with the fast development of wireless communication , spectrum resources have become increasingly scarce , motivating the development of technologies such as d2d communications  to improve spectrum utilization of the crowded spectrum bands .meanwhile , as an early study by federal communications commission suggests , some of the allocated spectrum is largely under - utilized in vast temporal and geographic dimensions  .cognitive radio , focusing on these bands , has attracted wide attentions over the past years as a promising solution to the spectrum reuse  . in cognitive radio networks ( crns ) , unlicensed secondary users ( sus )are allowed to access spectrum bands of the licensed primary users ( pus ) by two spectrum sharing approaches : underlay and overlay . in underlay spectrum sharing, the sus are allowed to operate if the interference caused to pus is below a given threshold with proper resource management  .overlay spectrum sharing , which is adopted in this paper , refers to the spectrum utilize technique that allows the sus to access only the empty spectrum for pus .thus , reliable identification of spectrum holes is required to protect the pus and maximize sus throughput .most existing works on overlay crns employ `` listen - before - talk '' ( lbt ) protocol on half - duplex ( hd ) radio , in which the traffic of sus is time - slotted , and each slot is divided into two sub - slots , namely sensing sub - slot and transmission sub - slot .sus sense the target channel in the sensing sub - slot to decide whether to access the spectrum in the following transmission sub - slot . in ,optimization of sensing and transmission duration has been discussed . in ,the authors considered general pu idle time distributions and imperfect sensing , and provide a tight upper bound of the performance in the lbt protocol .cooperative spectrum sensing has been studied in to achieve better sensing performance . though the conventional hd based lbt protocol is proved to be effective , it actually dissipates the precious resources by employing time - division duplexing , and thus , unavoidably suffers from two major problems as follow .  1 .the sus have to sacrifice the transmit time for spectrum sensing , and even if the spectrum hole is long and continuous , the data transmission need to be split into small discontinuous slots ; 2 .during the transmission sub - slots , the sus do not sense the spectrum .thus , if the pus arrive or leave during the transmission sub - slots , sus can not be aware until the next sensing sub - slot , which leads to long collision ( when the pus arrive ) and spectrum waste ( when the pus leave ) .   a more efficient way to utilize the spectrum holes and protectthe pu network should allow the sus to keep sensing the spectrum all the time .whenever a spectrum hole is detected , sus begin transmission , and once the pu arrives , the transmission ceases .this can be facilitated by full - duplex ( fd ) techniques  . in a fd system ,a node can transmit and receive using the same time and frequency resources .however , due to the close proximity of a given modem s transmitting antennas to its receiving antennas , strong self - interference introduced by its own transmission makes decoding process nearly impossible , which had been a huge impediment to the development of fd communications in the past .recently , there has been significant progress in the self - interference cancelation including proper hardware design and signal process techniques , presenting great potential for realizing the fd communications for the future wireless networks  .  motivated by the fd techniques , in this paper, we propose a `` listen - and - talk '' ( lat ) protocol , by which sus can simultaneously perform spectrum sensing and data transmission .we assume that the pu can change its state at any time and each su has two antennas working in fd mode .specifically , at each moment , one of the antennas at each su senses the target spectrum band , and judges if the pu is busy or idle ; while the other antenna transmits data simultaneously or keeps silent on basis of the sensing results .we remark that the ideas of this kind have also been mentioned by some other recent works .some of the papers such as have mentioned the simultaneous sensing and transmission briefly as a feasible application scenario of fd technology without further analysis , while some other papers study the similar topics .we concisely summarize them as follow .some works focus on deploying fd radios on cr users and the impact of some physical issues leading to residual self - interference and imperfect sensing .specifically , discussed the use of directional antennas , and showed that directionality of multi - reconfigurable antennas could increase both the range and rate of full - duplex transmissions over omni - directional antenna based full - duplex transmissions ; focused on comparing sensing error probabilities in the half - duplex , two - antenna full - duplex , and single - antenna full - duplex cognitive scenarios under energy detection ; and the impact of some physical issues leading to residual self - interference and imperfect sensing such as bandwidth , antenna placement error , and transmit signal amplitude difference was discussed in   in , the authors considered multiple su links with partial / complete self - interference suppression capability , and they could operate in either simultaneous transmit - and - sense ( ts ) or simultaneous transmit - and - receive ( tr ) modes .mode selection between the ts and tr mode and the coordination of su links were proposed to achieve high secondary throughput .the idea of the ts mode is similar to our protocol .however , in , one fixed threshold for energy detection was used in both so and ts modes , while in our work , a pair of sensing thresholds are designed to compensate for the imperfect self - interference cancelation . besides , the authors in only provided calculation of error sensing probabilities in series expressions in the analysis , and failed to present how well can the sus utilize the spectrum holes , which is addressed in our work .the authors in considered cooperation between primary and secondary systems . in their model, the cognitive base station ( cbs ) relays the primary signal , and in return it can transmit its own cognitive signal .the cbs was assumed to be fd enabled with multiple antennas .beamforming technique was used to differentiate the forwarding signal for primary users and secondary transmission .different from all the above works , throughout the paper , we focus on the following important issues :  * how to design the sensing strategy so that the benefits of the fd can be fully enjoyed ? * how to design the secondary transmit parameter , e.g. , transmit power , so that sus can achieve high throughput as well as satisfactory sensing performance ? *how well can the proposed lat perform in terms of spectrum utilization efficiency and secondary throughput ?we explore answers to these questions by both theoretical analysis and simulation results .the main contributions of this paper can be summarized below .* we clearly present the idea of simultaneous sensing and transmission , and design a `` listen - and - talk '' protocol indicating when and with what power should a su access the spectrum , and how to set the detection threshold .* we present theoretical analysis of the sensing performance and the spectrum utilization . especially , the closed - form expressions of the collision ratio at the primary network and the spectrum waste ratio are provided .* we report a power - throughput tradeoff , show the existence of a local optimal transmit power , with which the sus can achieve high throughput as well as satisfying sensing performance , and derive the theoretical expression of the local optimal transmit power .the rest of the paper is organized as follows .section ii describes the system model and the concept of simultaneous sensing and transmission . in section iii , we elaborate the proposed lat protocol and discuss the design of important parameters . in section iv , we investigate the analytical performance , including the spectrum utilization efficiency and secondary throughput .also , a power - throughput tradeoff is reported and analyzed .simulation results are presented to verify the analytical results in section v. we conclude the paper in section vi .in this section , the system model of the overall network is presented , and the concept of simultaneous sensing and transmission under imperfect self - interference suppression is elaborated .we consider a cr system consisting of one pu and one su pair as shown in the left part of fig .[ model ] , in which su is the secondary transmitter and su is the receiver . each su is equipped with two antennas ant and ant , where ant is used for data reception , while ant is used for data transmission .the transmitter su uses both ant and ant for simultaneous spectrum sensing and secondary transmission with the help of fd techniques , while the receiver su uses only ant to receive signal from su . )performs spectrum sensing , while the receiver ( su ) does not . this transmitter - only sensing mechanismis widely adopted in today s cognitive radio , considered that secondary transmitters and receivers can not continuously exchange sensing results without interfering the primary network .besides , we assume that su has two antennas for fairness and generality , since su does not always need to be the receiver . ]  the spectrum band occupancy by the pu is modeled as an alternating busy / idle random process where the pu can access the spectrum at any time . we assume that the probabilities of the pu s arrival and departure remain the same across the time , and the holding time of either state is distributed as the exponential distribution .we denote the variables of the idle period and busy period of the pu as and , respectively . and let  ] represent the average idle and busy duration . according to the property of exponential distribution ,the probability density functions ( pdfs ) of and can be written as , respectively ,   for sus , on the other hand , only the idle period of the spectrum band is allowed to be utilized . to detect the spectrum holes and avoid collision with the pu, su needs to sample the spectrum at sampling frequency , and make decisions of whether the pu is present after every samples , this makes the secondary traffic time - slotted , with slot length .considering the common case that can be sufficiently high and the state of pu changes sufficiently slowly , we assume that and is a sufficiently large integer .if we divide the pu traffic into slots in accordance with su s sensing process , the probability that pu changes its state in a stochastic slot can be derived as follows .* the pu arrives in a stochastic slot : where and we assume it to be a large integer . * the pu leaves in a stochastic slot : where is assumed to be a large integer .  notethat when and are sufficiently large , we have and .   with the help of fd technique , su can detect the pu s presence when it is transmitting signal to su .however , as shown by the dotted arrow in the system model in fig . 1, the challenge of using fd technique is that the transmit signal at ant is received by ant , which causes self - interference at ant .note that for ant , the received signal is affected by the state of the transmit antenna ( ant ) : when ant is silent , the received signal at ant is free of self - interference , and the spectrum sensing is the same as the conventional half - duplex sensing .thus , we consider the circumstances when su is transmitting or silent separately .  when su is silent , the received signal at ant is the combination of potential pu s signal and noise .the cases when the pu is busy or idle are referred to as hypothesis and , respectively .the received signal at ant under each hypothesis can be written as where denotes the signal of the pu , is the channel from the pu to ant of su , and denotes the complex - valued gaussian noise .we assume that is psk modulated with variance , and is a rayleigh channel with zero mean and variance .when su is transmitting to su , rsi is introduced to the received signal at ant .the received signal can be written as where and are the hypothesises under which the su is transmitting and the pu is busy or idle , respectively . in denotes the rsi at ant , which can be modeled as the rayleigh distribution with zero mean and variance , where denotes the secondary transmit power at ant and represents the degree of self - interference suppression , which is commonly expressed in dbs , and indicates how well can the self - interference be suppressed .spectrum sensing refers to the hypothesis test in either or .given that su has the information of its own state ( silent or transmitting ) , it can automatically choose one pair of the hypothesises to test and decide whether the pu is present or not .in this section , we first present the proposed lat protocol , and then discuss the key parameter design in spectrum sensing to meet the constraint of collision ratio to the primary network .   the right part of fig .[ model ] shows the lat protocol , in which su performs sensing and transmission simultaneously by using the fd technique : ant senses the spectrum continuously while ant transmits data when a spectrum hole is detected .specifically , su keeps sensing the spectrum with ant with sampling frequency , which is shown in the line with down arrows . at the end of each slot with duration , su combines all samples in the slot and makes the decision of the pu s presence .the decisions are represented by the small circles , in which the higher ones denote that the pu is judged active , while the lower ones denote otherwise .the activity of su is instructed by the sensing decisions , i.e. , su can access the spectrum in the following slot when the pu is judged absent , and it needs to backoff otherwise .  since the pu can change its state freely , there exist the following four states of spectrum utilization :  * state : the spectrum is occupied only by the pu , and su is silent . *state : the pu is absent , and su utilizes the spectrum .* state : the pu and su both transmit , and a collision happens . *state : neither the pu nor su is active , and there remains a spectrum hole .  among these four states ,state and state are the normal cases , and state and state are referred to as collision and spectrum waste , respectively .there are two reasons leading to state and state : ( a ) the pu changes its state during a slot , and ( b ) sensing error , i.e. , false alarm and miss detection .we adopt energy detection as the sensing scheme , in which the average received power in a slot is used as the test statistics : where denotes the sample in a slot , and the expression for is given in and .  with a chosen threshold , the spectrum is judged occupied when , otherwise the spectrum is idle .generally , the probabilities of false alarm and miss detection can be defined as , where and are the hypothesises when the pu is idle and busy , respectively .considering the difference of the received signal caused by rsi , we can achieve better sensing performance by changing the threshold according to su s activity .let and be the thresholds when su is silent and busy , respectively , and we can have two sets of probabilities of false alarm and miss detection accordingly , denoted as and , respectively .the most important constraint of the secondary networks is that their interference to the primary network must be under a certain level . in this article, we consider this constraint as the collision ratio between sus and the pu , defined as ] .the specific statistical properties and the description under each hypothesis are given in table [ fdgaussian ] , in which denotes the signal - to - noise ratio ( snr ) in sensing , and is the interference - to - noise ratio ( inr ) .detailed derivation of the distribution properties are provided in appendix a. + .properties of pdfs of lat protocol [ cols="^,^,^,^,^,^",options="header " , ]   based on table [ fdgaussian ] , the sensing error probabilities can be derived .* when su is silent and the test threshold is , the probability of miss detection ( ) and the probability of false alarm ( ) can be written as and respectively , where is the complementary distribution function of the standard gaussian distribution . * similarly , when su is transmitting with the threshold , the miss detection probability ( ) and the false alarm probability ( ) are , respectively , and   * state transition and overall collision probability : * different from the conventional lbt protocol in hd crns where each slot is independent , in the lat protocol , the selection of sensing threshold depends on sus activity , which is instructed by the sensing result in the previous slot .thus , the state of the system in each slot is no longer independent , and the collision ratio is not only related to sensing error probabilities in each slot , but also the state in the previous slots .thus , joint consideration of the transition among all kinds of slots is necessary . since the sensing error probabilities in the slots where the pu changes its presencecan be approximated by that in the other slots , in this part , we model the state transition of the system as a discrete - time markov chain ( dtmc ) , in which the system can be viewed as totally time - slotted with as the slot length . fig .[ transition ] shows the state transition diagram , where we denote state as  ( ) for simplicity .the probability that the system stays in the collision state is where , , , and .the probability for the system staying in each state can be calculated considering the steady - state distribution of the markov chain : where ^{\bf{t}}$ ] is the vector of steady probabilities , and is the state transition matrix abstracted from fig .[ transition ] , which is given at the top of next page .\nonumber\ ] ]  combining the constraint that , we have .\ ] ]  to have a check on the result , we consider the probability that the pu is busy and idle as and , which are consistent with the results when we consider the pu s traffic only .* collision ratio : * the collision of the pu and su occurs in the following two kinds of circumstances : ( a ) when the pu keeps occupying the spectrum and su fails to detect the presence of pu s signal in the previous slot , which is depicted in fig .[ transition ] as with the probability of .the collision length is .( b ) the certain slots in which pu arrives .su is very likely to be transmitting in these slots since the pu is likely to be absent in the previous ones .the occurrence probability of this circumstance is equal to the pu s arrival rate .  the average collision length under circumstance ( b ) , where the pu changes state , can be approximated by , when is large enough .  and the average collision length in this case can be calculated as where the approximation is valid when is sufficiently large .it is unavoidable in the lat protocol that when the pu arrives , a short head of the signal , with the length of a su s slot approximately , collides with the su s signal .combine the two circumstances , and the overall collision rate can be given by   * design of sensing thresholds : * for the parameter design , we have a maximum allowable as the system constraint , and all the parameters of the sensing process should be adjusted according to .note that and are only related to the pu s traffic , and and are closely related via thresholds and , respectively .thus , we actually have two independent variables of the secondary network to design to meet the constraint of .we choose and as the independent variables . with as the only constraint , there are infinite choices of ( ) pair . for simplicity ,we set , i.e. , to reduce the degree of freedom , and the constraint can be simplified as where , , , and are relevant only to the pu traffic , and can be derived from via test thresholds and .in the rest of this part , we calculate from the constraint of , from which the sensing thresholds and can be obtained .  combining and , and, we can obtain and as functions of as , respectively ,  from and , we can find a rise of the false alarm probability when the rsi exists .this result indicates that when interference increases , the sensing performance gets worse .  with and, can be expressed as .with given parameters of the pu s traffic and the slot length , can be solved from . since the analytical expression of is complicated , we only give some typical numerical solution in fig .[ numpm ] , where the sensing snr , inr , number of samples , and is set to be 6 to meet the real case that the typical spectrum occupancy is less than  .  ; = -10db , = 5db , , and ,width=345 ]  it is shown in fig .[ numpm ] that when goes down , becomes a fine approximation of . with the large- assumption , we regard as sufficiently small , and is determined by this indicates that with the same constraint and parameters of the pu s traffic , the required gets squeezed when su s slot length increases .  with ,the thresholds and can be obtained from and , respectively :  a lift of sensing threshold due to the rsi ( ) can be found from and , which is in accordance with the previous analysis .in this section , we first evaluate the sensing performance of the lat by the probabilities of spectrum waste ratio under the constraint of collision ratio . then , with the closed - form analytical secondary throughput , a tradeoff between the secondary transmit power and throughput is elaborated theoretically .* spectrum waste ratio : * similar to the analysis of the collision ratio , we combine the following two kinds of time slots to derive the spectrum waste ratio : ( a ) the slots when the spectrum remains idle ; and ( b ) the slots of the pu s departure. there exists waste of spectrum holes in ( a ) when the system is in the state in fig .[ transition ] , and the probability is given by in .every time when the su fails to find the hole , the waste length is . in ( b ) , the average waste length can be derived from the pu s traffic with the similar method in , and it also yields of the average waste length .the probability of the pu s departure is , which is same as its arrival rate .the ratio of wasted spectrum hole is then given by   * secondary throughput : * su s throughput can be measured with the waste ratio and the transmit rate .the achievable rate under perfect sensing is given as where represents the snr in transmission , with denotes the channel gain of the transmit channel from su to su , and su s throughput can be measured as    in the expression of su s throughput in , there are two factors : and .on one hand , is positively proportional to su s transmit power . on the other hand, however , the following proposition holds .the spectrum waste ratio increases with the secondary transmit power .firstly , the inr increases with the transmit power and in turn lifts , which can be seen from .then , we can rewrite as when increases , the third term decreases and increases monotonically .then the increase of su s transmit power results in greater waste of the vacant spectrum .thus , there may exist a power - throughput tradeoff in this protocol : when the secondary transmit power is low , the rsi is negligible , the spectrum is used more fully with small , yet the ceiling throughput is limited by ; when the transmit power increases , the sensing performance get deteriorated , while at the same time , su can transmit more data in a single slot .  * * local optimal transmit power : * * the analysis above indicates the existence of a mediate secondary transmit power to achieve both high spectrum utilization efficiency in time domain and high secondary throughput . to obtain this mediate value of transmit power , we differentiate the expression of the throughput to find the local optimal points of the secondary transmit power , which satisfies with detailed derivation presented in appendix b , we have the local optimal power satisfies where the notations are as follow : with as the only unknown variable , it can be calculated numerically .  to obtain better comprehension about the properties of the local optimal transmit power, we consider the case when is sufficiently small , and can be simplified as   now we provide the existence conditions of the local optimal transmit power .the left side of is a convex curve of with a single maximum .when goes to zero or infinity , the value of the left side goes to zero . the value of the right side changes from to .we can roughly say that when the maximum of the left side is larger than either or , there would be two solutions to . when the maximum of the left side is smaller than the minimum of the right side , on the other hand , no solution exists .* characteristics of the power - throughput curve : * given the discussions above , there exist two cases of the power - throughput curve regarding the existence of the local optimal power .we analyze these two cases separately in the following .  1 .when equation has no solutions , the curves of transmit power on the left and right sides never meet .since the right side of is always far above zero and the left can go to zero when the transmit power is extremely high or low , we can safely say that the left side is always smaller than the right , i.e. , substituting the inequation to , we have , which indicates that the secondary throughput would increase with the transmit power monotonously .2 .  when the solutions of exist , we discuss the sign of piecewise .when the power is low or high enough , the left side is small , while the right remains considerable .the solid red curve ( maximum of the left side of ) is below the dash - dotted blue one ( value of the right side of ) , and .when the power is between the two solutions , we have .thus , at the smaller solution , , and this is the local optimal transmit power to achieve local maximum throughput .similarly , the larger solution denotes the local minimum of the throughput .  as an example, we plot the curves of the maximum of the left side and the corresponding value of the right side in fig .[ checksolution ] .it is shown that when is smaller than 0.86 , the maximum of the left is larger than the corresponding value of the right , and will have solutions and power - throughput is likely to exist .when is greater than 0.85 , there may be no tradeoff between the transmit power and secondary throughput , which is verified by the thick solid line in fig .[ power_thput ] .in this section , simulation results are presented to evaluate the performance of the proposed lat protocol .monte carlo simulations are performed by varying channel conditions and the pu s state .we set the default values of the simulation parameters as follow : the sample number in each slot , the corresponding probability that the pu arrives in a stochastic slot , and the probability that the pu leaves in any slot as .the constraint of collision ratio is set as 0.1 , and snr in sensing is assumed to be -5db .   as is shown in fig .[ power_thput ] , we consider the throughput performance of the lat protocol in terms of secondary transmit power .the solid and dotted lines represent the analytical performance of the lat protocol , and the asterisks ( * ) denote the analytical local optimal transmit power .the small circles are the simulated results , which match the analytical performance well .the thin solid line depicts the ideal case with perfect rsi cancelation . without rsi ,the sensing performance is no longer affected by transmit power , and the throughput always goes up with the power .this line is also the upperbound of the lat performance .the thick dash - dotted , dotted and dash lines in the middle are the typical cases , in which we can clearly observe the power - throughput tradeoff and identify the local optimal power , which is calculated from . with the decrease of rsi ( from -10db to -20db to -30db ) , the local optimal transmit power increases , and the corresponding throughput goes to a higher level .this makes sense since the smaller the rsi is , the better it approaches the ideal case , and the deterioration cause by self - interference becomes dominant under a higher power .according to fig .[ checksolution ] , when is sufficiently large  ( 0.85 in the figure ) , there exists no power - throughput tradeoff .we verify this result by the thick solid line denoting the cases when  .no local optimal point can be found in this curve , and the numerical results show that the differentiation is always positive .  one noticeable feature of fig . [ power_thput ]is that when self - interference exists , all curves approach the thin dotted line when the power goes up .this line indicates the case that the spectrum waste is 0.5 .when the transmit power is too large , severe self - interference largely degrades the performance of spectrum sensing , and the false alarm probability becomes unbearably high .it is likely that whenever su begins transmission , the spectrum sensing result falsely indicates that the pu has arrived due to false alarm , and su stops transmission in the next slot .once su becomes silent , it can clearly detect the pu s absence , and begins transmission in the next slot again .and the state of su changes every slot even when the pu does not arrive at all . in this case , the utility efficiency of the spectrum hole is approximately 0.5 , which is clearly shown in fig .[ power_thput ] .also , it can be seen that the larger is , the earlier the sensing gets unbearable and the throughput approaches the orange line .    and 1/100 , departure , the sample number of a slot is 300 , normalized secondary transmit power , sensing snr , and the rsi factor varies between 0.1 and 0.01.,width=345 ]  in this subsection , we use the receiver operating characteristic curves ( rocs ) to present the sensing performance . in fig .[ roccurve ] , with the sensing snr fixed on , we have the relation between the collision ratio and spectrum waste ratio .the thick lines denote the cases when the pu changes its state very slowly , while the fine lines represent the cases when the pu changes comparatively quickly . in fig .[ roccurve ] , smaller area under a curve denotes better sensing performance .it can be seen that the thick lines are lower than the corresponding fine lines , which indicates that when the pu changes its state slowly , the spectrum holes can be utilized with higher efficiency .this is because the spectrum waste due to the state change , i.e. , re - access and departure of the pu happens less frequently .also , comparing the solid and dotted lines with the same , it can be found that smaller rsi leads to better sensing performance , and the impact of the rsi can be significant .it is noteworthy that the ratio of spectrum waste of the lat can be quite close to zero if the self - interference can be effectively suppressed , and the constraint of collision ratio is not too strict .however , recall the conventional listen - before - talk , the spectrum waste ratio can theoretically never be suppressed lower than the sensing time ratio in a slot .   ,in which the collision ratio is 0.1 , sensing snr , the normalized secondary transmit power varies between 10db to 20db , and the numbers of samples in a slot varies between 300 and 500 , with the probability of the pu s arrival per slot varies between 1/500 and 1/300 , departure probability per slot varies between 6/500 and 6/300.,width=345 ]  in fig .[ x_pw ] , we consider the impact of the rsi factor on the sensing performance .we fix the constraint of as 0.1 , and evaluate the spectrum waste ratio under various .it can be seen from fig .[ x_pw ] that with the increase of , the spectrum waste ratio increases from zero to approximately 0.5 .this is reasonable in the sense that with sufficiently small rsi factor , the rsi can be neglected compared with pu s signal and noise , and sus can fully utilize the spectrum holes .when the rsi factor is moderate or close to 1 , which indicates that the rsi can not be suppressed well , the secondary signal may overwhelm the pu s signal , leading to unreliability of sensing , and the sus are likely to stop communication due to false alarm .note that the asymptotic value of the spectrum waste ratio when the rsi is large is 0.5 , which is in accordance with the results in fig .[ power_thput ] .  besides, it can be seen that when the normalized power of rsi ( ) ranges from approximately [ 0.1 , 10 ] , the spectrum waste ratio changes fast , and when the normalized power of rsi is below 0.1 , the waste ratio remains at a low level .this feature can be utilized to design the protocol parameters to achieve full utilization of the spectrum holes .also , when the pu s state change rate remains unchanged while the slot length enlarges , it can be seen that the sensing performance becomes better , especially at the points when the normalized power of rsi ranges from approximately [ 0.1 , 10 ] , where more samples in a slot would help improve sensing performance significantly .in this paper , we proposed a lat protocol that allows sus to simultaneously sense and access the spectrum holes .taken the impact of the residual self - interference on sensing performance into consideration , we designed an adaptively - changed sensing threshold for energy detection .spectrum utilization efficiency and secondary throughput under the lat protocol has been provided in closed - form , based on which a unique tradeoff between the secondary transmit power and the secondary throughput has been reported , i.e. , the increase of transmit power does not always yield the improvement of su s throughput , and a mediate value is required to achieve the local optimal performance .simulation results have verified the existence of the power - throughput tradeoff , and shown that the sus can efficiently utilize the spectrum holes under the lat protocol .the proposed lat protocol has the potential to allow the fd sus to fully utilize the spectrum holes , given that the sus no longer need to periodically suspend their transmission for sensing , and can react promptly to the spectrum opportunity . besides the basic model considered in this paper , the lat protocol can be readily extended to many other cr scenarios , like the multi - user and multi - channel cases . with simultaneous sensing and transmission ,the collision between multiple sus is likely to be shorten , and the performance of the whole secondary network is likely to enjoy a significant improvement .we first provide the general properties of the test statistics . given that each in is i.i.d . , the mean and the variance of can be calculated as = \mathbb{e}\left[\left|y\right|^2\right];~{\rm{var}}\left[m\right ] = \frac{1}{n_s}{\rm{var}}\left[\left|y\right|^2\right].\nonumber\ ] ] further , if the received signal is complex - valued gaussian with mean zero and variance , we have = \sigma_y^2,\nonumber\ ] ] and = \frac{1}{n_s}\left(\mathbb{e}\left[\left|y\right|^4\right]-\sigma_y^4\right)=\frac{\sigma_y^4}{n_s}.\ ] ]  then we consider the concrete form of the received signal under each hypothesis . in the lat protocol ,given the pu signal , rsi , and i.i.d .noise , the received signal is complex - valued gaussian with zero mean .the variance of under the four hypothesises are as follow : by substituting them into , we can obtain the results in table [ fdgaussian ] .the optimal power satisfies    }^2 } } } \cdot \left ( { \left ( { \frac{1}{\mu } - 1 } \right ) \cdot \left ( { 1 - p_f^0 } \right ) + { p_m } } \right)\\  & - \frac{{\sigma _ t^2}}{{\ln 2 \cdot \left ( { { \gamma _ t } + 1 } \right ) } } \cdot \left ( { \frac{\mu } { 2 } + \frac{{\mathcal{q}\left ( \rho  \right ) \cdot \left ( { \frac{1}{\mu } - 1 } \right ) - { p_m } + 1}}{{\left ( { \frac{1}{\mu } - 1 } \right ) \cdot \left ( { \mathcal{q}\left ( \rho  \right ) - p_f^0 + 1 } \right ) + 1 } } - 1 } \right ) , \end{split}\ ] ]   }^2 } } } \cdot \left ( { \left ( { \frac{1}{\mu } - 1 } \right ) \cdot \left ( { 1 - p_f^0 } \right ) + { p_m } } \right)\\ & + \frac{\sigma _ t^2}{\gamma_t + 1 } \cdot \left ( { \frac{\mu } { 2 } + \frac{{\left ( { \frac{1}{\mu } - 1 } \right ) \cdot \mathcal{q}\left ( \rho  \right ) - { p_m } + 1}}{{\left ( { \frac{1}{\mu } - 1 } \right ) \left(\mathcal{q}\left(\rho\right)-p_f^0 + 1\right )  + 1 } } - 1 } \right ) = 0 , \end{split}\ ] ]          i. f. akyildiz , w. y. lee , m. c. vuran , and s.mohanty , `` next generation / dynamic spectrum access / cognitive radio wireless networks : a survey , '' _ computer networks _ , vol . 50 , no . 13 , pp . 2127 - 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37 , hongkong , china , mar . 2007 .y. cai , y. mo , k. ota , c. luo , m. dong , l. t. yang , `` optimal data fusion of collaborative spectrum sensing under attack in cognitive radio networks , '' _ ieee network _ , vol .28 , no . 1 , pp .17 - 23 , jan .2014 .m. jain , j. i. choi , t. kim , d. bharadia , s. seth , k. srinivasan , p. levis , s. katti , and p. sinha .`` practical , real - time , full duplex wireless , '' in _ proc .acm mobicom 2011 _ , new york , ny , sep .2011 .j. choi , s. hong , m. jain , s. katti , p. levis , and j. mehlman , `` beyond full duplex wireless , '' in _ ieee conference record of the forty sixth asilomar conference on signals , systems and computers _ , pp . 40 - 44 , pacific grove , ca , nov . 2012.  e. ahmed , a. eltawil , and a. sabharwal , `` simultaneous transmit and sense for cognitive radios using full - duplex : a first study , '' in _ ieee antennas and propagation society international symposium ( apsursi ) _ , pp . 1 - 2 , chicago , il , jul .t. riihonen and r. wichman , `` energy detection in full - duplex cognitive radios under residual self - interference , '' in _ ieee international conference on cognitive radio oriented wireless networks and communications ( crowncom ) _ , pp .57 - 60 , oulu , finland , jun . 2014 .w. afifi and m. krunz , `` exploiting self - interference suppression for improved spectrum awareness / efficiency in cognitive radio systems , '' in _ proc . of ieee infocom13_ , pp . 1258 - 1266 , turin , italy , apr . 2013.  g. zheng , i. krikidis , and b. ottersten , `` full - duplex cooperative cognitive radio with transmit imperfections , '' _ ieee transactions on wireless communications _ , vol . 12 , no . 5 , pp . 2498 - 2511 , may 2013 .

Abstract: in traditional cognitive radio networks , secondary users ( sus ) typically access the spectrum of primary users ( pus ) by a two - stage `` listen - before - talk '' ( lbt ) protocol , i.e. , sus sense the spectrum holes in the first stage before transmitting in the second . however , there exist two major problems : 1 ) transmission time reduction due to sensing , and 2 ) sensing accuracy impairment due to data transmission . in this paper , we propose a `` listen - and - talk '' ( lat ) protocol with the help of full - duplex ( fd ) technique that allows sus to simultaneously sense and access the vacant spectrum . spectrum utilization performance is carefully analyzed , with the closed - form spectrum waste ratio and collision ratio with the pu provided . also , regarding the secondary throughput , we report the existence of a tradeoff between the secondary transmit power and throughput . based on the power - throughput tradeoff , we derive the analytical local optimal transmit power for sus to achieve both high throughput and satisfying sensing accuracy . numerical results are given to verify the proposed protocol and the theoretical results .   cognitive radio , full - duplex , listen - and - talk , residual self - interference .


the ligo interferometers are part of a world - wide network of gravitational wave detectors constructed to detect waves from astrophysical sources such as spinning neutron stars , coalescing binary black holes and neutron stars , and supernovae .each of the two arms of the ligo interferometers forms a resonant fabry - perot cavity .this increases the effective arm - length and thus the sensitivity of the instruments .gravitational waves ( along with seismic and other noise ) change the relative length of the optical cavities and produce an external strain , h(t)= [ ht0 ] that is incident on the interferometer . here , and are the effective lengths of the and -arms , respectively and the effective length of the cavities in the absence of an external strain .  the output of the interferometers , however , is not .rather , the output is derived from light that escapes the anti - symmetric ( `` dark '' ) port and provides the gravitational wave read - out .it is often called the error signal or `` gravitational wave channel '' . herewe will refer to it as .the re - construction of the gravitational wave strain from the error signal is an essential part of the data analysis .since the ligo instruments are linear the gravitational wave strain incident on the interferometer is a a linear functional of , namely , h(t)=r(t - t)q(t)dt , where is a suitable convolution kernel .convolution in the time domain is simple multiplication in the frequency domain so that h(f)=r(f)q(f ) , [ eq2 ] where denotes the fourier transform of .the ratio of the gravitational wave strain to the error signal is called the response function .finding the response function , or alternatively the kernel , is the procedure referred to as the calibration .so far ligo has used a calibration implemented entirely in the frequency domain , i.e. using eq .( [ eq2 ] ) . on the other hand ,the british - german interferometric detector geo600 has always produced a real - time calibrated using time - domain filters .some progress toward the production of for the ligo interferometers has already taken place . in this workwe adapt the geo time - domain calibration method to the case of the ligo interferometers .in section [ simple ] we provide a simple description of the length sensing and control system of the ligo interferometers , review the frequency domain calibration procedure and formally show how the strain can be reconstructed from the error signal of the interferometer in the time domain . in section [ match ] we describe a method to track changes in the response using sinusoidal excitations injected into the instrument . in section [ pipe ] we explain how the digital filters used to reproduce and invert the responses of various parts of the length control system are created and show their performance in the case of the hanford 4 km interferometer during the second science run ( s2 ) . we also describe our signal processing pipeline .we conclude in section [ concl ] .  , which converts the residual strain into the error signal , the feedback filter , which converts the error signal into the digital control signal , and the actuation which converts the digital control signal into the control strain that is subtracted from the external strain .the digital signals and are recorded by the data acquisition system at . ,at low frequencies the external strain is dominated by seismic noise and a control strain , , is subtracted from it ( see figure [ simpleservo ] ) by physically moving the mirrors in the fabry - perot cavity to compensate for the seismic noise .this ensures that the residual strain that enters the optical cavity , , remains small at low frequencies and keeps the optical cavities in resonance .the optical cavity of the feedback loop is represented by the so - called sensing function , .it converts the residual strain into the digital error signal , also called the gravitational wave channel , which is sampled at and recorded by the data acquisition system .the error signal is measured in arbitrary units called counts . here, is a reference sensing function that is measured at some time and is an overall ( real ) gain that depends on the light power stored in the fabry - perot cavity which changes with the alignment of the mirrors .it is important to keep track of this gain and the method we use is explained in section [ match ] .the frequency dependence of the sensing function is determined primarily by the fabry - perot cavity in each arm and corresponds to a real pole at around .  a digital feedback filter , ,is applied to the error signal which produces a digital control signal .the digital control signal , like the residual signal , is measured in units of counts .it is recorded by the data acquisition system at . here , is the feedback filter at some reference time and is a real overall gain .the gain is also recorded by the data acquisition system and may vary in time .for the first and second science runs ( s1 and s2 ) , however , it was kept fixed . the filter consists of a double pole at dc and various low pass filters as well as additional filters that keep the optics aligned and the fabry - perot cavity in resonance .the control signal is converted to strain and used to adjust the length of the cavities via the actuation function .this is achieved by converting the digital signal into currents that run through coils placed near magnets glued on the back of the mirrors in the fabry - perot cavity .the frequency response of the actuation function is largely determined by the pendulum suspension of the mirrors and thus consists of a pair of complex poles at the pendulum frequency .the re - construction of the external strain from the output of the interferometer feedback loop is referred to as the calibration .calibration of ligo data has thus far been performed in the frequency domain by constructing a response function that acts on the error signal of the feedback loop , namely , h(f)=r(f)q(f ) .[ hf ] the response function can be derived from the feedback loop equations from these equations it is easy to show that in eq .( [ hf ] ) is given by r(f)= , [ respf ] where is the reference open loop gain .the optical gain typically varies on a time - scale of seconds .the digital gain may also vary but was kept fixed during s1 and s2 . hence eqs .( [ fback1]-[fback4 ] ) apply for frequencies above a few tens of hz .we would like to perform this procedure entirely in the time domain . to do this we reconstruct the strain time - series from the residual and control strains and as follows .the residual strain is given by x_r(t)=h(t)-x_c(t ) , [ xr1 ] so that [ ht1 ] h(t)=x_c(t)+x_r(t ) . the low frequency part of the external strain is dominated by the control strain and the high frequency part is dominated by the residual strain .the optical gain of the sensing function typically varies on much longer time - scales than the decay time of the impulse response of the inverse of the sensing function .thus , the residual strain can be formally re - constructed according to x_r(t)t_c_0 ^ -1[q(t ) ] , [ xr2 ] where is a linear time - invariant filter operator for the inverse of the sensing function .similarly the control strain can be constructed from by filtering it through the actuation function x_c(t)=t_a[d(t ) ] .[ xc1 ] therefore we can write the time series for the external strain in terms of the error and digital control signals , and respectively , as [ ht2 ] h(t)t_c_0 ^ -1[q(t)]+t_a[d(t ) ] .although is an output of the interferometer servo loop recorded by the data acquisition system it can be computed from the error signal again assuming that varies in time much more slowly than the decay time of the impulse response function of the feedback filter as it is currently defined this condition is never satisfied : the double pole at dc results in an ill - behaved digital filter that does not have a well - defined impulse response time .later we show how to construct a modified digital filter which does satisfy this condition . ] . in particular ,d(t)(t ) t_d_0[q(t ) ] .[ xc2 ] so in terms of just the error signal the external strain can also be written as [ ht3 ] h(t)t_c_0 ^ -1 + t_a .note that , because is constant we could have factored it out of the second term on the right hand side of eq .( [ ht3 ] ) .thus , in order to produce the time - series for the external strain we need 1 ) to track changes in the optical gain of the cavity , and 2 ) construct time - domain digital filters for the inverse of the sensing function , the actuation function and the feedback filter .these problems are dealt with in the next two sections .in order to track changes in the optical gain of the cavity , calibration lines are injected into the instrument by adding sinusoidal excitations to the control signal .this is a standard technique used in gravitational wave detectors . in the ligointerferometers three calibration lines are injected ; one at around , another one around near the unity gain frequency of the open loop transfer function , and a third one at a few tens of hz .the excitations may be added directly to the digital control signal .alternatively they may injected into one of the two arms , as shown in figure [ actuation ] . durings2 they were injected into the -arm of the interferometers and for the purposes of the following calculation we will assume this to be the case .generalisations to injections into the -arm or to the case where they are directly added to the digital control signal are trivial .figure [ actuation ] depicts a more detailed model of the actuation function that shows the injection point .the actuation function is given by [ actua ] a= k_y g_y a_y- k_x g_x a_x , where and are analog gains measured at dc that depend on the configuration of each of the arms and and are digital gains that are mainly used to compensate for differences in the analog gains .the functions and are the same for both arms except for digital notch filters that take out frequency components of the signal at the violin mode frequencies of the mirror suspensions .since the calibration lines are not injected at those frequencies , for our purposes the functions and are identical along both arms and we can write a_x = a_y a_*. [ actxyeq ]   in figure [ simpleservo ] . , width=9 ]  the excitation is added to the -arm signal after the digital gain is applied to the digital control signal . therefore rather than eq .( [ fback3 ] ) , the feedback equation for the control signal at the frequency of a calibration line reads [ xc_line1 ] x_c(f_c)=a(f_c)d(f_c)-g_xa_*(f_c)l(f_c ) . using eqs .( [ actua ] ) and ( [ actxyeq ] ) we can write this as [ xc_line2 ] x_c(f_c)=a(f_c)[d(f_c)-l(f_c ) ] . with . since the calibration line is injected with an amplitude that is large compared to the external strain , the residual strain is dominated by the control signal at the frequency of the calibration line .so to a good approximation we can write the feedback equation for the control signal , eq . ( [ fback1 ] ) , as [ xr_line1 ] x_r(f_c)-x_c(f_c ) . combining eqs .( [ xc_line2 ] ) and ( [ xr_line1 ] ) with the feedback loop equations , ( [ fback2 ] ) and ( [ fback4 ] ) which remain un - changed , we arrive at two independent expressions ; one for the product of and [ alphabeta ] ( t)(t ) where , and another for , [ alpha ] ( t ) where .  thus , the sensing function gain as well as its product with the feedback filter are given by expressions involving the complex functions , and as well as complex ratios of the error and digital control signals to the injected excitation .  the digital feedback filter gain is recorded by the data acquisition system and so eqs . ( [ alphabeta ] ) and ( [ alpha ] )should be regarded as two independent ways of calculating the optical gain .the ratios of the error and digital control signals to the injected excitation are computed as follows .we take a fixed amount of time for each of the three time - series , and , and apply a hann window .we then perform a complex heterodyne at the frequency of the calibration line and finally integrate over time .this procedure yields complex , and which are used to compute the ratios and .( solid curve ) and the modified digitised sensing function ( dotted curve ) . on the top right the absolute value of the relative error between the two magnitude responses and on the bottom right the difference in degrees of the two phase responses ., width=12 ]  since the quantities in eqs .( [ alphabeta ] ) and ( [ alpha ] ) are complex and the measurement contains noise from the external strain the optical gain has a small imaginary part .this imaginary part is small only to the degree to which eq .( [ xr_line1 ] ) is a good approximation .it should be noted however that eq .( [ xr_line1 ] ) can be made arbitrarily accurate by either making the amplitude of the injected line arbitrarily large or , since the external strain has no component that is correlated with the calibration lines , by integrating for an arbitrarily long amount of time .the optical gain is typically calculated on time - scales of a few tens of seconds which , given the typical injected amplitudes of the calibration lines , is sufficient to reduce measurement noise errors to an acceptable few percent .the three filters of the feedback loop can be described in terms of analog zero and pole models .this is the starting point of the digitisation procedure .( solid curve ) and the modified feedback filter ( dotted curve ) . on the top right the absolute value of the relative error between the two magnitude responses and on the bottom right the difference in degrees of the two phase responses ., width=12 ]  the sensing function , as we have already discussed , consists essentially of a cavity pole at around so the inverse of this filter is a zero at around .unfortunately , digitisation of a zero does not lead to a well behaved digital filter . in order to resolve this problemwe introduce a pole at high frequencies in the inverted sensing function prior to digitisation . to ensure the magnitude and phase response of the filter do not change in the frequency range of interest to .this range covers the frequency bands used in gravitational wave data analyses performed by the ligo scientific collaboration so far . ] the additional pole is placed at a sufficiently large value of the frequency .we then digitise the analog filter using a bi - linear transform at a sample rate consistent with the presence of the additional pole .the sensing function also contains a unity gain order anti - aliasing elliptic filter at .this filter introduces frequency dependent phase shifts in the data that need to be accounted for .the anti - aliasing filter contains a series of zeros on the imaginary axis , which in the inverted filter turn into poles . to ensure that we obtain a well - behaved inverse digital filter the zeros need to be moved away from the imaginary axis prior to digitisation .  for the hanford 4 km interferometer in the s2 run we generated a modified digital filter with a bi - linear transform at a sample rate times greater than the sample rate the of the time series ,i.e. . the zeros of the elliptic filter were moved away from the imaginary axis by approximately and a single real pole was added at .( solid curve ) and the modified actuation function ( dotted curve ) . on the top rightthe absolute value of the relative error between the two magnitude responses and on the bottom right the difference in degrees of the two phase responses .the large errors in the phase beyond a few hundred hz are un - important because the contribution to the external strain from the control signal is negligible at these frequencies . , width=12 ]  in figure [ sense ] we show the magnitude and phase responses and errors of the analog sensing function , as well as our modified digitised sensing function .the errors between and are less than about in magnitude and in phase .  the ligo digital feedback filters are constructed from bi - linear transformations at of a series of analog filters .the main one of these is a double pole at dc which , on its own outside a feedback loop , is inherently unstable . in order to stabilise it ,the analog filter can be modified by shifting the double pole away from zero by a small amount prior to digitisation .it should be noted that this procedure changes the sign of the feedback filter at dc and introduces errors in the response of the filter at low frequencies .the rest of the filters in can be used exactly as they are implemented in the instrument .figure [ servo ] shows the magnitude and phase responses , as well as the errors , of the original digital feedback filter and the modified digital feedback filter for the hanford 4 km interferometer during s2 .as expected the errors are larger at low frequencies. however , the errors between and are less than about in magnitude and in phase .  the frequency response of the actuation function is primarily determined by the pendulum suspension of the mirrors and thus consists of a pair of complex poles at the pendulum frequency .additionally , the actuation function contains an anti - imaging order elliptic filter at , a snubber used to suppress resonances in the electronics and a measured time - delay modeled using a order pade filter .the actuator also contains a series of digital notch filters that remove components of the digital control signal at the violin mode frequencies of the mirror suspensions .all the analog parts of the actuator have been digitised using a bi - linear transform at .we have used the existing digital notch filters as they are implemented in the instrument .figure [ act ] shows the result of this procedure .plotted are the magnitude and phase response of the original actuation function and the digitised actuation function used in the calibration procedure as well as the errors for the hanford 4 km interferometer during s2 .the errors in the magnitude and phase are less than about and respectively at frequencies up to a few hundred hz . at higher frequencies the errors in the phase become larger but this is un - important : the contribution to the external strain from the control signal is negligible above a few hundred hz .ligo data is available in so - called `` science segments '' . at these timesthe cavities are in resonance and the instrument is sufficiently stable to produce data of scientific quality .the length of science segments depends mostly on the level of seismic activity at the location of each of the sites and can last anywhere between a few tens of seconds to tens of thousands of seconds .  science runs typically consist of a few hundred science segments .this lends the calculation of the external strain for an entire science run amenable to a cost - effective beowulf cluster , which is an ensemble of loosely coupled processors with a simple network architecture . in our implementationeach processor computes the external strain for a science segment .the few hundred processes required can be scheduled to run on the cluster with a batch system such as condor .indeed , the pipeline described below was used to compute the external strain for the entire s2 run for all three ligo interferometers using condor on the 300-node medusa cluster at uwm .   and denote up - sampling and down - sampling respectively ., width=12 ]  the first step of our procedure is to compute a table of values for the optical gain and the digital feedback filter gain for an entire science segment . as we have described in section [ match ] , obtaining reliable values of the optical gain requires an integration time of a few tens of seconds , so each science segment will have anywhere from a few to a few thousand values of . for the hanford 4 km interferometer durings2 we chose to compute the optical gain every 60s .science segments can be up to a few tens of thousands of seconds long .therefore we can not keep all the data for a science segment in the memory of a single node . as a result , in our signal processing pipeline, each science segment must be divided into a number of shorter parts . in the current implementation of the pipelinethe data is calibrated in second sub - segments .figure [ pipeline ] shows the digital processing pipeline used in the computation of the external strain from the error signal , i.e. our implementation of eq .( [ ht3 ] ) .the top and bottom branches show the computations of the residual and control strains respectively .  to avoid problems involving the dynamic range of double precision floating point numbers , we first high - pass filter 16s of the error signal .for the hanford 4 km interferometer during s2 we implemented the high - pass filter using a order butterworth filter at . this is a conservative choice and the specific frequency and order of the filter that is necessary depends on the low frequency character of the data .generally , we expect these choices to depend on the specific science run and interferometer . for numerical stabilitythe high - pass filter is divided into second order sections .furthermore , to ensure the phase of the signal is not adversely affected by the filtering procedure , the high - pass butterworth filter is applied to each segment of the data once forward and once in reverse .it should be noted that this procedure is acausal but since we are not attempting to simulate a physical process this is unimportant .since the data is being calibrated in segments care must be taken at the boundaries between consecutive segments .the finite impulse response time of the butterworth filter produces a discontinuity at the beginning and also at the end of each segment ( because the high - pass filter is applied forward as well as in reverse ) . to ensure continuity across contiguous segmentswe must allow for extra time at the end of each segment .figure [ segments ] shows a schematic of the procedure we have implemented in the case of three consecutive 16s segments .we use one extra second of data at the end of each segment which allows sufficient time for the high - pass filter to settle . we then then apply the first half of the extra second to the start of the next segment of data .the shaded areas in figure [ segments ] contain data for which the filters have not settled and are replaced with data from the previous or next segment .it should be noted that this procedure does not remove the few hundred milliseconds of initial ringing of the butterworth filter at the start of each science segment .  to compute the residual signal we first multiply each sample of the high - pass filtered by a suitably interpolated value of the inverse of the optical gain .interpolations are necessary here because the filter gains are typically computed over a time - scale of tens of seconds , and are not available for every sample of the error signal .we use the table of values of the optical gain created for the entire science segment described above to compute an interpolated for each sample of using a cubic spline . in principlewe could also have used band - limited interpolation but in practice this makes little difference .the inverse sensing function filter has been digitised at , and therefore the signal must be up - sampled by a factor of . the up - sampling procedure is simple : zeros are added between samples and the resulting time - series is smoothed with a low pass filter at a frequency lower than the nyquist frequency of the original time - series . for the hanford 4 km interferometer durings2 we have chosen a order butterworth filter at which is applied to each data segment once forward and once in reverse in second order sections . to ensure continuity across contiguous segments we apply a procedure along the lines described above and in figure [ segments ] adding an extra second of data at the end of each 16s segmentwe then filter the up - sampled signal through the inverse of the modified sensing function .we keep the history of the digital filter across segments to ensure continuity . finally , to produce the residual signal sampled at , we down - sample by a factor of by picking out one out of every samples .  in our pipelinethe digital control signal is computed by multiplying each sample of the high - pass filtered time series with an interpolated value of the digital filter gain and then filtering the result by the modified digital feedback filter .up to differences between the original and modified digital feedback filter this produces the digital control signal . note that we could have skipped these steps and instead used the digital control signal that is read out from the instrument .the low frequency components of as read out from the instrument and that computed by our pipeline are different .this is mostly due to the high - pass filtering done on prior to filtering it through . in figure [ dplot ]we show a plot of the digital control signal as recorded by the data acquisition system as well as the one computed in our pipeline versus time for a segment of data taken by the hanford 4 km interferometer during s2 . to allow for easier comparison in figure [ dplot ] both the digital control signal as recorded by the data acquisition system andthe one produced by our pipeline have been high - pass filtered by applying second order butterworth filters at consecutively .this removes the low - frequency components and makes the agreement between the two signals manifest .   as recorded by the data acquisition system ( circles joined by a line ) and the digital control signal produced by our pipeline ( crosses ) versus time in seconds for a segment of data taken by the hanford 4 km interferometer during s2 .due to the high - pass filtering done on the error signal prior to filtering it through the modified feedback filter the low frequency components of these two signals are different . to allow for easier comparisonboth signals have been high - pass filtered using second order butterworth filters at . , width=12 ]  the digital control signal is then filtered through the modified actuation function which produces the control strain .the histories for both the digital feedback filter and the actuator are stored in memory to ensure continuity across 16s segments .the calibrated strain is then computed by summing the residual and control strains and .we have described a method to compute the strain from the output of the ligo interferometers .this method has been implemented off - line for all three interferometers for the s2 run and results have been presented here for the hanford 4 km interferometer .the codes that implement the pipeline as well as the digital filters are available under the lsc algorithm library ( lal ) and the procedure is fully automated under condor .the pipeline introduces errors associated with the measurement noise involved in the calculation of the optical gain as well as the digitisation and modifications of the filters .we expect these errors to be less than about in amplitude and in phase across the detection band .we have tested the pipeline by comparing fourier transforms of time - domain calibrated data with data calibrated in the frequency domain and found the differences to be well within our errors .we have also faithfully reproduced the digital control signal in the time domain from the error signal using our modified servo filter ( see figure [ dplot ] ) .  at this time a third science run ( s3 ) has taken place and a similar off - line procedure will be applied to generate strain data for that run .the current plan is to place an on - line system at the ligo sites to generate in the near future .we would like to thank rana adhikari , gabriela gonzalez and daniel sigg for valuable discussions and suggestions .x. s. would like to thank the hospitality and support of the albert einstein institute , at which part of this work was done .the work of b. a. , j. c. and x. s. was supported by national science foundation grants phy 0071028 , phy 0079683 and phy 0200852 .

Abstract: the conversion of the read - out from the anti - symmetric port of the ligo interferometers into gravitational strain has thus far been performed in the frequency domain . here we describe a conversion in the time domain which is based on the method developed by geo . we illustrate the method using the hanford 4 km interferometer during the second ligo science run ( s2 ) .


biological systems must process information about fluctuating environments , with a classical example being a cell computing an external ligand concentration by time - averaging . understanding the thermodynamics of cellular information processingis of central importance and has attracted much interest recently .particularly , the role of energy dissipation has been studied for _ _ e .coli and eukaryotic adaptation , a cell computing an external ligand concentration , biochemical sensing , and proofreading ( see also for older works ) .  in these studies ,energy dissipation is characterized by the familiar entropy production of stochastic thermodynamics .however , they do not consider an entropic rate characterizing information processing of a noisy external environment that is related to the thermodynamic entropy production through an inequality .in related work , we have shown that the rate of mutual information between an internal process , corresponding to chemical reactions inside the cell , and an external process is _ not _ bounded by the thermodynamic entropy production .  more broadly , the relation between information and thermodynamics is a very active topic , with theoretical studies focused on second law inequalities and fluctuation relations , as well as experimental works .of particular relevance to this paper , it has recently been realized that a bipartite markov process provides a simple realization of a maxwell s demon , where the entropic rate characterizing information is the entropy reduction rate of a subsystem due to its coupling to the other subsystem composing the bipartite system .this entropy rate has been studied for the first time by allahverdyan et al . for two brownian particles coupled to different heat baths , where it was interpreted as an information flow . for bipartite jump processes a closely related entropic rate , also named information flow , has been studied more recently in .moreover , in related work a relation between dissipation and information about an external signal has been obtained .  in this paper , we show that for a bipartite process , with an internal process learning about an unaffected external process , the rate at which the uncertainty about the external environment , represented by a conditional shannon entropy , decreases due to the dynamics of the internal process is bounded by the thermodynamic entropy production .we call this rate the learning rate .this concept allows for the definition of an informational efficiency for biological information processing .particularly , we consider three different models inspired by the _ _ e .coli chemotaxis signaling network , where , for simplicity , the external process is assumed to correspond to an external ligand concentration jumping between two values .we demonstrate how an informational efficiency can be defined and analyzed in a simple four - state model , for which the entropy production corresponds to atp consumption inside the cell .for this system , a comparison with the efficiency of molecular motors is possible .we then consider a system where the internal process corresponds to an equilibrium monod - wyman - changeux ( mwc ) model for a single _ _ e .coli receptor . in this case , even if there is no energy consumption inside the cell , the learning rate is bounded by the chemical work done by the external process .finally , we analyze a more complete model for _ _ e .coli receptors including adaptation .we show that the learning rate can even exceed the rate at which the cell consumes the free energy source if work done by the external process compensates for it .we also analyze a coarse - grained learning rate for the model with adaptation : comparing it with the mwc model not including adaptation , we find that the concentration range for which the learning rate is non - negligible increases with adaptation.  the paper is organized as follows . in sec .[ sec2 ] , we define the class of markov processes and observables we study in this paper . moreover ,we show that the learning rate is bounded by the thermodynamic entropy production . in sec .[ sec3 ] , the four state model is studied , while sec .[ sec4 ] contains the mwc like model . in sec .[ sec5 ] , the model with adaptation is analyzed .we conclude in sec .let us first introduce discrete bipartite markov processes , where a state is labeled by two variables .moreover , it is assumed that is an external process unaffected by the internal process .the transition rates , from to , are defined as where a transition changing both variables is not allowed .the fact that is an external process implies that the transition rates are independent of the internal variable . denoting the stationary probability distribution by , the thermodynamic entropy production is given by  where .the first ( second ) term of the right hand side of ( [ thermoent ] ) , related to external ( internal ) jumps , is denoted by ( ) .   in the stationary statethe conditional shannon entropy of given is = -\sum_{i,\alpha } p_i^\alpha \ln p(\alpha|i),\ ] ] where , with . this shannon entropy quantifies the uncertainty about the external process given the internal variable .therefore , the stationary rate at which the internal process reduces the uncertainty of the external process due to its jumps can be written as in other words , , which we will call the learning rate , is the rate at which through its dynamics learns about .note that the jumps do not change the stationary shannon entropy of , therefore , the learning rate is also equal to the rate of change in the stationary mutual information due to the jumps .the full time derivative of  ] is and the conservation law for the coarse - grained learning rate reads . from the log sum inequality , we obtain , showing that the coarse - grained variable can not learn more than the full variable . moreover , the coarse - grained entropy production is defined as  which provides a lower bound on . finally ,similar to the second law inequality for a subsystem we also have we point out that this coarse - grained learning rate and the above discussion about it are novel .we start with the simplest thermodynamically consistent model for which the inequality ( [ mainine ] ) can be studied , see fig .[ fig1a ] .the internal process corresponds to a protein inside the cell y , which can be activated and deactivated by phosphorylation and dephosphorylation reactions .the chemical potential difference driving the reactions is .these reactions are represented by {\kappa_+ } { \rm y}^*+{\rm adp } \xrightleftharpoons[\omega_-]{\omega_+ } { \rm y}+{\rm adp}+{\rm p}_i , \label{eqreaction}\ ] ] where is the phosphorylation rate and is the dephosphorylation rate , with and representing the rates of the respective reversed reactions .the reaction rates are related to the chemical potential difference through  ] , is bounded by the rate of atp consumption .this model shows tight coupling since and are proportional to the same probability current .it allows for an analogy with molecular motors .whereas molecular motors transform chemical energy obtained from atp hydrolysis into mechanical work , in our model atp is consumed so that the internal protein can learn about the external process at rate .there are two special limits related to the time - scales of the external process .for the current ( [ currsimp ] ) goes to zero and , implying maximal efficiency .therefore , this limit of zero learning rate and efficiency one corresponds to an adiabatic case .second , for the current tends to its maximal value for fixed  but , implying .this limit is very inefficient , with zero learning rate and maximal atp consumption for fixed .these two limits are represented in the efficiency diagram plotted in fig .[ fig1b ] . comparing again with a molecular motor, the second limit corresponds to the case where the mechanical force goes to zero , leading to a high velocity of the motor but no mechanical work extraction . the first case , , is related to the stall force case , where the mechanical force equals the chemical potential difference driving the motor and the velocity tends to zero with efficiency approaching one .  from these two limiting casesit is clear that , for fixed , there is an optimal time - scale for which the learning rate should be maximal .this can be seen in fig [ fig1c ] .the maximum learning rate increases with , while the efficiency at maximum power decreases with , where , as plotted in fig .[ fig1d ] .furthermore , the efficiency at maximum power is maximal near equilibrium where it tends to , a well known result from linear response theory for tightly coupled heat engines and molecular motors .  the entropy production corresponding to the rate of atp consumed inside the cell is specific to this model .as we show next the entropy production can also arise from work done by the external process ._ e. coli _ receptors sit at its membrane and external ligands molecules can bind to them .the kinase chea is connected to the receptor through a protein chew and its activity is influenced by the binding of external ligands . if chea is in the active form , it acts as an enzyme of the phosphorylation reaction of the protein chey .here we consider a mwc model for a single receptor accounting for the indirect regulation of the kinase activity by the binding events .even though this single receptor model contains indirect regulation , it does lack cooperativity , which is the other key concept in mwc models . to study cooperativity more binding sitesare needed which corresponds to a straightforward extension of the model . as we do not investigate cooperativity effects but rather the relation between learning rates and energy consumption , it is more convenient to keep a simpler model with a reduced number of states .more precisely , the internal process corresponds to the following four - state model , see fig .[ fig2a ] .the receptor can be either bound ( occupied by an external ligand ) or unbound .the free energy difference between the bound and unbound states is given by , where is the dissociation constant and the external ligand concentration .the quantities and represent the free energies of the receptor together with the external solution : in the term is related to a change in the free energy of the receptor and is the chemical potential of the particle taken from the solution in a binding event . moreover , the kinase attached to the receptor can be either inactive or active .a conformational change in the receptor is assumed to be an equilibrium process with the free energy difference between active and inactive given by .the interaction between the receptor and the enzyme attached to it is reflected in the dissociation constant depending on , being for and for .it is assumed that , with the activity increasing the dissociation constant . combining these parameters ,the free energy of an internal state can be written as the transition rates of this four - state model , from state to state , have to respect the detailed balance relation = f(a , b , c)-f(a',b',c)$ ] .the four - state model corresponding to the internal process is represented in fig .[ fig2a ] , where and set the time - scale of the conformational changes and binding events , respectively .the time - scale of the latter is assumed to be much smaller than the time - scale of the conformational changes , i.e. , . as this is an equilibrium model ,the cycle affinity of the four - state cycle is zero .the external process corresponds to the external ligand concentration jumping between the values and at rate . the resulting full eight - state model is shown in fig . [ fig2b ]denoting the stationary probability of the state by , and using schnakenberg s formula , where the terms in the entropy production are cycle affinities multiplying probability currents , it is possible to write the entropy production ( [ thermoent ] ) in the form this entropy production arises from cycles , as indicated in fig .[ fig2b ] , where a ligand molecule is taken from the solution at concentration and released in the solution at concentration , corresponding to a chemical potential difference . therefore , is the chemical work that is done by the external process .in contrast to the first toy model , where the cell consumes atp , the work done by the external process compensates for a nonzero learning rate , which follows from ( [ learnrate ] ) as we also consider the coarse - grained rate at which learns about , which is obtained from ( [ learnratecg ] ) as where .the rate characterizes how much the internal kinase chea learns about the external process .this is the central quantity assuming that other chemical reactions inside the cell that are influenced by the activity can only learn as much as .  numerical results for this model with the choice of transition rates shown in fig .[ fig2 ] are shown in fig .[ fig4 ] . comparing figs .[ fig4a ] and [ fig4b ] , we see that the coarse - grained learning rate is close to if is small . as increases , the difference between both quantities increases .more precisely , if the external environment is considerably faster then the internal variable ( ) , then can not track concentration changes ( ). however , the learning rate is still nonzero , with its main contribution coming from the variable .the efficiency plot in fig . [ fig4c ] demonstrates that if the external environment is very slow the model reaches an adiabatic limit with and . for is a region of lower efficiency and high learning rates . if is further increased beyond the values displayed in fig .[ fig4c ] , the efficiency decays , going to zero for . in this case , not even can track the fast external changes .choosing the external concentrations as and , in fig .[ fig5a ] we plot and as a function of for different values of the conformational free energy difference .besides the simple illustration that the learning rate is bounded by the chemical work , these graphs demonstrate that for every there is an optimal value of that maximizes the learning rate .this is also the case for the coarse - grained learning rate . comparing with , we see that the contribution of the variable to is more substantial for very low and very high .however , for the correlation between bound ( unbound ) and inactive ( active ) is high leading to .since binding events are faster than the conformational changes , it is possible to integrate out the variable  . in this case , the internal process reduces to the two - state model of fig [ fig3a ] . in the limit the coarse - grained transition rates become  where corresponds to the stationary conditional probability of the full internal process .the effective free energy difference resulting from a conformational change from to then reads the full system , including the concentration jumps , is the four - state model shown in fig .[ fig3b ] .the entropy production for this model is the coarse - grained entropy production ( [ thermoentcg ] ) , which reads , \label{coarw}\ ] ] where denotes the stationary probability of the coarse - grained model .this quantity provides a lower bound on the full chemical work ( [ chemicalw ] ) , which in the limit becomes therefore , with this time - scale separation the full chemical work can be calculated with the coarse - grained model by using this formula . in fig .[ fig5b ] we compare for , , and .the important result shown in this figure is that for the lower bound is close to the full chemical work . in this case , since , , and the affinity in ( [ coarw ] ) becomes .  in principle, the conformational change of the receptor could also involve atp consumption , which would lead to an internal process breaking detailed balance even without the external jumps . in this case, the entropy production of the full model would involve both energy consumed inside the cell and work done by the external process .in this section , we study a more complete model also including adaptation , where the free energy difference arising from conformational changes for the activity is taken from the coarse - grained model discussed in the previous section .   for adaptation , besides the kinase activity , the internal process must include the methylation level  . as in the previous model , the activity takes the values if chea is inactive and if it is active .the average value of is assumed to be around , independent of . whereas quickly responds to a change in at a time - scale , the methylation level guarantees adaption by returning the average activity to at a time - scale .more specifically , a step decrease in leads to a fast increase in .this change in generates a slow decrease in the methylation level which acts back on the activity slowly decreasing it to .the free energy difference for the conformational change of the activity is obtained from the coarse - grained mwc model expression ( [ coarsefree ] ) .explicitly , where the free energy dependence on the methylation level is this choice for the free energy difference leads to . for simplicity , we assume   two different chemical reactions control the methylation level . if , the receptor can be methylated with the reaction +{\rm sam } \rightleftharpoons [ m+1]_0+{\rm sah } ,  \label{eq : demethylation}\ ] ] where the subscript indicate the value of . heresam represents a s - adenosyl methionine molecule and sah represents a s - adenosyl - l - homocysteine molecule .the free energy difference of the reaction is . likewise ,if , the receptor can be demethylated with the reaction +{\rm h_2o } \rightleftharpoons  [ m]_1+{\rm ch_3oh } ,  \label{eq : methylation}\ ] ] for which the free energy difference is the chemical potential difference then drives the internal process out of equilibrium , corresponding to the affinity of the internal cycles shown in fig .[ fig6 ] , where our choice for the transition rates for the internal process is displayed .we note that if the internal process is in equilibrium . for system dissipates but there is no adaptation : increases instead of repressing it .adaptation happens only if overcomes the conformational free energy difference in a internal cycle .the variable defines the internal process , the external process is again the fluctuating concentration of external ligand , which jumps with rate between and .the full model thus has states , as shown in fig .[ fig6b ] . in general , its transition rates are denoted by and the stationary probability by .moreover , the internal probability current is and the marginal probability is    using schnakenberg s network theory , the entropy production ( [ thermoent ] ) can be conveniently written as where \vardelta\mu \label{sigmamu}\ ] ] corresponds to the consumption of sam inside the cell and \label{sigmac}\ ] ] is a lower bound on the chemical work done by the external process , as explained in the previous section .the term is related to the internal cycles in fig .[ fig6b ] . in principle, the full chemical work can only be obtained if we consider a process with states , for which the variable is not integrated out . however ,if there is time - scale separation ( faster than ) , we can adapt equation ( [ fullw ] ) to include methylation levels . for the present model ,the full work done by the external process then reads where , contrary to ( [ fullw ] ) , is now the marginal probability ( [ marg ] ) .furthermore , the learning rate ( [ learnrate ] ) , becomes   +  numerical results for this model are shown in fig .[ fig7 ] . from the plot of the thermodynamic entropy production and the learning rate as a function of in fig .[ fig7a ] , we see that if the external process is very slow the dissipation inside the cell due to sam consumption is much larger than the learning rate .therefore , if the bacterium swims in an environment with , it will dissipate much more than it learns about the environment .the sam consumption rate is nearly independent of , being related to probability currents in the internal cycles as expressed in eq .( [ sigmamu ] ) , while grows with .the learning rate reaches a maximum , similar to the behavior observed with the toy model from sec .[ sec3 ] and the mwc model from sec .[ sec4 ] .  fig .[ fig7b ] demonstrates that the learning rate can be larger than the rate of sam consumption inside the cell since the work done by the external process also contributes to the cost of the learning rate .the internal dissipation grows with , while the learning rate saturates for high .  in adaptation, the fast variable learns about the changes in external concentration whereas the function of the slow variable is to appropriately regulate the conformational free energy difference .with such a perspective , it is more meaningful to look at the coarse - grained learning rate , which from ( [ learnratecg ] ) follows as the relation between the learning rates and the external concentration is plotted in fig .[ fig7c ] . the external concentrationare fixed to and , with as free parameter .the learning rates and approach zero if increases ( decreases ) beyond ( ) . moreover , in fig .[ fig7c ] we see that the difference between and is small in the region .an interesting result is obtained in fig .[ fig7d ] where we compare with the coarse - grained learning rate obtained for the mwc model without adaptation plotted in fig .[ fig5a ] . for fixed ,i.e. , without adaptation , the maximal learning rate achieved at any given value of by fine - tuning represents the maximum the kinase dynamics can learn about the environment for given .as demonstrated in fig .[ fig7d ] , the learning rate for the model with adaptation is below . the advantage of adaptation , however , is that it increases the region at which is non - negligible ( see comparison between the blue solid and gray dotted curves in fig .[ fig7d ] ) , which is roughly .similarly , it has been found that adaptation maintains high sensitivity over a wide concentration range ( see for a different perspective ) .  a closely related study of the present model for _ _ e . coli adaptation has been recently performed by sartori et al .in contrast to our model , in their work the external concentration is suddenly changed from an initial to a final value with some preassigned probability .the mutual information between the internal system and the final concentration , which is related to measuring the final concentration , is analogous to our learning rate . in mutual information is shown to increase with time , reaching its maximal value at a time of the order of the methylation time - scale .this is in agreement with our result that the learning rate is much smaller than the internal dissipation for : after a time of order the correlation with the external concentration is saturated but the cell keeps dissipating .for markov processes that can be decomposed into an external process unaffected by an internal process , as represented by the transition rates in eq .( [ defrates2 ] ) , we have defined an entropic rate that is bounded by the thermodynamic entropy production and that characterizes how much the internal process learns about the external process .this learning rate allows for the definition of an informational efficiency that , as demonstrated with three different models related to _ _coli sensory network , can be used to study the thermodynamics of cellular information processing .we have analyzed a simple toy model of an internal protein tracking an external concentration , for which the learning rate is bounded by the rate of atp consumption inside the cell .this thermodynamically consistent model allows for a comparison with molecular motors , where the learning rate plays the role of extracted mechanical work .  for an internal process corresponding to an equilibrium mwc model for a single _ _ e .coli receptor with no internal dissipation , we have shown that a nonzero learning rate is possible if work done by the external process accounts for it . in this model ,the work comes from the chemical potential difference of binding a ligand at concentration and unbinding it at another concentration .if the external medium changes much faster than the activity , the full learning rate is nonzero because the binding and unbinding process , which is assumed to be faster than activity changes , can also learn about the external environment .however , the learning rate of the activity alone goes to zero , as the activity can not track an environment much faster than its own time - scale .our framework has also been applied to a model including adaptation .we have shown that a bacterium in an external environment that changes at a time - scale much larger than the time - scale of the activity , dissipates through sam consumption at a rate much higher than the one of learning about the external environment , corresponding to a quite inefficient situation .in general , the rate of sam consumption does not bound the learning rate , which can increase due to work done by the external process . using a coarse - grained learning rate ,we have analyzed how much the activity alone learns about the external process .we have shown that adaptation increases the concentration range for which this learning rate is non - negligible .we have assumed the external process to be an external ligand concentration jumping between two values . however , the framework from sec . [ sec2 ] is also valid for more elaborate external processes .likewise , more complete internal processes that include further elements of the _ _ e .coli sensory network could be studied in the future .we expect features like the thermodynamic entropy production having one contribution due to energy dissipation inside the cell and another one due work done by the external process , the learning rate being much smaller than the internal dissipation if changes in the external medium are slow , and the learning rate not necessarily being bounded by the dissipation inside the cell to be relevant also for more complex models. finally , it would be worthwhile to explore the relation between the learning rate studied here and quantities characterizing further aspects of a sensory system , like adaptation error and sensitivity .

Abstract: we show that a rate of conditional shannon entropy reduction , characterizing the learning of an internal process about an external process , is bounded by the thermodynamic entropy production . this approach allows for the definition of an informational efficiency that can be used to study cellular information processing . we analyze three models of increasing complexity inspired by the _ _ e . coli sensory network , where the external process is an external ligand concentration jumping between two values . we start with a simple model for which atp must be consumed so that a protein inside the cell can learn about the external concentration . with a second model for a single receptor we show that the rate at which the receptor learns about the external environment can be nonzero even without any dissipation inside the cell since chemical work done by the external process compensates for this learning rate . the third model is more complete , also containing adaptation . for this model we show _ _ inter alia that a bacterium in an environment that changes at a very slow time - scale is quite inefficient , dissipating much more than it learns . using the concept of a coarse - grained learning rate , we show for the model with adaptation that while the activity learns about the external signal the option of changing the methylation level increases the concentration range for which the learning rate is substantial .