Partially Overlapping Tones for Uncoordinated Networks
Alphan Sahin, Erdem Bala, Ismail Guvenc, Rui Yang, Huseyin Arslan
aa r X i v : . [ c s . I T ] D ec Partially Overlapping Tones for Uncoordinated Networks
Alphan S¸ ahin , Erdem Bala , ˙Ismail G¨uvenc¸ , Rui Yang , and H¨useyin Arslan Department of Electrical Engineering, University of South Florida, Tampa, FL, 33620 InterDigital Communications Inc., Huntington Quadrangle, Melville, NY 11747 Department of Electrical and Computer Engineering, Florida International University, Miami, FL, 33174Email: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] Abstract —In an uncoordinated network, the link performancebetween the devices might degrade significantly due to theinterference from other links in the network sharing the samespectrum. As a solution, in this study, the concept of partiallyoverlapping tones (POT) is introduced. The interference energyobserved at the victim receiver is mitigated by partially over-lapping the individual subcarriers via an intentional carrierfrequency offset between the links. Also, it is shown that whileorthogonal transformations at the receiver cannot mitigate theother-user interference without losing spectral efficiency, non-orthogonal transformations are able to mitigate the other-userinterference without any spectral efficiency loss at the expense ofself-interference. Using spatial Poisson point process, a tractablebit error rate analysis is provided to demonstrate potentialbenefits emerging from POT.
Index Terms —non-orthogonal schemes, partially overlappingtones, Poisson point process, uncoordinated networks, waveform.
I. I
NTRODUCTION
Traditional broadband wireless networks have been strainedwith emerging demands such as being always-connected to thenetwork and very high throughput to satisfy data-hungry appli-cations such as real-time video. Satisfaction of these demandsconstitutes the main driving force for heterogeneous networks(HetNets) in which multiple tiers with varying coverage co-exist over the same network. In HetNets, interference amongthe tiers or the devices might dominate the noise and createinterference-limited networks. The interference issues becomeprominent especially when dense and unplanned deploymentssuch as device-to-device (D2D) communications are taken intoaccount. Considering this issue, a new technique in whichcertain features of the waveform itself are used to mitigatethe interference is proposed.A waveform, which is one of the core elements determiningthe characteristics of a communication system, describes theformation of associated resources in signal space [1], [2].Robustness of the transmitted signal to dispersion in the trans-mission medium, channel access, and hardware complexity arejust few features affected by the selected waveform. Hence,waveform design should be able to address the requirementsspecified by the system. When the performance of the networkis limited by noise, main consideration for the waveformdesign can naturally be on individual link properties such asreducing the interference created by the time and frequencydispersion of the channel [3]. However, interference createdby other users is many times a major factor limiting the performance of a network and as such the impact of the other-user interference might be more significant compared to theinterference due to the channel dispersion.
Conventionally ,the interference between the devices are elaborated with theapproaches which question the amount of the interferencepower at the receiver location without including the impact ofthe waveform itself. Most of the solutions devised to addressthe interference problem rely either on media access control(MAC) based coordination or interference cancellation. Forexample, interference coordination mechanisms with properscheduling and resource allocation aim to minimize the in-terference power [4]. In physical layer, methods like interfer-ence cancellation [5], multiuser detection [6], and interferencealignment [7] handle the other-user interference by exploitingthe difference between desired and interfering signal strengths,codes, and multipath channel.As opposed to the conventional solutions, in this paper, anew concept based on utilizing the time-frequency character-istics of waveforms to reduce the other-user interference isproposed. The main contributions of this paper are: • We introduce the concept of partially overlapping tones(POT) in which it is allowed for subcarriers allocatedto interfering links to partially overlap. The overlap isachieved by introducing an intentional CFO between thelinks and its amount is controlled by appropriately de-signing the time-frequency utilization of the waveforms. • It is shown that with orthogonal waveforms, there isa tradeoff between other-user interference and spectralefficiency. Mitigation of the other-user interference canbe achieved at the expense of a loss in spectral efficiency. • It is further shown that with non-orthogonal waveforms,there is a tradeoff between other-user interference andself-interference. Mitigation of the other-user interfer-ence can be achieved at the expense of increased self-interference while spectral efficiency remains unchanged. • A tractable bit error rate (BER) analysis for an unco-ordinated network deployment is provided. The analysisallows to understand the system performance for variousnetwork densities and waveform designs.The rest of paper is organized as follows: Related workis discussed in Section II. The system model including thephysical layer parameters is provided in Section III while theconcept of POT for orthogonal and non-orthogonal waveformstructures is introduced in Section IV. Then, BER analysis is provided in Section V and numerical results evaluating theperformance of the proposed approach are provided in SectionVI. Finally, the paper is concluded in Section VII.II. R
ELATED W ORK
The concept of overlapping wireless channels exists withinthe several 802.11 families (e.g. Wi-Fi systems). However,the simultaneous access to the channels is usually avoideddue to interference. The utilization of overlapping channelsto improve throughput has been investigated in several papers[8]–[13]. In [8], it is emphasized that the channel separationbetween the two pairs of Wi-Fi nodes can be interpretedas the physical separation between the nodes. Therefore, ifpartially overlapping channels are used carefully, it can providegreater spatial re-use. These papers consider the total spectrumutilization of the transmission, and do not show the impactof the partial overlapping on individual subcarriers . To thebest of our knowledge, detailed time-frequency analysis onthe interference due to the partially overlapping pulse shapesis not available in the literature.Some of the challenging aspects of the other-user interfer-ence are its asynchronous nature and its statistical characteri-zation, which depend on the deployment model and waveformstructure utilized in the network. Orthogonal frequency divi-sion multiplexing (OFDM) is a well-investigated multicarrierscheme in case of asynchronous interference, e.g., femtocell-macrocell coexistence [14]–[16]. By providing some timingoffset between the tiers intentionally, the different types ofthe interference, i.e. inter-carrier interference (ICI) and inter-symbol interference (ISI), is converted into each other in[16]. Yet, the total other-user interference is kept constant.A theoretical BER analysis investigating ISI versus ICI trade-offs in OFDM downlink is provided in [17]. In [18], BERdegradation due to the adjacent channel interference is inves-tigated by emphasizing superiority of filter bank multicarrier(FBMC) based cellular systems over an OFDM based ap-proach. Although these investigations provide useful intuitionson the performance degradation, the analyses are performedfor idealistic assumptions, such as grid-based cell deploymentand uniform user density. In [19], it is emphasized that evenif the geographical user density is uniform, the distance ofthe users linked to the corresponding serving points mightnot be uniform due to the irregular base station deploymentand shadowing characteristics. In [20], [21], homogeneousPoisson point processes (PPPs) are considered to model thedeployment of the base stations. This approach, which ispessimistic compared to highly idealized grid-based modelsand real deployment scenarios, yields a tractable tool whichexploits the stochastic geometry. In the following studies,e.g., [22] and [23], analytical models for uplink and K -tierheterogeneous networks are provided using PPPs.Investigation on the impact of PPPs on physical layer islimited, but available. For example, coexistence between ultrawide band (UWB) and narrow band systems is investigatedusing PPPs and impact of pulse shape is emphasized foraggregate network emission [20]. In [24], error rate analysesare provided for quadrature amplitude modulation (QAM) and RP TP (cid:1870) (cid:3036)(cid:2878)(cid:2869) TP Interference TP (cid:1856) (cid:2987) (cid:21)(cid:19) (cid:21) P RP (cid:23) P (cid:1856) (cid:3036)(cid:2878)(cid:2869) (cid:1856) (cid:3036) (cid:1856) (cid:3036)(cid:2878)(cid:2870) (cid:1870) (cid:3036)(cid:2878)(cid:2870) (cid:1870) (cid:3036) Interference
Interference
Fig. 1. Illustration of interference in an uncoordinated network. phase shift keying (PSK) modulations using PPPs, excludingthe impact of waveforms.III. S
YSTEM M ODEL
Consider an uncoordinated network where transmissionpoints (TPs) and their corresponding reception points (RPs)are distributed in an area as a realization of homogeneous 2-D PPP of Φ with the intensity λ as in Fig. 1. Interfering TPsand the RP investigated are called as aggressors and victim ,respectively. Without any loss of generality, victim is located atthe origin of the polar coordinates (0,0). The distance betweenthe i th aggressor and the victim is given as r i . Minimumdistance between the aggressors and the victim is set to r min .While the distance between RP and its associated TP for i thaggressor link is denoted by d i , the same distance is expressedby d ǫ for the desired link for the victim. Also, it is assumedthat aggressors are farther away than d ǫ , i.e., r i > r min ≥ d ǫ ,which is widely considered for the interference analyses basedon PPPs [25].In the following subsections, signal model for transmissionand reception based on multicarrier schemes and channelmodel that includes large and small scale effects are givenfor further discussions on POT. A. Signal Model for Transmission
The transmitted signal from the desired TP and the trans-mitted signals from the i th aggressor can be expressed as s ǫ ( t ) = ∞ X n = −∞ N − X l =0 X ǫnl g ǫnl ( t ) , (1)and s i ( t ) = ∞ X n = −∞ N − X l =0 X inl g inl ( t ) , (2)respectively, where X ǫnl and X inl are the information symbolswhich are independent and identically distributed (i.i.d.) withzero mean on the l th subcarrier and n th symbol, N is thenumber of subcarriers, and g ǫnl ( t ) and g inl ( t ) are the synthesisfunctions which map information symbols into time-frequencyplane based on a rectangular lattice as g ǫnl ( t ) = g ǫ ( t − nτ ) e j πlν t (3) and g inl ( t ) = g i ( t − nτ ) e j πlν t . (4)The family of functions in (3) and (4) are often referred toas Gabor frame or Weyl-Heisenberg frame , where g ǫ ( t ) and g i ( t ) are the prototype filters employed at the transmitters, ν is the subcarrier spacing and τ is the symbol spacing [26],[27]. For the sake of notation simplicity, ν and τ are givenin units of F and T , respectively (e.g., ν = 1 . × F and τ = 1 . × T ), where F = 1 /T and F is a number based onthe design. Without loss of generality, the energy of g ǫ ( t ) andthe energy of g i ( t ) are normalized as k g ǫ ( t ) k L ( R ) = k g i ( t ) k L ( R ) = Z ∞−∞ | g ǫ ( t ) | d t = 1 , (5)where L ( R ) denotes the square-integrable function space over R and k·k is the L -norm of function. B. Large Scale Impacts
Considering various path loss models depending on theenvironment, the path loss is characterized by L m ( · ) = a + b log ( · ) where the path loss parameters a and b are scalarsand the argument is the distance in meters. The received in-terference power from the i th aggressor and the desired signalpower at victim location per subcarrier are denoted by P i and P ǫ , respectively. Impact of shadowing is not considered inthis study. Main reason for this issue is to give insights on thePOT rather than introducing extra complexity for the systemmodel. However, using the methodologies proposed for themoment generation function of the summations of lognormaldistributed lognormal variables [28] and [29], it is possible toinclude the impact of shadowing on the investigation.For the link transmission, open loop fractional power controlis applied and some amount of the path loss, i.e., β ( a + b log ( · )) , is compensated, where β ∈ [0 , is the path losscompensation parameter. Note that TP might transmit with themaximum transmit power in some cases. However, since linkdistances considered are small, the possibility of transmissionat maximum power is excluded. C. Small Scale Impacts
Time-varying multipath channel is taken into account be-tween all RPs and TPs. Channel impulse response is charac-terized by h ( τ, t ) = P L − ℓ =0 ̺ ℓ ( t ) δ ( τ − τ ℓ ) where L denotesthe total number of multipaths, ℓ is the path index, and τ ℓ isthe delay of the ℓ th path. It is assumed that the path gains, ̺ ℓ ( t ) , are independent and identically distributed variables andthe signals experience Rayleigh fading, which is a commonmodel for interference analysis. Also, the expected channelpower is considered as P L − ℓ =0 E h | ̺ ℓ ( t ) | i = 1 . For the sakeof notation, the channel between i th interfering TP and thevictim RP and the channel between desired TPs and the victimRP are expressed as h i ( τ, t ) and h ǫ ( τ, t ) , respectively. D. Synchronization
As discussed in [30] and [31], synchronization to thereceived signal in the presence of interference might bechallenging, especially at low signal-to-interference-plus-noiseratio (SINR)s. However, the impairments like timing offsetand carrier frequency offset (CFO) are often related to thepreamble structure rather than the data portion of the frame.Therefore, perfect synchronization at the pair of interest isassumed. Besides, timing misalignment between the aggres-sor’s signals and synchronization point of the victim is takeninto account. The timing misalignment of i th aggressor signalwith respect to the synchronization point of the victim RP isdenoted by ∆ t i and its distribution f ∆ t i (∆ t i ) is assumed asuniform between 0 and τ . Besides, intentional CFO between i th aggressor and the victim RP is given by ∆ f i in order togenerate POT which is discussed in Section IV. The impact ofCFO due to the hardware mismatches between the aggressor’ssignals and desired signal is ignored. This is because of thefact that the impact of CFO due to the hardware mismatchesis relatively smaller than ∆ f i for POT. For example, whencarrier spacing is set to 15 kHz and CFO is 500 Hz, normalizedCFO becomes 0.033 ( Hz / kHz). However, the amountof normalized ∆ f i for POT, throughout the study, is at least0.5, which is significantly larger than CFO due to the hardwareerror. E. Signal Model for Reception
Considering all interfering TPs, and assuming a wide-sensestationary uncorrelated scattering (WSSUS) channel model[32], the received signal at the victim is obtained as r ( t ) = p P ǫ Z τ Z ν H ǫ ( τ, ν ) s ǫ ( t − τ ) e j πνt d ν d τ | {z } Desired signal + X i ∈ Φ p P i Z τ Z ν H i ( τ, ν ) s i ( t + ∆ t i − τ ) e j πνt d ν d τ | {z } Interfering signals + w ( t ) | {z } Noise (6)where H ǫ ( τ, ν ) and H i ( τ, ν ) are the Fourier transformationsof h ǫ ( τ, t ) and h i ( τ, t ) , respectively, and w ( t ) is the additivewhite Gaussian noise (AWGN) with zero mean and variance σ . In order to get the information symbol on the k thsubcarrier and m th symbol, the received signal is correlatedby the analysis function where γ ǫmk ( t ) = γ ǫ ( t − mτ ) e j πkν t . (7) Then, the output of the correlator is sampled with the samplingperiod to obtain the received symbol as ˜ X ǫmk = h r ( t ) , γ ǫmk ( t ) i , Z t r ( t ) γ ǫ ∗ mk ( t ) dt = p P ǫ X ǫmk A ǫmkmk | {z } desired part + p P ǫ K − X n = − K +1 n = m N − X l =0 l = k X ǫnl A ǫnlmk | {z } self - interference part + X i ∈ Φ p P i K − X n = − K +1 N − X l =0 X inl A inlmk | {z } other - user interference + W k |{z} noise . (8)In (8), A ǫnlmk = Z τ Z ν H ǫ ( τ, ν ) Z t g ǫnl ( t − τ ) γ ǫ ∗ mk ( t ) e j πνt d t d ν d τ , (9) A inlmk = Z τ Z ν H i ( τ, ν ) Z t g inl ( t − ∆ t i − τ ) e j π ∆ f i ( t − ∆ t i − τ ) × γ ǫ ∗ mk ( t ) e j πνt d t d ν d τ , (10)and they show the correlation between the symbols ( n, l ) and( m, k ) including the dispersion due the channel. As it is seenin (8), while other-user interference is caused by aggressorlinks, self-interference can occur due to the time-varying mul-tipath channel, hardware impairments, or non-Nyquist filterutilization. Considering (8), SINR can be expressed as SINR = G ǫ z }| { | A ǫmkmk | K − X n = − K +1 n = m N − X l =0 l = k | A ǫnlmk | | {z } I self + X i ∈ Φ P i P ǫ K − X n = − K +1 N − X l =0 (cid:12)(cid:12) A inlmk (cid:12)(cid:12) | {z } G i | {z } I i | {z } I other | {z } I total + σ P ǫ , (11)where K is the filter length in terms of symbol spacing, I total is the total interference, I self and I other are the self-interference and other-user interference, respectively, I i isthe interference due to i th aggressor, G ǫ and G i are theinterference gains including fading and filter characteristics,and P i P ǫ = d ǫ b − βb d βb i r i − b . (12)Note that K is related to the representation of the filter intime domain. As long as K is selected properly, the filtertruncation has a minor impact on self-interference compared tothe interference due to the time-varying multi-path channel orhardware impairments at the RP and/or TP. While G ǫ is a ran-dom variable with unit mean exponential distribution becauseof the Rayleigh fading [17], [18], G i can be characterized for (cid:1858) (cid:3) (cid:1858) (cid:3) (cid:2021) (cid:2868) (cid:2021) (cid:2868) (cid:1858) (cid:3) (cid:2021) (cid:2868) (cid:2021) (cid:2868) (cid:1858) (cid:3) (cid:2021) (cid:2868) (cid:2021) (cid:2868) (cid:1858) (cid:3) (cid:2021) (cid:2868) (cid:2021) (cid:2868) (cid:1858) (cid:3) Intentional frequency ffset (cid:2021) (cid:2868) (cid:2021) (cid:2868) (cid:2021) (cid:2868) (cid:2021) (cid:2868)
Aggressor’s subcarriers
Aggressor’s subcarriersVictim’s receive filters
Victim’s receive filtersReceived interference power Received interference power (a) Fully overlapping tones. (cid:1858) (cid:3) (cid:1858) (cid:3) (cid:2021) (cid:2868) (cid:2021) (cid:2868) (cid:1858) (cid:3) (cid:2021) (cid:2868) (cid:2021) (cid:2868) (cid:1858) (cid:3) (cid:2021) (cid:2868) (cid:2021) (cid:2868) (cid:1858) (cid:3) (cid:2021) (cid:2868) (cid:2021) (cid:2868) (cid:1858) (cid:3)
Intentional frequency offset (cid:2021) (cid:2868) (cid:2021) (cid:2868) (cid:2021) (cid:2868) (cid:2021) (cid:2868)
Aggressor’s subcarriers Aggressor’s subcarriers
Victim’s receive filters
Victim’s receive filters Received interference power Received interference power (b) Partially overlapping tones.Fig. 2. Illustrations for full overlapping and partial overlapping. While full-overlapping tones cause significant other-user interference, the main portionof the interference is mitigated by the receive filter with the concept of POT. a given ∆ t i and ∆ f i by exponential distribution where itsmean is given by σ i (∆ t i , ∆ f i ) = K − X n = − K +1 N − X l =0 (cid:12)(cid:12) h g inl ( t − ∆ t i ) e j π ∆ f i t , γ ǫ ( t ) i (cid:12)(cid:12) , (13)Conventionally, σ i (∆ t i , ∆ f i ) is considered as for link-level analyses [21], similar to the mean of G ǫ . However,expressing it as in (13) gives flexibility to include the impactof transmit and receive filters and calculate interference whenan additional processing is performed to reduce other-userinterference. Finally, I self is also a random variable with ex-ponential distribution where, considering the Rayleigh fadingassumption [18], its mean is given by σ = K − X n = − K +1 n = m N − X l =0 l = k |h g ǫnl ( t ) , γ ǫ ( t ) i| . (14)Essentially, calculations of both σ and σ i (∆ t i , ∆ f i ) arebased on the projection operation onto receive filters, whichcan be derived via corresponding ambiguity functions [2].IV. P ARTIALLY O VERLAPPING T ONES
The main goal of the POT is to mitigate other-user in-terference given in (13) by using the waveform structure. Itrelies on intentional CFO between aggressor’s Gabor systemand victim’s Gabor system. For example, while one of thelinks operates at carrier frequency f c , the other link operatesat f c + ν / . By allowing this operation, instead of full-overlapping between the subcarriers of the links, POT isobtained. This approach also fits the asynchronous nature ofother-user interference as it does not introduce any timingconstraint between interfering signals. One can interpret theintentional CFO as an alignment strategy in frequency domain.In Fig. 2, a motivating example based on filtered multitone(FMT) is illustrated for POT. In FMT, each subcarrier isgenerated via a band-limited filter [33]. As opposed to the conventional understanding of OFDM, the subcarriers arenot overlapped in frequency domain. By providing additionalguard bands, orthogonality between subcarriers is maintained.Note that these guard bands are also useful to provide im-munity against self-interference due to the time-frequencyimpairments. In the provided example in Fig. 2, these guardbands are exploited further and they are used to mitigatethe other-user interference. By applying an intentional CFObetween two different links, other-user interference mitigationis provided in an uncoordinated network.POT is fundamentally related to the utilization of the time-frequency plane by the waveform structure. Transmit filter,receiver filter, and density of symbols in time-frequency planedetermine the available resource opportunities jointly for theother-user interference mitigation by using POT, as exempli-fied in Fig. 2. Besides, further utilization of the waveformstructure via non-orthogonal schemes along with POT lead toa trade-off for uncoordinated networks: other-user interferenceversus self-interference . This trade-off is desirable in an unco-ordinated network as long as self-interference is handled viaself-interference cancellation methods, e.g., equalization. Inthe following subsections, orthogonality of schemes is stressedin conjunction with POT. POT with orthogonal schemes andnon-orthogonal schemes are investigated theoretically alongwith numerical results and their potential drawbacks. A. Partially Overlapping Tones with Orthogonal Schemes
For orthogonal schemes, transmitter and receiver utilize thesame prototype filter, i.e., g ǫmk ( t ) = γ ǫmk ( t ) . In addition, innerproducts of the different basis functions derived from theprototype filter yield zero correlations, i.e., h g ǫnl ( t ) , γ ǫmk ( t ) i = δ nlmk . Many fundamental schemes, e.g., OFDM, FMT, andFBMC, rely on orthogonality. In digital communication, or-thogonality in a multicarrier scheme is generally perceivedas a necessary condition. It simplifies the receiver algorithmssignificantly and provides optimum signal-to-noise ratio (SNR)performance in AWGN channels. Besides these features, or-thogonal schemes have another fundamental property due toorthogonal basis functions at the receiver: the energy of asignal before the projection onto receive filters is equal tothe energy after the projection onto receiver filters. This istypically expressed through the Plancherel formula given by k s ( t ) k = X m,k |h s ( t ) , u mk ( t ) i| , (15)where s ( t ) is an arbitrary signal, and { u mk ( t ) } is a set oforthogonal basis functions. Assume that s ( t ) is the interfer-ing signal. When an orthogonal transformation, e.g., discreteFourier transformation (DFT), is applied to s ( t ) at the receiver,the total amount of the interference does not change after thetransformation. This issue leads to an undesirable result: onlyway to mitigate the other-user interference is to discard someof subcarriers or to construct an incomplete Gabor system,i.e., τ ν > [2], [3], which causes less spectrally efficientschemes. In other words, POT with orthogonal schemes wouldbe beneficial only when some of subcarriers are not utilized It corresponds to Parseval’s theorem for Fourier series. ∆ t i ) ( × T) E ne r g y C ap t u r ed b y t he R e c e i v e F il t e r s α = 0 . α = 0 . α = 0 . α = 0 . σ (Self-Interference) σ (Other-user Interference - Full Overlapping) α = 0 . α = 0 . σ (Other-user Interference - Partial Overlapping) (a) Impact of timing misalignment when root raised cosine filter is employedalong with FMT. ν ) ( × F) A v e r age O t he r − u s e r I n t e r f e r en c e α = 0 . α = ν /F − α = 0 . α = ν /F − (b) Trade-off between spectral efficiency and other-user interference.Fig. 3. Other-user interference mitigation without introducing self-interference, but loss in spectral efficiency. or τ ν > . Indeed, norm-preserving feature of orthogo-nal transformations at the receivers explain why orthogonalschemes do not directly provide immunity against the other-user interference .POT offers intentional CFO between the different linksbased on the fact that timing synchronization between TPsin an uncoordinated network is a challenging issue. However,the intentional CFO approach also introduces some constraintson the waveform structure. For example, orthogonal multi-carrier schemes which provide non-overlapping subcarriers infrequency domain, e.g., FMT, complies with the intentionalCFO approach introduced by POT. However, POT might notbe as beneficial as in the case of FMT to the schemes where theorthogonality is maintained strictly on certain localizations inthe time-frequency plane, as in OFDM. Considering this issue,analyses throughout the study are performed based on FMT.In Fig. 3, considering timing misalignment between oneaggressor and the victim, ∆ t i is swept for one symbol periodwhen ∆ f i = ν / . FMT is generated based on root-raised- cosine (RRC) filter. Note that RRC filter is a band-limitedfilter and the excess bandwidth of the RRC filter is controlledvia a roll-off factor of α , where ≤ α ≤ . In Fig. 3(a), σ i (∆ t i , ∆ f i ) is calculated numerically, based on (13). Incase of full overlapping, σ i (∆ t i , ∆ f i ) is mitigated maximallywhen ∆ t i = 0 . × T , τ = T , and ν = (1 + α ) × F .This is because of the reduction of the ICI componentsmaximally due to the additional guard bands, when timingmisalignment occurs. In case of partial overlapping, impactof ∆ t i is removed totally, and σ i (∆ t i , ∆ f i ) is significantlyreduced since the receive filters reject the main portion ofthe interference, depending on the utilized α . Assuming theaggressor interference has a uniform timing misalignmentcharacteristics, trade-off between spectral efficiency and other-user interference is given for two different FMT cases inFig. 3(b). When ν is set to (1 + α ) × F , σ i (∆ t i , ∆ f i ) decreases for both full overlapping and partial overlappingdue to the less ICI components with the timing misalignment,as given in Fig. 3(a). When α is fixed to . , other-userinterference is mitigated more via partial overlapping, sincethis approach provides more gap in frequency for other-userinterference mitigation.Major concern of using POT with orthogonal schemes mightbe having less spectral efficient transmission for the sakeof other-user interference mitigation. However, as indicatedbefore, it allows the devices interrupted by the interference toachieve a better BER performance with a simple approach. B. Partially Overlapping Tones with Non-orthogonal Schemes
Similar to the orthogonal schemes, transmitter and receiverutilize the same prototype filters for non-orthogonal struc-tures, i.e., g ǫmk ( t ) = γ ǫmk ( t ) . However, inner products ofthe different basis functions do not yield zero correlations,i.e., h g ǫnl ( t ) , γ ǫmk ( t ) i 6 = δ nlmk . For example, non-orthogonalfrequency division multiplexing (NOFDM) can be constructedby using the rectangular lattice of OFDM with non-Nyquisttransmit filters and receive filters, e.g., Gaussian functions.For non-orthogonal schemes, the utilized basis functions at thereceiver also corresponds to a nonorthogonal transformations,i.e., h γ ǫnl ( t ) , γ ǫmk ( t ) i 6 = δ nlmk . In that case, the condition givenin (15) is relaxed as A k s ( t ) k ≤ X m,k |h s ( t ) , u mk ( t ) i| ≤ B k s ( t ) k , (16)where { u mk ( t ) } is a set of non-orthogonal elements, A and B are the lower bound and upper bound, respectively, and < A ≤ B < ∞ . Based on (16), when a non-orthogonaltransformation is applied at the receiver, the energy of s ( t ) does not have to be preserved after the transformation. In otherwords, the non-orthogonal transformations at the receivers areable to alter the amount of the observed interference energy.Hence, when POT is taken into account with non-orthogonalschemes, it is possible to mitigate other-user interference evenwhen τ ν = 1 .In order to understand the utilization of POT with non-orthogonal schemes, assume that τ ν = 1 and the transmitpulse shape and the receive filter are Gaussian filters. Gaussian Time F r e qu e n c y More other-user Interference Less self Interference (cid:2028) (cid:2868) (cid:2021) (cid:2868) (cid:543)(cid:1858) (cid:3036) (cid:543)(cid:1872) (cid:3036) (Misalignment) (a) Less self-interference, but more other-user-interference.
Time F r e qu e n c y Less other-user Interference More self Interference (cid:2028) (cid:2868) (cid:2021) (cid:2868) (cid:543)(cid:1872) (cid:3036) (Misalignment) (cid:543)(cid:1858) (cid:3036) (b) More self-interference, but less other-user-interference.Fig. 4. Illustration for the trade-off between self-interference and other-userinterference with the concept of POT. The desired signal and interfering signalare represented as solid and dashed lines, respectively. filter is the optimally-concentrated pulse in time-frequencydomain and it is expressed as p ( t ) = (2 ρ ) / e − πρt , (17)where ρ is the control parameters for the dispersion of thepulse in time and frequency and ρ > . While the selection of ρ = 1 yields a Gaussian filter that has isotropic dispersion intime and frequency, smaller ρ causes more dispersion in timedomain and less dispersion in frequency domain. Since Gaus-sian filter is not a Nyquist filter, consecutive symbols overlapmore with smaller ρ , yielding more self-interference in time,i.e., ISI. However, introducing more ISI is also beneficial tomitigate the other-user interference, when POT is considered,as illustrated in Fig. 4(a) and Fig. 4(b) for ∆ f i = ν / . Inother words, non-orthogonal schemes yield a trade-off betweenthe other-user interference and self-interference by exploitingthe POT.Considering the density of the symbols on time-frequencyplane of the victim RP, it is important to emphasize thedifferences between faster-than-Nyquist (FTN) signaling [34] ∆ t i ) ( × T) E ne r g y C ap t u r ed b y t he R e c e i v e F il t e r s σ (Other-user Interference - Full Overlapping) σ (Self-Interference) σ (Other-user Interference - Partial Overlapping) ρ = 0 . ρ = 0 . ρ = 1 ρ = 1 ρ = 0 . ρ = 0 . ρ = 0 . ρ = 0 . ρ = 1 (a) Impact of timing misalignment when Gaussian filter is employed. A v e r age O t he r − u s e r I n t e r f e r en c e Partial OverlappingFull Overlapping ρ = 1 . ρ = 0 . ρ = 0 . ρ = 0 . ρ = 0 . ρ = 0 . ρ = 1 ρ = 1 ρ = 0 . ρ = 0 . ρ = 0 . ρ = 0 . (b) Trade-off between self-interference and other-user interference.Fig. 5. Other-user interference mitigation without loss in spectral efficiencyand power, but at the expense of self-interference. and POT with non-orthogonal schemes. In FTN signaling, thedensity of the symbols in time-frequency plane is increasedmore than Nyquist rate, i.e., τ ν < , intentionally. However,each individual link operates at the Nyquist rate, i.e., τ ν = 1 ,for POT. The time-frequency plane of the victim RP is packeddue to the aggressors’ signals, which is common in co-channelinterference problems. In addition, POT does not suggest astructured symbol packing into the time-frequency plane, asin FTN signaling. It allows timing misalignment among theindividual links.Similar to the investigations given in Section IV-A, ∆ f i is set to ν / and ∆ t i is swept for one symbol period.Impact of timing misalignment is given in Fig. 5(a). In caseof full overlapping, when ∆ t i = 0 , the receive filter hasfull correlation with the concentric symbol of the aggressorand partial correlations with the neighboring symbols. Hence,the total energy after the correlation becomes more than 1.In case of partial overlapping, receive filters only captureenergy from only the neighboring symbols of aggressors, which yields that σ < as in Fig. 5(a). In Fig. 5(b),the trade-off between self-interference and average other-userinterference is given, assuming uniform timing misalignment.As in Fig. 5(b), Gaussian filter provides flexible trade-offbetween self-interference and other-user interference.There are two potential drawbacks of this approach: 1)necessity for a self-interference cancellation method, e.g.,equalization, since the filters do not satisfy Nyquist criterionand 2) colored noise due to the non-orthogonal receiverfilters. For the first issue, the introduced complexity due toself-interference cancellation method might be preferable incomparison with the complexities of the methods for handling asynchronous other-user interference. For the second point,note that non-orthogonal transformations always introducecorrelation between samples [3]. If a sequence-based equalizer,e.g., maximum likelihood sequence estimator (MLSE), isemployed, a whitening filter should also be utilized to improvethe performance of the receiver. Note that assuming the smalllink distances for the pairs, noise might become a secondaryproblem when interference is a dominant issue.V. A VERAGE
BER A
NALYSIS
In this section, average BER analysis is provided for POTfor orthogonal schemes that do not introduce self-interferenceas discussed in Section IV-A. To obtain theoretical (buttractable) BER analysis, a useful method for BER calculationsintroduced in [35] is combined with spatial PPP approaches[20], [21], [24]. First, BER is expressed along SINR given in(11). Then, its expected value is obtained considering other-user interference. Its computation complexity is significantlyreduced by using spatial PPP and ambiguity function. For thetrade-off introduced in Section IV-B, investigation on BERperformance is performed through the numerical analysis inSection VI, since achievable BER performance depends highlyon the employed self-interference cancellation method at thereceiver.Closed-form expression for BER of a square M -QAM inAWGN channel is readily available in the literature and it isgiven by BER (
SN R ) = √ M − X q c q erfc (2 q + 1) r SN R ! (18)where M is the constellation size, c q are the constantsdepending on the modulation order and P √ M − q =0 c q = 1 / [36]. For instance, c q = { / } and q = { } for -QAMand c q = { / , / , − / } and q = { , , } for -QAM,respectively.By substituting (11) into (18), BER is obtained for given I total , G ǫ , and d ǫ as BER ( E b /N | G ǫ , I total , d ǫ )= √ M − X q =0 c q erfc q − √ vuuut G ǫ I total + M −
13 log M E b /N . (19) Since the target is to calculate average BER under interference,the terms, I total , and G ǫ , have to be averaged out. In order toobtain average BER, we refer to following lemma introducedin [35]: Lemma-I: Let x and y be unit-mean exponential and arbi-trary non-negative random variables, respectively. Then E x,y (cid:20) erfc (cid:18)r xay + b (cid:19)(cid:21) = 1 − √ π Z ∞ e − z (1+ b ) √ z L y ( az ) d z where L y ( z ) = E y [ e − yz ] is the moment generation function(MGF) with negative argument (or Laplace transformation) ofrandom variable y . If Lemma-I is applied to (19) (see e.g., [17], [18], [35],[37]), average BER is obtained as
BER ( E b /N | , d ǫ )= √ M − X q =0 c q − √ π Z ∞ e − z (1+ q − M −
13 log2 M E b /N ) √ z ×L I total (cid:18) z (2 q − (cid:19) d z (cid:19) , (20) = 12 − √ π √ M − X q =0 c q Z ∞ e − z (1+ q +1)2 M −
13 log2 M E b /N ) √ z × L I total (cid:18) z (2 q + 1) (cid:19) d z . (21)Therefore, the complexity introduced by (19) reduces tocalculate Laplace transformation of I total . In the followingsubsections, Laplace transformation of I total is calculated incases of single aggressor and multiple aggressors. A. Single Aggressor
If only i th aggressor is considered, the Laplace transforma-tion of the total interference is obtained as L I total ( z ) = E I total (cid:2) e - zI total (cid:3) ( a ) = E I self (cid:2) e - zI self (cid:3) × E I i (cid:2) e - zI i (cid:3) = E I i (cid:2) e - zI i (cid:3) ( b ) = Z τ f ∆ t i (∆ t i )1 + zd ǫ b − βb d βb i r i − b σ i (∆ t i , ∆ f i ) d∆ t i (22)where (a) follows from the independent assumption of randomvariables I i and I self and the assumptions of zero self-interference via orthogonal schemes, (b) is because of theexponential distribution of I other and the randomness of timingmisalignment. Considering the uniform timing misalignmentassumption and being a constant function of σ i (∆ t i , ∆ f i ) respect to ∆ t i , as in Fig. 3(a), (22) is simplified as L I total ( z ) = 11 + zd ǫ b − βb d βb i r i − b σ i (∆ f i ) (23) B. Multiple Aggressors
When multiple aggressors exist in the network, the choice of ∆ f i within the link affects the performance of POT. In orderto avoid the coordination, it is assumed that ∆ f i is selectedrandomly from the set Ω given by [ ψ , ψ , . . . , ψ r , . . . ] . Theselection is performed based on a probability mass function(PMF) where p r corresponds to the probability of r th inten-tional CFO. Based on this assumption, the Laplace transfor-mation of the total interference is obtained as L I total ( z ) = E I total (cid:2) e - zI total (cid:3) ( a ) = E I self (cid:2) e - zI self (cid:3) × E Φ ,I i h e - z P i ∈ Φ I i i ( b ) = E Φ ,I i h e - z P i ∈ Φ I i i = E Φ "Y i ∈ Φ E I i (cid:2) e - zI i (cid:3) (24) ( c ) = exp (cid:20) − πλ Z ∞ r min (cid:0) − E I i (cid:2) e - zI i (cid:3)(cid:1) ν d ν (cid:21) (25)where (a) follows from the independent assumption of ran-dom variables I other and I self , (b) is because of zero self-interference via orthogonal schemes, and (c) is caused bythe probability generating functional of PPP, which states E Φ (cid:2)Q i ∈ Φ f ( x ) (cid:3) = exp R R (1 − f ( x ))d x for an arbitraryfunction f ( x ) and the assumption of i.i.d. interference fromeach aggressor I i and independent Φ from other randomvariables in the interference function I other [21]. Consideringrandomness of aggressors’ distances d i , E I i (cid:2) e - zI i (cid:3) is obtainedas E I i (cid:2) e - zI i (cid:3) = E d i ,G i h e - z PiPǫ G i i = X r p r Z ∞ f u ( u )1 + zd ǫ b − βb u βb ν − b σ i ( ψ r ) d u (26)which is based on the Laplace transformation of an exponen-tially disturbed random variable, uniform timing misalignmentassumption, and being a constant function of σ i (∆ t i , ∆ f i ) respect to ∆ t i . In (26), the probability density function (PDF)of d i is given by f u ( u ) = 2 πλue − λπu [22]. Then, L I total ( z ) is obtained as L I total ( z ) = exp (cid:20) − πλ Z ∞ r min − X r p r Z ∞ πλue − λπu zd ǫ b − βb u βb ν − b σ i ( ψ r ) d u ! ν d ν (27)by substituting (26) into (25). Note that (27) does not alwaysyield a closed-form solution since R ∞ xe − ax bx c d x producesan expression in terms of standard mathematical functionsdepending on a, b , and c . Nonetheless, (27) does not requireMonte Carlo simulations.VI. N UMERICAL R ESULTS
Numerical results are given in order to validate analyticalfindings with simulations and to investigate the performance ofuncoordinated networks along with POT. In the simulations, −2 −1 BE R E b / N (dB) α = 0 . ν = 2 × Fα = 0 . ν = 1 . × F Partial & Full Overlapping (OFDM)Partial Overlaping (RRC/FMT) α = 0 . ν = 1 . × F No interference(OFDM - Rayleigh) (a) RRC/FMT/4QAM (Solid lines: Analytical results based on (21) and (23)). −2 −1 BE R E b / N (dB) ρ = 0 . ρ = 0 . ρ = 0 . ρ = 0 . ρ = 0 . ρ = 0 . (b) Gaussian/NOFDM/4QAM.Fig. 6. BER performance with partial overlapping when there is a singleaggressor. POT with orthogonal schemes and POT with non-orthogonalschemes are exhibited by utilizing FMT with RRC filter andzero forcing equalization and by using NOFDM with Gaussianfilter and symbol-spaced MLSE equalization, respectively. ForMLSE, 7 taps are utilized for each subcarrier and trace-back depth for MLSE is set to 20. Unless otherwise stated,numerical result are obtained for Rayleigh channels.In Fig. 6, impact of partial overlapping is presented inRayleigh channel for the aforementioned trade-offs when adominant aggressor interrupts the transmission with the equalreceived signal power (i.e., signal-to-interference ratio (SIR)is set to dB). In Fig. 6(a), α is set to . and the subcarrierspacing is swept from . × F to × F , referring to thePOT with orthogonal schemes. Also, simulation results areverified with the theoretical results based on (21) and (23).As it can be seen in Fig. 6(a), efficacy of POT in the BERperformance increases with the subcarrier spacing, which alsocauses less spectrally efficient schemes. In Fig. 6(b), thesame analysis is performed for NOFDM to address the POTwith non-orthogonal schemes. When other-user interference −2 −1 BE R E b / N (dB) d ǫ = 25 m d ǫ = 18 .
75 mPartial & FullOverlapping(OFDM)Partial Overlaping(RRC/FMT, α = 0 . ν = 1 . × F )No interference(OFDM - Rayleigh) (a) RRC/FMT/4QAM (Solid lines: Analytical results based on (21) and (27)). −2 −1 BE R E b / N (dB) d ǫ = 25 m ρ = 0 . ρ = 0 . d ǫ = 18 .
75 mPartial & Full Overlaping (OFDM) ρ = 0 . ρ = 0 . ρ = 0 . ρ = 0 . (b) Gaussian/NOFDM/4QAM.Fig. 7. BER performance with partial overlapping when there are multipleaggressors modeled with PPP. do not exist, orthogonal schemes reach the Rayleigh boundand introduce superior BER performance compared to non-orthogonal schemes. This is mainly because of the fact thatMLSE loses its optimality under the colored noise caused bythe non-orthogonal transformation at the receiver. However,when the other-user interference exists, orthogonal schemescapture the total amount of the other-user interference andBER performance deteriorates significantly. In contrast to or-thogonal waveforms, non-orthogonal schemes become notablewith the concept of POT under the other-user interference.By providing sufficient non-orthogonality, e.g., ρ = 0 . , BERperformance remains the same of the case without other-user interference for NOFDM for low to medium SNR, asit can be seen in Fig. 6(b). Essentially, the results show thatBER performance is enhanced without sacrificing the spectralefficiency at the expense of complexity at the receiver.In Fig. 7, impact of POT on BER performance is shownwhen there are multiple aggressors. In the simulation, the path loss is modeled with the parameters given in [38] as L ( d ) =11 . ( f c ) + 40 log ( d/ (28)where f c is the carrier frequency in MHz (3500 MHz) and d is the distance in meters. Using given parameters, the pathloss formula is calculated as L ( · ) = 51 . · ) wherethe argument is in terms of meters. Accordingly, a and b areset to . and , respectively. The intensity of TP and r min are set to / ( π ) and m, respectively. In order to seethe best possible BER performance, all aggressors’ signalsare partially overlapped with the desired signal. Then, BERcurves are obtained for different victim link distance d ǫ . Asexpected, BER is directly related to the user distance. Espe-cially, the degradation becomes severe for the users located atfar distances. In Fig. 7(a), it is shown that orthogonal schemesallow better BER performance with the concept of POTby losing their spectral efficiencies. Also, simulation resultsmatch with the theoretical results based on (21) and (27).In Fig. 7(b), the impact of non-orthogonal schemes on BERperformance are shown for the same scenario and better BERperformance is obtained for high E b /N without any spectralefficiency loss, but complexity at the receiver. ConsideringFig. 6(b) and Fig. 7(b), it is important to emphasize thatone may obtain the optimum ρ , considering the amount ofthe attainable self-interference and the amount of mitigatedother-user interference. Although the selection of ρ = 0 . significantly improve the BER performance when the amountof the other-user interference is equal to signal power, the samescheme might not yield optimum BER performance whenother-user interference becomes weaker due to the path loss.Essentially, this issue indicates that there is a point where non-orthogonality starts to be harmful. Therefore, the best selectionof ρ depends on the equalizer performance and the amount ofthe other-user interference.VII. C ONCLUDING R EMARKS
In this study, by allowing intentional CFO between theinterfering links, other-user interference is mitigated in anuncoordinated network without any timing constraints viaorthogonal or non-orthogonal schemes. For a well-coordinatednetwork, transmission over orthogonal schemes might lead tobetter performance compared to non-orthogonal schemes dueto the absence of self-interference. However, when other-userinterference is inevitable and significant in an uncoordinatednetwork, spectral efficiency has to be sacrificed for orthogonalschemes in order to allow other-user interference mitigation.Specifically, schemes which allow non-overlapping subcarri-ers in frequency, e.g., FMT, complies with the intentionalCFO approach to avoid timing misalignment problems withPOT. As opposed to orthogonal waveforms, non-orthogonalschemes come into the prominence along with POT for aninteresting reason; self-interference problem is easier thanother-user interference problem in an uncoordinated networks.By utilizing non-orthogonal waveforms, POT is able to changethe type of interference from other-user interference to self-interference. This is beneficial when the receiver has properself-interference cancellation mechanisms. Especially, it is promising when two pairs sharing the same spectrum are closeto each other.Throughout the study, POT is presented for two intentionalCFO levels, i.e., f c and f c + ν / . Although the POT withtwo intentional CFO levels heuristically matches to two-usersscenarios, it might be a suboptimum solution for the multiple-user scenarios. However, it is possible to utilize multiple CFOlevels to extend POT to multiple-user scenarios. In addition,when the difference between the power levels of interferingsignal and desired signal are significantly large, well-knowninterference cancellation methods, e.g. successive interferencecancellation (SIC), might provide better results than POT.However, the combination of POT and interference cancel-lation techniques can increase the performance substantially.Since POT is able to increase the difference between the normof interference and the norm of desired signal power, POT isalso able to increase the separability of the signals.A CKNOWLEDGMENT
This study has been supported by InterDigital Communica-tions Inc. R
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