Spectrum Sensing and Throughput Analysis for Full-Duplex Cognitive Radio with Hardware Impairments

In Full-d uplex Cognitiv e Radios, the silen t period of the Secondary User (SU) during the Spectrum Sensing can be elimina ted by appl ying the Self -Interf erence Cancella tion (SIC). Due to the channel estima tion error and the hardw are imperf ections (the Phase Noise and the Non-Linear Distortion (NLD)) SIC is not perf ectl y perf ormed and resul ts in the Resid ual Self -Interf erence (RSI) which a ﬀ ects the Spectrum Sensing reliability . In this paper , the e ﬀ ect of RSI on Spectrum Sensing is anal yticall y deriv ed by expressing the detection ( p d ) and false alarm ( p f a ) probabilities under FD in terms of the ones under Half -Duplex (HD) (where SU shoul d remain silen t during the Spectrum Sensing period). In addition, an algorithm is proposed to suppress the NLD and improv es the Spectrum Sensing perf ormance. Hereinafter , the SU throughput under FD is anal ysed comparing to HD by deriving the upper and lower bounds to be respected by p f a and p d respectiv ely in order to make FD beneficia rela tive to HD.


Introduction
Recen tl y, the Full Duplex (FD) transmission has been introd uced in the context of Cognitiv e Radio (CR) to enhance the Data-Ra te of the Secondary (unlicensed) User (SU).In FD systems, SU can sim ul taneousl y transmit and sense the channel.In classical Half Duplex (HD) systems, the SU shoul d stop transmitting in order to sense the sta tus of the Primary (licensed) User (PU).Recen t adv ancemen ts in the Self -Interf erence Cancella tion (SIC) make the applica tion of FD in CR possible.Due to man y imperf ections, a perf ect elimina tion of the self -interf erence cannot be reached in real world applica tions [2,3].In CR, the SU makes on Cognitiv e Radio Orien ted Wireless Netw orks (CROWNCOM 2016) [1]   * Corresponding author Email: abbas.nasser@univ-brest.fra decision on the PU sta tus using a Test Statistic [4][5][6].This Test Statistic depends on the PU signal and the noise.Any resid ual interf erence from the SU signal can affect the Test Statistic norm and leads to a wrong decision about the presence of PU.
In wireless systems, the FD is considered as achiev ed if the Resid ual Self Interf erence (RSI) pow er becomes lower than the noise lev el.For tha t an importan t SIC gain is required (around 110 dB for a typical WiFi system [2]).This gain can be achiev ed using a passiv e suppression and an activ e cancella tion.The passiv e suppression is rela ted to man y factors tha t red uce the Self Interf erence (SI) such as the transmission direction, the absorption of the metals and the distance betw een the transmitting antenna, T x , and the receiv e antenna, R x .The activ e cancella tion red uces the Self -Interf erence (SI) by using a copy of the known transmitted signal.The estima tion of channel coefficien ts becomes an essen tial factor in the activ e cancella tion process.Any error in the channel estima tion leads to decreasing the SIC gain.Experimen tal resul ts show tha t hardw are imperf ections such as the non-linearity of am plifier and the oscilla tor noise are the main limiting perf ormance factors [2,[7][8][9].Theref ore, the SIC shoul d also consider the receiv er imperf ections.The authors of [2] modify their previous method of [10 ] to estima te the channel and the Non-Linearity Distortion (LND) of the receiv er Low -Noise Amplifie (LNA).Their method requires tw o training symbol periods.During the firs period, the channel coefficien ts are estima ted in the presence of the NLD.
The non-linearity of the am plifie is estima ted in the second period using the already estima ted channel coefficien ts.It is worth men tioned tha t the estima tion of the NLD par ameters in the second phase depends on the one of the channel coefficien ts done in the firs phase.How ev er the estima tion of the channel coefficien ts in the firs phase can be depending on unknown NLD par ameters.To sol ve the previous dilemma, we propose hereinafter an estima tion method of the NLD in such way tha t the estima tion of the channel cannot be affected by the NLD.
The works of [11 -16 ] deal with the applica tion of FD in CR.In [11 -13 , 15 ], the RSI is modeled as a linear combina tion of the SU signal without considering hardw are imperf ections.In [13 , 16 ] the Energy Detection (ED) is studied in a FD mode and the probability of detection, (p d ), and false alarm, p f a , are found anal yticall y.According to our best knowledg e, there is no anal ytic rela tionship betw een the RSI, p d and p f a for both HD and FD mode.Further , due to the RSI, p d and p f a are highl y affected.This fact impacts neg ativ el y the SU throughput.In some circumstances, the SU throughput under HD can be higher than the one of FD.This can be occurred when the false alarm rate is high.On the other hand, PU transmission is consider abl y disturbed by the SU activities when the detection rate becomes low.For tha t reason, upper bound of false alarm rate and lower bound of detection rate are importan t in order to be abided by SU in FD mode in order to enhance the throughput without increasing the interf erence rate to PU comparing to HD mode.This paper deals with the Spectrum Sensing in real world applica tions.At firs , we anal yticall y determine the impact of the RSI pow er on the detection process.For tha t objectiv e, we deriv e a rela tion betw een the RSI pow er, the probabilities of detection and false alarm under HD and FD modes.Secondl y, we anal yse the NLD impact on the channel estima tion and the Spectrum Sensing Perf ormance.Hereinafter , nov el al gorithms are proposed to suppress the NLD of LNA.
In addition, the effect of FD mode on the SU throughput is compared to the HD mode.As the throughput is rela ted to p d and p f a , the upper bound of false alarm rate and the lower bound of the detection rate are deriv ed.These tw o bounds char acterize the required limit in FD in order to enhance the SU throughput comparing to HD without making an additional interf erence to PU.The rest of this paper is presen ted as foll ows, in section (2), an overview of OFDM receiv er is presen ted by focusing on the circuit imperf ections.The Spectrum Sensing hypothesis in Full-Duplex Cognitiv e Radio is presen ted in section (3), where the effect of RSI is anal ysed anal yticall y.In section (4), an al gorithm of the NLD mitig ation in RF domain is proposed with its corresponding numerical resul ts.Throughput anal ysis in terms of detection and false alarm probabilities is presen ted in section (5).At the end, section (6) concl udes the work by providing new perspectiv es.
Throughout this paper , uppercase letters represen t frequency -domain signals and lower-case letters represen t signals in time-domain.

Receiver chain Imperfection Analysis
In this section, we anal yse the imperf ections in a classical OFDM receiv er.By ref erring to figur (1), which represen ts a typical Ordinary Chain (OC) of an OFDM receiv er bl ock-diagr am, the SU signal receiv ed at R x is modelled as: where h(t) is the channel effect betw een T x and R x and * stands for the convolution oper ator.After tha t, y(t) is am plifie using the LNA which introd uces a NLD.The output of LNA can be modelled in a gener al form as a pol ynomial of odd degrees: The firs coefficien t a 1 , stands for the linear componen t which represen ts the am plifie signal.The other coefficien ts (a 2i+1 , i ≥ 2) stand for the non-linear componen ts contributing in NLD.Since the higher componen t are of negligible pow er, the NLD of the output of the LNA is limited to the third order pol ynomial output: In digital domain, y a (n) can be expressed as foll ows [3,17 ]: After the am plific tion process, the receiv ed signal is down-con verted to the base-band form.As shown in figur (1), the oscilla tor can introd uce a mul tiplica tiv e noise exp After converting to the base-band, the Anal og-to-Digital Converter (ADC) digitizes the receiv ed signal and introd uces a unif orm noise w q (n), which has a pow er inversel y proportional to the number of used bits.Consequen tl y, the receiv ed time-domain base-band signals at OC is presen ted as foll ows: Where w(n) is an additiv e zero mean white Gaussian noise.w(n) is the internal noise in the receiv er circuit and it is rela ted to the input signal lev el and to the bl ocks of the receiv er chain.FFT and CP remov al oper ations will be applied on y a (n) to obtain Y a (m).By assuming tha t w q (m) is domina ted by w(n), Y a (m) can be presen ted as foll ows: Where is the NLD of the LNA in frequency domain and In frequency domain, the channel estima tion is done in order to perf orm SIC, as well as the circuit imperf ection mitig ation.The frequency domain SU signal S(m), contributing in the channel estima tion and the imperf ection mitig ation is coming from an Auxiliary Chain (AC), which is designed in a way to mitig ate the imperf ections and to help estima te the channel.
After the SIC and the circuit imperf ections mitig ation, the obtained signal, Ŷ (m), can be presen ted as foll ows: Where ξ(m) is the RSI and define as: Ĥ(m) and D(m) are the estima ted channel and the NLD respectiv el y.
Ideall y Ĥ(m) = H(m) and D(m) = D(m), theref ore equation (7) becomes: Ŷ (m) = W (m) + ηX(m), which corresponds to an HD mode.Any mistake in the cancella tion process may lead to a wrong decision about the PU presence.

The RSI effect on the Spectrum Sensing
In the spectrum sensing context , we usuall y assume tw o hypothesis: H 0 (PU signal is absen t) and H 1 (otherwise).
In our works, we assume tha t PU signal and SU signals are wideband signals such as OFDM.By focusing onl y on the additiv e receiv er distortion which is domina ted by the NLD of the LNA [2], the receiv ed signal can be modeled as foll ows: H(m) is the channel betw een the SU transmitter antenna T x and the SU receiv e antenna R x , S(m) is the SU signal, W (m) is an Additiv e White Gaussian Noise (AWGN), D(m) represen ts the NLD of the LNA, X(m) is imag e of the the PU signal on R x and η ∈ {0, 1} is the channel indica tor (η = 1 if PU is activ e and η = 0 otherwise).
In order to decide the existence of the PU, sev eral al gorithms have been proposed in the liter ature [4].The most commonl y used one is the Energy Detector (ED), which compares the receiv ed signal energy , T ED , to a predefine threshol d, λ.
Contrary to HD, where Onl y the noise variance shoul d be estima ted prior to establishing the Spectrum Sensing, in FD the pow er of the RSI shoul d be also taken into accoun t.By assuming the i.i.d property of (n) (the time domain version of ξ(m)), w(n) and x(n), then ξ(m), W (m) and X(m) become i.i.d.(See Appendix (A.1 )).In this case, the distribution of T ED shoul d asym ptoticall y foll ow a normal distribution for a larg e number of sam ples, N , according to the cen tral limit theorem.Consequen tl y, the probabilities of False Alarm, p f f a , and the Detection, p f d , under the FD mode can be obtained as foll ows (See Appendix (A.2 )):   Let us defin the Probability of Detection Ratio (PDR), δ, for the same probability of false alarm under FD and HD modes, as foll ows: Where p h d and p h f a are the probabilities of detection and false alarm under HD respectiv el y, 0 ≤ α ≤ 1 and 0 ≤ δ ≤ 1.As with an excellen t SIC, the ROC can mostl y reach in FD the same perf ormance of HD.In order to show the effect of RSI on δ, let us defin the RSI to noise ratio γ d as foll ows: Using ( 11) and ( 13), the threshol d, λ, can be expressed as foll ows: By replacing (15 ) in (12 ), γ d can be expressed as foll ows: If δ = 1, then we can prov e tha t γ d becomes zero, which means tha t the SIC is perf ectl y achiev ed.

The Non-Linear Distortion of the Low Noise Amplifier
In real world applica tions, the full duplex transceiv er seems hard to be depl oyed due to hardw are imperf ections: the non-linearity of the am plifiers the quan tization noise of ADC, the phase noise of the oscilla tor, etc.
The NLD of LNA is an importan t perf ormance limiting factor [2,[7][8][9][10].According to NI 5791 da tasheet [18 ], the NLD pow er is of 45 dB bel ow the pow er of the linear am plifie componen t.A new efficien t al gorithm is proposed in this section, in order to mitig ate as possible the NLD of LNA and to make the channel estima tion more reliable.The proposed al gorithms are anal ysed by neglecting the quan tific tion and the phase noises.

Estimation of the Non-Linearity Distortion of LNA
The LNA output can be written as a pol ynomial of odd degrees of the input signal [17 ].The NLD stands for the degrees grea ter than one.By limiting to the third degree and neglecting the higher degrees pow er [19 ], the NLD componen t can be written as foll ows: Where β is the NLD coefficien t.The estima tion of β can be helpful to suppress the LNA output.In this case, the channel estima tion is no long er affected by the NLD.The overall output signal of the LNA, y a (t), can be expressed as foll ows: By deriving J with respect to a and b we obtain: Using equa tions (20 ) and ( 21), a linear system of equa tions can be obtained: Where: Once the non-linearity coefficien t, β, is estima ted, the non-linearity componen t can be subtr acted from the output signal of the am plifie .the noise pow er to -72 dBm [18 ].As shown in figur (4), the resid ual pow er of NLD decreases with an increasing of N e when the method of [2] is applied.How ev er our method keeps a constan t val ue of this pow er.Our technique outperf orms significa tl y the method proposed in [2].To show the impact of NLD on the channel estima tion and the RSI pow er, figur (5) shows the pow er of Ŷ (m) obtained in FD under H 0 .The channel is estima ted according to the method previousl y proposed by [20 ] as foll ows:

Numerical Results
Where IDFT stands for the inverse discrete Fourier transf orm and n tap is the channel order .The number of training symbols, N e , is fixe to 4 symbols.The number of sub-carrier is 64, the transmitted signal is of -10 dB (i.e.20 dBm), and the noise f oor is -102 dB (i.e.-72 dBm) [18 ].The transceiv er antenna is assumed to be omni-directional with 35 cm separ ation betw een T x and R x , so tha t a passiv e suppression of 25 dB is achiev ed [3].According to the experimen tal resul ts of [3], in a low reflectio environmen t, 2 channel taps are enough to perf orm the SIC when the passiv e suppression is bell ow 45 dB.Furthermore, the line of sight channel is modelled as a Rician channel with K-factor about 20 dB.The non-line of sight componen t is modeled by a Rayleigh fading channel.Figure (5) shows tha t our method leads to mitig ate almost all the self interf erence, so tha t the pow er of Ŷ (m) becomes very closed to the noise pow er.How ev er, with the method of [2], the RSI pow er increases with the NLD pow er beca use the NLD pow er is a limiting factor of the channel estima tion which leads to a bad estima tion of the channel.
To show the impact of the NLD on the Spectrum Sensing, figur (6) shows the ROC in various situa tions under γ x = −10 dB.The sim ula tions par ameters in this figur are similar to those of figur ( 5), onl y the NLD pow er is set to 45 dB under the linear componen t according NI 5791 indica tions [18 ].The method of [2] leads to a linear ROC, which means tha t no meaningful inf orma tion about the PU sta tus can be obtained.By ref erring to figur (5), the RSI pow er is of -82 dB for a NLD pow er of -45 dBc, which means tha t γ d in this case is about 20 dB.This high RSI pow er leads to a harmful loss of perf ormance (see figur (3)).From the other hand, our method makes the ROC in FD mode almost colinear with tha t of the ROC of HD mode, which means tha t all SI and receiv er impairmen ts is mitig ated.Figure (7) shows the PDR for a targ et α = 0.1 and P H d = 0.9.The ratio δ increases with the SNR.At a low SNR of -10 dB, δ becomes closed to 1, so tha t a negligible

Throughput Analysis
In this section, we anal yse the SU perf ormance by discussing from throughput viewpoin t.Although FD-CR is introd uced to enhance the SU da ta rate, the RSI affects both false alarm and detection probabilities as discussed in section (3).Consequen tl y, the throughput of SU under FD can be lower than the throughput under HD for some circumstances.First , let us presen t the activity period, T , of SU in HD mode (see figur ( 8)).We assume tha t this activity can be divided into sensing period (T s ) and transmitting period (T t ).The throughput of SU (R SU ) is affected by the sensing perf ormance since the detection of the PU is not perf ect and it can be done up to a targ et pair (p f a , p d ).After the sensing period, SU can contin ue transmitting in one of the tw o foll owing scenarios: 1. Missed Detection: SU decides tha t there is no activ e PU in the bandwid th, while PU is trul y activ e.In this case, SU becomes activ e and interf eres with PU 2. Reject: SU rejects the hypothesis H 1 by detecting correctl y the absence of PU.Theref ore SU becomes activ e As SU is activ e under Missed Detection and Reject cases, R SU becomes the sum of the tw o throughputs: R 0 under Reject case and R 1 under Missed Detection case.
Let us denote by p 0 the probability tha t PU is trul y absen t, and p 1 the probability tha t PU is trul y transmitting, with p 0 + p 1 = 1.Accordingl y, R 1 and R 0 are giv en by [21 ]: where τ = T t T is the ratio of the transmission period with respect to the overall activity period, and C 0 (resp.C 1 ) is the throughput of SU when it oper ates under H 0 (resp.(H 1 )).By assuming tha t SU and PU signals are sta tisticall y independen t, white and Gaussian, C 0 and C 1 are giv en by: where γ r is the SNR and ζ r and the Signal to Noise and Interf erence Ratio (SNIR) of SU signal at the Secondary receiv er.R 1 represen ts the interf ering throughput of SU transmission to PU.This throughput shoul d be minimized as possible by increasing p d in order to do not affect the PU transmission.In contrast , R 0 represen ts the gainful throughput of SU to be maximized by decreasing p f a , since it resul ts from a true decision on the absence of PU.On the other hand, both R 0 and R 1 are rela ted to τ.In FD-CR the ratio τ is equal to one since no silence period is required.Even though this fact is importan t for the SU throughput it can affect the Spectrum reliability , by increasing p f a (i.e.decreasing the gainful throughput R 0 ) and decreasing p d (i.e.increasing the interf erence throughput R 1 ).To establish the minim um requiremen t of SU transmission in FD in order to exceed the throughput in HD; tw o conditions have to be respected: the interf ering throughput under HD mode represen ts the upper bound of the interf ering throughput under FD mode.This means the interf erence amoun t under FD mode must not exceed tha t under HD mode.
2. The gainful throughput under HD mode represen ts the lower bound of the FD gainful throughput.
The firs condition can be expressed as foll ows: The left -hand side of the equa tion (30 ) represen ts the interf ering throughput in FD mode, whereas the righthand side represen ts the interf ering throughput under HD mode.Accordingl y, p f d becomes: Reg arding the gainful throughput (2nd condition), R 0 , the condition resul ting in increasing the SU throughput in FD mode rela tiv el y to HD can be introd uced by: equa tion resul ts in the upper bound of the false alarm rate of SU to be respected in order to increase the gainful throughput of FD rela tiv el y to HD: Figure (10 ) shows the upper bound of p f f a , bell ow which the FD gainful throughput exceeds the one of HD mode.p h f a and p f f a are linear ly correla ted.On the other hand, it is clear from this figur tha t the upper bound of p f f a grows inversel y to τ. Increasing p f a does not affect the PU transmission, but it prev ents SU from using efficien tl y the spectrum opportunity .
In figur (11 ), we examine the gainful throughput (R 0 ) of SU in terms of SNR for various val ues of p f a .Note tha t p h f a is fixe at 0.1, τ = 0.75 and p 0 = 0.9.As shown in this figure the throughput decreases with the increase of p f a , this is due to the spectrum opportunity loss.On the other hand, for p f f a = 0.4, the FD throughput becomes lower than the HD one.This means the functioning in HD mode becomes much interesting to SU.Based on [22 ], tw o normal variables are independen t if and onl y if (if f ) they are uncorrela ted.The correla tion, C(m 1 , m 2 ), of R(m 1 ) and R(m 2 ) ∀ m 1 m 2 is giv en as foll ows: As C(m 1 , m 2 ) = 0; ∀ m 1 m 2 , theref ore R(m 1 ) and R(m 2 ) become uncorrela ted and independen t since they are Gaussian.

A.2. Probability of Detection and Probability of False Alarm
As by our assum ption ξ(m), W (m) and X(m), are asym ptoticall y Gaussian i.i.d., then Ŷ (m) is also Gaussian and i.i.d.Theref ore, the Test Statistic, T ED , of equa tion (10 ) foll ows a normal distribution according to CLT for a larg e N.Under H 0 (i.e.X(m) does not exist), the mean, µ 0 , and the variance, V 0 of T ED can be obtained as foll ows: w + σ 2 d ) 2 .Back to equa tion (A.4 ), the variance, V 0 becomes: By foll owing the same proced ure, µ 1 and V 1 can be obtained as foll ows under H 1 (X(m) exists):

2 w.
If the SIC is perf ectl y achiev ed, i.e. σ 2 d = 0, p f f a and p f d take their expressions under the HD mode.

Figure ( 2 )
shows the required number of sam ples to reach p d = 0.9 and p f a = 0.1 under the HD and FD modes for di fferen t val ues of SNR.In FD mode, we set σ 2 d = σ 2 w as the targ et val ues of σ 2 d in digital comm unica tion.Figure (2) shows tha t the number of required sam ples slightl y increases under the FD modes.For exam ple, if γ x = −5 dB, then 85 sam ples are enough to reach the targ et (p d ; p f a ) under the HD mode while under FD mode, around 300 sam ples are needed.

Figure ( 3 )Figure 3 .
Figure 3. Evolution of γ d with respect to γ x for various values of δ, (p h f a ; p h d ) = (0.1 ; 0.9)

Figure 4 .
Figure 4.The effect of the number of training symbols on the NLD residual power

Figure ( 4 )
Figure(4) shows the resid ual pow er of the NLD cancella tion.The NLD pow er is fixe to 45 dB under the linear componen t[18 ].This pow er is red uced to less than -300 dB after the applica tion of our method.The method of[2] red uces the NLD pow er by around 50 dB.β is estima ted using various number of training symbols, N e .In this sim ula tion, OFDM mod ula tions are used with 64 sub-carriers and a CP length equal to 16.The receiv ed pow er is fixe to -5 dBm and

Figure 6 .Figure 7 .
Figure 6.The ROC curve after applying the proposed technique of NLD suppression

Figure 8 .
Figure 8. Activity period of SU under HD functioning is divide into Sensing sub-period (T s ) and Transmission sub-period (T t ).

)Figure 9 .
Figure 9. Variation of p f d in terms of p h d for various values of τ

Figure 10 . 4 Figure 11 .
Figure 10.Variation of p f f a in terms of p h f a for various values of τ