Generalized Self-Interference Model for Full-Duplex Multicarrier Transceivers
Gustavo J. González,Member, IEEE,Fernando H. Gregorio, Juan Cousseau,Member, IEEE, Taneli Riihonen,Member, IEEE,Risto Wichman, Member, IEEE,
Abstract—In the modeling of full duplex (FD) transceivers using multicarrier modulations, it is usually assumed that the intercarrier (ICI) interference of the coupling path is negligible.
Therefore, the system can be analyzed on a subcarrier basis, because the model reduces to a set of orthogonal flat channels.
However, for a more general kind of FD transceivers, or when hardware components introduce nonlinear distortion, the orthog- onality is lost and a more rigorous model is needed. In this work, we introduce a model for the coupling interference of FD systems which allows the uplink and the downlink to have different time and frequency offsets, and different parameters, such as the number of subcarriers and symbol length. Additionally, we present a linear approximation for the nonlinear coupling based on a generalization of the Bussgang’s theorem, for the case of transmitters with nonuniform power allocation. We validate the proposed model with measurements and show numerically that it improves the accuracy of the signal to interference plus noise ratio (SINR) expression, and the formulation of optimization problems. Therefore, the proposed model paves the way to more realistic performance analysis and the derivation of more accurate power allocation strategies than is possible with the classic model.
Index Terms—Full-duplex, Generalized self-interference model, Intercarrier and intersymbol interference.
I. INTRODUCTION
Full-duplex (FD) communications are changing the concep- tion about wireless communication systems. The ability to perform simultaneous co-channel transmission and reception, not only leads to the direct benefit of theoretically doubling the spectral efficiency, but it also changes the network protocols since new functionalities are now available. Some examples are self-backhaul networks [1], simultaneous sensing and transmission in cognitive radio systems [2], among others.
Due to the simultaneous transmission and reception, FD systems suffer from self-interference (SI), caused by the coupling of the transmitted signal into the receiver chain. In compact receivers, the SI is usually much higher than the
The work of G. González, F. Gregorio, and J. Cousseau was supported by the Agencia Nacional de Promoción Científica y Tecnológica (PICT-FONCYT
#2016-0051), and the Universidad Nacional del Sur (PGI 24/K080 and K081), Argentina. The work of T. Riihonen was supported by the Academy of Finland under Grants 310991 and 315858. The work of R. Wichman was supported by the Academy of Finland under Grant 288249.
G. González, F. Gregorio, and J. Cousseau are with CONICET-Department of Electrical and Computer Engineering, Universidad Nacional del Sur, Argentina (e-mail: {ggonzalez, fernando.gregorio, jcousseau}@uns.edu.ar).
T. Riihonen is with the Faculty of Information Technology and Communi- cation Sciences, Tampere University, Finland (email: taneli.riihonen@tuni.fi).
R. Wichman is with the Department of Signal Processing and Acous- tics, Aalto University School of Electrical Engineering, Finland (email:
risto.wichman@aalto.fi).
signal of interest [3] and depends on the transmitted signal power, the isolation between transmitter and receiver chains, and the surrounding environmental reflectors. The performance of FD transceivers relies on an effective reduction of the aforementioned interference. The SI mitigation is carried out first in the analog domain, by using antenna [4], [5], [6] and RF cancellation [7], [8], and then in the digital domain, by means of residual suppression filters [9], [10], [11], [12].
Multicarrier modulation schemes, as orthogonal frequency division multiplexing (OFDM), are widely used in most mod- ern communication systems [13], [14]. Mainly, this is because the channel or interference equalization is straightforward in an ideal condition, using one coefficient per subcarrier. In other words, data transmitted in a subcarrier are not affected by those transmitted in another subcarrier, i.e., there is no intercarrier interference (ICI).
Many power allocation algorithms to maximize the sum data rate, throughput, or to improve security have been proposed in the literature for multicarrier FD systems [15], [16], [17], [18], [19]. Other resource allocation problems considering energy efficiency, maximum achievable rate, channel uncertainties, and quality-of-service requirements are treated in [20], [21], [22], [23], [24]. Then, [25] presents expressions for bounding the optimal input back-off setting that maximizes the signal- to-interference-plus-noise ratio (SINR). In these results for FD systems using multicarrier modulations, it is assumed there is no ICI. In other words, if we consider that a source transmits to the FD receiver and the FD device transmits simultaneously to a destination, the SINR at subcarrier k at the receiver is expressed as
Ψ(k)= α(k)Ps(k)
β(k,k)Pt(k)+γ(k)+σr2 (1) whereγ(k) is an ICI term. Trivially,γ(k)=0in the simplified classic model. Ps(k) and Pt(k) are respectively the power of the source and FD transmitter signals, α(k) and β(k,k) are scaling factors depending on the communications channels and hardware impairments, and σ2r is the Gaussian noise at the receiver. The term β(k,k)Pt(k) represents the SI from subcarrier k in the transmitter to subcarrier k at the receiver, due to the FD operation. From this model, withγ(k) =0, it can be understood that to increase the SINR at subcarrier k, it is enough to decrease the transmitted power on the same subcarrier Pt(k). However, in a more general scenario when ICI is present,γ(k),0 and the performance of the proposed algorithms and the accuracy of obtained performance bounds are seriously compromised.
Some works have considered ICI in FD transceivers caused by a single hardware impairment. The interference of mirror subcarriers produced by the I/Q imbalance is considered in [26]. Then, [27] considers the ICI produced by the carrier frequency offset (CFO), and [28], [29] the interference due to phase noise. However, beyond these efforts, the composite effect of different impairments sources, such as different modulation parameters, time and frequency offsets, as well as nonlinearities in the transceiver, are not taken into account in these scenarios.
In this work we develop a generalized interference model for FD systems that use multicarrier modulation, where the ICI in the coupling path is taken into account. Particularly, the SINR at subcarrier k at the receiver is inherently affected by SI from all the other subcarriers such that
γ(k)= X
l,k
β(k,l)Pt(l) (2)
instead of γ(k) =0 that is assumed by the classic modeling.
Here, the term β(k,l)Pt(l) represents the SI from subcarrierl in the transmitter to subcarrier kat the receiver. In the gener- alized case, if we switch off the subcarrierkat the transmitter, the interference power does not disappear at subcarrier k in the receiver since the termγ(k) depends on the power in other subcarriers. This generalized model can be used to describe the behavior of more general FD systems. There, the transmitter and receiver links operate at the same time and frequency band, but they do not share the same parameters, such as:
time offset (TO), CFO, number or subcarriers, cyclic prefix (CP) length, etc. The proposed formulation is able to describe the behavior of, e.g., heterogeneous inter-cell communications, where a FD transceiver can be attached to two different cells without requiring the same synchronization parameters. An- other case are self-backbone networks, in which the backbone network operates with different modulation parameters than the access network. An example of particular importance is the high speed train (HST) FD relayed link [30]. To make the relay–train link more robust against the low channel coherence time, the intercarrier spacing is increased. On the other hand, as the base station–relay link is the backhaul, the intercarrier spacing remains the same, resulting in a FD device with different modulation parameters in the transmitter and the receiver. Furthermore, the generalized model can describe the behavior of the transceiver in more realistic scenarios, when hardware impairments (like phase noise, IQ imbalance, PA non linearity, sampling jitter, etc.) are present.
In this article, we cover the linear and the nonlinear coupling path interference cases. A list of the main contributions is as follows:
• We derive a rigorous linear SI model that takes into account the ICI produced by different OFDM parameters, TO and CFO. The model shows there are two sources of ICI:
frequency shift due to resampling and leakage.
• We present simplified expressions for the synchronization offsets when the uplink and the downlink have the same symbol and CP lengths.
• We derive a Gaussian approximation for nonlinear memory- less SI, in the case of nonuniform power allocation. We show
that the power allocation strategy affects the interference level and produces power leakage.
• We validate the proposed linear and nonlinear models with measurements.
• We present simulation results to give more insight into the structure of the coupling interference and take the HST as a study case to show the applicability of our model.
• We conclude that the classic model is not able to predict the system behavior when ICI is present in the coupling path, but the generalized model provides an accurate estimate.
The rest of the work is organized as follows. In Section II, we present the system model and introduce the generalized SINR formulation. The coupling interference in a FD system with TO and CFO, and different modulation parameters is derived in Section III-A, whereas some simplifications of the general case when the links have the same symbol and CP lengths are presented in Section III-B. The Gaussian approx- imation of the nonlinear interference is derived in Section IV. In Section V, we include some examples that illustrate the interference structure and present the measurement results.
Finally, in Section VI we conclude the paper.
II. SIGNALMODEL OF THEFULL-DUPLEX SYSTEM
In this section we first introduce the signal notation of the FD system and then present our generalized interference model. We consider a general FD transceiver, where the source (S) communicates with the receiver (R) and the transmitter (T) with the destination (D), forming respectively the links S–R and T–D, as depicted in Fig. 1. We additionally assume that time and frequency references as well as modulation parameters of the S–R and T–D links are not necessarily the same. Due to the FD operation there is a coupling path between T and R.
The i-th block transmitted by T and S is denotedxt(i) = [Xt(0,i), . . . ,Xt(N1−1,i)]Tandxs(i)=[Xs(0,i), . . . ,Xs(N2− 1,i)]T, with length N1 and N2, respectively. Subindex 1 denotes parameters of the T–D link, whereas subindex 2 to those of the S–R link. Data symbols Xt(k1,i) and Xs(k2,i) are taken from a M-QAM modulation, with power Pt(k1) = E{|Xt(k1,i)|2} and Ps(k2) = E{|Xs(k2,i)|2}, ∀i. After the inverse discrete Fourier transform (IDFT), a CP Ncp1 (Ncp2) is added in the transmitter (source), giving a total symbol of Nt1 = N1+Ncp1 (Nt2 = N2 +Ncp2). Considering that the sampling period of the T–D (S–R) link isT1/N1 (T2/N2), we can define the time length of the CP and the total symbol as Tcp1 = Ncp1T1/N1 (Tcp2 = Ncp2T2/N2) andTt1 =Tcp1 +T1 (Tt2 = Tcp2 +T2). As the antialiasing filter at the receiver limits the bandwidth of the received signal, we assume that the bandwidth of the transmitted signal is lower than or equal to the bandwidth in the receiver. Note that the opposite case, the bandwidth of the transmitted signal being larger than the received one, reduces to the case of equal bandwidths due to the antialising filter. Finally, after removing the CP and applying the discrete Fourier transform (DFT), the signal at the receiver is denoted byy(i)=[Y(0,i), . . . ,Y(N2−1,i)]T.
The residual SI channel of length Lc, after antenna, RF, and digital cancellation, is denoted as hc(n). It is worth to
S R
T D
+
Transceiver
+
ht(n)
hc(n) )
hs(n)
wr(i)
wd(i)
xt(i)
xs(i) y(i)
Fig. 1. FD transceiver with different numerology, and time and frequency references in the S–R and T–D links.
mention that due to imperfections in the devices and errors in the channel estimation process it is not possible to cancel completely the SI. Although the SI channel is usually modeled as Rician, we assume that the cancellation stages are able to remove the strong main coupling path, and only the residual reflections are present inhc(n) [4], [7], [31]. When the SI level is high, the RF canceler may introduce nonlinear distortion.
We assume that this imperfection can be modeled by the linear approximation introduced in Section IV. The T–D and S–R channels are denoted as ht(n) and hs(n), with lengths Lt and Ls, respectively. All the channels are considered static and frequency-selective. Note that the sampling frequency of channel ht(n) is T1/N1, whereas that of channels hs(n) and hc(n) is T2/N2. We assume that channel models include the effect of pulse shaping, the coefficients are uncorrelated, and channel delay spreads are less thanTcp1 andTcp2. TheN1×1 (N2×1) thermal noise vector at the destination (receiver) input iswd(i) (wr(i)), with covariance matrixσ2dIN1 (σr2IN2), where IN is the N×N identity matrix.
Considering the same time and frequency references and the same OFDM parameters in both links (N1=N2,Ncp1=Ncp2, andT1 =T2), thei-th received symbol can be expressed as
yid(i)=FPrHsPaFHxs(i)+FPrHcPaFHxt(i)+wr(i) (3)
=Ksxs(i)+Kcxt(i)+wr(i) (4) where F is the DFT matrix with elements [F]p,q = 1/√N2exp(−j2πpq/N2) for 0 ≤ p,q ≤ N2−1, and [A]p,q
denotes the element in row p and column q of the ma- trix A. Pa = [P,IN2]T is the CP append matrix, with P the Ncp2 × N2 matrix formed by the last Ncp2 columns of IN2. Alternatively, Pr = [0N2×Nc p2,IN2] is the CP re- moval matrix, where 0N2×Nc p2 is the N2×Ncp2 null matrix.
Superindices (·)T and (·)H denote transpose and Hermitian transpose, respectively. TheNt2×Nt2Toeplitz channel matrices are defined as Hc = T {hca,hcb} and Hs = T {hsa,hsb}, where the operator T {a,b} builds a Toeplitz matrix, with a being the first column and b the first row1, and channel vectors are hca = [hc(0), . . . ,hc(Lc −1),0, . . . ,0]T, hcb = [hc(0),0, . . . ,0]T, hsa = [hs(0), . . . ,hs(Ls−1),0, . . . ,0]T, and hsb = [hs(0),0, . . . ,0]T, all of length Nt2. Finally, the diagonal channel matricesKs andKcare defined respectively
1The first element ofaandbmust coincide.
as D{Hs(0), . . . ,Hs(N2−1)}andD{Hc(0), . . . ,Hc(N2−1)}, where D{a} builds a diagonal matrix from vector a and Hx(k)=P
nhx(n)exp(−j2πnk/N2) is the DFT of the channel hx(n), forx∈ {s,c}. In (4) we use the diagonalizing property of circulant matrices to simplify the expression. As a conse- quence, we can see that there is no ICI and the system behaves as a set of parallel channels.
The SINRΨ(k2) without ICI (γ(k2)=0) is defined by (1), where α(k2) = E{|Hs(k2)|2} and β(k2,k2) = E{|Hc(k2)|2}. When the orthogonality in the SI signal is lost due to impair- ments between the S–R and T–D chains, (4) can be rewritten as
y(i)=Ksxs(i)+ X∞ ℓ=−∞
C∆(ℓ)xt(ℓ)+wr(i) (5) where the N2 × N1 matrices C∆(ℓ) consider the coupling interference produced by the transmitted symbolsxt(ℓ) in the received symbol y(i), due to the impairment ∆. We assume that the elements ofC∆(ℓ) are independent of the transmitted data. The terms [C∆(ℓ)]k2,k2 corresponds to the self-carrier interference (SCI), whereas[C∆(ℓ)]k1,k2fork1,k2, to the ICI.
Evidently, the challenge behind the model (5) is to calculate matricesC∆(ℓ) given the impairment∆.
Using the independence of SI terms and the transmitted data, we can express the interference power as
E
X∞ ℓ=−∞
C∆(ℓ)xt(ℓ)
2
= X∞ ℓ=−∞
D∆(ℓ)et (6) where et = [Pt(0), . . . ,Pt(N1 −1)]T and D∆(ℓ) are the SI power coefficient of symbol ℓ. Then, we can combine the contribution of each transmitted symbol and define the overall coupling coefficients as
β(k2,k1)= X∞ ℓ=−∞
[D∆(ℓ)]k2,k1 (7) Finally, the coupling ICI term can be expressed as
γ(k2)=
N1−1
X
k1=0 k1,k2
β(k2,k1)Pt(k1) (8)
It is important to highlight that our contribution models more accurately the redistribution of SI due to the ICI than the classic model. Note that β(k2,k2) is not equal in the proposed and the classic models. Furthermore,γ(k2)=0 in the classic model, because β(k2,k1)=0for k1,k2.
In Section III, we obtain an expression of D∆(ℓ) when impairments in the uplink and downlink produce linear inter- ference, whereas in Section IV we handle the case of nonlinear distortion.
III. COUPLINGINTERFERENCEPRODUCED BYLINEAR
IMPAIRMENTS
In this section, we introduce a generalized linear signal model that describes the coupling interference of FD systems with TO and CFO, and different modulation parameters, such
that number of subcarriers, symbol length, and CP. These mismatches may occur when the links of the FD device have different parameters. A practical case of study is presented in Section V. Additionally, a TO is produced when a user close to the cell edge requires a timing advance larger than the CP in the uplink2. Another source of delay between the links can be introduced by a software defined radio (SDR) implementation with processing restrictions. First, in Section III-A we treat the general case, and then in Section III-B, we introduce some simplified expressions for systems with the same symbol and CP length in the links.
A. General case: Time and Frequency Offset with Different Modulation Parameters
As in the general caseT1,T2, several transmitted symbols can interfere the received symbol. With this consideration, we specialize C∆(ℓ) to define the interference matrices for TO and CFO, and different modulation parameters as
CM(ℓ)=
FPrEξHcS(ℓ)FHmEt(ℓ) if Nin f(i)≤ℓ≤Nsup(i)
0 otherwise
(9) where Nin f(i) = ⌊(iTt2 +(Ncp2 −Lc −n0 +1)T2/N2)/Tt1⌋ is the first transmitted symbol that interfere y(i), Nsup(i) =
⌊((i + 1)Tt2 −t0)/Tt1 − ǫ⌋ is the last interfering symbol,
⌊·⌋ is the floor operation, n0 is the discrete TO, t0 = n0T2/N2 is the TO in seconds, ǫ is an infinitesimal3, Eξ = D{1,exp(j2πξi Nt2/N2), . . . ,exp(j2πξ(Nt2+i Nt2−1)/N2)},ξ is the CFO relative to the intercarrier spacing at the receiver, [Fm]p,q =1/√
N1exp(−j2πT2pq/(N2T1)) for 0 ≤p ≤N1−1 and 0 ≤ q ≤ Nt2 −1 is a modified DFT matrix, Et(ℓ) = D{1,exp(j2π(iTt2−ℓTt1−Tcp1 −t0)/T1), . . . ,exp(j2π(N1− 1)(iTt2−ℓTt1−Tcp1−t0)/T1)}, andS(ℓ)=D{s(iTt2−ℓTt1− t0),s(T2/N2+iTt2−ℓTt1−t0), . . . ,s((Nt2−1)T2/N2+iTt2− ℓTt1−t0)}, with
s(t)=
1 if 0≤t<Tt1
0 otherwise. (10)
The matrixEt(ℓ) takes into account the different delays when adding interference symbols in (5), Fm is a modified DFT matrix that considers a different sampling frequency from the transmitter and different number of subcarriers between the S–R and T–D links,Eξtakes into account the frequency shift, andS(ℓ) is a selection matrix that picks the samples in xt(ℓ) that interfere in y(i).
Rearranging the terms in (9) and solving for the DFT, the matrixCM(ℓ) can be expressed in a simplified form as in (26).
The details of the derivation are included in Appendix A. Note that a simplified expression for the instantaneous interference is obtained by replacing (26) into (5). From the simplified expression (26) and using the independence between channel
2For example, the maximum timing advance in LTE is 667.66µs, whereas the CP length is 4.69µs [13], [32].
3ǫ prevents thatNs u p(i)=(i+1)Tt2/Tt1, which is forbidden according to the definition ofs(n).
coefficients and transmitted data, we can express the interfer- ence power matrix defined in (6) as
[DM(ℓ)]=
Lc−1
X
p=0
Phc(p)G(ℓ,˜ p) if Nin f(i)≤ℓ≤Nsup(i)
0 otherwise
(11) wherePhc(p)=E{|hc(p)|2},
[ ˜G(ℓ,p)]k1,k2 =
1 N1N2
sin2(θ(k1,k2)Ng/2)
sin2(θ(k1,k2)/2) if k2 ,|k1T2/T1+ξ|N2
1
N1N2Ng2 if k2 =|k1T2/T1+ξ|N2
(12) θ(k1,k2)=−2πN2
k2−k1T2 T1 −ξ
, and the expression of Ng is defined in Table II, in the Appendix A.
An analysis of the structure of the power interference matrix DM(ℓ) reveals that there are two main sources of interference: frequency shift due to resampling and leakage.
The power transmitted at k1 appears at the receiver around k2 = |k1T2/T1 +ξ|N2 (more details in Appendix A). If we define the sampling rate ratio asλ=(T2/N2)/(T1/N1), we can rewrite the condition in terms of the normalized bandwidth (in the interval [0,1]) as κ2 =|κ1λ+ξ/N2|1, where κ1 =k1/N1 and κ2 = k2/N2. It is interesting to note that if ξ = 0 and λ=1 (same sampling frequency), the interference appears at the same normalized frequency band at the receiver (κ2=κ1).
On the other hand, we have that there is no power leakage when DM(ℓ) is diagonal. This condition is met, e.g., when ξ =0,0 ≤n0 ≤ Ncp2−Lc+1,T1 =T2, andTcp1 =Tcp2. In the general case, power leakage occurs and the orthogonality between subcarriers is lost.
Interference matrices DM(ℓ) are in general non stationary, since they depend on i. If a stationary approximation is preferable, the instantaneous model can be replaced by its time average, i.e.:
D˜M = 1 Nav
Nav−1
X
ℓ=0
DM(ℓ) (13) provided that the transmitted symbol power do not change in time. If the interference matrix is periodic, due toT1,T2 and Tcp1,Tcp2 are not relative prime, Nav can be defined as the period. In the case of aperiodic DM(ℓ), Nav can be defined as the pseudo-period or a sufficient high value. We show in Section V that this approximation is quite accurate.
B. Transmitters and Receivers with the Same Symbol and Cyclic Prefix Length
The complete interference model presented in (6) and (11) simplifies notoriously when both links in the FD device have the same symbol and CP lengths, i.e.,T1=T2andTcp1=Tcp2. In this section, we use the generalized model to analyze the behavior of a FD system under time and frequency mismatches
between the transmitter and receiver links, considering differ- ent number of subcarriers and CP samples between S–R and T–D links.
Without loss of generality, we assume the TO spans−Nt2+ 1≤n0 ≤Nt2−1, i.e., we only consider those TOs from which at least one sample ofxt(i) affects y(i). The model for other ranges can be easily obtained by shifting the time base of the transmitted signal xt(i). Under these assumptions, only three transmitted blocks interfere at the receiver,ℓ∈ {i−1,i,i+1}, for different values ofn0. More details in Appendix B.
When the system experiments CFO and no TO (n0 = 0), the previous result can be further simplified. In this case, we have Nsup(i)=Nin f(i)=iand therefore only one transmitted symbol interfere at the receiver. With these conditions, we can rewrite (11) as
[Dξ(i)]k2,k1 =
1 N1N2
sin2(θ(k1,k2)N2/2) sin2(θ(k1,k2)/2)
Lc−1
X
p=0
Phc(p) if k2,|k1+ξ|N2 N2
N1
Lc−1
X
p=0
Phc(p) if k2=|k1+ξ|N2 (14) For more details of the derivation, refer to Appendix B. In the case of no CFO (ξ =0), (14) reduces to a diagonal matrix, which means that no ICI occurs.
It is worth noting that the ICI power produced by the TO and CFO is also approximated in [33], [34], after assuming that γ(k) follows a Gaussian distribution, for the TO expression, or that the channel average power at each subcarrier is one with uniform power loading, for the CFO approximation.
The formulation presented in this article is more general, since it does not make any assumption about the statistics of channel taps (only independence between channel coefficients) or the power loading of the OFDM symbols. Additionally, we consider the case of different OFDM parameters.
If the transmitter and the receiver in a FD transceiver can operate with different synchronization parameters, it allows the device to be attached to different cells or networks at the same time, opening up a wide range of new applications. Ad- ditionally, it is important to note that we focus the derivation in this article on those impairments that have not been studied, as the different numerology between the receiver and transmitter.
We also include those imperfections that although have been extensively studied in previous works, have a different effect in the context of FD systems, as the TO and CFO. The presented model can be also used to describe the behavior of the FD system for other mismatches between the links, such as phase noise, I/Q imbalance, sampling jitter, etc. First, matricesC∆(ℓ) are obtained considering the mismatch∆, and then; the interference energy is calculated following a similar procedure that in Section III-A.
The structure of the power interference for the linear case is further illustrated in Section V. In the next section, we derive an approximated expression for the received signal when nonlinear distortion is present in the coupling path.
IV. COUPLINGINTERFERENCEPRODUCED BYNONLINEAR
IMPAIRMENTS
In this section, we introduce an approximated signal model that describes the coupling interference produced by mem- oryless nonlinear distortions, considering nonuniform power allocation in the transmitter T. The most important sources of nonlinear distortion are the PA in the transmitter, the low noise amplifier at the receiver and the RF canceler. We consider the nonlinearity in the transmitter, but the procedure presented here can also be used to approximate nonlinearities at the receiver. In order to simplify the derivation, we assume that the nonlinearity is the only impairment of the system. Then, in the sake of notation simplicity, we drop the use of subindices 1 and 2.
Considering a memoryless nonlinear function F {·}, CN×1
→CN×1, the received signal y(i) results
y(i)=Ksxs(i)+FPrHcF {PaFHxt(i)}+wr(i) (15) As the nonlinearity is memoryless, the coupling term can be rewritten asFPrHcF {PaFHxt(i)} =KcFF {FHxt(i)}. The signal at the output of the nonlinearity can be approximated as a Gaussian multivariate vector, as described in Appendix C.
Then, using the affine transformation to solve for the DFT and the channel matrixKc, we can express the received nonlinear signal as
y(i)≈Ksxs(i)+CNxt(i)+wpa(i)+wr(i) (16) where CN = KpaKc and wpa(i) results a complex Gaus- sian vector N(0,KcFRwFHKHc), with elements [wpa(i)]k = Wpa(k).
Finally, from (16) we can calculate the terms of the SINR as
β(k,k)=|Kpa|2E{|Hc(k)|2} (17) and
γ(k)=[KcFRwFHKHc]k,k (18) Note that although the definition ofγ(k) does not follow (8), the proposed SINR model in (1) remains valid.
The proposed nonlinear SI model is a simple but effective way to model the ICI power due to nonlinearities in the FD transceiver. The accuracy of the approximation is verified in the next section.
V. NUMERICAL SIMULATIONS AND MEASUREMENTS
In this section, we evaluate numerically the coupling in- terference terms β(k2,k2) and γ(k2) of (1), considering S–R and T–D links with time and frequency shifts, and different sampling frequencies. We also include an application example based on the high speed train (HST) FD relayed link, to show the applicability of our model. Finally, we validate our linear and nonlinear models derived in Sections III and IV using measurements.
Before presenting numeric results, it is important to summa- rize a procedure to use our generalized model. For the linear
case, first matricesD∆(ℓ) in (11) or (14) are evaluated using the system numerology, the channel estimation, TO, and CFO.
Then, β(k2,k1) is obtained fromD∆(ℓ) in (7), andγ(k2) from (8). Finally, β(k2) and γ(k2) are replaced in the SINR (1).
Additionally, an instantaneous expression for the SI can be found by replacing (24) into (5). For the nonlinear case, we calculate directly β(k,k) from (17) andγ(k) from (18), using system parameters. Then, again, those values are replaced in SINR (1).
We consider a FD system where the T–D link has N1=64 subcarriers and a CP of Ncp1 =4, whereas the S–R link has N2=128andNcp2 =8, except other parameters are specified.
Both links use 64-QAM symbols. The transmitter power of T and S is set to 0 dBm and hc(n) has relative power decay profile [0,−5,−8] dB with Lc = 3. It also includes a total coupling attenuation corresponding to a signal to interference ratio (SIR) at the receiver of 10 dB, taking into account antenna, RF, and digital domain cancellation. A signal to noise ratio (SNR) of 20 dBis considered at the receiver.
1- Coupling interference due to time and frequency offset for the same symbol length
We begin the analysis considering the simplest scenario, a system where S–R and T–D links have the same symbol (intercarrier spacing) and CP lengths (T1 = T2 and Tcp1 = Tcp2), as it is discussed in Section III-B. First, we consider the case of CFO only (n0=0), where a single transmitted symbol interfere at the receiver. In Fig 2, we evaluate the interference in two scenarios: i) only one subcarrier is active, and ii) the transmitter uses interleaved carrier allocation scheme (ICAS) [34]. In the first scenario only the 50th subcarrier is modulated, whereas in the second, we consider an ICAS of two users, with a resource slot of 8 subcarriers. In Fig 2(a) we plotγ(k2) for the case i), in Fig. 2(b) β(k2,k2) for ii), and Fig. 2(c) γ(k2) also for the case ii). As expected, we note from Fig. 2(a) that the interference moves from subcarrier 45 to subcarrier 55 when the CFO varies between −5 ≤ ξ ≤ 5. There is no ICI for integer ξ except in the carrier 50+ξ. In Fig. 2(b), we observe the SCI components of the coupling interference, which correspond to the interference produced by transmitted subcarrier k2 into the receiver subcarrier k2. The interference is only present for active carriers at T. There is no interference for subcarriers 64 to 128 due to the oversamplig. On the other hand, in Fig. 2(c) we observe the leakage contribution in the ICI power interference term, as described in Section III-A. In this case, the interference is no longer zero for integer ξ at subcarriers belonging to the ICAS resource block. Note that there is no frequency shift sinceT1=T2.
Now, we analyze the TO interference considering the same two scenarios as in the CFO case. From Fig. 3(a), we note again that the interference appears around subcarrier 50.
Additionally, there is zero interference around −Nt2,0, and Nt2, except for k2 =50. From Fig. 3(b), we note again that the SCI appears at modulated subcarriers and has a minimum for n0 = −N2/2 and N2/2+Ncp2. On the other hand, from Fig. 3(c), we can observe again the effect of the leakage.
Contrary to the SCI, the ICI has a maximum forn0=−N2/2 and N2/2+Ncp2.
5
0
-5 100 50 -60 -40 -20 0
0
γ(k2)[dB]
k2 ξ
(a)γ(k2) for a single subcarrier
5
0
-5 100 50 -40 -30 -20 -10 0
0
β(k2,k2)[dB]
k2 ξ
(b)β(k2,k2) for ICAS
5
0
-5 100 50 -40 -30 -20 -10 0
0
γ(k2)[dB]
k2 ξ
(c)γ(k2) for ICAS
Fig. 2. Section 1: Coupling interference power at R due to CFO for the same symbol length. a) ICIγ(k2) when only the 50th subcarrier is modulated, b) SCIβ(k2,k2) for ICAS, and c) ICIγ(k2) for ICAS. System parameters are T1=T2=1s,N1=64,N2=128,Nc p1=4, andNc p2=8.
2- Coupling interference due to different system param- eters
In Fig. 4, we consider the interference at the receiver for a FD system with different modulation parameters, in the same two cases as in the previous figures, but without synchronization offsets (ξ = 0 and n0 = 0). We consider a fixed symbol length at the receiver T2 = 1s and a varying symbol length at the transmitter T2N2/N1 ≤ T1 ≤ 2s, to avoid aliasing. In Fig. 4(a), we note how the interference appears in different subcarriers as the transmitter sampling timeT1/N1changes. We call this effect frequency shift, and it is produced by the change in the sampling frequency. When T1 =T2 (same intercarrier spacing), there is no interference
100 50 0 -50 -100 100
50 -20
-60 -40 0
0
γ(k2)[dB]
k2 n0
(a)γ(k2) for a single subcarrier
100 50 0 -50 -100 100
50 -5 -3 -2 -1
-4
0
β(k2,k2)[dB]
k2 n0
(b)β(k2,k2) for ICAS
100 50 0 -50 -100 100
50 0
-20
-30
-40 -10
0
γ(k2)[dB]
k2 n0
(c)γ(k2) for ICAS
Fig. 3. Section 1: Coupling interference power at R due to TO for the same symbol length. a) ICIγ(k2) when only the 50th subcarrier is modulated, b) SCIβ(k2,k2) for ICAS, and c) ICIγ(k2) for ICAS. System parameters are T1=T2=1s,N1=64,N2=128,Nc p1=4, and Nc p2=8.
except for k2 = 50, as in the previous cases. For λ = 1, T1 = N1T2/N2 = 0.5s, and the interference appears at the same normalized subcarrier κ2 =κ1 ≈0.78 at the receiver or k2=k2T2/T1=100. Fig. 4(b) reveals a wither interference (T1
axis) for lower carriers, since subcarriers near zero interfere itself no matter the value ofT1. As expected, the interference appears only in active carriers. In Fig. 4(c), we can distinguish the contribution of the leakage and the frequency shift.
3- Time evolution of the power interference matrix In Fig. 5, we show the time evolution ofγ(k2) for different i, forT1=1s andT2 =0.75s. We see that the interference is periodic, with periodNav =4. So, we can use this fact to find
2 1.5 1 0.5 100 50 -60 -40 -20 0
0
γ(k2)[dB]
k2 T1
(a)γ(k2) for a single subcarrier
2 1.5 1 0.5 100 50 0
-40
-60 -20
0
β(k2,k2)[dB]
k2 T1
(b)β(k2,k2) for ICAS
2 1.5 1 0.5 100 50 0 20
-40 -20
0
γ(k2)[dB]
k2 T1
(c)γ(k2) for ICAS
Fig. 4. Section 2: Coupling interference at R due to different modulation parameters. a) ICIγ(k2) when only the 50th subcarrier is modulated, b) SCI β(k2,k2) for ICAS, and c) ICIγ(k2) for ICAS. System parameters areT2= 1s,T2N2/N1≤T1≤2s,N1=64,N2=128,Nc p1=4, and Nc p2=8.
a time invariant approximatedγ(k2), as defined in (13).
4- Application example: The high speed train (HST) FD relay link
To show the applicability of our approach, we use the generalized model to study the performance of an HST link, composed by a base station (BS), a FD relay, and a receiver node in the train. The scenario is similar to the one presented in [30]. In the downlink case, and taking the definitions in Fig. 1, the BS is the source S, the relay receiver is R, the relay transmitter is T, and the train is the destination D. FD relays are deployed along the railway to ensure a near line- of-sight link with the train node and provide a backhaul to the BS. The BS serves the FD relays and other users in its
20 15 10 5 0 100 50 0
-10
-20
-30
-40 0
γ(k2)[dB]
k2 i
Fig. 5. Section 3: Time evolution of the power interferenceγ(k2) as a function ofi.
range. Due to the high speed of the train, the relay–train link is highly time-variant. One possible solution to increase the robustness with respect to the low channel coherence time, high CFO, and phase noise, is to increase the intercarrier spacing in the relay–train link. At the same time, it is desirable to maintain the bandwidth constant so it is necessary to reduce the number of carriers in the same proportion. On the other hand, the numerology of the BS–relay should remain the same to keep the compatibility with other users. We consider that both links are synchronized in time and frequency, i.e. the relay compensates for the CFO and TO produced by the train movement and oscillator drift. However, as the BS–relay link is static, the coupling signal is not synchronized with the receiver. We use our model to analyze the performance of the HST relayed link in two scenarios: a) throughput dependence on the SI due to CFO, and b) the use of our model in a throughput maximization problem.
Figure 6 shows the throughput of the FD relay for the de- scribed HST scenario as a function of the CFO. For the down- link, N2 =1024, Ncp2 = N2/16, N1 = {1024,512,256,128}, Ncp1 =N1/16,T2=71.43µs, andT1=T2N1/N2. We consider the BS transmits an OFDMA signal, ICAS with only one active user, and a resource block of 8 subcarriers. Thus, the subcarriers {0, . . . ,7,16, . . . ,23, . . .} are active and the other subcarriers are not used. On the other hand, the train transmits a resource block of 8N1/N2 subcarriers to maintain the same spectral occupancy. Channel lengths are Ls =5 and Lc =1.
The throughput is calculated as C= 1
Nit
Ni t
X
i=0 N2−1
X
k2=0
log2(1+Ψ(k2)) (19) where Nit =100is the number of simulated channel realiza- tions and Ψ(k2) is the SINR in (1) given that the coupling channel is known. The SINR for the generalized model is obtained by replacing (11)4in (7) and (8), and the result in (1).
For the classic model, γ(k2) =0, β(k2,k2)=|Hc(k2)|2, and Pt(k) is calculated from the receiver resource block structure to approximate the spectral occupancy in the transmitter. For both models,α(k2)=|Hs(k2)|2. Even considering a high train
4As the SINR is conditioned on the coupling channel,Ph c(n) is replaced by |hc(n)|2.
speed, the dominant CFO is due to oscillation inaccuracy. For a 10 ppm error, a frequency of 2.5 GHz, and an intercarrier spacing of15 kHz, the resulting CFO w.r.t. intercarrier spacing is around 1.6 [35]. Therefore, we consider0≤ξ≤1.5 in the simulation.
From the figure, we can note that our proposal is able to model the interference very accurately. On the other hand, the classic model only represents the system accurately for ξ=0, when there is no ICI. The classical model overestimates the SI and predicts a lower throughput, because it does not consider the power leakage to neighbor resource blocks. It must be noted that models for the ICI when the links have the same numerology have been proposed, but these models are not valid for different OFDM parameters. Large subcarrier spacing (N=128) in relay–train link increases the throughput of the source–relay link for low CFO. However, although the large subcarrier spacing makes the relay–train link more robust, it decreases the throughput of the source–relay link for large CFO. With narrow subcarrier spacing (N =1024), the throughput increases with CFO, because self-interference spreads out to the unused resource blocks more than with the large subcarrier spacing.
0 0.5 1 1.5
1200 1250 1300 1350 1400 1450 1500 1550
Throughput[bits/s/Hz]
CFO
N11024 N1512 N1256 N1128 Class.N11024
Fig. 6. Section 4: Throughput of the S-R link in the HST network for different CFO values,N2=1024,N1=1024,512,256,and128. Solid curves correspond to the generalized SINR model and dashed curves to the signal model.
Fig. 7 shows the throughput of the S–R link of a FD relay with N1 = 128, N2 = 1024, and the same ICAS as in the previous example. The source S allocates the power on active subcarriers according to the water filling algorithm [36], using the classic and generalized interference models when calculating the SINR. Throughput curves based on signal models are obtained by computing numerically the SINR terms from (5). Alternatively, throughput curves based on the SINR models are obtained from (19) as in the previous example. The classic SINR model predicts much lower throughput than the signal model, because it overestimates the SI power falling on the active subcarriers in case of different OFDM parameters.
Classic signal model gives slightly lower throughput than the generalized model, because the water filling power allocation is based on the incorrect model. In contrast to the classic model, the generalized SINR model estimates the throughput accurately and provides better throughput than the classic
TABLE I
SYSTEM PARAMETERS FOR THE MEASUREMENT SETUP. Linear setup Nonlinear setup
N1 64 64
N2 32, 64, 128 64
Nc p1 16 16
Nc p2 8, 16, 32 16
Active carriers {2-5, 10-13, 18-21, ... {5-8, 50-53}
45-48, 53-56, 61-64 }
Preamble 2 OFDM symbols 10 OFDM symbols
Carrier frequency 2.4 GHz 2.4 GHz
N1/T1 20 MHz 20 MHz
N2/T2 20 MHz 20 MHz
Tx power −25 dBm 20 dBm
AGC output −15 dBm −15 dBm
Antenna canc. FD patch [6] (≈ −55 dB) −50 dBattenuator
model due to more efficient power allocation.
0 0.5 1 1.5
1150 1200 1250 1300 1350 1400 1450 1500
Throughput[bits/s/Hz]
CFO
Signal model (Class) Signal model (Gen) SINR model (Gen) SINR model (Class)
Fig. 7. Section 4: Throughput of the S-R link using water filling for different CFO values, and N2=1024and N1=128. Solid curves correspond to the SINR model and dashed curves to the signal model.
5- Experimental setup: Validation of the linear and nonlinear models
A measurement campaign based on the WARP v3 kit [37]
was implemented to validate the linear (5) and nonlinear (16) interference models. To capture the SI without distortion, we do not include the signal of interest in the setup. A summary of the system parameters is presented in Table I 5.
First, we measure the SI for different modulation parameters between the transmitter and the receiver. The transmitting power is set to a low value to minimize the nonlinear effects of the power amplifier. We use the FD antenna designed in [6]
that provides a SI cancellation of approximately 55 dB. The results are averaged over 200 realizations to reduce the effects of the thermal noise. As the channel does not vary significantly between realizations, we average the instantaneous expression (5) to calculate the SI power. The SI channel is estimated using a pseudo-random preamble inserted at the beginning of the frame. In Fig. 8, we compare the proposed SI model with the measured SI power at the receiver, considering a transmitter with N1 = 128 and a receiver with N2 = 32, 64, and 128
5As the sampling frequency of the WARP kit is 40 MHz, we interpo- late/decimate2×.
subcarriers. WhenN2=64, the SI is orthogonal to the received signal and there is no ICI. For the other cases, it is evident that ICI is present. It must be noted that the apparent random structure of the interference for N2 = 128is not because of the lack of averaging, but due to the actual structure of the SI. In this case, the SI is produced by the cyclic prefix of the transmitted OFDM symbols that are not correctly discarded at the receiver due to the different numerology. We note from the figure, that the model is able to predict accurately the interference power.
-10 -5 0 5 10
-40 -35 -30 -25 -20 -15 -10 -5 0
SIpower[dBm]
BW
N2=128 N2=64 N2=32
Fig. 8. Section 5: Validation of the linear model for different parameters in the transmitter and receiver.N1=64andN2=32,64,and 128. The proposed model is in solid line and the measurements in dashed lines.
Then, we validate the proposed nonlinear model for the SI considering the same numerology in the transmitter and the receiver. In this case, we set the device to its maximum trans- mitting power. The SI channel is estimated using a low peak- to-average-power-ratio pseudo-random preamble to decouple its estimate from theKpa estimation. Note that the energy of the preamble sets a trade-off between channel estimation errors due to the nonlinear distortion and errors due to thermal noise.
As the output of the power amplifier is not available, Kpa is calculated using the transmitted signal and the received signal after channel equalization, as described in (30). In order to isolate the disturbance and obtain a better channel estimation, we use a calibrated attenuator (R&S RDL50 1035.1700.52) instead of the FD antenna. The histogram of the measurements and the Gaussian approximation proposed by our model are shown in Fig. 9. Subcarriers 5 to 8 are modulated while the subcarriers from 9 to 12 are unloaded. We note from the figure that the predicted probability density function (pdf) slightly deviates from the measurements for the modulated subcarri- ers, although they are Gaussian. Analyzing the measurement results, we find that the noise variance is quite sensitive to errors in the estimation of the SI channel and Kpa. On the other hand, the model matches well to the measurements for unmodulated subcarriers.
It is clear that the FD operation increases the bandwidth efficiency but also introduces a penalty in the performance of the system due to the coupling interference. Our analysis describes rigorously the SI for a general system model and parameters, and it enables to derive more efficient power
