On the Suboptimality of Rate--Distortion-Optimal Compression: Fundamental Accuracy Limits for Distributed Localization

1 On the Suboptimality of Rate–Distortion-Optimal Compression: Fundamental Accurac y Limits for Distrib uted Localization Amir W eiss Abstract —W e derive fundamental accuracy limits for dis- tributed localization when a fusion center has access only to in- dependently rate–distortion (RD)-optimally compressed versions of multi-sensor observations, under a line-of-sight propagation model with a Gaussian wideband wavef orm. Using the Gaussian RD test-channel model together with a Whittle spectral Fisher- information characterization, we obtain an explicit frequency- domain Cram ´ er –Rao lower bound. A two-band, two-le vel special- ization yields closed-form expressions and re veals a rate-induced regime change: RD-optimal compression under a squared-error distortion measur e can eliminate localization-informati ve spectral content. A simple band-selective scheme can outperf orm RD compression by orders of magnitude at the same rate, motivat- ing localization-aware compression for networked sensing and integrated sensing and communication systems. Index T erms —Distributed localization, rate–distortion theory , task-oriented compression, Cram ´ er –Rao lower bound. I . I N T RO D U C T I O N The proliferation of distrib uted sensing and communica- tion platforms—including Internet-of-Things deployments [ 1 ], sensor networks [ 2 ], wearable devices [ 3 ], and edge-assisted cyber –physical systems [ 4 ]—has accelerated the need to oper- ate under stringent resource constraints. In such settings, raw wa veform streaming from multiple nodes to a fusion center (FC) is often infeasible due to limited uplink rates, ener gy bud- gets, latency requirements, or shared-spectrum constraints. As a result, modern architectures increasingly rely on compressed observations [ 5 ], where each node transmits a rate-constrained representation of its measurements [ 6 ], while inference is performed centrally or cooperatively . Understanding the fun- damental impact of compression on estimation fidelity [ 7 ] is therefore critical for principled system design. A particularly important inference task is distributed lo- calization , which underpins numerous applications such as navigation, robotics, and industrial monitoring [ 8 ]. It has also emerged as a key capability in integrated sensing and communications (ISAC) [ 9 ]–[ 11 ], where distrib uted recei vers collect observ ations that must often be relayed o ver rate- limited links. This makes the interplay between compression and localization accuracy a first-order design consideration. Despite extensi ve work on localization and on compression for estimation in relative isolation (e.g., [ 12 ]–[ 16 ]), includ- ing task-oriented quantization perspectiv es [ 6 ], a Cram ´ er– Rao lower bound (CRLB) characterization for time-delay estimation (TDE) in dispersed-spectrum settings [ 17 ], and A. W eiss is with The Alexander K ofkin Faculty of Engineering, Bar-Ilan Univ ersity , Ramat Gan, 5290002, Israel, e-mail: amir .weiss@biu.ac.il. CRLB–versus–rate tradeof fs for rate-constrained distrib uted TDE from quantized observations [ 18 ], the literature that provides analytically e xplicit accuracy limits for localiza- tion fr om compr essed multi-sensor observations remains rela- tiv ely sparse, especially in wideband waveform models where time-of-arriv al structure is central. In particular , while rate– distortion (RD) theory [ 19 ] offers a tractable lens for modeling optimal compression under a squared-error distortion crite- rion, it is unclear how RD-optimal compression for signal r econstruction translates into localization performance, and whether compress-then-estimate design is well aligned with the information content most relev ant for localization. This letter addresses the follo wing fundamental question: What ar e the accuracy limits of distrib uted localization when the FC has access only to independently compressed ver - sions of the sensor observations? Under a line-of-sight wide- band Gaussian wa veform model, we deri ve frequency-domain CRLBs for localization from compressed observ ations by com- bining a spectral Fisher-information (FI) characterization with a Gaussian test-channel model of compression. W e specialize the bound to a two-band, two-le vel spectral model, yielding closed-form expressions that re veal nontrivial rate-dependent behavior , including regime changes induced by water-filling. Finally , we construct an explicit counterexample showing that compression optimized for mean-square signal reconstruction can be strictly suboptimal for localization accuracy , which mo- tiv ates localization-aware (task-oriented) compression beyond compress-then-estimate designs. I I . P RO B L E M F O R M U L A T I O N W e consider a distributed sensing system with M ≥ 2 spatially separated sensors. Let p m ∈ R d × 1 denote the known position of sensor m ∈ { 1 , . . . , M } , and let p ∈ R d × 1 denote the unknown source location, where typically d ∈ { 2 , 3 } . Assuming line-of-sight propagation with known speed c > 0 , the propagation delay to sensor m is τ m ( p ) ≜ ∥ p − p m ∥ 2 c ∈ R + , m ∈ { 1 , . . . , M } . (1) As will be evident from the second-order statistics below , and as is standard in passive localization, the dependence on p is only through the time-delay differences ∆ mℓ ( p ) ≜ τ m ( p ) − τ ℓ ( p ) . Thus, no “global time-shift” parameter is introduced. Each sensor observ es a noisy , time-shifted version of a common wideband source ov er the interval t ∈ [0 , T ] , x m ( t ) = s  t − τ m ( p )  + n m ( t ) ∈ R , m ∈ { 1 , . . . , M } , (2) 2 where s ( t ) is a zero-mean wide-sense stationary (WSS) Gaus- sian process with (two-sided) power spectral density (PSD) function S s ( f ) supported on | f | ≤ B for some known B > 0 . The noises { n m ( t ) } M m =1 are mutually independent, independent of s ( t ) , zero-mean WSS Gaussian processes with PSDs { S n m ( f ) } M m =1 (supported on | f | ≤ B ). Define the sensor vector process x ( t ) ≜ [ x 1 ( t ) · · · x M ( t )] T . Its (matrix-valued) cross-spectral density admits the form S x ( f ; p ) = S s ( f ) v ( f ; p ) v H ( f ; p ) + S n ( f ) ∈ C M × M , (3) where S n ( f ) ≜ diag  S n 1 ( f ) , . . . , S n M ( f )  and v ( f ; p ) ≜ î e − ȷ 2 π f τ 1 ( p ) · · · e − ȷ 2 π f τ M ( p ) ó T ∈ C M × 1 . (4) In particular , for m  = ℓ , [ S x ( f ; p )] mℓ = S s ( f ) e − ȷ 2 π f ∆ mℓ ( p ) ∈ C , (5) which shows explicitly that p af fects the observation law only through time-difference-of-arri val terms. In the distributed setting with communication constraints, each sensor m communicates to a FC over a rate-limited link of rate R m bits per second. Thus, the FC does not ha ve direct access to x m ( · ) , but rather to a compressed version thereof, denoted as b x m ( · ) , produced at sensor m under the rate constraint R m . Let b x ( t ) ≜ [ b x 1 ( t ) · · · b x M ( t )] T denote the vector process of the compressed signals available at the FC. Based on b x ( · ) ov er [0 , T ] , the FC constructs an estimator b p of the unknown location p . The focus of this letter is to characterize the fundamental localization limits from RD- optimally compressed observations by deriving the Cram ´ er– Rao lower bound (CRLB) on the mean-square error of any unbiased estimator of b p based on b x ( · ) over [0 , T ] . In addition, and perhaps surprisingly , we will sho w that it is easy to de- sign a strictly RD-suboptimal compression gi ving significantly higher localization accuracy , thus highlighting the need for dev eloping joint compression-localization schemes. I I I . C R L B F O R R D - O P T I M A L LY C O M P R E S S E D S I G N A L S Under the Gaussian signal model in Section II and the squared-error distortion measure, the RD optimal compression of a WSS Gaussian process admits a con venient test-channel representation (e.g., [ 19 ]). In particular , the m -th compressed wa veform a vailable at the FC can be modeled as b x m ( t ) = ( b m ∗ x m ) ( t ) + z m ( t ) ∈ R , (6) where b m ( t ) is a linear time-in variant (L TI) filter with (real- valued) frequency response B m ( f ) , and z m ( t ) is a zero-mean WSS Gaussian “compression-noise” process, independent of x m ( t ) , with PSD S z m ( f ) . Denoting by S x m ( f ) the PSD of x m ( t ) , the RD water -filling solution [ 20 ] implies that there exists a water lev el λ m ≥ 0 such that B m ( f ) = ï 1 − λ m S x m ( f ) ò + , S z m ( f ) = λ m B m ( f ) , (7) where [ u ] + ≜ max { u, 0 } . The water le vel λ m is uniquely determined by the rate constraint R m via R m = 1 2 ∞ Z −∞ ï log 2 Å S x m ( f ) λ m ãò + d f , (8) where the specialized integral ( 8 ) in our bandlimited signal setting is effecti vely ov er | f | ≤ B . Since each b x m ( t ) in ( 6 ) is obtained from a linear transforma- tion of x m ( t ) with an addition of independent WSS Gaussian noise, the compressed vector process b x ( t ) remains WSS Gaus- sian. Consequently , its statistical law is fully characterized by its cross-spectral density matrix S b x ( f ; p ) , given by S b x ( f ; p ) = S s ( f ) B ( f ) v ( f ; p ) v H ( f ; p ) B ( f ) + Σ ( f ) , (9) where we hav e defined B ( f ) ≜ diag( B 1 ( f ) , . . . , B M ( f )) ∈ R M × M + , (10) and using { S w m ( f ) ≜ | B m ( f ) | 2 S n m ( f ) + S z m ( f ) } M m =1 , Σ ( f ) ≜ diag( S w 1 ( f ) , . . . , S w M ( f )) ∈ R M × M + . (11) A. CRLB via a spectral FI Since b x ( t ) is WSS Gaussian with a cross-spectral density matrix S b x ( f ; p ) , the localization information can be expressed in the frequency domain. Specifically , under standard regular- ity conditions for purely non-deterministic stationary Gaussian processes, 1 the FI rate (i.e., per unit time) matrix admits the Whittle spectral representation [ 21 ], i.e., lim T →∞ 1 T [ J RD ( p )] ij (12) = 1 2 ∞ Z −∞ tr Ä S − 1 b x ( f ; p ) S ′ b x ,i ( f ; p ) S − 1 b x ( f ; p ) S ′ b x ,j ( f ; p ) ä d f , (13) where, for all 1 ≤ i, j ≤ d , [ J RD ( p )] ij denotes the ( i, j ) -th element of the FI rate matrix for the RD-optimally compressed signals b x ( · ) , S ′ b x ,i ( f ; p ) ≜ ∂ S b x ( f ; p ) ∂ p i , and in our bandlimited model the integral reduces to f ∈ [ − B , B ] . For an observation horizon T , the CRLB is (e.g., [ 22 ]) C o v ( b p − p ) ≜ E î ( b p − p ) ( b p − p ) T ó ⪰ J RD T − 1 ( p ) , (14) where J RD T ( p ) denotes the FI matrix (FIM) for p based on b x ( · ) ov er [0 , T ] . Moreover , J RD T ( p ) grows linearly with T , and J RD ∞ ( p ) ≜ lim T →∞ 1 T J RD T ( p ) (15) exists and is given by ( 12 ). Thus, J RD T ( p ) = T J RD ∞ ( p ) + o ( T ) . T o ev aluate ( 13 ), it remains to compute S ′ b x ,i ( f ; p ) . Recall ( 9 ), and that Σ ( f ) is independent of p . Thus, the dependence in p is only through the phase terms e − ȷ 2 π f τ m ( p ) in v ( f ; p ) . Differentiating ( 1 ) yields, for all 1 ≤ i ≤ d , ∂ τ m ( p ) ∂ p i = 1 c p i − [ p m ] i ∥ p − p m ∥ 2 , m ∈ { 1 , . . . , M } , (16) and consequently , for each m , ∂ [ v ( f ; p )] m ∂ p i = − ȷ 2 π f ∂ τ m ( p ) ∂ p i [ v ( f ; p )] m . (17) Using the notation v ′ i ( f ; p ) ≜ ∂ v ( f ; p ) ∂ p i , by the product rule, S ′ b x ,i ( f ; p ) = S s ( f ) B ( f ) Q i ( f ; p ) B ( f ) , (18) 1 For example, it is sufficient to assume that S b x ( f ; p ) is uniformly bounded and positiv e definite on [ − B , B ] , and is continuously differentiable in p . 3 where Q i ( f ; p ) ≜ v ′ i ( f ; p ) v H ( f ; p ) + v ( f ; p ) v ′ H i ( f ; p ) , and which together with ( 12 ) yields an explicit CRLB expression as a one-dimensional integral over f ∈ [ − B , B ] . W e next spe- cialize this expression to a two-band, two-lev el spectral model to obtain fully closed-form bounds and to show , with a simple RD-suboptimal compression, that RD-optimal compression for reconstruction can be strictly suboptimal for localization. I V . R D I S N O T L O C A L I Z A T I O N - O P T I M A L : A T W O - B A N D C O U N T E R E X A M P L E The CRLB expression deriv ed in Section III is explicit, but still in volves a frequency integral over the (possibly arbitrary) source PSD S s ( f ) , and the RD water-filling induces rate- dependent changes in the effecti ve spectrum seen at the FC. It is therefore generally unclear whether RD-optimality for wa veform reconstruction transfers to optimality in terms of localization accuracy (or even preserves the most localization- informativ e spectral components). In this section, we adopt an explicit two-band, two-lev el spectral model that (i) yields closed-form expressions for the CRLB; and (ii) already cap- tures the key phenomenon: RD-optimal compression for signal reconstruction can be strictly suboptimal for localization. A. T wo-band, two-level spectral model Fix 0 < f L < f H ≤ B and define the two disjoint bands B L ≜ { f : | f | ≤ f L } , B H ≜ { f : f L < | f | ≤ f H } . (19) W e assume the follo wing piece wise-constant source spectrum S s ( f ) =      S L , f ∈ B L , S H , f ∈ B H , 0 , | f | > f H , (20) with S L > S H > 0 . For simplicity , we also assume identical sensor noises with flat PSD in the relev ant band, S n m ( f ) = ® N 0 , | f | ≤ f H , 0 , | f | > f H , ∀ m ∈ { 1 , . . . , M } , (21) so that S x m ( f ) = S s ( f ) + N 0 for | f | ≤ f H . Under ( 20 )–( 21 ), the RD water-filling solution ( 7 ) is also piecewise constant ov er the two bands. In particular , letting λ m denote the water le vel at sensor m , we ha ve B m ( f ) =          î 1 − λ m S L + N 0 ó + , f ∈ B L , î 1 − λ m S H + N 0 ó + , f ∈ B H , 0 , | f | > f H , (22) and S z m ( f ) = λ m B m ( f ) for | f | ≤ f H . As R m decreases, λ m increases and a r e gime change occurs at λ m = S H + N 0 : • High compression rates : when λ m < S H + N 0 , both bands are activ e and B m ( f ) > 0 on B L ∪ B H . • Intermediate compr ession rates : when S H + N 0 ≤ λ m < S L + N 0 , then B m ( f ) > 0 on B L but B m ( f ) = 0 on B H . Critically , the RD solution dr ops the high-fr equency band . • Low compression rates : when λ m ≥ S L + N 0 , B m ( f ) ≡ 0 and no information is con ve yed. Thus, in this two-band two-le vel spectral model, RD-optimal compression can eliminate an entire band as the rate decreases. Howe ver , recall from ( 17 ) that the geometry dependence enters through the phase terms e − ȷ 2 π f τ m ( p ) , and differentia- tion introduces a factor of f . Consequently , the integrand in ( 13 ) contains an intrinsic fr equency-squared weighting: each deriv ativ e matrix S ′ b x ,i ( f ; p ) is proportional to f , hence the trace term tr  S − 1 b x S ′ b x ,i S − 1 b x S ′ b x ,j  scales as f 2 (up to other spectral factors). Intuitiv ely , high-frequency components can therefore be substantially more informative for localization, ev en when they carry less signal energy . B. Closed-form CRLB under the two-band model W e now specialize ( 13 ) under ( 20 )–( 22 ) and deri ve a closed- form e xpression. F or clarity of exposition (and since the phenomenon is per -sensor), we assume the symmetric setting R m = R ⇒ ® λ m = λ, B m ( f ) = B ( f ) , ∀ m ∈ { 1 , . . . , M } . (23) For b ∈ { L , H } , define the per-band effecti ve SNR level γ b ( R ) ≜ S b B 2 b ( R ) S w ,b ( R ) , S w ,b ( R ) ≜ B 2 b ( R ) N 0 + λ ( R ) B b ( R ) , (24) where S L , S H are giv en in ( 20 ), λ ( R ) is the RD water level satisfying ( 8 ), and B b ( R ) ≜ B ( f ) for any f ∈ B b (which is constant ov er each band under ( 22 )). Define the band information weights, w b ( R ) ≜ 2 M γ b ( R ) 2 1 + M γ b ( R ) , b ∈ { L , H } , (25) and with it the lower and higher frequency information terms, J L ( R ) ≜ f 3 L w L ( R ) , J H ( R ) ≜  f 3 H − f 3 L  w H ( R ) . (26) Finally , define (entrywise) the geometry matrix G ( p ) ∈ R d × d , [ G ( p )] ij ≜ e g T i ( p ) e g j ( p ) , e g i ( p ) ≜ Pg i ( p ) , (27) where P ≜ I M − 1 M 11 T ∈ R M × M is a projection matrix and g i ( p ) ≜ ï ∂ τ 1 ( p ) ∂ p i · · · ∂ τ M ( p ) ∂ p i ò T ∈ R M × 1 . (28) W e are now ready to state our main result. Theor em 1 (Closed-form FI rate and CRLB under the two- band model): Under the two-band, two-le vel spectrum model ( 20 )–( 21 ), RD-optimal compression ( 22 ) and the symmetric setting ( 23 ), the FI rate for p based on b x ( · ) ov er [0 , T ] reads J RD ∞ ( p ) = 4 π 2 3 ( J L ( R ) + J H ( R )) G ( p ) . (29) Consequently , for any unbiased estimator b p of p based on b x ( · ) ov er [0 , T ] , it holds that C o v( b p − p ) ⪰ J RD − 1 T ( p ) = 1 T J RD − 1 ∞ ( p ) + o Å 1 T ã (30) = 3 4 π 2 T 1 J L ( R ) + J H ( R ) G − 1 ( p ) + o Å 1 T ã , (31) whenev er G ( p ) is nonsingular . Pr oof: See Appendix A . Remark 1: Note that the dependence on p is entirely through the geometry-dependent centered delay gradients e g i = Pg i . 4 Under ( 20 )–( 21 ), the PSD of x m ( · ) is S x m ( f ) = S s ( f ) + N 0 for | f | ≤ f H , hence the rate constraint ( 8 ) becomes R = f L  log 2  S L + N 0 λ  + + ( f H − f L )  log 2  S H + N 0 λ  + . (32) Define the critical rate (corresponding to λ = S H + N 0 ), R crit ≜ f L log 2 Å S L + N 0 S H + N 0 ã . (33) Then λ ( R ) is explicit: • High-rates ( R > R crit ): both bands are activ e ( λ < S H + N 0 ) and λ ( R ) = 2 − R f H ( S L + N 0 ) f L f H ( S H + N 0 ) f H − f L f H . (34) Thus B b ( R ) = 1 − λ ( R ) / ( S b + N 0 ) for b ∈ { L , H } , and w b ( R ) follows from ( 24 )–( 25 ). • Intermediate-rates ( 0 < R ≤ R crit ): only B L is acti ve ( S H + N 0 ≤ λ < S L + N 0 ) and λ ( R ) = ( S L + N 0 ) 2 − R/f L , (35) B L ( R ) = 1 − 2 − R/f L , B H ( R ) = 0 , (36) so that w H ( R ) = 0 and accordingly J H ( R ) = 0 . Along with ( 33 )–( 36 ), ( 31 ) provides an explicit closed-form CRLB parameterized by the physically meaningful quantities ( S L , S H , N 0 , f L , f H , M , R, T , { p m } , c ) . Specifically , when R drops below R crit , RD-optimal compression for signal recon- struction eliminates B H (hence w H ( R ) = 0 ), ev en though the Fisher integrand scales as f 2 , indicating that higher-frequenc y components can carry more localization-related information. This mechanism is generally localization-suboptimal and can lead to a catastrophic degradation in localization accurac y . T o see this more clearly , consider the following example. Fix a rate R ≤ R crit , for which RD-optimal compression satisfies B H ( R ) = 0 and hence J H ( R ) = 0 . Now , consider instead the follo wing (RD-suboptimal) band-selective com- pression scheme: each sensor suppresses B L and applies the Gaussian test channel only on B H , i.e., B sel ( f ) = ( 0 , f ∈ B L , î 1 − λ sel S H + N 0 ó + , f ∈ B H , (37) such that S sel z ( f ) = λ sel B sel ( f ) on B H and zero otherwise. Imposing the same rate R ov er bandwidth ( f H − f L ) giv es R = ( f H − f L ) ï log 2 Å S H + N 0 λ sel ãò + , (38) λ sel ( R ) = ( S H + N 0 ) 2 − R/ ( f H − f L ) . (39) Hence on B H , B sel H ( R ) , B sel L ( R ) = 0 . (40) Define γ sel H ( R ) and w sel H ( R ) by the same formulas as in ( 24 )– ( 25 ), but with ( B b , λ ) replaced by ( B sel b , λ sel ) . Then, the FI rate under the band-selectiv e scheme satisfies J sel ∞ ( p ) = 4 π 2 3 J sel H ( R ) G ( p ) , J sel H ( R ) ≜ ( f 3 H − f 3 L ) w sel H ( R ) , (41) while under RD (for R ≤ R crit ) we hav e J RD H ( R ) = 0 , hence J RD ∞ ( p ) = 4 π 2 3 J RD L ( R ) G ( p ) , J RD L ( R ) = f 3 L w L ( R ) . (42) If J sel H ( R ) > J RD L ( R ) , then J sel ∞ ( p ) ≻ J RD ∞ ( p ) and, consequently ,  J − 1 T ( p )  sel ii  J − 1 T ( p )  RD ii = J RD L ( R ) J sel H ( R ) < 1 , 1 ≤ i ≤ d, (43) i.e., the CRLB for the band-selectiv e scheme is strictly smaller . As a more concrete example, consider a mmW ave/ISA C- like wideband regime [ 23 ], [ 24 ] with M = 4 sensors, wherein f H = 200 MHz , and further fix f L = 5 MHz , S L = 100 , S H = 20 , and N 0 = 1 . For these v alues, R crit ≈ 11 . 3 Mb / s . For a rate R = 10 Mb / s < R crit (a plausible per-sensor backhaul budget), using the expressions from Theorem 1 , we obtain in this setting  J − 1 T ( p )  sel ii  J − 1 T ( p )  RD ii = J RD L ( R ) J sel H ( R ) ≈ 9 . 97 × 10 − 3 , 1 ≤ i ≤ d, (44) i.e., at the same rate R the band-selectiv e scheme yields a reduction of two orders of magnitude in the CRLB. V . D I S C U S S I O N A N D O U T L O O K This letter deriv ed fundamental localization limits when a FC has access only to rate-constrained (compressed) versions of wideband multi-sensor observations. Le veraging a Gaussian line-of-sight wa veform model and the Gaussian RD test- channel representation, we obtained an explicit frequency- domain characterization of the Fisher information and the associated CRLB. Specializing further to a simple, yet insight- ful two-band, two-lev el spectrum model yielded closed-form expressions that transparently expose the interaction between rate allocation, spectral content, and localization accuracy . Beyond providing an analytically tractable CRLB, the two- band specialization highlights a key conceptual message: under the Gaussian per-sensor RD benchmark, compression opti- mized for wav eform reconstruction can be poorly aligned with localization. In particular , under RD-optimal compression, decreasing the rate can induce a sharp regime change in which high-frequency components are eliminated when the compression rate is sufficiently low , despite the fact that localization information is inherently weighted tow ards higher frequencies due to the phase sensitivity . This rev eals an explicit mechanism by which compress-then-estimate designs based on reconstruction-optimal per-sensor RD compression may incur a dramatic loss in localization accuracy . This observation, in turn, motiv ates task- or goal-oriented compression strategies that preserve the most localization-informativ e signal features. Naturally , there are many related important questions that remain to be addressed. While our analysis focused on RD- optimal compression under the quadratic distortion criterion, a central direction is to characterize and design localization- awar e compression rules, e.g., by formulating rate-allocation problems that maximize localization fidelity subject to rate constraints. It is also of both theoretical and practical interest to extend the framework beyond line-of-sight to multipath and cluttered en vironments, and to integrate the resulting bounds into ISA C-oriented system design, including bandwidth alloca- tion, and joint sensing–communication resource management. 5 A P P E N D I X A D E R I V A T I O N O F T H E C R L B F O R T H E T W O - BA N D M O D E L Fix a band b ∈ { L , H } and a frequency f ∈ B b . Under ( 20 )–( 21 ) and the symmetric assumption ( 23 ), the compressed cross-spectral density ( 9 ) reduces on B b to S b x ( f ; p ) = S b B 2 b v ( f ; p ) v H ( f ; p ) + S w ,b I M , (S1) where S b ∈ { S L , S H } is the source PSD lev el on B b , B b ≜ B ( f ) is constant on B b , and recall ( 24 ), so that S b x ( f ; p ) = S w ,b  I M + γ b v ( f ; p ) v H ( f ; p )  . (S2) Since v H ( f ; p ) v ( f ; p ) = M , the Sherman–Morrison formula giv es S − 1 b x ( f ; p ) = 1 S w ,b K b ( f ; p ) , (S3) K b ( f ; p ) ≜ I M − γ b 1+ M γ b v ( f ; p ) v H ( f ; p ) . (S4) Next, recall ( 16 ) and ( 28 ), and with it define the diagonal matrix D i ( p ) ≜ diag( g i ( p )) . Then, ( 17 ) becomes v ′ i ( f ; p ) = − ȷ 2 π f D i ( p ) v ( f ; p ) . (S5) From ( 23 ), B ( f ) = B b I M on B b , hence ( 18 ) giv es S ′ b x ,i ( f ; p ) = S b B 2 b  v ′ i v H + vv ′ H i  (S6) = ȷ 2 π f S b B 2 b  vv H D i − D i vv H  , f ∈ B b , (S7) where we suppress ( f ; p ) and ( p ) for brevity . W e recall that, with the projection matrix P = I M − 1 M 11 T , we have e g i ( p ) = Pg i ( p ) ( 27 ), and by further defining ¯ g i ( p ) ≜ 1 M 1g T i ( p ) , we have e g i ( p ) = g i ( p ) − ¯ g i ( p ) 1 . Note also that v H D i v = P M m =1 g m,i = M ¯ g i since | [ v ] m | = 1 . T o prove the theorem, we will use the follo wing key lemma. Lemma 1: For each b ∈ { L , H } and f ∈ B b , tr  S − 1 b x ( f ; p ) S ′ b x ,i ( f ; p ) S − 1 b x ( f ; p ) S ′ b x ,j ( f ; p )  = (2 π f ) 2 w b ( R )[ G ( p )] ij . (S8) Pr oof: Fix b and f ∈ B b , and suppress ( f ; p ) for bre vity . Using ( S3 ) and ( S7 ), tr  S − 1 S ′ i S − 1 S ′ j  =  ȷ 2 π f S b B 2 b  2 S 2 w ,b tr  K b A i K b A j  = − (2 π f ) 2 γ 2 b tr  K b A i K b A j  , (S9) where A i ≜ D i vv H − vv H D i and we used ȷ 2 = − 1 . Define ‹ D i ≜ D i − ¯ g i I M and u i ≜ ‹ D i v . Then v H u i = v H ‹ D i v = M ¯ g i − M ¯ g i = 0 , hence u i ⊥ v , and A i = u i v H − vu H i . (S10) Moreov er , since u i = diag( e g i ) v and | [ v ] m | = 1 , u H i u j = M X m =1 e g m,i e g m,j = e g T i e g j = [ G ( p )] ij , (S11) where we recall the definition ( 27 ) of the geometry matrix. Next, note that K b v = 1 1+ M γ b v and K b u i = u i (since u i ⊥ v and K b is a rank-one modification along span { v } ). Therefore, by ( S10 ) we hav e K b A i K b = 1 1+ M γ b A i , hence tr  K b A i K b A j  = 1 1+ M γ b tr( A i A j ) . (S12) Finally , using ( S10 ) and v H u i = u H i v = 0 , A i A j = ( u i v H − vu H i )( u j v H − vu H j ) (S13) = − u i ( v H v ) u H j − v ( u H i u j ) v H , (S14) so that tr( A i A j ) = − M tr( u i u H j ) − ( u H i u j ) tr( vv H ) = − M ( u H j u i ) − M ( u H i u j ) = − 2 M u H i u j . (S15) Combining ( S9 ), ( S11 ), ( S12 ) and ( S15 ) yields tr  S − 1 S ′ i S − 1 S ′ j  = (2 π f ) 2 γ 2 b 2 M 1 + M γ b [ G ( p )] ij , (S16) which prov es the lemma. Using Lemma 1 in ( 13 ) and summing the two bands gi ves [ J RD ∞ ( p )] ij = 2 π 2 Ñ X b ∈{ L , H } Z B b f 2 w b ( R ) d f é [ G ( p )] ij . (S17) or , equiv alently , in matrix form, J RD ∞ ( p ) = 2 π 2 Ñ X b ∈{ L , H } Z B b f 2 w b ( R ) d f é G ( p ) . (S18) Since the scalar term is constant ov er each band, we obtain J RD ∞ ( p ) = 4 3 π 2 ( J L + J H ) G ( p ) , (S19) where lo wer and higher frequency information terms, J L and J H , respectiv ely , are defined in ( 26 ). Using the relation ( 15 ) and in verting the FIM J RD T ( p ) concludes the proof. R E F E R E N C E S [1] L. Da Xu, W . He, and S. Li, “Internet of things in industries: A survey , ” IEEE T rans. Ind. Informat. , vol. 10, no. 4, pp. 2233–2243, 2014. [2] D. Wu and J. Liebeherr , “ A low-cost low-power LoRa mesh network for large-scale en vironmental sensing, ” IEEE Internet Things J. , vol. 10, no. 19, pp. 16700–16714, 2023. [3] T . Sztyler and H. 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