidirectional link and the SIC capability. Starting from a general channel model characterized by Rayleigh frequency selective fading, we have investigated the role of the main system parameters from an analytical and simulation points of view, in order to highlight the potential of the PD solution.
Abbreviations
 AWGN:

Additive white Gaussian noise
 BPD:

Block partialduplex
 EVM:

Error vector magnitude
 FD:

Fullduplex
 HD:

Halfduplex
 ITUVehA:

International Telecommunication Union  vehicular A
 OFDM:

Orthogonal frequency multiplexing division
 PD:

Partialduplex
 PDP:

Partialduplex parameter
 SIC:

Selfinterference cancelation
 SNR:

Signaltonoise ratio
 SPD:

Selective partialduplex
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Barzegar, H.R., Reggiani, L. & Dossi, L. Capacity gain and design tradeoffs for partialduplex OFDM wireless communications. J Wireless Com Network 2018, 288 (2018) doi:10.1186/s1363801813014
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