relaying systems achieve higher spectral efficiency than one - way relaying systems . in two - way relay channel ( twrc ) , two source nodes exchange messages with each other via a relay , using various multiple access broadcast ( mabc ) protocols consisting of the multiple access ( mac ) phase and the broadcast ( bc ) phase .basically , in the mac phase , two source nodes simultaneously send their signals to the relay node , and then the relay node amplifies and forwards the signal received in the mac phase to the two source nodes in the bc phase . with a prior knowledge of its own transmitted signal ,each source node is able to cancel the propagated self - interference ( si ) back to it and decode the desired signal .because full - duplex ( fd ) systems allow concurrent transmission and reception in the same frequency band , they usually achieve higher spectral efficiency than half - duplex ( hd ) systems only if the loopback si due to the concurrent transmission and reception is properly suppressed . using a circulator which connects antenna to transceiver, a single antenna can be shared between the transmitter and the receiver in fd systems .the merits of fd systems and two - way relaying systems in terms spectral efficiency motivated two - way relaying with a fd relay which outperforms both two - way relaying with a hd relay and one - way relaying with a fd relay . in , the two - way fd amplify - and - forward ( af ) relay systems with two sources and one relay with a single antenna each were studied . power allocation to maximize the achievable ratewas explored in but residual si at each node was not considered assuming perfect si suppression . however , given loopback channel estimation errors and limitation of analog - to - digital converter ( adc ) in practical environments , the assumption of perfect si suppression is too ideal to evaluate practical gains of two - way fd relaying .contrary to , power allocation addressing residual si due to imperfect si cancelation was investigated under a sum power constraint in . to maximize the sum rate in the presence of residual si , power allocation under individual power constraints and relay mode selection , among one - way hd , two - way hd , one - way fd , and two - way fd ,were proposed in . as another performance metric in the presence of residual si , in , the outage probability of the two - way fd af relay system was derived . the application of physical layer network coding to the two - way fd af relay was investigated in terms of bit error rate when there exists residual si .a two - way fd af relay system with multiple relays was addressed in and an optimal relay selection scheme was proposed considering residual si .although residual si severely hurts the merit of fd , addressing the residual si only with single antenna at a fd af relay has limitations . on the other hand ,beamforming with multiple antennas enables to suppress the residual si , which motivated the study of a two - way fd af relay system with a relay having multiple antennas .however , to the authors best knowledge , there are only few studies on the two - way fd af relay system with multiple antennas . in , a two - way fd af relay system with multiple antennaswas studied , where two sources have a single antenna each and the relay has multiple antennas . to maximize the achievable sum rate ,beamforming at relay and power allocation at sources were jointly optimized with a zero forcing ( zf ) constraint such that the residual loopback si is perfectly canceled at the relay .however , a closed form solution was not available and thus an iterative algorithm was proposed in .contrary to the beamforming matrix which perfectly cancel out the residual si with a zf constraint , beamforming at a fd relay is required to balance between the residual si suppression and the desired signal transmission . in this context , in a two - way fd af relay system with multiple antennas with circulator at the relay as well as the two sources , we design the beamforming matrix balancing them in terms of minimum mean square error ( mmse ) under a power constraint at the relay . because multiple antennas at the two sources are considered unlike , the beamforming matrix also need to be designed to address the inter - stream interference in the multi - input multi - output ( mimo ) channels . moreover , to optimally balance the residual si suppression and the desired signal transmission , the beamforming matrix should be updated every time slot because the residual si in the previous time slot is amplified by the current beamforming matrix and added to the received signals from the two sources in the current time slot .  in this paper , we analytically identify this coupled effect and show that the optimal beamforming matrix at each time slot is determined as a function of not only the current channels but also all the channels in the past time slots from the first transmission time slot .using the lagrangian method , we derive the optimized relay beamforming matrix and receive beamforming matrices at sources in closed form . to account for beamforming design with limited memory , we also propose a beamforming method using the channels in a limited number of the latest time slots , not from the first time slot , and show that the performance degradation is marginal relative to the beamforming method based on all the channel matrices from the first time slot to the current time slot . finally , from our numerical results , we identify the signal - to - noise ratio ( snr ) and interference - to - noise ratio ( inr ) region where the achievable sum rate of the proposed scheme is greater than that of the hd system . based on this result, we propose a duplex mode selection scheme in terms of achievable sum rate according to snr and inr .the rest of this paper is organized as follows .section ii describes the system model . in section iii, we derive the optimal beamforming matrix in terms of mmse and explore beamforming matrix design with limited memory size . in section iv , we present numerical results .finally , conclusions are drawn in section v.  _ notations _ : matrices and vectors are denoted , respectively , by uppercase and lowercase boldface characters ( e.g. , * a * and * a * ) .the transpose , hermitian , and inverse of * a * are denoted , respectively , by , , and .an identity matrix and matrix consisting of all zero entries are denoted , respectively , by and .the operators  ] for and =p_r\textbf{i}_{n_r} ] for and =\sigma_{n , r}^2\textbf{i}_{n_r} ] and =\sigma_{e , r}^2\textbf{i}_{n_r^2} ] . with imperfect loopback si cancellation , the received signal at the relay in time slot is given by substituting into leads to   at source in time slot , subtracting the propagated si back to the source node , and the loopback si , the received signal at source is given by where means the index of the other source ( i.e. , if and if ) . source multiplies by a receive beamforming matrix .  accordingly , considering imperfect loopback si cancellation , the received signal at the relay in time slot ( ) is given by which shows that the current received signal is affected by the beamforming matrices in the past time slots . in timeslot , the forwarded signal from the relay to the source nodes is given by where is the relay beamforming matrix at time slot and .subtracting the propagated si back to the source node and the loopback si , the received signal at source in time slot ( ) is obtained as after subtracting si , source multiplies by a receive beamforming matrix .the problem for joint design of relay beamforming matrices and source receive beamforming matrices is non - convex and basically intractable .therefore , we propose an iterative algorithm which decouples the primal problem into two subproblems and solve them alternately ; one is for relay beamforming design and the other is for receive beamforming design at sources .  as identified in , the current received signal is a function of the beamforming matrices , the channel estimation error matrices , and transmitted signals at source in the past time slots , which implies that to optimally balance the residual si suppression and the desired signal transmission , the beamforming matrix should be updated every time slot . without loss of generality ,we design the relay beamforming matrix based on the latest time slots ; if , the beamforming matrix in time slot is based on all the time slots from the first time slot .otherwise if , the beamforming design corresponds to the scenario when the memory size at the relay is limited .unlike one - way relay systems , transmitted signals at each source during the past time slots become interference signals at each source in the current time slot in two - way relay systems . in addition , the relay transmits the received noise signals during the past time slots to the sources .therefore , the minimum mean square error ( mmse ) criterion is an effective metric enabling a compromise between interference suppression and noise reduction .moreover , the mmse criterion allows tractable analysis and optimization of the two - way fd af relay system . for similar reasons ,mean square error ( mse ) has been popularly used for beamforming design in two - way relay systems in the literature such as .  to balance the residual si suppression and the desired signal transmission , the relay beamforming matrix and receive beamforming matrices for in time slot determined based on the mse criterion under a transmit power constraint at the relay .the corresponding optimization problem is formulated as = n_rp_r \label{}\end{aligned}\ ] ] where the sum of mse is defined as + \mathbb{e}\big[||\textbf{x}_1^{(t-1)}- \alpha^{(t)^{-1}}\textbf{r}_2^{(t)^h}\hat{\textbf{y}}_2^{(t ) } ||^2 \big ] \label{}\end{aligned}\ ] ] and is a positive scaling factor .note that is multiplied by the positive scaling factor since the relay beamforming matrix is split into two components , as in , such that to facilitate the derivation of a closed form solution , where and thus and represent power amplification and steering directions , respectively .the lagrangian of the optimization problem is defined as -n_rp_r)\\  & = n_s(p_1+p_2 ) - \alpha^{(t)^{-1 } } \{ \text{tr}(\textbf{w}_{f,0}^{(t)^h}\textbf{f}^{(t ) } ) + \text{tr}(\textbf{w}_{f,0}^{(t)}\textbf{f}^{(t)^h } ) \ } \\ & \quad+ \alpha^{(t)^{-2 } } \ { \mathbb{e}\big[|| \textbf{r}_1^{(t)^h}\hat{\textbf{y}}_1^{(t ) } ||^{2}\big ] + \mathbb{e}\big[|| \textbf{r}_2^{(t)^h}\hat{\textbf{y}}_2^{(t ) } ||^{2}\big ] \ } + \lambda^{(t)}(\mathbb{e}\big[||\textbf{x}_r^{(t)}||^{2}\big]-n_rp_r ) , \label{l22 } \end{split}\ ] ] where is a lagrangian multiplier in time slot , ,   & = \mathbb{e } \big [ \text{tr } \big\ { \textbf{r}_1^{(t)^h}\textbf{h}_{r,1}^{(t)}\textbf{f}^{(t ) } ( \textbf{g}_0^{(t ) } + p_2\textbf{h}_{2,r}^{(t-1)}\textbf{h}_{2,r}^{(t-1)^h } + \sigma_{n , r}^2\textbf{i}_{n_r } ) \textbf{f}^{(t)^h}\textbf{h}_{r,1}^{(t)^h}\textbf{r}_1^{(t ) } \big\ } \big ] \\ & + p_1 \mathbb{e } \big [ \text{tr } ( \textbf{r}_1^{(t)^h}\boldsymbol{\delta}_{1,1}^{(t)}\boldsymbol{\delta}_{1,1}^{(t)^h}\textbf{r}_1^{(t ) } ) \big ] + \sigma_{n,1}^2\text{tr}(\textbf{r}_1^{(t)^h}\textbf{r}_1^{(t ) } ) , \\ \mathbb{e}\big[|| \textbf{r}_2^{(t)^h}\hat{\textbf{y}}_2^{(t ) } ||^{2}\big ] & = \mathbb{e}\big [ \text{tr } \big\ { \textbf{r}_2^{(t)^h}\textbf{h}_{r,2}^{(t)}\textbf{f}^{(t ) } ( \textbf{g}_0^{(t ) } + p_1\textbf{h}_{1,r}^{(t-1)}\textbf{h}_{1,r}^{(t-1)^h } + \sigma_{n , r}^2\textbf{i}_{n_r } ) \textbf{f}^{(t)^h}\textbf{h}_{r,2}^{(t)^h}\textbf{r}_2^{(t ) } \big\ } \big ] \\ & + p_2 \mathbb{e } \big [ \text{tr } ( \textbf{r}_2^{(t)^h}\boldsymbol{\delta}_{2,2}^{(t)}\boldsymbol{\delta}_{2,2}^{(t)^h}\textbf{r}_2^{(t ) } ) \big ] + \sigma_{n,2}^2\text{tr}(\textbf{r}_2^{(t)^h}\textbf{r}_2^{(t ) } ) , \\\mathbb{e}\big[|| \textbf{x}_r^{(t ) } ||^{2}\big ] & = \mathbb{e}\big [ \text{tr } \big\ { \textbf{f}^{(t ) } ( \textbf{g}_0^{(t ) } + p_1\textbf{h}_{1,r}^{(t-1)}\textbf{h}_{1,r}^{(t-1)^h } + p_2\textbf{h}_{2,r}^{(t-1)}\textbf{h}_{2,r}^{(t-1)^h } + \sigma_{n , r}^2\textbf{i}_{n_r } ) \textbf{f}^{(t)^h}\big\ } \big ] , \label{exp_y1_y2_xr } \end{split}\ ] ]  where , note that is the number of past time slots considered in the beamforming matrix design and the expectations in are with respect to for .let be an random matrix with =\sigma^2\textbf{i}_{n^2} ] assume holds when and then we prove that it also holds when .  when , the left hand side of is rewritten as \label{lemma_proof_1}\\ & \overset{(b)}{=}\mathbb{e}\bigg [ \text{vec}(\boldsymbol{\delta}_{k-1})^h \bigg\ { \bigg ( \textbf{v}_{k-1 } \prod_{j=1}^{k-2 } ( \boldsymbol{\delta}_{k-1-j}\textbf{v}_{k-1-j } ) \prod_{j=1}^{k-2 } ( \textbf{v}_j^h\boldsymbol{\delta}_j^h ) \textbf{v}_{k-1}^h\bigg)^t \otimes ( \textbf{v}_k^h\textbf{v}_k ) \bigg\ } \text{vec}(\boldsymbol{\delta}_{k-1 } ) \bigg]\label{lemma_proof_2}\\ & \overset{(c)}{= } \text{tr}\left ( \mathbb{e}\bigg [ \text{vec}(\boldsymbol{\delta}_{k-1})^h \bigg\ { \bigg ( \textbf{v}_{k-1 } \prod_{j=1}^{k-2 } ( \boldsymbol{\delta}_{k-1-j}\textbf{v}_{k-1-j } ) \prod_{j=1}^{k-2 } ( \textbf{v}_j^h\boldsymbol{\delta}_j^h ) \textbf{v}_{k-1}^h\bigg)^t \otimes ( \textbf{v}_k^h\textbf{v}_k ) \bigg\ } \text{vec}(\boldsymbol{\delta}_{k-1 } ) \bigg]\right)\\ & \overset{(d)}{=}\text{tr } \bigg\ { \mathbb{e}\bigg [ \bigg ( \textbf{v}_{k-1 } \prod_{j=1}^{k-2 } ( \boldsymbol{\delta}_{k-1-j}\textbf{v}_{k-1-j } ) \prod_{j=1}^{k-2 } ( \textbf{v}_j^h\boldsymbol{\delta}_j^h ) \textbf{v}_{k-1}^h\bigg)^t \otimes ( \textbf{v}_k^h\textbf{v}_k ) \bigg ] \mathbb{e}\bigg [ \text{vec}(\boldsymbol{\delta}_{k-1 } ) \text{vec}(\boldsymbol{\delta}_{k-1})^h \bigg ] \bigg\ } \label{lemma_proof_3}\\ & \overset{(e)}{=}\sigma^2\text{tr}(\textbf{v}_k\textbf{v}_k^h ) \mathbb{e}\bigg [ \text{tr } \bigg\ { \textbf{v}_{k-1 } \prod_{j=1}^{k-2 } ( \boldsymbol{\delta}_{k-1-j}\textbf{v}_{k-1-j } ) \prod_{j=1}^{k-2 } ( \textbf{v}_j^h\boldsymbol{\delta}_j^h ) \textbf{v}_{k-1}^h \bigg\ } \bigg ] \label{lemma_proof_4}\end{aligned}\ ] ] where ( a ) is due to ; ( b ) is due to ; ( c ) is because when is a constant ; ( d ) follows from )=\mathbb{e}[\text{tr}(\textbf{a})] ] , =\sigma^2\textbf{i}_{n^2} ] ,  ] are obtained , respectively , as & = \text{tr } \big\ { \textbf{h}_{r,1}^{(t)^h}\textbf{r}_1^{(t)}\textbf{r}_1^{(t)^h}\textbf{h}_{r,1}^{(t)}\textbf{f}^{(t ) } ( \textbf{g}_c^{(t ) } + p_2\textbf{h}_{2,r}^{(t-1)}\textbf{h}_{2,r}^{(t-1)^h}+\sigma_{n , r}^2\textbf{i}_{n_r } ) \textbf{f}^{(t)^h } \big\ } \\ & + ( n_sp_1\sigma_{e,1}^2 + \sigma_{n,1}^2)\text{tr}(\textbf{r}_1^{(t)^h}\textbf{r}_1^{(t ) } ) , \\\mathbb{e}\big[|| \textbf{r}_2^{(t)}\hat{\textbf{y}}_2^{(t ) } ||^{2}\big ] & = \text{tr } \big\ { \textbf{h}_{r,2}^{(t)^h}\textbf{r}_2^{(t)}\textbf{r}_2^{(t)^h}\textbf{h}_{r,2}^{(t)}\textbf{f}^{(t ) } ( \textbf{g}_c^{(t ) } + p_1\textbf{h}_{1,r}^{(t-1)}\textbf{h}_{1,r}^{(t-1)^h}+\sigma_{n , r}^2\textbf{i}_{n_r } ) \textbf{f}^{(t)^h } \big\ } \\ & + ( n_sp_2\sigma_{e,2}^2 + \sigma_{n,2}^2)\text{tr}(\textbf{r}_2^{(t)^h}\textbf{r}_2^{(t ) } ) , \\\mathbb{e}\big[|| \textbf{x}_r^{(t ) } ||^{2}\big ] & = \text{tr } \big\ { \textbf{f}^{(t ) } ( \textbf{g}_c^{(t ) } + p_1\textbf{h}_{1,r}^{(t-1)}\textbf{h}_{1,r}^{(t-1)^h } + p_2\textbf{h}_{2,r}^{(t-1)}\textbf{h}_{2,r}^{(t-1)^h}+\sigma_{n , r}^2\textbf{i}_{n_r } ) \textbf{f}^{(t)^h}\big\ } , \label{exp_y1_y2_xr_w } \end{split}\ ] ] where \\ + & \mathbf{\gamma}_3 \sum_{i=2}^{t-1 }  \bigg [ ( \sigma_{e , r}^2)^i\textbf{i}_{n_r } \prod_{j = t+1-i}^{t-1 } \text{tr}(\textbf{f}^{(j)}\textbf{f}^{(j)^h } ) \\ & \text{tr } \ { \textbf{f}^{(t - i)}(p_1\textbf{h}_{1,r}^{(t-1-i)}\textbf{h}_{1,r}^{(t-1-i)^h}+p_2\textbf{h}_{2,r}^{(t-1-i)}\textbf{h}_{2,r}^{(t-1-i)^h}+\sigma_{n , r}^2\textbf{i}_{n_r})\textbf{f}^{(t - i)^h } \ } \bigg ] \label { } \end{split}\end{aligned}\ ] ] and , , and are defined as    the problem of relay beamforming matrix design in each time slot is formulated as = n_rp_r \label{}\end{aligned}\ ] ] using and substituting , the lagrangian in is reduced to where   for the lagrangian in , we have three conditions of optimality as , , and . using the linear and nonlinear properties of the complex matrix derivative and the cyclic permutation and linearity of the trace function , these conditions are reduced to from and , we have and substituting into , we obtain to solve , we use the following lemma from .let , , and for .a matrix is a solution of the general linear matrix equation . if and only if is a solution of the equation with and .  using _ lemma 2 _ , the optimal is expressed as where and .note that the optimal in time slot involves the past channels from time slot to time slot since it is a function of , , and in .once the optimal is obtained , the optimal can be obtained from such that substituting into , the optimal is given by then , the optimal relay beamforming matrix in time slot is obtained as . in the next time slot ,the optimal relay beamforming matrix is derived in the same way ._ remark _ : the proposed relay beamforming design method can be applied to one - way fd af relay systems .unlike the two - way fd af relay systems , using matrix inverse operation , the relay beamforming matrix solution can be obtained without _lemma 2_.   the problem of receive beamforming matrices design in each time slot is formulated as where the sum of mse is defined as where by solving for , the receive beamforming matrices at the sources are given by where means the index of the other source ( if and if ) .an iterative algorithm is proposed for the joint relay and receive beamforming design .the relay beamforming matrix and receive beamforming matrices are optimized alternately until convergence in each time slot .the proposed iterative algorithm is shown in algorithm 1 .  [cols="<",options="header " , ]in this section , we numerically evaluate the sum of mse and the achievable sum rate for the proposed scheme , conventional scheme , and the hd two - way relaying scheme .the proposed scheme updates the relay beamforming matrix and receive beamforming matrix for every time slot , which is based on the channels in the latest time slots . unlike the proposed scheme , the conventional scheme updates the relay beamforming matrix based only on the channels in the current time slot , i.e. .the hd two - way relaying scheme is used as a referential scheme .the sum of mse and achievable sum rate are obtained by averaging over 100 channel realizations .the number of iteration for relay and receive beamforming design is 30 .the channel matrices for are set to follow rayleigh fading , i.e. , the elements of each channel matrix are independent complex gaussian random variables with zero mean and unit variance .the channel estimation error matrices for are set to follow rayleigh fading , i.e. , the elements of each channel matrix are independent complex gaussian random variables with zero mean and variance . for simplicity , it is assumed that all the nodes transmit with power , i.e. , the power is set to be .the noise variance and the loopback channel estimation error variance at each node are set to be as and .snr and inr are defined as and , respectively .[ th ! ]  fig .[ fig_result_iteration ] shows the sum of mse and the achievable sum rate versus number of iterations for different inr values in the 10th time slot when snr , , , and , respectively .the proposed scheme addresses two sub - problems for relay beamforming design and receive beamforming design at sources .since the solution for each sub - problem is optimal , the sum of mse decreases with each iteration .[ t ! ]  fig .[ fig_result_snr_pro ] shows the sum of mse and achievable sum rate versus snr for different inr values in the 10th time slot when , , and , respectively .as shown , the proposed joint beamforming design achieves better performance than the proposed design of relay beamforming only . as snr increases and inr decreases , the sum of mse decreases and the achievable sum rate increases for both the proposed schemes .t ]  fig .[ fig_result_m ] shows the sum of mse and the achievable sum rate versus the time slot index for different values of and inr when snr db , , and .the proposed scheme with and are the methods that the relay and receive beamforming matrices are designed with all the beamforming and channel matrices from the first time slot and with only those in the latest time slots , respectively . if is too small , the performance is unstable after time slot because it can not handle the propagated effect of the residual loopback si .for instance , when under inr db and under inr db , both of the sum mse and the sum rate oscillate and the oscillations become larger as either inr increases or decreases .on the other hand , if under inr db , the performance gap from the case of is marginal .  [ t ! ]  for stability in the fd mode.,title="fig:",width=377 ]  fig .[ fig_result_range ] shows the criteria for the duplex mode selection and the selection of for stability in the fd mode when and . in the region above the solid line ,the achievable sum rate of the proposed scheme in fd mode is greater than that of hd mode , so fd is preferred to hd .however , below the solid line , hd is better than fd .this figure also reveals that as inr increases , the snr required for selection of fd grows . for each pair of snr and inr , the values of for stabilityare presented , too . for the region where hd is preferred , not specified because the residual loopback si in the hd mode does not exist .  [ !t ]  fig .[ fig_result_snr_procon_mse ] shows the sum of mse versus snr for different inr values in the 10th time slot when and .it is shown that the proposed scheme of joint beamforming design provides a lower mse than the conventional scheme .as snr increases and inr decreases , the mse decreases for both the proposed and conventional schemes . the performance difference between the two schemes decreases as snr increases and inr decreases .this is because the proposed scheme updates the relay and receive beamforming matrices based on the covariance matrix of loopback channel estimation error .this figure also verifies that provides almost the same mse performance as .[ fig_result_snr_procon_rate ] compares the achievable sum rates of the proposed scheme of joint beamforming design and the conventional scheme at the 10th time slot when and . in either low snr or high inr regime ,fd is not beneficial relative to hd because the cost of handling the residual loopback si is higher than the hd loss in those regimes , which strongly suggests adaptive selection between hd and fd according to snr and inr ; if inr is less than db when snr  db or less than db when snr  db , fd with the proposed beamforming scheme is preferred to hd .if we focus on the scenarios where fd is preferred , offers almost the same achievable sum rate as , as in fig .[ fig_result_snr_procon_mse ] .[ fig_result_snr_procon_gain ] shows the achievable sum rate gain versus snr for different inr values at the 10th time slot when and .the achievable sum rate gain is defined as the ratio between the achievable sum rate of the proposed joint design scheme and that of the hd scheme .for example , the achievable sum rate gain is 2 and 1.9 when inr and inr is db , respectively .the gain increases as snr decreases or inr increases .t ]  fig .[ fig_result_ts ] shows the sum of mse and achievable sum rate versus the time slot index for different inr ( i.e. , inr db , db , and db ) when snr db , , and .the performance gap between the proposed joint design scheme and the conventional scheme is shown to be larger as the time slot index increase , for both mse and sum rate .the error propagation effect of residual loopback si over time slots is not properly addressed in the conventional scheme and accordingly the loss due to the error propagation increases with time slot index .interestingly , the performance gap between the first and the second time slots is greater than that between the second and the third time slots . this is because the residual si at relay starts to affect from the second time slot whereas the residual si at source affects from the first time slot . as in the previous figures ,it is verified again that offers almost the same performance as .[ fig_result_nr ] presents the sum of mse and achievable sum rate versus snr for different values of at the time slot when and inr db . as the increases , the mse decreases and achievable sum rate increases for both the proposed joint design scheme and the conventional scheme .the performance gap between the proposed and conventional schemes increases with because the multiple antennas at relay are more efficiently used in the proposed scheme .this paper investigated joint design of relay beamforming and receive beamforming at sources in fd two - way af relay systems under a relay transmit power constraint .we analyzed the coupled effect of beamforming matrix design across time slots and derived the optimal bemforming matrices in closed form at each time slot in terms of mmse , considering the coupled effect .to reduce the burden on the beamforming design incorporating all the time slots from the first time slot to the current time slot , we also proposed the beamforming design method based only on the latest time slots .it was shown that if is not too small , the performance degradation due to a limited number of was marginal . with numerical analysis, we revealed when full - duplex operation is beneficial compared to hd operation , and proposed adaptive selection between fd and hd according to inr and snr .f. s. tabataba , p. sadeghi , c. hucher , and m. r. pakravan , `` impact of channel estimation errors and power allocation on analog network coding and routing in two - way relaying , '' , vol .61 , no . 7 , pp. 3223 - 3239 , sep .

Abstract: we consider the full - duplex ( fd ) two - way amplify - and - forward relay system with imperfect cancelation of loopback self - interference ( si ) and investigate joint design of relay and receive beamforming for minimizing the mean square error under a relay transmit power constraint . due to loopback channel estimation error and limitation of analog - to - digital converter , the loopback si can not be completely canceled . multiple antennas at the relay can help loopback si suppression but beamforming is required to balance between the residual si suppression and the desired signal transmission . moreover , the relay beamforming matrix should be updated every time slot because the residual si in the previous time slot is amplified by the current beamforming matrix and added to the received signals from the two sources in the current time slot . we derive the optimally balanced relay beamforming and receive beamforming matrices in closed form based on minimum mean square error , taking into account the propagation of the residual loopback si from the first to the current time slot . we also propose beamforming design using only the channels of the latest time slots , not from the first time slot . based on our numerical results , we also identify when fd is beneficial and propose selection between fd and half - duplex according to signal - to - noise ratio and interference - to - noise ratio .


random graphs were studied since the middle of the 20th century with the initiative works of mathematicians pl erds and alfrd rnyi . in 1960they published their paper _ on the evolution of random graphs _, with the first thorough study of the graph theory  .however , these results were strictly theoretical , since tools to measure real - world random graphs at that time were not available . with the evolution of the personal computers , we have now the possibility to study this topic in practice , and , interestingly , the computers themselves provide one of the most exciting real random graphs : the computer networks .as these networks evolved , their properties were analyzed , and soon it was clear , that the erds - rnyi models ( er ) were not appropriate for these graphs  .the most important difference was the _ scale free _property , that the computer networks have , but the classical models lack .it turned out , that this feature is shared by other types of networks , like social  and metabolic  ones .it means that the distribution of the degrees of the nodes follows a power - law distribution , while the classic graphs has a poisson degree distribution , with an exponential tail .it shows that , in the real - world graphs , nodes with a high number of connections are much more likely than expected .therefore new models were necessary to describe these kind of random graphs .one of these models was developed by lszl barabsi and rka albert ( ba model )  .their method has two key features : _ incremental growth _ and _ preferential attachment_. incremental growth means that the graph is constructed by adding nodes to the existing graph , and connecting them according to a construction rule , contrary to the original erds - rnyi picture with a static graph .preferential attachment means the the likelihood of a connection depends on the degree of a node , again lifting the classical assumption of equal probabilities .this model has more variations , and they are able to describe a wide class of random graphs .there is another class of models study such networks without incremental growing , purely on their statistical properties  .  in this paper, we investigated network properties of ba - like models with different parameters and raise the question whether non - classical graphs may perform better transmitting data over them .certain value of parameters yield a graph with an exponential tail degree distribution , hence allowing us to compare these classical type models to the ones with power - law distribution . in sectionii we analyze the properties of the extended ba models , estimating their node distribution .next , in section iii we construct networks based on these models , and simulate a network traffic on such a graph in a simplified model estimating the traffic load on the nodes .the scaling of the load with nodal degree is presented , allowing to estimate the proper bandwidth allocation when upgrading a node . in sectioniv we study a more realistic setup with tcp dynamics and compare the theoretical result of the previous section to the simulated ones , while in section v we discuss the overall performance of the different simulations . finally , we conclude out analysis .in the original ba - model the graph is constructed as follows . starting from a small initial graph , we extend it by adding a new node in each step and connecting it to randomly selected existing nodes .the probability of choosing a particular node is proportional to its degree , p_i=. this model constructs a scale free graph , where the cumulative degree distribution has a power - law decay with an exponent 2  . in the following ,we modify the construction law , and use the more general form p_i= , weighting the probability with a power of the nodal degree , similarly to ref .  .the ba model corresponds to =1 , while for the preferential connectivity is cancelled , and we are back to a classical er - like graph model with a uniform distribution , leading to exponential node distribution .while this model is similar to the original er model , it differs in some aspects , such as it has a minimal guaranteed degree , and the ordering of the nodes presents nonzero correlations in the degrees  . with ,the models provide a smooth transition between the classical and the scale free models .the degree distribution for these models can be derived following the method introduced in  .here we give a fast estimate on the distribution , for the exact result see  .first , we estimate the rate of growth of the degree at each node , assuming that the growth of the degree is continuous in time . at time are exactly nodes and links between them .hence , the expected degree value of node is growing as _t k_i(t ) = .[ ba - ext ] for , the denominator is simply conunting the number of nodes , and is equal to , while for , it is ( double ) counting the number of links , and thus is , both being a linear function of the time .numerical simulations showed that for the denominator is well approximated by a linear function , where is the dependent measured slope .hence , generally our differential equation can be written as _t k_i(t ) = .[ weib - appr ] fortunately , these equations are solvable for each in the choosen range . specifically , for we get k_i(t)= , and more generally for k_i(t)=((1- ) ) ^. [ kalpha ] finally , for the scale free case the structure changes , and we arrive at k_i(t)=t^1c .[ kpower ] in order to get a formula for the degree distribution , note that for each , , since at the time the node is added it has exactly connections .the growth process for node has a similarity to the growth of the previous nodes , and this suggests that k_i(t ) = k_1 ( ) .the cumulative distribution counts the number of nodes with , i.e. , which leads to > i. normalizing back with the total number of nodes at time ( being also ) we get the approximate cumulative probability p(k_i > x ) 1k_1 ^ -1(x ) .inverting eq .( [ kalpha ] ) we find p(k_i > x ) e^-x^1- , [ weib ] the weibull - distribution for with scale parameter , and shape parameter .we note , that the exact result is of the form  p(k_i > x ) x^- e^-x^1- ... , where denotes higher order correction terms.  for ( ) one recovers the classical limit .in the region the distribution still vanishes exponentially , so strictly speaking no heavy tails are present , however , the probability of finding nodes with large degree increases dramatically .numerical studies revealed , that the distribution indeed follows the form ( [ weib ] ) , however , the parameters are slightly different from the predicted ones , due to the approximation used in ( [ weib - appr ] ) . in the limit ( ) we arrive at the genuine scale free heavy tail result p(x ) ~x^-c .we note , that the analysis can be extended to , however , such a distribution actually tends to the degenerate case with one node being connected with all others in a star - like topology .for the original ba model results in , however , different growth stategies are able to vary the value of , hence the decay exponent . in this workwe restrict ourselves to the original ba model .in the following , first we generate random graphs on a computer according to the extended ba model ( [ ba - ext ] ) and study the load in a simplified network model .we assume , that each node generates a constant traffic ( data stream ) to all the other nodes .the amount of traffic is the same for all connections , and furthermore , we assume that the link capacity can handle the accumulated traffic .the properties of the network are analysed tuning the geometrical parameter .for each value of we performed measurements on 8 different graphs ( generated with the same statistical properties ) , and then averaged .each topology was made out of nodes and links per node .the traffic was routed using the standard shortest - path strategy , and the number of data streams , passing a node was counted for each node . since by construction at least data channel are open per node , we define the load , to be the number of data channels divided by , the number of nodes , at node . in the large limit this gives the number of links , if no `` foreign '' connection is going through the selected node .one can regard this quantity as `` weighted degree '' , with weights equal to the link loads .  to study the effect of congestion , we assume , that for nodes a data stream occupies part of the link capacity .with connections there will be certainly `` overused '' links , where the link bandwidth capacity constrains the througput of the node .thus we define the throughput of a link to be the minimum of the load and the link capacity , while the throughput of the node to be the sum of the link troughputs over the links connected to the given node .figure 1 shows the tails of the cumulative distribution functions ( ) for degrees , load and ( node ) throughput on graphs with =0 , 0.5 and 1 .the distribution of the _ degrees _ show the expected tails , exponential ( , er graph ) , weibull ( ) , and the power - law ( , ba graph ) , respectively .however , the distribution of load shows interesting deviations from the one of the nodal degree . for the weighted nodal degree ( load )distribution can be approximated much better by a weibull distribution than an exponential one , describing the nodal degree distribution .this is a clear indication , that the distribution of the network traffic even in a simulated `` uniform '' situaton does not follow one - by - one the underlying network topology , rather developes a heavier tail .for the load remains weibull , however , the shape parameter ( theoretically ) , changes from the numerical value to . in case of power - law decay survives , but with changing cdf tail exponent , descreasing from the numerical value to .we may conclude , that the load distribution has considerably fatter tails than the underlying nodal distribution .the per - node _ throughput _ shows a transition between the nodal degree and the load distribution . for low throughput valuesit follows the load distribution ( the bandwidth is enough to hold the traffic ) , however , at higher values it approaches the nodal degree distribution , simply counting the number of links .the transition is governed by the link capacities , in a real network environment with large available capacity one expects the througput to follow the load distribution , however , in an underdesigned network it follows the degree distribution , as we show in the next section .     the change of the distribution from exponential to weibull , and the parameter changes in the weibull and in the power - law case in the load distribution suggests to examine the correlation between the degree and the loadthis is shown in figure 2 ( left ) on a scatter - plot .since it is log - log scaled , the linear clusters indicate a power - law correlations , . the dependence of the scaling exponent , on the network geometry parameter , is shown in figure 2 ( right ) .the load is pushed to have fatter tails than the degree distribution , and the more `` classical '' is the network the larger the deviation .for the ba geometry the the load distribution decays 1.6 times faster than the corresponding degree distribution .this means that not only the distribution of the load is more heavy - tailed than the distribution of the degrees , but also means that this dependence is quite strictly a power - law function. it also explains the deviations from the degree distributions , since a power of an exponentially distributed ( ) random variable is weibull , while in the case of weibull or power - law degree distributon the transformation results only in a parameter change for the distribution .it seems that the traffic pushes the distribution into heavier tails , from the exponential distribution , and converging to a power - law through weibulls .the initial push at is extremly high , with an exponent 1/2 in the weibull distribution .the load distribution is much less sensitive to the underlying network as the degree distribution .since the nodal throughput is bounded from above by the degree , it limits the throughput on the high - degree nodes , where the links are already fully utilised .therefore the throughput - degree joint distribution is different from the load - degree distribution , the power - law correlation is only valid for low and medium - degree nodes .but in real life the congestion at the overloaded nodes also affects the other parts of the network , since every data flow through these nodes is jammed .furthermore , the tcp dynamics is also known to be chaotic  which may change the scalings observed in a simpler model . to simulate the real life situation , and compare them with the results of the simplified model , next, we simulate a realistic traffic on a computer network , too .in order to compare the theoretical results from the previous section to a more realistic setup , we simulated a tcp / ip network with nodes , links per node , and a uniform link bandwidth of mb / s .every node communicated with a randomly selected target node . to study the effect of congestion we modelled three scenarios : a low , a medium , and a high traffic one , using constant bitrate data flows of kb / s , kb / s , and kb / s , respectively .the simulation was ran using the berkeley network simulator package  .the link throughput was calculated as the number of packets sent through that link , and the nodal values as the sum of the throughput of the incident links .     to our surprise , in each scenario the throughput showed a good scaling through the whole degree range .since the throughput at the high - degree nodes is obviously limited , it must mean that the congestion limits the throughput of the other nodes in such way that the power - law throughput - degree correlation remains valid .it also means that the congestion affects each other node proportionally to its degree .the correlation exponent was , however , different in the three simulation .in the low - traffic scenario the measured distribution is exactly the same as obtained from the numerical simulations of the load in the previous section .the congestion still has not set up , and the throughput is identical to the load .as the traffic intensity grows , exponent decreases , and flattens .it shows that the heavy traffic is more evenly spreads through the network , but its dependence on the degrees remains a power - law function .the next quantity we studied is the total number of transferred packets in the simulated network .each node transmits constantly tcp packects to its randomly choosen partner , and if a packet arrives , an acknowledgement is sent back to the originating node .we counted the number of packets for which the acknowledgement was received and the difference betweeen the number of transmitted packets and acknowledged packets is the loss .the source of the loss of packets in the network is the congestion : whether the link capacity can not handle the amount of traffic , or the nodes in between can not cope with the routing of the packages .one would expect , that in a network with smaller shortest pathes between two randomly choosen nodes the load on the links and routers is higher , hence using the same devices it would drop the packets more often as a network with larger shortest path .it is also known , that scale free network have smaller shortest pathes connecting two arbitrary nodes  , _ i.e. _ a scale free network uses less routing devices , however , the load on them is higher . indeed , numerical simulation , performed in the previous section also showed , that with increasing traffic the performance of the scale free ( ba ) network downgrades , for example , with a drop rate of 24% already at medium traffic , while the classical ( er ) network show a downgrade only of 6% for the same traffic ..the performance of the networks in millions of packets successfully sent [ cols="<,^,^,^",options="header " , ]in this paper we showed , that the distribution of the network traffic even in a simulated `` uniform '' situaton does not follow one - by - one the underlying network , rather it developes fatter tails than the nodal degree distribution of the network . a scaling between the nodal degree distribution and the load on a nodewas established , showing a power like patter , , where the scaling exponent , is a decreasing function of the network parameter , . for networks with fatter nodal distributionthe exponent is smaller . as a consequence ,attaching a new connection to the node requires less upgrade in the bandwidth for a scale free network to keep the performance , as for a classical ( er ) one .the above theoretical result was confirmed by simulation , the per - node throughput is still _ scaling _ with nodal degree .the scaling dependes on the amount of the traffic , for a completely congested situation the throughput distribution by definition agrees with the nodal degree distribution , hence the scaling exponent is 1 , however , for partly congested or congestion free networks this scaling approaches the load distribution with scaling exponents in the range 1.4 ( partially congested ba network ) to 2 ( congestion free er ) .a scale free network requires less resource upgrade when a new node is added .the overall performance of a scale free network is decreasing rapidly with the traffic , where the classical network still has almost no loss .it is due to the feature , that the er network uses more routers along the shortest connection , and hence the traffic is distributed more evenly .99 p. erds , a. rnyi , publ .inst . hung .* 5 * ( 1960 ) 17 .h. jeong , r. albert , a.l .barabsi , nature * 401 * ( 1999 ) 130 .s. strogatz , nature * 410 * ( 2001 ) 268 .h. jeong _ et al ._ , nature * 407 * ( 2000 ) 651 .barabsi and r. albert , science , * 286 * , ( 1999 ) 509 .r. albert , a.l .barabsi , rev .* 74 * ( 2002 ) 47 .z. burda , j.d .correia , a. krzywicki , phys.rev . * e64 * ( 2001 ) 46118 .p. l. krapivsky , s. redner , phys.rev .* e 63 * ( 2001 ) 066123 .d. s. callaway _ et al ._ , phys.rev .* e 64 * ( 2001 ) 041902 .i. csabai , journal of physics * a27 * ( 1994 ) l417 .network simulator v2 , _ http://www.isi.edu/nsnam/ns _a. fronczak , j.a .holyst , m. jedynak , j. sienkiewicz phys . * a 316 * ( 2002 ) 688 .

Abstract: we study the performance of weibull and scale free internet - like networks and compare them to a classical random graph based network . the scaling of the traffic load with the nodal degree is established , and confimed in a numerical simulation of the tcp traffic . the scaling allows us to estimate the link capacity upgrade required making and extra connection to an existing node .


in conventional sampling , the analog signal is sampled at twice the highest frequency of the signal , the so - called nyquist rate .this makes inefficient use of the available samples because frequency components below the nyquist rate are oversampled .this uniform , frequency - independent sampling causes two predicaments : ( i ) it limits the maximum frequency that can be captured with a given sampling rate ( to half of the sampling rate ) and ( ii ) when the signal has redundancy , it results in a record length that is much larger than necessary ( since low frequencies are oversampled ) .time - stretching performed in the analog domain prior to sampling [ 1 - 7 ] overcomes the first problem by reducing the signal bandwidth ( see fig .1 ) . in this method, the signal is modulated on a chirped optical carrier and then subjected to dispersive fourier transform ( dft ) , which causes the signal , now represented by the modulation intensity of the carrier , to be stretched in time ( its bandwidth compressed ) .since the photodiode measures the modulation intensity this reduces the bandwidth requirements of the photodiode and the analog to digital converter . herethe time - bandwidth product ( tbp ) remains constant because when the modulation intensity bandwidth is compressed by a factor m , the signal s time duration is increased by the same factor .fast features are suitably slowed down for the digitizer to sample and quantize them at the nyquist rate ; however , the slow features are oversampled .this redundant oversampling results in a needlessly large record length. it would be highly desirable to compress the bandwidth without this proportional increase in the time duration ; in other words , a reduction of the modulation tbp .this requires feature - selective time - stretch . in principle, this should be possible when fast features occur sparsely .the benefit would be similar to that offered by compressive sensing [ 8,9 ] , albeit , through warping of the signal as opposed to modification of the sampling process .  in this paper ,we propose and experimentally demonstrate a new transformation that compresses the time - bandwidth product of the modulation by reshaping the signal s complex field in the analog domain before sampling and digitization .this variable sampling is performed by reshaping the signal with a phase filter with a nonlinear group delay .because of the analogy to anamorphism in graphic arts ( discussed later ) , we refer to this operation as the anamorphic transformation .we identify the specific group delay vs. frequency profile using the modulation intensity distribution , a two - dimensional function that unveils the signal s modulation bandwidth and its dependence on the group delay .the signal reshaping operation is then combined with complex field detection followed by digital reconstruction at the receiver .the net result is that the modulation bandwidth is reduced without the aforementioned expense of a proportional increase in temporal duration ( see fig .our technique makes it possible to capture an ultrafast signal in real - time with a digitizer that would otherwise have insufficient sampling rate . at the same time , the number of samples needed for digital representation , and hence the digital data size , is reduced .our technique measures both the time domain and the spectrum of ultrafast signals in real - time . for application to optical waveforms , the nonlinear group delay filter operation can be performed with dispersive elements with engineered group velocity dispersion such as chirped fiber bragg gratings ( cfbg ) , chromo modal dispersion [ 10 ] or free space gratings .while this paper focuses on applications in capturing ultrafast analog signals where the anamorphic transformation is performed in the analog domain , our mathematical transformation can also be performed in digital domain on digital data .this all - digital implementation would be a data compression algorithm that may prove useful in dealing with the storage and transmission bottlenecks of `` big data '' .   by reshaping the signal prior to digitization, the proposed anamorphic transformation causes the digitizer s uniformly spaced samples to be nonuniformly distributed .our method is inspired by the fovea centralis in human eye .the fovea centralis is a part of the eye located near the center of the retina .it has a much higher density of photoreceptors than the rest of the retina and is responsible for the high resolution of central vision , necessary for humans to read , watch , drive , and other activities where visual detail is of primary importance . while the fovea comprises less than of the retina , it uses over of the brain s visual processing power .since photoreceptors perform sampling , the fovea causes nonuniform sampling of the field of view .although the physical sample density ( sample rate of the digitizer ) in our system is uniform , we achieve a non - uniform distribution of samples across the signal by reshaping the signal prior to sampling in the temporal domain .while there is no nonlinear filter in the eye , the eye achieves similar nonuniform sampling by using nonuniform photoreceptor density provide by the fovea .hence our technique can be interpreted as biomimicry of the human eye .the reshaping of the signal in our technique evokes comparison to anamorphic image transformation techniques used to create optical illusion and art [ 11 ] .fundamental differences exist between our technique and the conventional anamorphic imaging .first , our technique warps the frequency domain ( fourier domain ) , whereas in anamorphic imaging the image is warped in its original spatial domain .second , the image transformation in anamorphic imaging is arbitrary and chosen for artistic considerations , or to change the aspect ratio of the image . in contrast , in our technique the transformation self - adapts to the frequency content of the signal causing fast time features to be slowed down more than the slower ones .this self - adaptation occurs naturally and is a consequence of the frequency dependence of optical dispersion ( used to create the transformation as discussed later ) .third , our technique works in the time domain , in particular it is applied to the digitization ( e.g. adc and dac ) and processing of temporal waveforms such as communication signals .because of the non - uniform warping of signal s spectrum in our technique , albeit executed in the frequency domain , our technique may be referred to as the anamorphic transform , or warped dispersive fourier transform .a passband analog signal can be represented by an modulation ( baseband ) waveform modulated on a carrier .adcs usually detect the modulation of the input signal , i.e. after downconversion . herewe derive a mathematical algorithm that describes the optimum analog transformation that reshapes the spectrum of the signal such that its modulation can be captured with an adc that would otherwise be too slow .unlike the conventional uniform time - stretch processing , the new transformation minimizes the record length and number of samples .this transformation is implemented via a filter with an engineered group delay .temporal group velocity dispersion can be represented by a filter with quadratic phase , i.e. one with the transfer function .we generalize the problem by allowing the phase to be an arbitrary function of frequency ( see fig . is the spectral response of a filter with phase and group delay ( gd ) of /\partial\omega$ ] .the complete list of parameters and acronyms we have used in this paper is given in table 1 in the appendix .the modulation intensity spectrum of the input signal can be described in terms of the complex amplitude :    where is the fourier transform operator and is the modulation ( sideband ) frequency measured with respect to the carrier frequency , .it is easy to show that the modulation spectrum can be written as a correlation function :    where is the spectrum of the input signal .equation ( 2 ) describes the correlation of the electric field with its frequency - shifted copy and calculates the spectrum of the modulation intensity . after passing through the filter , the modulation spectrum of the output signal ,can be calculated as follow :    here we define a new transform called anamorphic stretch transform ( ast ) [ 12 ] , that relates the input _ carrier _ ( field ) spectrum to the output _modulation _ spectrum ( ) :   } d\omega \end{split}\ ] ]  as seen later , for time - bandwidth compression , the shape of the optimum group delay is a sublinear function resembling the letter `` s '' .therefore , one may refer to this particular implementation as the s - transform ( st ) , although it should be noted that the anamorphic stretch transform is more general than this particular group delay function .  for filters operating in the far field ( i.e. filters with large group velocity dispersion ( gvd ) ) ,the signal is stretched in time by a large amount , hence its modulation frequency and the bracketed term in the exponent of equation 4 is reduced to the group delay , .thus , in the far field condition , ast is simplified to :     anamorphic transform gives the modulation spectrum of the signal at the output of the filter . since our objective is to simultaneously minimize the modulation bandwidth and time duration , we require a mathematical tool that describes both the modulation spectrum and its temporal duration .the following 2d distribution describes the modulation intensity spectrum and its dependence on time .we refer to this as the modulation intensity distribution ( mid ) :   }  e^{j \omega t } d\omega \end{split}\ ] ]  the modulation spectrum and time duration of a signal subject to an arbitrary group delay is obtained from this 2d distribution .this information is then used to design a filter with the right group delay .the mid can be mathematically described as the cross - correlation of the output signal spectrum with its temporally shifted waveform . at ( i.e. time shift of zero )the mid becomes the autocorrelation of the output signal spectrum ( i.e. the output modulation spectrum ) .thus the trajectory at in the mid represents the output modulation spectrum ( i.e. ast ) and its width determines the output modulation bandwidth .also the maximum absolute amount of the temporal shift that the cross - correlation has non - zero values is given by the time duration of the output signal .hence , the output signal duration can be measured from the mid as half of the time range over which the mid has non - zero values .  the mid plot for an arbitrary signal ( see fig .2(b ) ) subjected to a filter with an s - shaped gd ( see inset of ( c ) ) is shown in fig .mid belongs to a class of time - frequency distribution functions that also includes the `` ambiguity function '' [ 15 ] and `` wigner distribution '' . the 3d plot shows the dependence of the modulation amplitude ( color ) at the output on time and modulation frequency .the inset shows the same for the input signal .it relates the bandwidth and temporal length of the modulation to the filter phase response ( group delay profile ) . by choosing the proper filter, we can engineer the modulation bandwidth of the signal to match the sampling rate of the adc and its time duration to minimize the number of samples needed to represent it .as an example , the horizontal arrow shows the modulation bandwidth and the vertical arrow designates the time duration .it should be mentioned that the output signal has both amplitude and phase information requiring complex - field detection .the time domain signal can then be reconstructed from the measured complex field .a number of complex field detection techniques can be employed here [ 16 20 ] .while filters with arbitrary gd profiles can be considered for ast operation , here we are particularly interested in filters with general gd profiles that compress the modulation tbp . as suggested by the mid plot in fig . 2, such filters should have a sublinear group delay profile .the function provides a simple mathematical description of such filters :    where and are arbitrary real numbers . using eq .( 7 ) , a wide range of filter gd profiles can be generated requiring only two parameters to represent them ( see section 5 for more information ) .parameter in eq .( 7 ) is the amount of group delay dispersion and determines whether the filter is in the near field or far field regime . in the near field , is on the order of the input signal duration , whereas in the far field , is much larger than the duration .parameter is related to the degree of anamorphism .section 5 provides more information about the choice of function .the mid function shows that the modulation bandwidth is given by a trajectory through of the mid , that is the horizontal axis .this property deserves an explanation as it is central to the utility of this new distribution function in identifying the optimum filter ( group delay profile ) that compresses the time - bandwidth product .the filter applies a phase shift that is an increasing function of frequency . referring to eq .( 3 ) , higher frequencies in the argument of the integral become highly uncorrelated and the integral over these fast oscillations vanishes . thus the modulation bandwidth is governed by the central portion of the mid .mathematically , this property is similar to the stationary phase approximation in dispersive fourier transform [ 21,22 ] .note that the modulation bandwidth defined in the mid ( fig .2 ) is the passband ( double sideband ) bandwidth whereas after the photo detector , we would be concerned with the baseband ( single sideband ) bandwidth which would be half of the former .time - stretch dispersive fourier transform ( dft ) [ 3,17,21 ] can be considered a special case of the anamorphic transformation . in other words , in the limit when the group delay is linear , the system operates in the far field and detector performs intensity detection , anamorphic transform and dispersive fourier transform are related via fourier transform .anamorphic transform , ast , can be described as generalized dispersive fourier transform .dft relies on linear gvd to perform time dilation and fourier transformation on the input signal in real time ( fig .3(a ) ) [ 3 ] .dft relates the input _ carrier _ ( field ) spectrum to the output _modulation _ in the time domain through the following transformation :    there are important differences between ast and dft .first , dft maps the input field spectrum to the output modulation in the time domain but ast maps the input field spectrum to the output modulation in the spectral domain ( compare eqs .( 4 ) and ( 8) ) .second , dft occurs in the far field only whereas ast spans both far field and near field ( see section 3 ) .third , in dft the filter has quadratic phase profile but in ast the filter has arbitrary phase profile . in other words , ast can be described as warped dispersive fourier transform that also spans the near field [ 12 ] .  in the limitwhen the filter has a quadratic phase profile , with large phase change , i.e. , ast is related to dft through fourier transform . here is the duration of input signal .this case refers to time stretch dft , which is a special case of the anamorphic stretch transform .when the mid is applied to a system with linear gd ( quadratic phase ) , it provides valuable insight into how dispersion affects the time bandwidth product of signals in such a system .3(b ) is the mid for such a system . to show the analytical power of the mid , we have considered a system in the near field .time stretch dft has been shown to be a powerful method for real - time high - throughput spectroscopy [ 22 - 24 ] and imaging [ 25 27 ] .owing to their high - throughput streaming operation , these instruments generate massive amounts of data , the storage and processing of which becomes challenging . compared to dft , the proposed anamorphic transformation reduces the record length and hence digital data size , easing the problem of big data in dft - based real - time instruments .photonic time stretch [ 1 - 7 ] is a dft based method that compress the input signal ( modulation ) bandwidth so it can be captured using a photodiode and analog - to - digital converter that otherwise would have insufficient bandwidth . with digital reconstruction, our method can capture an ultrafast signal , in the same spirit as the photonic time stretch system .there are important differences between our method and conventional time stretch concept .first , in conventional time stretch the signal is modulated on a chirped optical carrier using a modulator ( mixer ) and then is subjected to large amount of dispersion causing the signal to be stretched in time ( its bandwidth compressed ) .the present method does not use any modulator .second , in conventional time stretch the modulation tbp does not change .this means that when the signal ( modulation ) bandwidth is compressed m times , its duration is also increased m times . in our methodthe modulation tbp is compressed . when the signal modulation bandwidth is compressed m times , its duration is not increased proportionally ( as depicted in fig .1 ) . in temporal imaging( see e.g. [ 28 ] ) , tbp is similarly conserved .the present technique can be used to realize a warped temporal imaging system with added benefit of tbp compression .in the first example on engineering the mid , we discuss the optimum group delay ( gd ) profile for a filter operating in the far field condition .  the far field and near field regimes of group velocity dispersion can be understood in terms of the stationary - phase approximation .the far field corresponds to having sufficient dispersion to satisfy the stationary phase approximation while the near field refers to the regime before the approximation is satisfied [ 3,21 ] .we aim to compress the modulation bandwidth of the input analog signal while minimizing its duration . as an example , we consider an input signal with modulation bandwidth of 1 thz ( field bandwidth 0.5 thz ) and duration of 180 ps , see fig .the mid of the input signal without any filter in the system is shown in fig .we aim to compress the input signal modulation bandwidth to 8 ghz , i.e. a compression factor of 125 .the filter transfer function is chosen such that gd for higher frequencies is less than the case of linear gd .this is because to achieve the same output modulation bandwidth , the gd required to compress the bandwidth of the high frequency portion of the spectrum is less , achieved using eq .( 7 ) with and .4 compares the nonlinear gd with a linear gd that would have resulted in the same 8 ghz output modulation bandwidth .notice that the frequency axis in this figure shows the frequency deviation , i.e. filter s zero dispersion ( origin in the plot ) is at the input signal s carrier frequency .  as seen in fig .6(a ) , the modulation bandwidth is 8 ghz in both cases .however , the temporal duration ( see fig .6(b ) ) is 18 ns vs. 30 ns , i.e. reduction .6(c ) compares the recovered input signal using ast method ( red dotted curve ) with the input signal ( blue solid curve ) . captured signal with the same 8 ghz adc but without astis also shown with a black dash - dot curve .6 shows that using ast the input signal can be captured accurately with an adc that has lower bandwidth than the input signal .ast also minimizes the record length in comparison to the case of using a filter with linear gd .figure 7 compares the mid plots for the case of linear gd and the nonlinear gd used here .these mid plots were used to design and analyze the optimized bandwidth compression system in this example .the distribution is characterized by a well - defined , sharp , trajectory because the system is operating in the far field .as another example we discuss the optimum gd profile for time - bandwidth compression using a filter operating in the near field .this would be important for cases where far field regime can not be achieved because of insufficient available gd or limited bandwidth of the input signal . in this example, the input signal has an modulation bandwidth of 40 ghz ( field bandwidth 20 ghz ) and a 4 ns time duration ( cf .the mid of the input signal is shown in fig .we aim to compress the input signal modulation bandwidth to 16 ghz , i.e. a compression factor of 2.5 .the filter transfer function is chosen such that for frequency components ranging from dc to 8 ghz a larger gd is applied to higher frequencies than the case of linear gd .the gd for frequency components above 8 ghz is designed to be less than the case of linear gd .this is because to achieve the same output modulation bandwidth , less group delay is required for fast features .specifically , the chosen parameters for the filter s group delay profile given by eq . ( 7 ) is and .9 compares the nonlinear gd used with a linear gd that would have resulted in the same 16 ghz output modulation bandwidth .  as seen in fig .10(a ) , the modulation bandwidth is 16 ghz in both cases .however , the temporal duration ( see fig .10(b ) ) is 13 ns vs. 20 ns , i.e. reduction .10(c ) compares the recovered input signal using the ast method ( red dotted curve ) with the original input signal ( blue solid curve ) , while the captured signal with the same 16 ghz adc but without ast is also shown with black dash - dot curve . fig .10 shows that using ast input signal can be captured accurately with an adc that has lower bandwidth than the input signal .ast also minimizes the record length for bandwidth compression in comparison to the case of using a filter with linear gd .figure 11 compares the mid plots for the case of linear gd and the nonlinear gd used here .these plots were used to identify the optimum gd profile .the complex interference patterns in the mid plots arise because the system is operating in the near field .ast can be considered the generalized ( or nonlinear ) time - wavelength mapping .it reduces the modulation bandwidth so the signal can be captured with an adc with a bandwidth that would otherwise be insufficient . at the same time , it minimizes the number of samples needed for a digital representation of the signal ; in other words , it reduces the record length or the digital data size .a valid question is whether this time - bandwidth compression results in a loss of information . as a consequence of ast, a portion of the information contained in the signal modulation is transferred into the phase of the carrier .hence no information is lost and the compression is lossless .because some of the information is now contained in the phase , complex field detection is necessary in order to recover the original signal .ast uses an all - pass filter to add phase shift to the input signal the amount of which increases with frequency in a prescribed manner .the proposed modulation intensity distribution ( mid ) shows that , in order to compress the time - bandwidth product , the filter must have a nonlinear group delay profile , with proper slope at the origin ( at carrier central frequency ) .the slope at the origin is inversely proportional to the modulation bandwidth .the relation between the filters with linear and nonlinear gds can be represented by an all - pass filter with a rational polynomial function . in the region of interest ,close to the origin , the lowest order polynomial gives the tan-1 function in eq .the proof of this is beyond the scope of this paper .a tailored dispersion profile can be obtained by a number of techniques such as chirped fiber bragg grating ( cfbg ) with custom chirp [ 29 ] , chromo modal dispersion ( cmd ) [ 10 ] or diffraction gratings [ 30 ] .cfbg offers great flexibility in dispersion profile and low insertion loss . at the same time, it exhibits group delay ripples which are problematic .the recently demonstrated technique for mitigating these gd ripples [ 31 ] may be employed in our technique .we aim to compress the modulation bandwidth of the input analog signal while minimizing its duration .12(a ) shows one possible implementation of the ast system used in our experiments . the experimental setup used for demonstration of time - bandwidth compressionis shown in figs .12(b ) and ( c ) .the experiments compare the time - bandwidth of the signal for ( b ) nonlinear ( inverse tangent ) group delay ( gd ) and ( c ) for linear group delay , and the results validate the time - bandwidth compression using the specifically designed nonlinear gd .the specific gd profile was obtained using the mid function , see eq .the nonlinear gd is experimentally realized using a custom chirped fiber bragg grating , and the linear gd is realized using dispersion compensating fiber ( dcf ) . to reconstruct the input signal from the measured waveform , output complex - field recoveryis required followed by digital back - propagation technique . in this demonstrationwe used the stars [ 16 ] technique for complex field measurement , although other complex field recovery methods may also be used .the test input signal was generated using mode - locked laser ( mll ) and an optical waveshaper ( finisar 1000s ) .the input signal was designed using numerical simulations and had a field modulation bandwidth of 1 thz and duration of 50 ps , see fig . 13 .its field spectrum is shown in the inset .we aim to compress the input signal electrical bandwidth from 1 thz to 2 ghz , i.e. a compression factor of 500 .to show the effectiveness of our technique , we compare the case of ast using filters with linear gd to the case of nonlinear , specifically inverse tangent ( ) gd .the linear case , in the far field , corresponds to the well - known time - stretch dft .14 shows comparison of the different filter gd used in the experiment . for time - bandwidth compression ,the ast uses an inverse tangent profile . to show time - bandwidth compression ,this results were compared to those that use linear gd ( realized using dispersion compensating fiber ( dcf ) modules .two different modules were used : `` small gd '' has total gd equal to that of ast filter , and `` large gd '' has the same gd slope at the origin . specifically , large and small . for the inverse tangent group delay , we used the following profile :    where and .the ast filter with gd was implemented using a cfbg with customized grating chirp profile . referring to fig .14 , we note that the frequency axis is the frequency deviation , i.e. filter s zero dispersion ( origin in the plot ) is at the input signal s carrier frequency .15(a ) compares the measured modulation intensity spectrums . in order to compare the true bandwidth of the waveforms , the 2 ghz low pass filter shown in fig12 was not used in these measurements .as clearly seen , the electrical bandwidth in case of small gd is 5.5 ghz and in cases of large gd and gd is 2 ghz ( i.e. the target electrical bandwidth ) .15(b ) compares the measured output temporal intensity profiles . here, the 2 ghz electrical low pass filter emulates a system with only 2 ghz analog input bandwidth . in the case of small gd ,the output electrical bandwidth is 5.5 ghz so after the 2 ghz low pass filter , the measured signal has lost its higher frequency features . in cases of large gd and gd , electrical bandwidths are reduced from 1 thz to the target 2 ghz .however the temporal length , and hence the number of samples needed to represent the signal , is 2.73 times smaller with the gd .this clearly demonstrates time - bandwidth compression .15(c ) compares of the measured phase profiles .the dynamic range of the phase in the case of gd is less than that of for large gd .it should be noted that this reduction in time duration results in higher peak power making the detection easier . also , the in the case of large gd , the loss of the dispersive element is about 18db compared to about 1db for the inverse tangent filter .in fact , to observe the signal in the case of large gd , the signal had to be averaged 4000 times .therefore , while the large gd results is equally effective in reducing the electrical bandwidth , it has much lower signal to noise ratio than the inverse tangent gd .  for complex - field recovery, we have used stars technique [ 16 ] .spectral amplitude and phase response of the stars filter is shown in fig .fig 16(b ) shows the oscilloscope traces of the two stars outputs , for the gd .the complex - field is recovered using these two measurements and the algorithm we proposed in [ 16 ] . using the complex field , the input signal is reconstructed via digital back propagation .16(c ) compares the recovered input signal with the original signal programmed into the waveshaper . in the case of small gd, the input signal can not be recovered because fast features are lost . in cases of large gd and the input signal is properly recovered , however the temporal record length is 2.73 times lower with the gd . as noted earlier , the large losses in the case of large gd necessitated signal averaging .therefore , the results for this case are not real - time .in this work we introduced a new mathematical transform that can be used to compress the modulation time - bandwidth of signals .this analog grooming is performed prior to digitization and is aimed ( i ) to overcome the bandwidth limitation of data converters ( ii ) to reduce the digital record length , and ( iii ) to enable real - time digital processing .unlike in traditional time - stretching , the bandwidth compression is achieved without a proportional increase in the temporal record length .the proposed anamorphic transformation can be employed to engineer the modulation bandwidth of an ultrafast signal to match the sampling rate of the adc while minimizing the number of samples needed to represent it .this physics - based grooming of the analog signal allows a conventional adc to perform variable resolution sampling .the net result is that more samples are allocated to higher frequencies , where they are needed , and less to lower frequencies , where they are redundant .we would like to acknowledge valuable discussions with dr .george valley , ata mahjoubfar , brandon buckley and peter devore .m. h. asghari was supported by a canadian nserc .the work was partially supported by the nsf cian engineering research center .> p2.5 cm | > p5 cm  +  + & time + & carrier frequency + & modulation intensity frequency , i.e. sideband frequency measured with respect to carrier frequency , + & digitizer sampling rate + & modulation intensity bandwidth + & modulation intensity bandwidth compression factor + & number of samples ( discrete - time record length ) + & complex amplitude in time domain + &electric field spectrum + & filter transfer function + & filter phase response + gvd & group velocity dispersion + & total 2nd order dispersion ( gvd ) coefficient + & group delay profile + & intensity + ft & fourier transform + adc & analog to digital converter + ast & anamorphic stretch transform + mid & modulation intensity distribution + dft & time - stretch dispersive fourier transform +   y. han , and b. jalali , `` photonic time - stretched analog - to - digital converter : fundamental concepts and practical considerations , '' journal of lightwave technology 21 , 3085 - 3103 ( 2003 ) .a. m. fard , s. gupta , and b. jalali , `` photonic time - stretch digitizer and its extension to real - time spectroscopy and imaging , '' laser and photonics reviews 7 , 207 - 263 ( 2013 ) .k. goda , and b. jalali , `` dispersive fourier transformation for fast continuous single - shot measurements , '' nature photonics 7 , 102 - 112 ( 2013 ) .g. c. valley , `` photonic analog - to - digital converters , '' optics express 15 , 1955 - 1982 ( 2007 ) .a. khilo , s. j. spector , m. e. grein , a. h. nejadmalayeri , c. w. holzwarth , m. y. sander , m. s. dahlem , m. y. peng , m. w. geis , n. a. dilello , j. u. yoon , a. motamedi , j. s. orcutt , j. p. wang , c. m. sorace - agaskar , m. a. popovic , j. sun , g. zhou , h. byun , j. chen , j. l. hoyt , h. i. smith , r. j. ram , m. perrott , t. m. lyszczarz , e. p. ippen , and f. x. kartner , `` photonic adc : overcoming the bottleneck of electronic jitter , '' optics express 20 , 4454 - 4469 ( 2012 ) . j. stigwall and s. galt , `` signal reconstruction by phase retrieval and optical backpropagation in phase - diverse photonic time - stretch systems , '' journal of lightwave technology 25 , 3017 - 3027 ( 2007 ) .w. ng , t. d. rockwood , g. a. sefler , and g. c. valley , `` demonstration of a large stretch - ratio ( m=41 ) photonic analog - to - digital converter with 8 enob for an input signal bandwidth of 10 ghz , '' ieee photonics technology letters 24 , 1185 - 1187 ( 2012 ) . e. j. candes and m. b. wakin , `` an introduction to compressive sampling , '' ieee signal processing magazine 25 , 2130 ( 2008 ) .g. c. valley , g. a. sefler , t. j. shaw , `` compressive sensing of sparse radio frequency signals using optical mixing , '' optics letters 37 , 4675 - 4677 ( 2012 ) .e. d. diebold , n. k. hon , z. tan , j. chou , t. sienicki , c. wang and b. jalali , `` giant tunable optical dispersion using chromo - modal excitation of a multimode waveguide , '' optics express 19 , 23809 - 23817 ( 2011 ). j. l. hunt , b. g. nickel , and c. gigault , `` anamorphic images , '' american journal of physics 68 , 232 - 237 ( 2000 ) .m. h. asghari and b. jalali , `` anamorphic transformation and its application to time - bandwidth compression , '' applied optics , in press . m. h. asghari and b. jalali , `` warped dispersive transform and its application to analog bandwidth compression , '' ieee photonic conference ( ipc 2013 ) , paper tug 1.1 , sep 2013 .m. h. asghari and b. jalali , `` anamorphic time stretch transform and its application to analog bandwidth compression , '' accepted for 2013 ieee globalsip symposium .p. m. woodward , probability and information theory , with applications to radar , pergamon press , new york , 1953 .m. h. asghari , and b. jalali , `` stereopsis - inspired time - stretched amplified real - time spectrometer ( stars ) , '' ieee photonics journal 4 , 1693 - 1701 ( 2012 ) .d. r. solli , s. gupta , and b. jalali , `` optical phase recovery in the dispersive fourier transform , '' applied physics letters 95 , 231108 ( 2009 ) .m. h. asghari and j. azana , `` self - referenced temporal phase reconstruction from intensity measurements using causality arguments in linear optical filters '' , optics letters 37 , 3582 - 3584 ( 2012 ) .f. li , y. park , and j. azana , `` linear characterization of optical pulses with durations ranging from the picosecond to the nanosecond regime using ultrafast photonic differentiation , '' journal of lightwave technology 27 , 46234633 ( 2009 ) . c. dorrer and i. kang , `` complete temporal characterization of short optical pulses by simplified chronocyclic tomography , '' optics letters 28 , 1481 - 1483 ( 2003 ) .k. goda , d. r. solli , k. k. tsia , and b. jalali , `` theory of amplified dispersive fourier transformation , '' physical review a 80 , 043821 ( 2009 ) .d. r. solli , j. chou , and b. jalali , `` amplified wavelength - time transformation for real - time spectroscopy , '' nature photonics 2 , 48 - 51 ( 2008 ) .d. r. solli , c. ropers , p. koonath , and b. jalali , `` optical rogue waves , '' nature 450 , 1054 - 1057 ( 2007 ) .b. wetzel , a. stefani , l. larger , p. a. lacourt , j. m. merolla , t. sylvestre , a. kudlinski , a. mussot , g. genty , f. dias and j. m. dudley , `` real - time full bandwidth measurement of spectral noise in supercontinuum generation , '' scientific reports 2 , article number : 882 ( 2012 ) .k. goda , k. k. tsia , and b. jalali , `` serial time - encoded amplified imaging for real - time observation of fast dynamic phenomena , '' nature 458 , 1145 - 1149 ( 2009 ) .f. qian , q. song , e. tien , s. k. kalyoncu , o. boyraz , `` real - time optical imaging and tracking of micron - sized particles , '' optics communications 282 , 46724675 ( 2009 ) .c. zhang , y. qiu , r. zhu , k. k. y. wong , and k. k. tsia , `` serial time - encoded amplified microscopy ( steam ) based on a stabilized picosecond supercontinuum source , '' optics express 19 , 15810 - 15816 ( 2011 ) .m. a. foster , r. salem , d. f. geraghty , a. c. turner - foster , m. lipson and a. l. gaeta , `` silicon - chip - based ultrafast optical oscilloscope , '' nature 456 , 81 - 84 ( 2008 ) .t. erdogan , `` fiber grating spectra '' , journal of lightwave technology 15 , 1277 - 1294 ( 1997 ) .m. g. f. wilson and m. c. bone , `` theory of curved diffraction gratings '' , workshop on integrated optics , technical university berlin , 85 - 111 , may 1980 .g. a. sefler and g. c. valley , `` mitigation of group - delay - ripple distortions for use of chirped fiber - bragg gratings in photonic time - stretch adcs , '' journal of lightwave technology 31 , 1093 - 1100 ( 2013 ) .

Abstract: a general method for compressing the modulation time - bandwidth product of analog signals is introduced and experimentally demonstrated . as one of its applications , this physics - based signal grooming performs feature - selective stretch , enabling a conventional digitizer to capture fast temporal features that were beyond its bandwidth . at the same time , the total digital data size is reduced . the compression is lossless and is achieved through a same - domain transformation of the signal s complex field , performed in the analog domain prior to digitization . our method is inspired by operation of fovea centralis in the human eye and by anamorphic transformation in visual arts . the proposed transform can also be performed in the digital domain as a digital data compression algorithm to alleviate the storage and transmission bottlenecks associated with `` big data '' . +  + keywords : anamorphic stretch transform ; photonic time stretch , time - stretch dispersive fourier transform ; time stretch analog - to - digital conversion ; feature selective sampling ; feature - selective time stretch ; same - domain transform ; warped dispersive fourier transform ; generalized dispersive fourier transform ; ambiguity function , wigner distribution function ; warped temporal imaging ; generalized time - wavelength mapping ; time frequency distribution , ultrafast processing ; digital data compression ; big data .