Remote Estimation of Correlated Sources under Energy Harvesting Constraints

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1 Remote Etimation of Correlated Source under Energy Harveting Contraint Donloaded from: :42 UTC Citation for the original publihed paper (verion of record): Ozcelikkale, A., McKelvey, T., Viberg, M. (208) Remote Etimation of Correlated Source under Energy Harveting Contraint IEEE, 7(8): N.B. When citing thi ork, cite the original publihed paper. 208 IEEE. Peronal ue of thi material i permitted. Hoever, permiion to reprint/republih thi material for advertiing or promotional purpoe or for creating ne collective ork for reale or reditribution to erver or lit, or to reue any copyrighted component of thi ork in other ork mut be obtained from the IEEE. Thi document a donloaded from here it i available in accordance ith the IEEE PSPB Operation Manual, amended 9 Nov. 200, Sec, ( (article tart on next page)

2 Remote Etimation of Correlated Source under Energy Harveting Contraint Ayça Özçelikkale, Toma McKelvey, Mat Viberg Abtract Remote etimation ith an energy harveting enor ith a limited data and energy buffer i conidered. The enor node oberve an unknon temporally correlated field and communicate it obervation to a remote fuion center uing the energy it harveted. The fuion center employ linear minimum mean-quare error (LMMSE) etimation to recontruct the unknon field. We provide performance guarantee for the etimation error under a block tranmiion cheme, here at each tranmiion block, data and energy buffer are completely emptied. Our bound provide inight into ho tatitical propertie of the energy harveting proce and buffer ize may affect the etimation error. In particular, thee bound ugget inenitivity of the performance to buffer ize for ignal ith lo degree of freedom and ugget performance improvement ith increaing buffer ize for ignal ith relatively higher degree of freedom. Depending only on the mean, variance and finite upport of the energy arrival proce, thee reult provide inight for the energy and data buffer ize for deployment in future energy harveting irele ening ytem. Index Term energy harveting, irele enor netork, ditortion minimization, correlated ignal I. INTRODUCTION With the ever increaing number of connected device, here over 6 billion device are expected to be connected by 2022, poering of thee device and enabling energy autonomou netorked ytem i a central concern []. Here, energy harveting provide a promiing approach. In energy harveting (EH) ytem, device are equipped ith capabilitie to collect energy from reneable ource, uch a olar poer. EH capabilitie not only enable efficient uage of energy ource but alo offer enhanced mobility and prolonged netork life-time [2 5]. Feaibility of energy harveting approache have been invetigated and favourable reult are obtained for variou cenario, including harveting from olar energy, mechanical energy ource and radio-frequency (RF) energy [3 5]. Device ith variou energy harveting modalitie have recently become commercially available including olar [6], thermoelectric [7] and vibration [8]. Deployment that utilize energy harveting olution have already tarted to appear for a ide range of application, including mart building and indutrial ite [6 8]. Relevant tandardiation and commercial olution effort A. Özçelikkale acknoledge the funding upport from the European Union Horizon 2020 reearch and innovation programme under grant agreement No and Sedih Reearch Council under grant A. Özçelikkale i ith the Diviion of Signal and Sytem, Uppala Univerity, Seden. T. McKelvey i ith the Dept. of Electrical Engineering, Chalmer Univerity of Technology, Seden. M. Viberg i ith the Dept. of Architecture and Civil Engineering, Chalmer Univerity of Technology, Seden. ayca.ozcelikkale@angtrom.uu.e, {toma.mckelvey, mat.viberg}@chalmer.e. that target lo poer conumption for enor node for internet of thing olution, uch a EnOcean irele tandard (ISO/IEC X), LoRa ( and SigFox ( have recently emerged. In parallel to thee promiing development, there ha been a ignificant effort to undertand information tranfer capabilitie of communication ytem ith EH capabilitie. In the cae of energy harveting from RF ource, the main challenge lie in deigning the optimal trategie at the tranmitter [9 2]. In the cae of ytem energy harveting from natural ource, uch a olar poer, the key iue i the intermittent nature of the energy upply. The main challenge in thee ytem i to provide reliable and efficient operation even hen the energy upply i unreliable. In thi ork, e focu on thi intermittent nature of EH ource and it effect on the performance of remote etimation ytem. A. Prior Work An important ditinction in the energy harveting literature i the one beteen the offline optimization cheme and the online optimization cheme [2]. In the offline (or determinitic) cheme, profile of the harveted energy i aumed to be knon non-caually. In contrat, in the online (or tochatic) cheme, only tatitical knoledge about the future energy arrival i aumed to be knon. The offline optimization cheme i relatively ell-tudied, epecially in term of formulation that adopt communication rate a the performance metric. Analytical reult exit for variou cenario, uch a point-to-point channel [3], [4], broadcat channel [5] and multiple-acce channel [6]. In contrat, online cheme i conidered to be le tractable analytically. A typical numerical method here i dynamic programming approach, hich utilize a earch over a quantized tate pace. Unfortunately, thi approach not only ha high computational complexity, hich limit it applicability in lo-complexity EH enor, but it alo fall hort of providing ytematic inight into the effect of ytem parameter [2]. On the other hand, reult that directly provide analytical inight for the online cheme are available only for a limited number of cenario [7 2]. Structural reult for capacity and rate optimization under intermittent energy arrival are provided in [7 20]. Under a binary deciion cheme, here at each time intant the enor make a deciion to tranmit or not, threhold-baed policie are proven to be optimal for remote etimation of Markov ource [2]. A learning theoretical approach, here optimal tranmiion trategie are learned over time ithout knoledge of tatitical parameter (c) 208 IEEE. Peronal ue i permitted, but republication/reditribution require IEEE permiion. See for more information.

3 2 of energy and data arrival procee, i invetigated [22]. Here, e contribute to the analyi of online cheme by providing performance guarantee for ignal recovery under a block tranmiion trategy. We further dicu our approach in Section I-B. Etablihing a cloe connection ith etimation of unknon phyical field and in particular degree of parity, hence varying degree of correlation of unknon ignal, i an important apect of performance evaluation for ening ytem. Here parity, or equivalently degree of freedom of a ignal family, refer to the effective lo dimenionality of the unknon ignal [23]. In addition to providing a reaonable model for phyical field, parity can be utilized to compenate for the unreliable nature of the energy ource in an EH ytem. Yet, for EH ytem, tructural reult that directly exploit parity or correlation characteritic are available only for a limited number of cenario, uch a etimation of a ingle parameter [24 26], Markov ource [2], [27], [28], circularly ide-ene tationary ignal [29], [30], to correlated Gauian variable [3], and i.i.d. Gauian ource, a a reult of the finding of, for intance, [7 20], [32], [33]. A e ill further dicu in Section I-B, here e contribute to thi apect by providing performance guarantee for etimation error hich depend on the parity of the unknon ignal and tatitical propertie of the energy arrival proce. B. Contribution In thi ork, e conider an EH enor hich oberve an unknon correlated field and communicate it obervation to a remote fuion center uing the energy it harveted. The fuion center employ linear minimum mean-quare error (LMMSE) etimation to recontruct the unknon ignal. We conider thi problem under a limited data and energy buffer contraint uing a block tranmiion cheme here, at each tranmiion frame, the data buffer and the battery are completely emptied. Motivated by the high complexity and the high energy cot of ource and channel coding operation, e conider an amplify-and-forard trategy a in [25], [26], [32]. We focu on the cheme here the energy ued at each tranmiion i modelled a a random variable, i.e. the online cheme. A preliminary verion of thi etup i conidered in [34], here energy arrival proce i retricted to be a Bernoulli proce and ignal model i retricted to circularly ide-ene tationary ignal. An important contribution of our ork tem from our focu on the correlated ignal model. Due to thi correlated ignal model, calculation of the mean-quare error require evaluation of a matrix invere. Hence the performance criterion, in general, cannot be ritten a a ummation of utilitie over time in contrat to the cae of formulation baed on throughput [4], [5]. Uing random matrix theory and compreive ening tool, e provide performance bound for thi correlated ignal et-up under a block tranmiion cheme. Our reult provide inight into ho tatitical propertie of the energy harveting proce and buffer ize affect the etimation error. Conitent ith compreive ening (CS) reult, our bound ugget inenitivity of the performance to the buffer ize for ignal ith lo degree of freedom, and poible performance gain due to increaing buffer ize for ignal ith relatively higher degree of freedom. Thee performance guarantee, hich depend on the parity of the ignal to be oberved and the firt and econd order tatitical propertie of the energy arrival proce, provide inight into buffer and battery ize choice. An important pecial cae e conider i the cae of circularly ide-ene tationary (c...) ignal, hich are a finitedimenional analog of ide-ene tationary ignal [35], [36]. In addition to the above block tranmiion cheme, e alo conider the trategy of tranmiion of equiditant ample for the lo-pa c... ignal. The equiditant ample tranmiion cheme i motivated by the ampling theorem for c... ignal [23], [36]. Our performance guarantee ugget that for lo-pa c... ignal imilar performance can be obtained by both trategie of block tranmiion (i.e. preading the energy a much a poible on all ample in the buffer) and ending only equiditant ample ith all the energy in the battery at each tranmiion frame. Our reult here complement the reult of Ref. [30], here the focu i on the off-line cheme and no high probability reult are preented. Together ith the off-line reult of [30], the performance bound preented here upport the poible flexibility in energy management for ening of lo-pa c... ignal under energy harveting contraint. The ret of the paper i organized a follo. In Section II, ytem model i decribed. Our performance guarantee are preented in Section III. We conider the cae of c... ignal in Section IV. Dicuion of connection to compreive ening i provided in Section V. In Section VI, numerical illutration are provided. The paper i concluded in Section VII. Notation: We denote a column vector a C N by a = [a ;...;a N ] C N here emi-colon ; i ued to eparate the ro. Complex conjugate tranpoe of a matrix A i denoted by A. Spectral norm of a matrix A i denoted by A. The l th ro, k th column element of a matrix A i denoted by [A] lk. Poitive emi-definite (p..d.) partial ordering for Hermitian matrice i denoted by. I N denote the identity matrix ith I N R N N. The l 2 norm of a vector a i denoted by a. We denote the diagonal matrix hoe diagonal element are the element of the vector a by diag(a). Statitical expectation i denoted by E[.]. We denote expectation ith repect to (.r.t.) ignal involved ith E S [.] and expectation.r.t. energy arrival ith E E [.] for the ake of clarity hen needed. A. Signal Model II. SYSTEM MODEL The aim of the remote etimation ytem i to etimate the unknon complex proper zero mean field x = [x ;...;x N ] C N. Here x C N denote a field that i defined over time and x t denote the field value at time t, here t =,...,N. The covariance matrix K x = E[xx ] model the poible correlation of the field value in time. Let be the number of non-zero eigenvalue ofk x, i.e. rank ofk x. Let K x = U Λ x, U be the (reduced) eigenvalue decompoition (c) 208 IEEE. Peronal ue i permitted, but republication/reditribution require IEEE permiion. See for more information.

4 3 Battery E Qe E Qe+ Et Data Queue x k J k EH Senor pk x k Energy Arrival x... x Qd x Qd + x N Fig. : Energy Harveting Senor (EVD) of K x, here Λ x, C i the diagonal matrix of non-zero eigenvalue and U C N i the ub-matrix of unitary U C N N correponding to non-zero eigenvalue. Let = tr[k x ] = tr[λ x, ]. We conider Λ x, of the form Λ x, = Px I. Here give the number of degree of freedom (d.o.f.), i.e. parity level of the ignal family. We note that thi model cover ignal familie ith a ide range of correlation tructure. In particular, ignal ith rank one correlation matrice here the ignal component have a correlation coefficient of one, and hite ignal ith K x = Λ x,n = I n are covered ith thi model. By varying and U, thi model can be ued to repreent ignal ith different correlation tructure in beteen. Thi type of model have been ued to repreent ignal familie that have a lo degree of freedom in variou ignal application, for intance a a pare ignal model in compreive ening literature [23], [37]. B. Sening and Communication to the Fuion Center We conider an energy harveting enor a hon in Fig.. We focu on a lotted dicrete-time etting here at each time lot t, the enor oberve the field value at time lot t, i.e. x t. The obervation are held in a buffer of finite ize Q d before tranmiion. We conider a block tranmiion cheme, here time lot t atifying (k )Q d + t kq d belong to tranmiion frame k a hon in Fig. 2. Hence, the buffer content at the end of tranmiion frame k, i.e. at the end of time lot kq d, i given by x k = [x (k )Qd +;x (k )Qd +2;...;x (k )Qd +Q d ] C Q d. For convenience, N T = N/Q d i aumed to be an integer, here N T give the number of tranmiion frame. At the end of tranmiion frame k, the enor tranmit the data in it buffer to a fuion center uing an amplify-and-forard block tranmiion trategy a follo ȳ k = p k x k + k, k =,...,N T, () here p k, k and ȳ k denote the amplification factor, channel noie and the received ignal at the fuion center for tranmiion frame k, repectively. The channel noie = [ ;...; NT ] C N i modeled a complex proper zero mean ith K = E[ ] = 2 I N. The above type of block tranmiion cheme allo u to pread the energy over multiple ignal ample and facilitate connection ith uniform poer allocation trategie hich are optimal for hite ource in the offline cheme [7], [30], and the poer allocation trategie hich match the average arrival rate of the EH proce and optimal for hite ource in the online cheme [9]. It i alo upported by the fact that for device ith lo poer budget, it i more energy efficient Data Tranmiion Tranmiion Frame p x Tranmiion Frame N T pnt x NT Fig. 2: Time Schedule for the Energy Harveting Senor to end relatively larger amount of data at each tranmiion [38]. A a econd-order characterization of the dynamical range of x t, e aume that P(x t / [ α r xt,α r xt ]) ǫ r 0, here 2 x t i the variance of x t and α r > 0 i a given contant. For intance, hen x t are modeled a uniform random variable over [ a t, +a t ], one ha an exact equality, i.e. ǫ r = 0 ith α r = 2. For the cae here x t are modelled a Gauian random variable, one ha ǫ r 0 for a large enough choice of α r, here α r = 3 i a common practical choice ued a the effective idth of a Gauian random variable. Hence, e aume that x t are delimited according to their effective dynamical range ith a uitably choenα r and ignore the poible aturation effect. We aume that energy cot of ending a enor meaurement cale ith the variance of the random variable but not ith it realization. We note that thi i conitent ith the fact that modern enor typically provide meaurement uing analog-to-digital converter and the output of uch enor are repreented by the ame number of bit regardle of the realization of the phyical quantity meaured. Hence, the energy ued by the enor for communication at frame k can be ritten a follo Q d J k = β p k x 2, (2) (k )Qd +t t= here β i a proportionality contant that include the time duration per ymbol and α r. For convenience,β i normalized aβ= in the ret of the paper. We note that even hen enor output are repreented by a fixed number of bit, one may need to cale the channel input according to the dynamic range of the random variable to have an effective ignal-to-noie ratio on the channel that i conitent ith the variable dynamic range. Hence, e aume that the energy cot of a tranmiion cale ith 2 x t. Note that for the cenario ith 2 x t contant, uch a c... ignal, the energy cot only change ith p k. A more cloer invetigation of tranmiion of enor output here there i no channel coding but fixed-bit enor reading are tranmitted require invetigation of different modulation, quantization cheme, header; hich i beyond the cope of (c) 208 IEEE. Peronal ue i permitted, but republication/reditribution require IEEE permiion. See for more information.

5 4 thi ork. Here, e ue () together ith (2) a a limited but neverthele analytically tractable model for tranmiion of enor output over noiy channel. We note that our formulation alo cover the folloing related cenario here the energy cot of tranmiion are aumed to directly cale ith the energy of the realization: Suppoe that during the time lot t, N R tatitically independent realization of x t arrive. Hence, the data buffer hold Q d N R value. For intance, ith Q d = 2 for tranmiion frame k, the data buffer hold 2N R value: N R realization of x (k )Qd + and N R realization of x (k )Qd +2. At the end of frame k, enor end Q d N R value uing the caling p k. If N R i large enough, average energy of the realization ill be proportional to Q d t= p k 2 x (k )Qd +t regardle of the value of Q d, ince e then have lim NR NR N R E[ x i (Q d )k+t 2 ]=x 2, here xi (Qd )k+t (Q d )k+t i= xi (Q d )k+t 2 i the ith realization of x (Qd )k+t and E[x (Qd )k+t]=0. Hence, our formulation alo cover thi cae ith a poibly different contant β in (2). C. Energy Contraint at the Senor: We conider a battery-aided operation here energy i tored at a battery and ued in regular time interval. Let the initial energy tored at the battery be 0, i.e. the battery i empty. At time lot t, an energy packet of 0 E t < arrive to the enor, here the harveted energy proce i an i.i.d. dicrete-time tochatic proce ith mean µ E and variance E and E t E u <, here 0 < E u < denote the maximum value of the energy packet. At the end of frame k, total energy that ha arrived to the battery during frame k i given by Ēk a follo Q e Ē k = E (k )Qe+t. (3) t= We aume that the time frame for the data buffer and the battery i ynchronized and Q e = Q d = Q. We aume that battery capacity C atifie C E u Q, Q o that a total energy of E u Q can be tored in the battery. In general, the enor ha to operate under energy neutrality k l= J l k l=ēl,k =,...,N T. Thee condition: condition enure that the energy ued at each tranmiion frame doe not exceed available energy. Here, e focu on the cae here at each tranmiion all the energy at the battery i ued, i.e. J k = Ēk, k =,...,N T. (4) Here the left-hand ide of (4) depend on the poer amplification factor p k at tranmiion frame k through (2). The right-hand ide give the available energy, i.e. realization of the total energy tored at the battery at the end of tranmiion frame k. Performance of linear tranmiion trategie under uch poer contraint here available energy i modeled a a determinitic variable have been conidered before, ee for intance [39 4] for formulation ith total energy contraint and [25], [26], [32] for energy harveting formulation. In thi ork, e provide performance guarantee under a tochatic energy arrival model ith block tranmiion. D. Etimation at the Fuion Center: After N T tranmiion frame, i.e. obtaining y = [ȳ ;...;ȳ NT ] C N, the fuion center form an etimate of x uing Linear Minimum-Mean Square Error (LMMSE) Etimation. Let u conider a fixed E t, t =,...,N realization, here p k are determined through (4). Hence the LMMSE etimate conditioned on the energy arrival E t can be found a [42, Ch2] ˆx = K xy K y y. (5) Thi i the tandard linear mean-quare error etimator here the fuion center ue the econd-order tatitic of the ource and noie to form a linear etimate of the unknon variable [42, Ch2]. Let E S [.] denote the tatitical expectation ith repect to noie and ignal tatitic, including x,, but not ith repect to energy realization (and hence not ith repect to p k hich are alo a function of the energy realization). Then the mean-quare error, ε = E S [ x ˆx 2 ], can be expreed a follo [42, Ch2] [ ε = tr ( I + ] 2 U GU ), (6) here G = diag(g) = diag([p Qd ;...;p NT Qd ]) R N N, g = [g ;...;g t ;...;g N ] R N and Qd = [;...;] R Q d i the vector of one. Hence y = G /2 x+. We note that the poible additional ditortion due to the dynamic range limiter for x t ith unbounded upport i omitted here. A tudy of thi apect in the context of etimation under energy harveting contraint can be found in [24]. For the above tandard LMMSE etimation, p k are aumed to be knon at the fuion center a in [25], [26]. Determination of p k can be een a a part of the channel etimation proce in the communication link beteen the enor and the fuion center. We note that here e focu on the recontruction of the unknon field and the energy cot of thi channel etimation operation i not accounted for in our ork. Here G i a random vector due to random energy arrival; hence our etting i different from the offline cheme here the performance i evaluated under knon energy value. Furthermore, calculation of the mean-quare error in (6), in general, require evaluation of a matrix invere a oppoed to a direct um of utility function over time, uch a in the cae of throughput baed formulation. Our block tranmiion cheme provide a poibly ub-optimal but lo-complexity trategy for thi correlated ignal etting. III. PERFORMANCE BOUNDS Let u define ( f bt (µ,,r) 2exp µ 2h f bn (µ,,r) 2exp ( )) µr ) ( r2 /2 µr/3+ ith h(a) (+a)ln(+a) a, a 0. (7) (8) (c) 208 IEEE. Peronal ue i permitted, but republication/reditribution require IEEE permiion. See for more information.

6 5 We no preent our main reult, i.e. guarantee on the error performance that hold ith high probability: Theorem 3.: Let u i C denote the i th column of the matrix U. Let η L = min i u i 2, and = max i u i 2. Let E u be parametrized a E u = r E µ E, r E. Performance of the EH ytem atifie the folloing bound I. P(ε < ε I ) f bt (µ I, I,r) f bn (µ I, I,r) (9) for r (0, ], here II. ε I = + µ 2 E ( r) (0) µ I = max{r E,}min{Q,} η L () I = E µ 2 E ηl 2 Q min{q,} (2) P(ε < ε II ) f bt (µ II, II,r) f bn (µ II, II,r) (3) for r (0, ], γ [0,Qr E ], here ε II = + E( 2 r) (4) µ II = max{ p η,}min{q,} L (5) II = ηl 2 ( p )min{q,} (6) p = P(Ēk γµ E ) (7) Proof: The proof i preented in Section VIII. Both Bound I in (9)-(2) and Bound II in (3)-(7) provide performance guarantee (i.e. upper bound) for the meanquare error performance. For intance, Bound I tate the folloing: The mean-quare error ε i guaranteed to be loer than ε I ith probability greater than f bt (µ I, I,r), here the parameter ε I,µ I, I,r are related through (0)-(2). Bound II ha a imilar form ith the parameter ε II,µ II, II,r. We note that hether Bound I or Bound II i tighter depend on the ytem parameter. Thi apect i illutrated in Fig. 3 and Fig. 4 in Section VI. Bound I and Bound II can be een a performance guarantee baed on average number of ample that can be tranmitted. Thee dicuion, together ith connection ith compreive ening, are provided in Section V. Remark 3.: For Q =, energy E t that arrive to the enor at time t i immediately ued to end the ample x t. A the buffer ize Q > get larger, the probability of ending the ample in the buffer (ith non-zero poer) increae ince the probability of the battery being charged ith nonzero energy alo increae hile aiting for the data buffer to be full. On the other hand, the poer ued to end each ample may be loer compared to the cae here the energy i ued to end a feer number of ample, for intance compared to the cenario of directly ending the ample x t ith energy E t if an energy packet of E t > 0 arrive (Q = ). Hence, the bound preented here can be interpreted a an exploration of the trade-off beteen uing a mall number of ample ith high ignal-to-noie ratio (SNR), i.e. high poer, and a high number of ample ith lo SNR in the etimation proce. Remark 3.2: Energy allocation hich are a uniform a poible, or alternatively a balanced a poible are optimal for hite ource [7], [9], [30]. In our formulation, the data buffer and energy buffer allo u to mimic thee uniform-like allocation, here larger buffer ize allo energy allocation that are more uniform over the hole time duration of interet, i.e. t N. Hence, varying Q value allo u to tudy the effect of different buffer ize, or equivalently the effect of balanced energy allocation for ignal that are not necearily exactly hite. Note that e aume the battery capacity i large enough o that C E u Q, Q. Hence, the obervation here are in comparion to the maximum ize of the energy packet that can arrive at each time lot. In particular, a large Q value mean the device ha a large enough battery o that Q of the energy packet can be tored. We note that the block tranmiion cheme conidered in thi ork i poibly ub-optimal in the ene that there may be other tranmiion trategie that ue only tatitical knoledge of future energy arrival, but can guarantee maller error value for a given fixed probability. A. Comparion ith the average performance For comparion purpoe, e no preent a loer bound on the average error performance over different realization of the energy arrival proce E t. Lemma 3.: The folloing loer bound on the average error hold ith η L = min i u i 2 E E [ε] +. (8) µ 2 E η L The proof i provided in Section IX. We note that thi bound doe not depend on Q. Remark 3.3: Comparing (8) and the error expreion in Thm. (3.) e oberve that both expreion provide error expreion in the form +SNR here SNR eff take the eff form SNR D eff = 2 η L µ E for (8) and it take the form SNR P eff = 2 µ E ( r), for intance, for (0). Hence the error expreion in Thm. (3.) provide different operating point for ho cloe one can operate to (8) and ith hich probability through the variable r. B. Comparion ith the off-line cheme ith a total energy contraint A a benchmark for our bound in Thm. 3., e no conider an aociated off-line cheme [30]. In particular, e conider the cae here amplification factor are not modeled a random variable that depend on the energy arrival but determinitic variable to be optimized. Let u conider the cae here each component x t i ent a follo: y t = b t x t + t, t =,...,N. (9) (c) 208 IEEE. Peronal ue i permitted, but republication/reditribution require IEEE permiion. See for more information.

7 6 Here e introduced the notationb t 0 to denote amplification factor to emphaize that thee are modeled a determinitic variable a oppoed to random variable. In contrat to the etting of Section II and hence the etting of Thm. 3., here a block tranmiion contraint i not impoed onto the et of admiible enor trategie (hence Q = ). Let u denote the error a follo [ ε(b) = tr ( I + ] 2 U BU ), (20) here b t 0, t and B = diag(b) = diag([b ;...;b N ]) R N N. We conider the folloing optimization problem ε d = min B ε(b).t. N b l x 2 l = E tot. (2) In thi determinitic cheme, the enor ha a total energy of E tot and it can freely ditribute thi energy on the ample in order to minimize the error. We note the folloing reult: Lemma 3.2: [30, Lemma 3.5] An optimal trategy for (2) i given by uniform b t ith b t = E tot /, t. The optimum value i given by ε d = + E tot. 2 Conider the cae ith E tot = µ E N, hich i the total energy that ould have been obtained if an energy packet of µ E ere harveted at each time lot. We note that µ E i the mean of the energy arrival proce. In thi ene, Lemma 3.2 may be ued a a determinitic benchmark. Hence, the benchmark become ε d = l= + P N x. (22) µ 2 E Similar to the loer bound of (8), (22) i a benchmark for Thm. 3. in term of ho cloe one can operate to thi value. We note that, in general, η L N, hence (8) and (22) provide different benchmark. IV. CIRCULARLY WIDE-SENSE STATIONARY SIGNALS In thi ection, e pecialize to the cae of circularly ideene tationary ignal, hich contitute a finite dimenional analog of ide-ene tationary ignal [35], [36]. Covariance matrice of c... ignal are circulant by definition, i.e. covariance matrix of a c... ignal i determined by it firt ro a [K x ] tk = [K ] modn(k t), here K C N i the firt ro of K x [35], [36]. The unitary matrix U in the EVD of covariance matrice of c... ignal i given by the Dicrete Fourier Tranform (DFT) matrix [35], [36]. Let F N denote the DFT matrix of ize N N, i.e. [F N ] tk = (/ N)exp( j 2π N (t )(k )), t,k N, here j =. Hence, the reduced EVD of K x i given by K x = FΩ NΛ x,fω N, here Λx, = diag(λ k ) = Px I R and FΩ N CN i the matrix that conit of column of F N correponding to the non-zero eigenvalue. Due to the circulant covariance matrix tructure, the variance of the component of a c... ignal atify x 2 t = x 2 = /N, t. Hence, J k = Q d t= p k 2 x (k )Qd +t = p kq /N, and by (3), (4), e have the folloing p k = N Q E (k )Q+t. (23) t= For c... ignal, e have η L = min i u i 2 = N, and = max i u i 2 = N due to the DFT matrix. Hence, (0)- (2) can be expreed a ε I = + N µ 2 E ( r), (24) µ I = max{r E,}min{Q,}, (25) N I = E µ 2 E Q min{q,}, (26) N here r (0,]. Here e have caled r,µ I, I hile going from Eqn. (0)-(2) to Eqn. (24)-(26), ince f bt (.) and f bn (.) only depend on the ratio beteen r,µ I, I. Eqn. (4)-(6) can be pecialized to the cae of c... ignal, imiliarly. We note that thee bound alo hold for other ignal familie for hich u i 2 i contant for all i, uch a unitary Hadamard matrice. A. Equiditant ampling of lo-pa c... ignal We no focu on the cae of lo-pa c... ignal, i.e. c... ignal for hich eigenvalue of K x that correpond to the lo frequency component indexed by Ω = {0,,..., } are poibly non-zero and the ret of the eigenvalue are zero. Hence, for c... ignal only eigenvalue that are poibly non-zero are the one aociated ith the frequencie exp( j 2π N r), r = 0,...,. Such ignal can be recovered from their uniformly taken ample ith zero mean-quare error hen the total number of (complex-valued) ample i larger than the number of non-zero eigenvalue [23]. Thi property, hich i conitent ith the determinitic ampling theorem and the ampling theorem for ide-ene tationary ignal [36], [43], motivate u to tudy trategie that end equiditant ample under our EH frameork. In particular, e are intereted in undertanding hich of the folloing i a better trategy: i) ending all of the obervation of the enor ith a equal energy a poible a uggeted by the energy harveting literature; or ii) ending only the equiditant ample a uggeted by the ampling theorem. The block tranmiion cheme of Section II provide a lo-complexity approach for implementing trategie imilar to (i). We tudy the trategy in (ii) belo. In particular, e conider trategie that end one ample out of everyq = N/ ample a follo: Lett d {0,...,Q } be the fixed initial delay before ending the firt ample and N T = N/Q Z be the number of tranmiion a before. Recall that y t = g t x t + t. Hence, under uniform ampling e have g t 0, if t = Q(k )+t d +, k N T, and g t = 0 otherie. Hence, the received ignal at tranmiion frame k i the ingle ample x Q(k )+td + a follo y k = p k x Q(k )+td + + k, (27) (c) 208 IEEE. Peronal ue i permitted, but republication/reditribution require IEEE permiion. See for more information.

8 7 here p k denote the amplification factor, k C, E[ k k ] = 2 denote the i.i.d. complex proper zero-mean channel noie, a before. Energy ued by the enor for communication at tranmiion frame k can be ritten a follo: J k =p k x 2 = p (k )Qd +t d + k N, (28) here e have ued the fact that for c... ignal 2 x t = 2 x = /N, t. By (4), e again have J k = Ēk, k. At each tranmiion frame k, thi cheme ue all the energy in the battery to end only one x t value and dicard all the other ample in the data buffer. We obtain the folloing bound for the performance of thi ytem: Theorem 4.: Performance of the equiditant ample tranmiion trategy of (27) for lo-pa c... ignal atifie the folloing bound ith r (0,) P(ε < ε u I) f bt (µ u I, ui,r u ) f bn (µ u I, ui,r u ) (29) ε u I = + N µ 2 E ( r) (30) µ u I = max{r E,} (3) ui = E µ 2 E N. (32) Proof: The proof i preented in Section XI. Comparing (30)-(32) ith (24)-(26) for Q = N reveal that for lo-pa c... ignal, both the trategy of Thm. 3., hich pread the energy accumulated in the battery evenly on the ample in the buffer, and the equiditant ample tranmiion trategy of Thm. 4., hich ue the energy only on one ample from the buffer, reult in the ame performance guarantee. Thi property i conitent ith the performance of the aociated trategie in the off-line cenario under a total energy contraint a dicued belo: Comparion ith the off-line cheme under equiditant ample tranmiion trategy: Let u conider the equiditant ample tranmiion cheme under a total energy contraint: ε de = min B u ε(b u ).t. N b l x 2 l = E tot, (33) under the condition b t 0, if t = Q(k ) + t d +, k N T, and b t = 0 otherie; and Q = N/, B u = diag([b ;...;b N ]) R N N. Lemma 4.: [30, Corollary 3.3] An optimal trategy for (33) i given by b t = Etot N, if t = Q(k )+t d +, k N T and b t = 0 otherie. The optimum value i given by ε de = + E tot. 2 Hence, under the off-line cheme ith a total energy contraint, performance of the uniform poer allocation over all x t, hich i given by Lemma 3.2, and performance of the equiditant ample tranmiion trategy given by Lemma 4. are the ame. In thi ork, e have hon that performance bound in the online cae for block tranmiion cheme of Thm. 3. (pecialized to c... ignal in (24)-(26)) and the l= performance bound for the equiditant ample tranmiion cheme of Thm. 4. are alo the ame. Thee to et of reult together ugget flexibility in energy allocation for etimation of lo-pa c... ignal in energy harveting ytem. Neverthele, e note that Thm. 3. and Thm. 4. provide upper bound, i.e. guarantee for ignal recovery ith a given error ith a given probability. Hence, inight and guideline derived from thee reult hould take thi point into conideration. V. CONNECTIONS TO COMPRESSIVE SENSING The cenario of Q = in () i cloely related to the claical compreive ening etting. In particular, conider the cae here the energy arrival proce can be modeled a an i.i.d. Bernoulli random proce. A typical compreive ening et-up i the cenario here the meaurement proce i modeled a an i.i.d. Bernoulli proce here a meaurement i made, for intance, hen the Bernoulli random variable i and i dropped hen the Bernoulli random variable i 0. Hence for Q =, the bound preented here are cloely related to the eigenvalue bound provided in compreive ening literature [44, Ch.2]. In particular, conider the cenario of Q = ith tatic x 2 t = x, 2 (uch a in the cae of circularly ideene tationary ignal) and Bernoulli energy arrival. Then, the bound in Thm. 3. can be een a a conequence of the eigenvalue bound in the CS literature, ee for intance [44, Ch.2], [45, Thm..2], [23]. For Q > or non-uniform x 2 t, Thm. 3. provide a et of novel eigenvalue bound for the formulation introduced in Section II. To further elaborate on connection to compreive ening, e no focu on the cenario here there i no noie on the channel, i.e. 2 = 0. Hence, the ytem model become y = G /2 x, (34) here G i the diagonal matrix of amplification factor a defined in Section II-D. Let u conider the folloing quetion: For hich energy arrival rate, parity level and queue ize, can e recover x from the obervation y ith zero meanquare error (ith high probability)?. Recall that, K x = U Λ x, U, hence x C N belong to a ignal family of lo degree of freedom x = U x here x C and the covariance matrix K x = Λ x,. Hence, e have a etting that i imilar to typical compreive ening etup. Neverthele, note that in typical CS cenario, upport of the ignal i not knon during the ignal recovery herea here e conider a cenario here the upport i knon. Note that location of the meaurement (i.e. hich ro of G are non-zero) are modelled a random both in our etting and in compreive ening cenario. Thm. 3. ha the folloing corollary: Corollary 5.: Fix N. Conider the energy arrival proce E t ith E t = κ t E b, κ t Bernoulli(p); E b > 0. Let 2 = 0, 0 p /2, Q N/, p = ( p) Q, p /2, δ (0,] and U be the DFT matrix. Suppoe that at leat one of the folloing condition i atified C e,i (Q/3+) ln(2/δ) N p (35) C e,ii Q ln(2/δ) N p (36) (c) 208 IEEE. Peronal ue i permitted, but republication/reditribution require IEEE permiion. See for more information.

9 8 here C e,i > 2, and C e,ii > 8/3, are numerical contant. Then, the mean-quare error ε i zero ith probability at leat δ. The proof i provided in Section X. In the above, C e,i 2 and C e,ii 8/3, pleae ee Section X for detail. We note that the right hand ide of (35)-(36) increae ith increaing energy arrival ucce rate and can be interpreted a the average number of tranmitted ample. Hence, (35)-(36) aert that if i mall enough and the energy arrival rate i high enough, then the ignal can be recovered from it ample ith high probability. Thi i analogou to the compreive ening reult here ufficient number of meaurement for recovery of pare ignal are preented. In particular, conider the folloing ufficient condition from [45, Thm..] C c ln(n/δ) M (37) and C c ln2 (N/δ) M here C c and C c are fixed numerical contant and M i the number of meaurement hoe location (i.e. hich ro of G are non-zero ) are choen randomly. If (37) hold, then ith probability at leat δ, an arbitrary ignal (ith random ign) ith upport of ize can be recovered from randomly elected M meaurement [45, Thm..]. Note that Q = here. Comparing (35)-(36) and (37), e oberve that both condition give the (average) number of obervation that guarantee ignal recoverability. An important tep in the derivation of compreive ening reult i the derivation of eigenvalue bound. In particular, conider the folloing type of ufficient condition [45, Thm..2], [44, Thm. 2.2], C c ln(2/δ) M, (38) here C c > 8/3 i a numerical contant and M i, again, the number of meaurement. If (38) hold, the matrix U GU i invertible ith probability at leat δ. Hence, the meanquare error ill be zero in (34). Note that (38) i derived under the aumption that upport i fixed and knon, a in our et-up. A dicued in the beginning of thi ection, for Q =, (35)-(36) and (38) are the ame. Different from (38), our reult in (35)-(36) reveal ho the eigenvalue bound depend on the queue ize parameter Q, hich i included in the ytem formulation due to the energy harveting apect. We note that (35), hich a derived from Bound I, ugget maller buffer length are preferable (in the ene that for fixed, p, N value, larger Q value ill not atify (35)). On the other hand, p that appear on the right hand ize of (36) alo depend on Q. Note that (36) a derived from Bound II. Whether (36) (and Bound II) favor maller or larger Q value depend on the ytem parameter. Further invetigation of thi point i provided in Section VI. A. Dicuion In a ide range of ening application, there exit unknon phyical quantitie that e ould like to etimate, uch a temperature value in a mart building application or flo rate in an indutrial application. Typically, enor make meaurement of thee parameter and thee meaurement are collected at a remote central deciion center irelely. puc I, Q=2 II, Q=2 I, Q=4 II, Q= MSE Bound 0-3 Fig. 3: p uc veru Bound I and Bound II, Bernoulli energy arrival, = 4, p = 0.5. puc I, Q=2 II, Q=2 I, Q=4 II, Q= MSE Bound Fig. 4: p uc veru Bound I and Bound II, Bernoulli energy arrival, = 8, p = 0.3. Practical enor deployment that utilize energy harveting enor for uch remote etimation tak have tarted to emerge for variou application, including oil and ga indutrie, conumer electronic, chemical proceing, teel manufacturing [6 8]. Neverthele, fundamental performance limit of thee ytem from an etimation theoretic frameork are not fully invetigated. Our ork here contribute to thi apect by tudying uch a remote etimation problem under a LMMSE frameork. Compreive ening baed approache provide u an attractive et of tool for invetigating thee remote etimation problem. In particular, concept of parity allo u to tudy a large cla of ignal including correlated ignal. Moreover, the tool developed for tudying the effect of random meaurement in compreive ening literature provide promiing candidate for tudying the unreliable nature of available energy in EH ytem, a illutrated in thi ork. Hence, e believe that compreive ening baed approache ill be intrumental to tudy fundamental ening trade-off for future EH ening ytem. VI. NUMERICAL RESULTS We no illutrate our bound by preenting the trade-off beteen the guaranteed MSE and the probability of obtaining (c) 208 IEEE. Peronal ue i permitted, but republication/reditribution require IEEE permiion. See for more information.

10 Fme Q = Q = 2 Q = 4 Q = 8 Fme Q = Q = 2 Q = 4 Q = MSE MSE 0-3 Fig. 5: Empirical CDF of MSE (F me ), Bernoulli energy arrival, = 2, p = 0.5. Fig. 6: Empirical CDF of MSE (F me ), Bernoulli energy arrival, = 6, p = 0.3. that MSE. The horizontal axi correpond to the error bound a provided by ε I or ε II and the vertical axi correpond to the probability on the right-hand ide of (9)/(3), hich i referred a p uc. We ue f bt (.) for the calculation of p uc. Hence, the horizantal axi value of each point on the plot ho a particular error value and the vertical axi value ho the probability ith hich e can guarantee the MSE to be maller than that particular error value. We normalize the error bound ith the total uncertainty in the ignal, i.e. e report ε I / and ε II /. Unle otherie tated, e conider the energy arrival proce E t i.i.d. ith E t = κ t E b, κ t Bernoulli(p); E b = and U i the DFT matrix. Hence, the benchmark of (8)/(22) i equal to ε d = + 2 p N. Comparion of Bound I and Bound II: We note that both Bound I and Bound II are upper bound. Which bound i tighter (i.e. hich bound guarantee a given error value ith the highet probability) depend on the ytem parameter. We no illutrate thi point. Let N = 256, = N, 2 = 0 4. Both Bound I and Bound II are preented in Fig. 3 and Fig. 4, for = 4,p = 0.5 and = 8,p = 0.3, repectively. In Fig. 3, Bound I i tighter herea in Fig. 4 Bound II i tighter. Thi behaviour i conitent ith our other numerical invetigation here Bound I i oberved to be typically tighter for mall /N ratio and high energy arrival rate and vicea-vera for Bound II. For intance, if e decreae p value for Fig. 4, Bound I can no longer provide a guarantee ith non-zero probability. In the ret of thi ection, hile plotting the bound, for a given probability value p uc e preent the tightet of Bound I and Bound II, i.e. the bound that guarantee a given error value ith the highet probability. Comparion ith Empirical Performance: For comparion purpoe, e firt preent the empirical cumulative ditribution function (CDF) of the mean-quare error in (6). In thee experiment, e fix the upport (i.e. location of the nonzero eigenvalue) and look at the empirical CDF of the meanquare error ith random energy arrival over N im = 2000 realization. Let N = 52, =N, 2 =0 4. Empirical CDF value are preented in Fig. 5 and Fig. 6 for = 2, p = 0.5 and = 8, p = 0.3 repectively. The correponding bound are preented in Fig. 7 and Fig. 8. The benchmark of puc Q = Q = 2 Q = 4 Q = MSE Bound 0-4 Fig. 7: p uc veru MSE Bound, Bernoulli energy arrival, = 2, p = 0.5. (8)/(22) i ε d i for Fig. 5 and ε d i for Fig. 6. Comparing Fig. 5 and Fig. 7, e oberve that both the empirical reult and the bound ho that e ill operate cloe to thee benchmark ith high probability for thi cenario, here /N i mall and energy arrival rate p i high. On the other hand, the gap beteen the bound and the empirical reult, and alo the gap beteen the empirical reult and the benchmark of (8)/(22) are larger for Fig. 6 and Fig. 8 here the ratio /N i larger and p i maller. Thee obervation are conitent ith compreive ening literature here ignal recovery guarantee are provided only for pare ignal (lo /N value). Effect of Sytem Parameter on Performance Guarantee: In both Fig. 7 and Fig. 8, a the target performance become more demanding, i.e. the error value decreae, the probability that thi error can be guaranteed become maller. When the degree of freedom of the ignal i ufficiently lo (=2, Fig. 7), the performance bound i oberved to be relatively inenitive to the buffer ize. On the other hand, hen the degree of freedom i higher and energy arrival rate i maller, (=6, Fig. 8) the bound become more enitive to the buffer ize. For =6, ith Q =, the bound cannot provide any guarantee that hold ith probability higher than 0.9; herea ith higher buffer ize, relatively mall value of error can be guaranteed ith (c) 208 IEEE. Peronal ue i permitted, but republication/reditribution require IEEE permiion. See for more information.

11 0 puc Q = Q = 2 Q = 4 Q = MSE Bound Fig. 8: p uc veru MSE Bound, Bernoulli energy arrival, = 6, p = 0.3. puc MSE bound Q = Q = 2 Q = 4 Q = 8 Fig. 9: p uc veru MSE Bound, uniformly ditributed energy packet, = 6. high probability (for intance ith probability higher than 0.9). We oberve that a become larger, ignal can be aid to be more cloe to a hite ource, ith the limiting cae of =N correponding to an exactly hite ource. Hence, thee reult are conitent ith the reult of [7], [9], [30] dicued in Remark 3.2. We no conider the cenario ith E t i.i.d. ith uniformly ditributed energy packet, i.e. E t Uniform[0,E u ], E u = 0.6 in Fig. 9. Here E u i choen o that the uniform arrival cae here ha the ame average energy ith the Bernoulli arrival cenario of Fig. 8. Comparing Fig. 8 and Fig. 9, e oberve that in the uniform energy arrival cae of Fig. 9, long buffer length do not offer performance gain a they provide in the Bernoulli energy arrival cae of Fig. 8. Thi i conitent ith the fact that in the cae of uniformly ditributed arrival the variance of energy packet i maller and the need to pread the energy over ample by the ue of a buffer i expected to be le prominent compared to Bernoulli energy arrival cae. VII. CONCLUSIONS We have conidered remote etimation of an unknon field ith an EH enor ith a limited data and energy buffer. In contrat to much of the exiting ork, e have focued on a correlated ignal model. We have provided tructural reult in term of performance guarantee on the achievable ditortion under random energy arrival ith a block tranmiion cheme. Our performance guarantee provide inight into the trade-off beteen the ize of buffer, tatitical propertie of the energy arrival proce, degree of freedom of the ignal and the achievable ditortion. Thee reult alo have the advantage that their calculation require only knoledge of the mean, variance and finite upport about the energy arrival proce, hoe exact probability ditribution can be difficult to reliably etimate in practice. Generalization of our approach into etting that allo energy aving beteen tranmiion block and application to fading environment are conidered a important future reearch direction. VIII. PROOF OF THM. 3. The proof relie on the Matrix Berntein Inequality, a fundamental random matrix theory tool ued in compreive ening [44, Ch.8]. We firt prove the firt family of bound indexed by I in (9)-(2). We firt note that ε = i= λ i ( I + U GU 2 ), (39) + 2 λ min (U GU ). (40) In the remaining of the ection, e let k =,...,N T, t =,...,Q and ue the indexing z k,t = z (k )Q+t for any variable z i, i =,...,N. Let S k x 2 k,t (4) t= here 2 x k,t = 2 x (k )Q+t. By (4) and (4), e have p k = S k Ē k = S k E k,l, (42) here E k,l = E (k )Q+l. Let u i C denote the i th column of the matrix U. Let Y k,t u k,t u k,t C, ith u k,t = u (k )Q+t. Let u conider k= k= l= p k p k E[p k ], (43) W k Y k,t (44) t= Z k p k W k (45) Hence Q Z k = p k Y k,t E[p k ] Y k,t, (46) t= k= t= = U GU U ḠU, (47) here G = diag([p Qd,...,p NT Qd ]) R N N, Ḡ = diag([e[p ] Q,...,E[p NT ] Q ]) R N N and Q = [,...,] R Q i the vector of one. We ill no ue the Matrix Berntein Inequality on Z k to find loer bound for the eigenvalue of the firt term in (47). We ill then ue thee in (40) to bound the etimation error (c) 208 IEEE. Peronal ue i permitted, but republication/reditribution require IEEE permiion. See for more information.

12 Lemma 8.: [Matrix Berntein Inequality [44, Ch.8]] Let V,..., V M C be independent zero-mean Hermitian random matrice. Aume that V l µ V, l {,...,M} almot urely. Let V M ]. Then, for t > 0 l= E[V l 2 M P( V l t) f bt (µ V, V,t) f bn (µ V, V,t) (48) l= ith f bt (.) and f bn (.) a defined in (7)-(8). We note that Z k in (45) are tatitically independent Hermitian random matrice ith E[Z k ] = 0. We bound the pectral norm of Z k a follo ( ) Z k = p k W k max p k W k. (49) k We obtain the folloing bound for W k here W k = u k,t u k,t, (50) t= We alo have the folloing Qmax u k,tu k,t k,t, (5) = Qmax k,t 2, k,t (52) = Q, (53) max k,t u k,t 2. (54) W k W k = I =, (55) k= here (55) follo from the fact that for A 0 and B 0, λ max (A) λ max (A + B). By (53) and (55), e have the folloing W k min{q,}. (56) We no conider the term ith p k = p k E[p k ] in (49) max p k E[p k ] maxmax{p k E[p k ],E[p k ]} (57) k k max k max{ QE u Qµ E, Qµ E S k S k } (58) Qmax{E u µ E,µ E } (59) mins k µ E max{r E,} η L, (60) here e have ued E[p k ] = QE[E k ] = Qµ E, p k QE u and E u = r E µ E. Here (60) follo from S k = t= here 2 x k,t = Px u k,t 2 and 2 x k,t Qmin k,t 2 x k,t = Qη L, (6) η L min k,t u k,t 2. (62) Hence by (49), (56) and (60) Z k µ E η L max{r E,}min{Q,} µ I, k. We no conider the variance term, i.e., N T E[Zk 2 ] = N T k= k= max k (63) E[ p 2 k ]W2 k, (64) N T E[ p 2 k ] k= Wk 2, (65) here e have ued E[ p 2 k ]W2 k (max k E[ p 2 k ])W2 k and NT k= E[ p2 k ]W2 k (max ke[ p 2 k ]) N T k= W2 k. Here (65) follo from the fact that for Hermitian A, B ith A B, e have λ k (A) λ k (B), here λ k (.) denote the ordered eigenvalue [46, Cor ]. The pectral norm term in (65) can be bounded a N T k= Wk 2 max W k k k= W k, (66) min{q,} (67) here (66) follo from the fact that W k 0, ee for intance [47, Sec. 2], and (67) follo from (56) and (55). We no conider E[ p 2 k ] in (64). We have the folloing E[ p 2 k ] = S 2 k E[(E k,l E[E k,l ]) 2 ] (68) l= = Q S 2 ke[(e k,l E[E k,l ]) 2 ] (69) = Q E Sk 2 (70) E QηL 2 ( ) 2, (7) here E i the variance of the energy arrival proce a defined before, (68) follo from the fact that p k i a um of tatitically independent zero mean variable and (7) follo from Sk 2 Q2 (min k,t x 2 k,t ) 2 = Q 2 ηl 2(Px )2. Hence the variance term in (64) can be bounded a follo N T k= E[Zk] 2 E QηL 2 ( ) 2 min{q,} I. (72) Uing (63), (72) and the Matrix Berntein Inequality reveal that for r > 0, N T k= Z k < r hold ith probability greater than p bt = f bt ( µ I, I, r). We note that for Hermitian A,B, A B < r implie λ min (A) > λ min (B) r. Therefore, uing (47), ith probability greater than p bt λ min (U GU ) > λ min (U ḠU ) r (73) min k E[p k ] r (74) = µ E r (75) (c) 208 IEEE. Peronal ue i permitted, but republication/reditribution require IEEE permiion. See for more information.