Angular Spectral Plane-Wave Expansion of Nonstationary Random Fields in Stochastic Mode-Stirred Reverberation Processes

Angular Sp ectral Plane-W a v e Expansio n of Nonstationary Random Fields in Sto c hastic Mo de-Stirred R ev erb eration Pro cesses Luk R. Arnaut Time, Quantum and Ele ctr omagnetics Div ision, National Physic al L ab or atory, T e ddington TW11 0L W, Unite d Kingdom and Dep artment of Ele ctric al and Ele ctro nic Engine ering, Imp erial Col le ge of S cie nc e, T e chnolo gy and Me dicine, South Kensington Campus, L ondon SW7 2AZ, Unite d Kingdom Septem b er 13, 2018 Abstract W e derive an in tegral expre ssion for the plane-wav e expansion of the time-v arying (nonsta tion- ary) rando m field inside a mo de-stirre d r ev erb eration cham b er. It is s ho wn that this expansio n is a so - called oscilla to ry pr o cess, whos e kernel can b e express ed explicitly in clo sed form. The effect of nonstationar it y is a mo dulation of the sp ectral field on a time scale that is a function of the cavit y relax ation time. It is also s ho wn how the contribution b y a nonze r o initial v alue of the field can be incorp orated int o the expans ion. The results are extended to a sp ecial class of s econd-order pro cesses, relev a n t to the p erception of a mo de-stirred r e v erb eration field by a device under test with a first-order (r e laxation-type) frequenc y res p onse. 1 In tro duction In recent y ears, complex scenarios and ap p lica tions in electromagnetism h a ve sparke d a renewed in- terest in the theory and charac terization of r andom electromagnetic (EM) fi elds in static and d ynamic en vironments. Examples include c haracterization of the imp edance and radiation efficiency of electri- cally sm all an tennas [1], multi -terminal wireless comm unications systems in multi -path high-mobility 1 propagation environmen ts [2], dynamic atmospheric p ropagat ion effects [3] on th e group dela y for satellite -based na vigation systems, radio-frequency and optical sp ec kle and scint illation asso ciated with interac tion of EM w av es with rough surfaces or random media [4], p arametric ally v arying mi- cro w av e b illia rd s and ca vities [5]–[7], electromagnetic compatibilit y (EMC) including testing of emis- sions or immunit y of electronic equipment to irr adiati n g EM fields [8], etc. Con tinuously ev olving m ulti-scattering ele ctromagnetic en vironments (EMEs) are ubiquitous but difficult to c haracterize and analyze accurately , ev en using full-wa v e numerical sim ulation metho ds, b ecause of their large size relativ e to the w a ve length and consequent extreme s ensivit y to small p ertu r bations or configurational uncertain ties, and b ecause of complicated transient effects o ccurring in the pr opaga ting signals th at result fr om dynamically c hanging b ound ary conditions, esp ecially in resonan t multi-sca tter environ- men ts [7]–[8]. Because of the spatial and/or temp oral flu ctuations of th e wa ve prop erties in such an EME, their c haracteristics can often b e considered as quasi-random and can b e efficien tly mo delled using sto c hastic metho ds. F rom a p h ys ics p oint of view, the task is then to c haracterize the statistical parameters and distr ib utions of the random field in terms of deterministic physical q u an tities of the problem of interest. A canonical E ME for generating statistical ly rand om v ector fields is the mod e-tuned or mo de- stirred rev erb eration c h amb er (M T/MSRC) [9], [10]. It consists of an e lectrically large ca vit y furnish ed with a mechanical stirring mec hanism to alter the b ound ary and/or excitation conditions (e.g., a large reflectiv e paddle wheel or moving walls) or a v arying (h op p ing or sw ept) excitation frequency or n oise source for mixing the mo dal fields. Suc h ca vities are imp ortan t devices for studing the temp oral, spatial, str uctural, and p olarizati on prop erties of lo cal instan taneous scalar and vecto r w a ve fields, as w ell as c haracteristics of their spatio-temp oral coherence (correlation) prop erties. F or a static or quasi-static MT/MSRC, the assu mption of wide-sense quasi-stationarit y of the field is an excellen t and often-made approxima tion. Ho w ev er, when the rate of c hange of the EME (expressed by the correlation time in space or time) is relativ ely f ast compared to the c haracteristic time or length scale of the field in th e corresp onding static EME (e.g., frequency , relaxation time), this assump tio n b r eaks d o wn [11]. In this case, more sophisticated field m odels and methods of analysis are necessary . Th e purp ose of the present pap er is to develo p a framewo rk f or the th eo retical c haracterization of su ch nonstationary fields . The issue b ears resemblance to the fi ltered observ ation of dynamic scenes through natur al vision. The limited image r eso lution time of the human nak ed ey e ( ∼ 30 ms) and pro cessing time of vision b y the brain ( ∼ 150 m s) limits the amoun t and capacit y of visual information th at can b e faithfu lly captured in r ea l time. In particular, u ltra- sh ort-t erm transien t natural phenomena ma y th er efore b e p erceiv ed as distorted (blurred ) or eve n completely maske d . This is a m an if estation of (nonlinear) 2 nonstationary filtering of the actual scene. Using high-sp eed cameras, the image fr ames and dynam- ics of the hidden realit y can b e reconstructed. P arenthetic ally , comp ound ey es allo w f or detecting transien ts more rapidly , through high temp oral r esolutio n of fl ick er across th e ommatidia, but at the exp ense of p o orer sp at ial resolution compared to single-c h am b ered ey es [12], [13]. 2 Metho ds for expansion of v ector EM fields in dynamic complex confined en vironmen ts F or the repr esen tation of fields in side a MSRC or in a complex E ME exhib iting multiple scattering or m ultipath fading, t wo m etho ds h a ve f ound app lic ation. Under idealized cond itions, b oth metho ds should lead to equiv alen t results, but eac h has its particular adv an tages and disadv ant ages. Mo dal ex- p ansions (MEs) consider the EME as a m ulti-mo de system c haracterized by a set of vecto r eigenmo des ψ mnp at d iscrete resonance frequencies (eigenfrequencies) ω mnp at time t . These eigenmod es form a set of basis f unctions for expand ing the lo cal s tationary interio r field, with expansion co efficien ts a mnp . In a d ynamic EME, the eigenmo des and exp ansion co efficien ts b oth dep end on time [14]: E ( r , t ) = X m,n,p a mnp ( t ) ψ mnp ( t ) . (1) MEs are exact, in the sense of inherently incorp orating the E M b oun dary cond itio ns and ca vity geometry int o the description. Since m odes are n onlocal, the description inherently in cludes the spatial inhomogeneit y of the field caused by the vicinity of b ound aries and ob jects. Since a ME do es not presume quasi-randomness of the field, it can b e used in principle at an y f requency , although the in crea sin g mod al coun t and mo dal o v erlap (mo dal coupling) in high-Q c hambers f or increasing frequencies ma y mak e their imp lemen tation prohibitiv e at v ery high frequencies. A greater b urden is th e fact that realistic c hamb ers often exhibit complex-shap ed time-v arying b oundaries (diffractors, e.g., corrugations and mo de stirrers), w hic h h ampers the pr ac tical implemen tation of MEs that require the calculati on and/or functional form of the eigenmo des. F urthermore, a correct description requ ires the ca vity to b e d escribed as a partially op ened system [15], due to the transmitter and r ec eive r in tro ducing ap ertures to the unb ou n ded exterior region of the ca vit y . This requir es mixed (Robin) b oundary conditions and complicates the analysis and eigenmo de charact erization. In a second metho d, angular sp e ctr al plane-wave exp ansions (ASP WEs) ha ve b een used f or d ete r- ministic [16], [17] and r andom [18], [19], [20] fields. I n ASPWEs, the lo ca l field with a s u ppressed exp(j ω t ) time d ep end ence for its carrier is expanded as a discrete sum (for b ound ed closed finite-sized ca v ities) or in tegral (for un b ound ed op ened regions supp orting quasi-random fields) of an angular 3 sp ectrum of p lane wa ve s [16], [21], [22], [23]: E ( r , t ) = 1 Ω Z Z Ω E (Ω , t ) exp [ − j k ( t ) · r ] dΩ (2) where Ω = 2 π sr or 4 π sr is the solid angle asso ciated with a half-space or entire space for the inciden t fi eld, r esp ec tive ly . Unlik e MEs, ASPWEs provide c haracterizations of the lo cal 1 field, thus a v oiding the need for determin in g eigenmo des explicitly . They lend thems elves easily to statistical c haracterization of this lo cal fi eld and, when extente d to spatial correlation functions, to nonlo calit y within a finite region. With the aid of prop osed axioms f or the a v erage and co v ariance of the amplitude sp ectrum [3], [24], [25] and sp ectral densit y [21] of the w av e comp onen ts, explicit calculation of a ve rage and co v ariance of the expanded random field is p ossible. By a p osteriori imp osing EM b oundary conditions, the AS PWE ev en enables a complete d eriv ation of prob ab ility dens ity functions based on the statistical moment s [23], [26]. In b oth MEs and ASPWEs, in tegral repr esen tations are us ed as approximati ons to discrete sums in ca v ities that are sufficien tly large relativ e to the wa v elength. Strictly , b oth metho ds are applicable only for static or qu asi- statically p erturb ed ca vities, for wh ich the concept of mo des and their decomp osition in to s tand ing plane w a ves are prop erly defi ned. When b oundaries are mov ed, complicatio ns arise relating to n onstati onary transient fields and dynamics of eigenmo des, as w ell as the ve ry concept and definition of eigenmo des in suc h circum stance s. Note that, in order for the effect of th is m ot ion to b e significant, this do es n ot necessitate relativisitic sp eeds, u nlik e in u n b oun ded EMEs: the relev an t time scales are the r atios of the correlation time of the mo de-stirring pro cess to the deca y constant of the ca vit y (at a giv en frequ ency), to the resp onse time of th e ante nn a or to th e c haracteristic time (e.g., cycle p erio d) of the device und er test (DUT), and of course to the p erio d of the source wa ve itself [27]. F or example, it is well known that ev en small Doppler shifts (cf. Sec. 3.1), frequen cy c hanges of the order of a few hundred Hz at 1 GHz may cause degradation of p erformance of wireless comm unication systems, b ecause they giv e rise to considerable jitter and d isto rtion in impulse radio and other w ideband signals. In this pap er, w e devel op an extension of the ASPWE for steady-state fields to nonstationary fields. The analysis is for temp oral n onstat ionarity of scalar fields ; corresp onding results for s p atia l nonstationarit y and vect or fields follo w mutatis m utandis. 1 If the field is statistically homogeneous (spatially uniform), then the characterization applies of course throughout the volume. 4 3 Rev erb eration pro cess with stirring slip 3.1 Sp ectral expansion A mo del of a slippin g rand om field based on a random-w alk m o del wa s present ed in [28], to whic h we refer for details. Here we briefly review and summarize the main results of this mo del. W e consider the evol ution of a nons ta tionary detected statistical mean field Y ( t ), resu lting fr om a stationary sour ce field X ( t ) (Fig. 1). Here, X ( t ) is tak en to b e a p erio dic field with deterministic w a vefo rm , as emitted inside a static ca vity . (By extension, a wide-sense stationary n oi se could b e treated.) In the application to EMC in Sec. 3.5, X ( t ) and Y ( t ) are mo dulated complex RF fi elds. The field Y ( t ) is pro duced by rand om step c hanges δ X ( t ) in duced by a c han ge in b oun dary conditions and weig hted in th e mean b y a relaxati on time, i.e., it is go verned b y the first-order Langevin–Itˆ o sto c hastic differen tial equ at ion (S DE): d Y ( t ) d t + 1 τ Y ( t ) = 1 τ X ( t ) . (3) This equation of motion for the r an d om fi eld sp an s a v ariet y of cases of field dyn amics across a time in terv al ∆ t , ranging from instant aneous resp onse (∆ t/τ → + ∞ ) to Brownian m ot ion (∆ t/τ → 0). Within eac h infinitesimal time in cremen t δ t , Y ( t ) ev olve s fr om its initial v alue at t 0 in the direction of its asymptotic steady-state v alue th at wo uld b e reac hed for ∆ t → + ∞ . In Fig. 1, Y (0) ( t i ) for t i ∈ [ t 0 , t 1 ] r ep resen ts the mean outgoing (fading) fi eld for t i , i.e., represent ed b y the set of p lane- w a ve comp onen ts asso ciated with the “previous” state of the sys tem and ev aluated at t = t i − . This “previous” field d eca ys at a rate go v ern ed by the deca y time τ for eac h plane-wa v e comp onen t. By con trast, Y (1) ( t i ) at an y t i ∈ [ t 0 , t 1 ] represents the mean incoming (emerging) fi eld at t i , i.e., asso ciated with the set of plane-wa v e comp onen ts f or the “next” state, as ev aluated at t = t i + . The fact that the outgoing field requires a nonzero time to v anish giv es rise to a “slipp ing” fi eld, i.e., a pro cess with a gradually fading memory , s u c h that Y ( t ) is a nonuniformly w eigh ted mixture of b oth con tribu tio ns , viz., Y ( t 0 + ∆ t ) = Y (1) ( t 0 + ∆ t ) + c (∆ t ) Y (0) ( t 0 ) with 0 ≤ | c (∆ t ) | ≤ 1 (cf. [28] for a detailed classification and discussion). The fact that this ev olution can b e c haracterized, in th e mean, by a single time constan t τ go ve rn ing the rate of c hange of the instant aneous mean env elop e of the transient fi eld (mean effectiv e relaxation time τ ) has b een amply sup p orte d by numerical and practical exp eriment s [9], [29], [30]. In actualit y , eac h comp onen t of th e field, whether angular-sp ectral or mo dal, has its own particular relaxation time constant asso ciate d with it. Ho wev er, their v alues are relativ ely close within a narrow band of frequencies, in the sense that | τ mnp − τ m ′ n ′ p ′ | τ mnp + τ m ′ n ′ p ′ ≪ 1 (4) 5 0 0.5 1 1.5 2 2.5 3 3.5 4 −0.5 0 0.5 1 X(t), Y (0) (t), Y (1) (t), Y(t) t (units δ t) X(t) Y (0) (t) Y (1) (t) Y(t) Figure 1: (color online) Discretized s a mple-and-hold input pro cess X ( t ) and re s ulting w eig h ted mean field Y ( t ) = Y (0) ( t ) + Y (1) ( t ) for a fir st-order pro cess, with time-indep enden t av erag e relaxa tion time τ = 0 . 3 δ t (nonlinear weigh ting). The actua l contin uo us-time stir ring pro cess is obtained as the limit δ t → 0 . for any pair of mo des (or stand ing plane w a ves w ith opp ositely directed w av e v ectors) ( m, n, p ) and ( m ′ , n ′ , p ′ ) lo cate d within this f requency band. F u rthermore, str on g mo dal coupling, b oth in time and frequency , causes equalization of their different v alues τ mnp ( f mnp , t i ). F or these reasons, a one- parameter description is often sufficien tly ad equ ate , at least in a mean-square sense, i.e., wh en the field arises as some w eigh ted a verag e, as observ ed in a coupled multi -mo dal incoheren t system. In an y case, extensions of the m odel are p ossible. F or examp le, a second-order mo del go v ernin g the resp onse to step transitions wo uld incorp orate not only the a v erage rate of Y ( t ) app roac hing its steady-state v alue, but also the a verag e leve l of o v ersho ot of Y ( t ) ab o ve steady-state du ring suc h transitions. In [28, Eq . (9)], the resultan t Y ( t ) in a discretized inte rv al [ t i , t i +1 ] was expressed iterativ ely , as a function of X ( t i ). F or the p resen t pu rp ose, it is m ore b eneficial to use the follo wing equ iv alen t closed-form expression for Y ( t ) Y ( t m ≤ t ≤ t m +1 ) = Y ( t 0 ) exp  − t − t 0 τ  + exp  − t − t m τ  ×  1 − exp  − δ t τ  m − 1 X i =0 exp  − t m − 1 − t i τ  X ( t i ) +  1 − exp  − t − t m τ  X ( t m ) (5) for m ≥ 1, where t m ∆ = t 0 + mδ t . F or m = 0, th e same exp ression holds, but without the s econd term. Neither Y ( t ) nor its increments δ Y m ( t ) ≡ Y ( t m +1 ) − Y ( t m ) are in general stationary , b ecause b oth 6 dep end on t m . Since w e shall fur ther b e taking the limit ∆ t/τ → 0 and b ecause we shall b e mainly in terested in late times of observ ation su c h that ( t − t 0 ) /δt ≫ 1, we restrict further ev aluation to the discrete time instances t = t m +1 , wh en Eq. (5) simp lifi es to Y ( t m +1 ) = Y ( t 0 ) exp  − ( m + 1) δ t τ  + exp  − m δ t τ  ×  1 − exp  − δ t τ  m X i =0 exp  + i δ t τ  X ( t i ) (6) Up on taking the limit δ t/τ → 0 in E q . (6), the d iscrete sum reduces to an integ ral from t 0 to t m , and w e obtain Y ( t m ) → Y ( t 0 ) exp  − t m +1 − t 0 τ  + lim δt → 0 1 − exp  − δt τ  δ t × Z t m t 0 exp  − t m − s τ  X ( s )d s (7) A t this p oint , w e introdu ce the F ourier–Stieltjes sp ectral expansion of the stationary source field X ( t ) [31], giv en b y X ( s ) = Z + ∞ −∞ exp (j ω s ) d Z X ( ω ) (8) in whic h Z X ( ω ) for an ideal mo de-tuned field is assigned the follo wing prop erties [24], [31]: h d Z X ( ω ) i = 0 , (9) h d Z X ( ω 1 )d Z ∗ X ( ω 2 ) i = δ ( ω 1 − ω 2 )d F X ( ω 1 , 2 ) , (10) i.e., Z X ( ω ) is orthogonal. [Note that Z X ( ω ) represents the accum ulated (in tegrated) sp ectrum of X , not to b e confu sed with its differentia l, i.e., th e F ourier sp ectrum.] With the substitution u ∆ = ( t m − s ) /τ and letting t m +1 → t m ∆ = t in Eq. (7), we arrive after some manipu lat ion at Y ( t ) = Y ( t 0 ) exp  − t − t 0 τ  + Z + ∞ −∞ exp (j ω t ) × 1 − exp  − (1 + j ω τ ) t − t 0 τ  1 + j ω τ d Z X ( ω ) . (11) T urning attent ion to th e first term in Eq. (11), and u s ing general pr operties of the F ourier transf orm and the Dirac delta distribu tion [32, Eqs. (2.24) and (2.40)] for X ( t 0 ) = X ( t ) δ ( t − t 0 ), this term can b e assigned a sp ectral expansion, viz., X ( t 0 ) exp  − t − t 0 τ  = Z + ∞ −∞ X ( t 0 ) τ exp [j ω ( t − t 0 )] 1 + j ω τ d ω (12) 7 = Z + ∞ −∞  Z + ∞ −∞ X ( ω ′ ) exp  j ω ′ ( t − t 0 )  d ω ′  × τ exp [j ω ( t − t 0 )] 1 + j ω τ d ω . (13) Note that X ( t 0 ) ≡ Y ( t 0 ) b ecause nonstationarit y has not yet manifested itself at the start time t 0 , whence the output pr ocess can then b e completely id entified with th e inpu t p r ocess at this instance. With an in terc hange of the order of int egration, this yields the general result Y ( t 0 ) exp  − t − t 0 τ  = Z + ∞ −∞ θ ( ω ; t ) exp(j ω t )d Z X ( ω ) (14) where θ ( ω ; t ) ∆ = τ exp( − j ω t 0 ) Z + ∞ −∞ exp [j ω ′ ( t − t 0 )] 1 + j ω ′ τ d ω ′ . (15) In the particular case where w e imp ose the initial cond itio n Y ( t 0 ) = 0, for simplicit y , only the second term in Eq . (11) r emains and can b e rep r esen ted as an oscil lator y pr o c ess [33], i.e., Y ( t ) = Z + ∞ −∞ φ ( t ; ω , τ ) exp (j ω t ) d Z X ( ω ) , ( 16) whose k ernel φ ( t ; ω , τ ) ∆ = 1 − exp  − (1 + j ω τ ) t − t 0 τ  1 + j ω τ (17) is a complex amp litud e mo dulation function expr essed in exp lici t, i.e., closed form. It is seen f rom Eq. (16) that th e effect of nonstationarit y on the plane-wa v e expansion is a mo dulation of th e sp ectral field on a time scale that is a function of th e ca vit y relaxation time. Fig. 2 sho ws the time evol u tion of Eq. (17) at selected v alues of τ = Q/ω for an o vermoded ca vit y , for narr owband op eration at an excitation frequency ω = 2 π × 10 9 rad/s. It is seen that larger v alues of τ (and, hence, Q ) result in longer trans iti on times b efore φ r ea ches a regime, as is intuitiv ely clear. The strongest v ariabilit y of φ ( t ) o ccurs w hen t ∼ 2 π /ω , as exp ected. As is we ll known, oscillatory pr ocesses pro vide an evolutio nary sp ectral repr esen tation for a sp ecial class of nonstationary pro cesses [33, Eq. (3.9)], [37]. They apply when nons ta tionarit y is weak (i.e., sufficien tly slo wly ev olving), so th at the ev olution can b e considered as a m o dulation of the original pro cess X ( t ). F or ω τ → 0 and t/τ → + ∞ , we r etrieve from Eq. (11) the classica l sp ectral expansion for temp orally stationary (bu t spatially homogeneous as well as inhomogeneous) fields, viz., Y ( t ) = Z + ∞ −∞ exp (j ω t ) d Z X ( ω ) ≡ X ( t ) . (18) On the other hand, for τ ≫ 1 /ω w e obtain a sinc( ω t )-t yp e mo dulation, in view of the s ymmetric in tegration limits. 8 3.2 Statistics The ensem ble mean v alue of Y ( t ) is h Y ( t ) i = Y ( t 0 ) exp  − t − t 0 τ  (19) b ecause h d Z X ( ω ) i = 0. The co v ariance function follo ws from Eq. (11) as h Y ( t 1 ) Y ∗ ( t 2 ) i = | Y ( t 0 ) | 2 exp  − t 1 + t 2 − 2 t 0 τ  + Z + ∞ −∞ exp [j ω ( t 1 − t 2 )] 1 + ω 2 τ 2 ×  1 + exp  − t 1 + t 2 − 2 t 0 τ  exp [ − j ω ( t 1 − t 2 )] − exp  − t 1 − t 0 τ  exp [ − j ω ( t 1 − t 0 )] − exp  − t 2 − t 0 τ  exp [j ω ( t 2 − t 0 )]  d F X ( ω ) . (20) This general resu lt is imp ortan t in the inv estigation of th e effect of non s ta tionarit y on the trans f or- mation of correlation charac teristics an d effectiv e num b er of degrees of freedom. F or t 1 = t 2 ∆ = t , the v ariance follo ws as σ 2 Y ( t ) = h| Y ( t ) | 2 i − |h Y ( t ) i| 2 = Z + ∞ −∞ h 1 + ω 2 τ 2 i − 1  1 − 2 exp  − t − t 0 τ  × cos [ ω ( t − t 0 )] + exp  − 2( t − t 0 ) τ  d F X ( ω ) , (21) whic h describ es the evo lution of fluctuation levels under n onstati onarity . W e now consid er tw o im- p ortan t sp ecial cases for the inpu t pro cess. 3.2.1 Ideal white noise If X ( t ) is ideal white noise, i.e., d F X ( ω ) = f X ( ω )d ω = σ 2 X d ω , then from Eqs. (19), (21), and [[38], Eq. (3.723.2 )], σ 2 Y ( t ) = π σ 2 X τ  1 − exp  − 2( t − t 0 ) τ  . (22) In the limits t/τ → 0 and t/τ → + ∞ , w e retriev e σ 2 Y → 2 π σ 2 X ( t − t 0 ) /τ 2 and σ 2 Y → π σ 2 X /τ , resp ectiv ely , as exp ected. 9 3.2.2 Ornstein-Uhlen b ec k pro cess If X ( t ) is a first-order pro cess that is wide-sense stationary and exp onen tially correlated with fi n ite correlation time T , i.e., ρ X ( t − t 0 ) = exp  − t − t 0 T  (23) so that d F X ( ω ) = f X ( ω )d ω = 2 σ 2 0 T / [ π  1 + ω 2 T 2  ]d ω , then Eq. (21) reduces to σ 2 Y ( t ) = 2 σ 2 0 T π  1 + exp  − 2( t − t 0 ) τ  × Z + ∞ −∞ d ω (1 + ω 2 τ 2 )(1 + ω 2 T 2 ) − 4 σ 2 0 T π exp  − ( t − t 0 ) τ  × Z + ∞ −∞ cos [ ω ( t − t 0 )] d ω (1 + ω 2 τ 2 )(1 + ω 2 T 2 ) . (24) The firs t inte gral in Eq. (24) is easily calculate d with th e aid of [[38], Eqs. (2.124.1 ) and (2.161.1) ], yielding π / ( τ + T ), i.e., pro viding a p ositiv e contribution irresp ectiv e of whether τ ≤ T or τ ≥ T . The second integral in E q . (24) is obtained by cont our in tegration of φ ( z ) = exp [ − z ( t − t 0 )] (1 − z 2 τ 2 )(1 − z 2 T 2 ) (25) across the left half of th e complex z -plane (Fig. 3 ). T he residues in z 1 = 1 /τ and z 2 = 1 / T are R 1 = τ exp  − t − t 0 τ  2 ( T 2 − τ 2 ) and R 2 = T exp  − t − t 0 T  2 ( τ 2 − T 2 ) , (26) resp ectiv ely , whence σ 2 Y ( t ) = 2 σ 2 0 T τ + T  1 − exp  − 2( t − t 0 ) τ  . (27) Again, this resu lt h olds ir resp ect ive of w hether τ ≤ T or τ ≥ T . It is ve rifi ed that for T /τ ≪ 1, Eq. (27) redu ces to Eq. (22) with σ 2 X = 2 σ 2 0 T /π . F or ( t − t 0 ) /τ ≪ 1, we r etriev e the lin ea r dep endence as in the case of ideal white noise un der the same condition. F urthermore, for T /τ → 0, it is verified that Eq. (27) approac hes Eq. (22) with 2 σ 0 T = π σ 2 X . 3.3 Instan taneous energy densit y The evo lution of the nonstationary energy densit y is of fundamental imp ortance for incoheren t fields. When considering the instantaneo us field as pur e functions of time (i.e., not as an analytic fu nction 10 or p hasor), the follo wing definition for the asso ciate d energy density applies [39, p. 14], [40]: W e ( t ) = Z t −∞ w e ( t ′ )d t ′ ∆ = Z t −∞ Y ( t ′ ) · ∂ [ ǫ ( t ′ ) Y ( t ′ )] ∂ t ′ d t ′ = 1 2 ǫ Y 2 ( t ) (28) where the latter equalit y holds for a homogeneous time-inv arian t medium [ D ( t ) ∆ = ǫ ( t ) ⋆ E ( t ) = ǫ E ( t )], as w e shall fur ther assume. The p revious analysis for a mean complex field Y ( t ) holds exactly if Y ( t ) ins tead sy mb olize s th e mean intensit y of a complex field. If we maint ain the original defin itio n, h o wev er, then for anharmonic mo dulated fields that are b eing p ertu rb ed relativ ely s lowly with resp ect to a cen tral fr equency ω , we can use the Gab or analytic field representati on Y ( t ) = Y ( t ) exp( − j ω t ) to write w e ( t ) = Y ( t ) · ∂ [ ǫY ∗ ( t )] ∂ t . (29) Therefore, for disp ersionless ǫ , the quantit y w e ( t ) for analytic fields is prop ortional to the field intensit y | Y ( t ) | 2 . T he general expression of W e ( t ) for nonstationary r andom fields is deriv ed in the App end ix and is given b y (51)–(59). F or the magnetic energy d ensit y W h , s imila r expressions follo w by replacing ǫ by µ , with Y ( t ) no w represent ing th e magnetic analytic field. 3.4 Structure function Random field incremen ts pro vide a transition b et wee n stationary systems and fully-devel op ed nonsta- tionary systems (Bac h elie r–Einstein–Wiener–L ´ evy pro cesses). They ma y also b e us efu l f or describing undermo ded fields, in whic h sm al l r apid v ariations “rid e on top” of a slo wly v arying mean v alue. Their second-order prop erties are charact erized in general b y th e structur e fun ctio n, whic h describ es the co v ariance of field incremen ts [3, 25]. I t follo w s up on sub s tit u tion of Eq. (6) as D ( m, n ) ∆ = h [ Y ( t 0 + m ∆ t ) − Y ( t 0 )] [ Y ( t 0 + n ∆ t ) − Y ( t 0 )] ∗ i = | Y ( t 0 ) | 2  1 − exp  − t m − t 0 τ  ×  1 − exp  − t n − t 0 τ  + Z ∞ −∞ h 1 + ( ω τ ) 2 i − 1 ×  exp ( j ω t m − 1 ) − exp  − t m − 1 τ  ×  exp ( j ω t n − 1 ) − exp  − t n − 1 τ  d F X ( ω ) . (30) 11 In w riting the second term in Eq. (30), we made use of the fact that t m − 1 ≃ t m when ∆ t/t 0 → 0, and lik ewise for t n − 1 . F or m = n , Eq. (30) reduces to D ( m ) = | Y ( t 0 ) | 2  1 − exp  − t m − t 0 τ  2 + Z ∞ −∞ h 1 + ( ω τ ) 2 i − 1 ×  1 − 2 exp  − t m − 1 τ  cos ( ω t m − 1 ) + exp  − 2 t m − 1 τ  d F X ( ω ) , (31) whic h is form all y equiv alen t to (21 ). F or example, for an Orns tein–Uhlenb ec k pr ocess, (31) b ecomes D ( m ) = 2 σ 2 0 T τ + T  1 − exp  − 2 t m − 1 τ  . (32) 3.5 Example: EMC immu nity t esting using nonstationary mo de-stirred vs. quasi- stationary mo de-tuned reverberation fields The ab o ve analysis can b e applied to determine the resp onse of a linear DUT that is c haracterized by an imp ulse r esp onse function h ( t ) or a frequency c haracteristic H ( ω ) and whic h has b een placed inside a mo de-stirred rev erb eration cham b er. A t any lo cation inside the ca vity , the lo cal fi eld consists of m ultipath r eflect ions, i.e., p lane wa ve s arr ivin g from isotropically distribu ted directions with randomly p olarized direction of p olarization and uniform ly distribu ted phases. Within a time in terv al δ t d uring a mo de stirr ing pr o cess, eac h plane wa ve und ergoes a tr an s iti on of its d ir ect ions of arriv al, direction of p olarizati on and absolute p hase. The transition of eac h parameter can b e rep resen ted by a tr a jectory in state s p ace . The actual (i.e., physical) rate of fl uctuation d uring this tran s itio n is a fu nction of the rate of c h an ge of cavit y p erturbation an d degree of c haoticit y of the ca vit y . The p erceiv ed rate is, in addition, a f u nction of the c haracteristic time (resp onse time) of the DUT to changes of the excitation field. In general, the sp ecific form of H ( ω ) for an DUT is often unknown. Here, as in [11], we simply consider a class of DUTs and inv estigate to what exten t a mo de-stirred (i.e., con tin uously v arying) vs. mo de-tuned (i.e., discrete, step wise c hanging) excitatio n fi eld has a difference in effect on the resp onse of the DUT. Lik e the excitation, this resp onse is a random function, hence comparison b et w een b oth cases requires determination of one or more statistical metrics, suc h as the mean, η %-confidence in terv als, statistics of the maximum-to- mean ratio, up ward thresh old crossing frequ en cy , excursion length, etc. W e shall consider the case of an Or nstein–Uhlen b eck pro cess, as a canonical case, for which explicit results can b e calculated and from which results for wh ite noise X ( t ) will follo w as a sp ecial case. Since 12 the (real) p o w er sp ectrum is d F X ( ω ) = f X ( ω )d ω = F X, 0 1 + ω 2 T 2 d ω , (33) where F X, 0 = Z X, 0 Z ∗ X, 0 , the (complex) amplitude sp ectrum is d Z X ( ω ) = z X ( ω )d ω = Z X, 0 1 + j ω T d ω . (34) Hence, Y ( t ) = Y ( t 0 ) exp  − t − t 0 τ  + Z + ∞ −∞ exp (j ω t) × 1 − exp  − ( 1 + j ω τ ) t − t 0 τ  (1 + j ω τ )(1 + j ω T ) Z X, 0 d ω . (35) Then the “output” field W ( t ) of the DUT du e to a mo de-stirred excitation field, i.e., as measured or p erceiv ed at one of its test p orts, a critical internal comp onent, etc., is then [33, Sec. 6] W ( t ) = Z + ∞ −∞ h ( u ) Y ( t − u ) exp [ − j ω 0 ( t − u )] d u (36) = Z + ∞ −∞ H ω + ω 0 ( ω ; t ) A ( ω + ω 0 ; t ) exp (j ω t )d Z X ( ω + ω 0 ) (37) = Z + ∞ −∞ H ω + ω 0 ( ω ; t ) exp (j ω t ) ×  Y ( t 0 ) exp  − t − t 0 τ  δ ( ω + ω 0 ) + 1 − exp  − [1 + j( ω + ω 0 ) τ ] t − t 0 τ  [1 + j( ω + ω 0 ) τ ] [1 + j( ω + ω 0 ) T ] Z X, 0 # d ω (38) where ω 0 is an y constan t frequency and H λ ( ω ; t ) = Z + ∞ −∞ h ( u ) A ( λ ; t − u ) A ( λ ; t ) exp( − j ω u )d u (39 ) at λ = ω + ω 0 . 4 Second-order pro c esses The resp onse of an ov ermo ded system can only b e c haracterized appro ximately and in the mean, as a fi rst-order system. In realit y , the large n umber of participating mod es (i.e., those w ithin the instan taneous r esonance band of the ca vit y) and their strong inte rmo dal coupling causes the resp onse to b e more ir regular, r equiring a h igher-ord er description. T o inv estigate the effect of increasing th e 13 order, we here analyze a second-order s y s te m. Because of th e resonant nature of the mo des, this is exp ected to p ro vide more accurate r esu lts. An y higher-order pro cess then f ollo ws from cascading of first- and second-order systems. F or th e second-order system charact erized by d 2 Y d t 2 + 2 ζ d Y d t + ω 2 n Y = K 0 ω 2 0 , (40) the step resp onse is the inv erse Laplace transform of Y ( t ) = K 0 ω 2 n / [ s  s 2 + 2 ζ s + ω 2 0  ]. T he residues of the simple p oles s 0 = 0 an d s 1 , 2 = − ζ ± j q ω 2 0 − ζ 2 lead to the solution Y ( t ) = K 0    1 − ω 0 exp ( − ζ t ) q ω 2 0 − ζ 2 × cos  q ω 2 0 − ζ 2 t − sin − 1  ζ ω 0  (41) for 0 < ζ < ω 0 [F or ov erdamp ed regime ( ζ > ω 0 ), the step resp onse inv olve s hyp erb olic fu nctions.] In principle, the pr evio us analysis of the first-order mo del can b e rep eated. T o this end, Eq. (5) could b e written as Y ( t m ≤ t ≤ t m +1 ) = Y ( t 0 ) φ m (∆ t ) φ ( t − t m ) + φ ( t − t m ) × [1 − φ (∆ t )] m − 1 X i =0 φ m − 1 − i (∆ t ) X ( t i ) + [1 − φ ( t − t m )] X ( t m ) (42) no w with φ ( t ) = ω 0 exp ( − ζ t ) q ω 2 0 − ζ 2 cos  q ω 2 0 − ζ 2 t − sin − 1  ζ ω 0  (43) in view of ω n ≃ ω , τ n ≃ τ . T o obtain the sp ectral expans ion, it is adv ant ageous to consider the Gab or analytic signal representa tion Y ( t ) = Y ′ ( t ) + j Y ′′ ( t ). The step resp onse is then Y ( t ) = K 0    1 − ω 0 exp ( − ζ t ) q ω 2 0 − ζ 2 × exp  j  q ω 2 0 − ζ 2 t − sin − 1  ζ ω 0  . (44) Because of the presence of the factor 1 / q ω 2 0 − ζ 2 in Eq. (41), this approac h do es not lend itself to express Y ( t ) easily as an oscillatory or r ela ted pr ocess, in the spirit of Eqs. (16)–(17) for the fi rst-order 14 system. Therefore, w e wr ite Eq. (40) instead as a system of t wo coupled fi r st-order equations:          d Y ( t ) d t + 1 τ 1 Y ( t ) = 1 τ 1 X ( t ) d Z ( t ) d t + 1 τ 2 Z ( t ) = 1 τ 2 Y ( t ) , (45) thereb y assu ming that τ 1 ≪ τ 2 ≪ 1 (46) where the r esp on s e is now giv en b y Z ( ω ), with Y ( ω ) d en ot ing an auxiliary int ermed iate pro cess. F or an o v erd amp ed second-order s y s te m ( ζ > ω 0 ), such a decomp osition is alw a ys p ossible. Th e p rocess Y ( t ) has a physical meaning: for a first-order system resp ond in g to a nonstationary field whic h, in turn, is go verned b y a firs t-o rd er p rocess, the excitatio n of this s y s te m is giv en by Y ( t ). The sp ectral represent ation follo ws by replacing X ( s ) in Eq. (7), as giv en by Eq. (8) and Y ( s ) given b y Eq. (11): Z ( t ) = Z ( t 0 ) exp  − t − t 0 τ 2  + Y ( t 0 ) 2 ζ τ 2 exp  −  t τ 2 − t 0 τ 1  × [ exp ( − 2 ζ t 0 ) − exp ( − 2 ζ t )] + Z + ∞ −∞    exp (j ω t ) − exp  − t − t 0 τ 2  exp (j ω t 0 ) (1 + j ω τ 1 ) (1 + j ω τ 2 ) + exp (j ω t 0 ) 1 + j ω τ 1 τ 1 τ 2  exp  − 2 ζ t + t 0 τ 1  − exp  − t τ 2  × d Z X ( ω ) . (47) F or th e case where t 0 = 0, Y ( t 0 ) = Z ( t 0 ) = 0, this expr ession reduces to Z ( t ) = Z + ∞ −∞    exp (j ω t ) − exp  − t τ 2  (1 + j ω τ 1 ) (1 + j ω τ 2 ) + 1 1 + j ω τ 1 τ 1 τ 2 ×  exp ( − 2 ζ t ) − exp  − t τ 2  d Z X ( ω ) . (48) F or an Or nstein–Uhlen b eck pr ocess, con tour integ ration yields Z ( t ) = − j 2 σ 2 0 T 2 τ + T  T T + τ 2  exp  − t T  + exp  − t τ 2  − τ 1 τ 2  exp ( − 2 ζ t ) − exp  − t τ 2  . (49) Note th at , eve n though the original ω 0 and ζ ha ve b een r eplac ed by τ 1 and τ 2 , exh ibiting a one-to-one relationship, the pro cess of replacing the second-order SDE by tw o coupled first-order SDEs requires for one time constan t to b e muc h smaller than the other one, in order to main tain the v alidit y of the Langevin–Itˆ o formulatio n. 15 5 Conclusions Angular sp ectral plane-w av e expansions p ro vide a framew ork for the s toc hastic c h aracterization of dynamic random EM fi elds that result from moving b ound aries, spatial scanning, and/or c han ging excitatio n frequen cy w ith consequent c hanges in the pattern of participating eigenmo des. It w as sho wn that the sp ectral expansion can b e exp ressed as an oscillatory pro cess Eq. (16 ) with ke rn el Eq. (17) (or, more generally , as Eqs. (11), (14), (15)). The fact that the expansion can still b e represented by an oscillatory pro cess (whic h is u sually asso ciated with narrowband, i.e., slowly mo dulating (quasi- stationary) pr ocesses, but now with no su c h limitation on the rate of flu ctuat ions) can b e attributed to the linearit y of the s ystem. Th e results for highly o v ermo ded resonances apply in the mean (incoherent sup erp osition), i.e., f or th e collection of ca vity mo des participating in form in g the instanta neous field at an y one time. A u seful extension w ould b e the ”microscopic” p lane-w a ve sp ectral expansion for a single second- order resonan t system (ca vity mo de), without an y limiting condition on the separation of the t wo time constan ts inv olved in decomp osing int o t wo first-order pro cesses. Such a complete second-order c haracterization could form the basis for a coherent su p erp osit ion of excited ca vit y mo des. 16 10 −2 10 −1 10 0 10 1 10 2 10 3 10 −6 10 −4 10 −2 10 0 t (ns) | φ (t; ω , τ )| τ =1ns τ =10ns τ =100ns τ =1000ns (a) 10 −2 10 −1 10 0 10 1 10 2 10 3 −150 −100 −50 0 t (ns) ∠φ (t; ω , τ ) (deg) τ =1ns τ =10ns τ =100ns τ =1000ns (b) Figure 2: (co lor online) Mo dulating function φ ( t ; ω , τ ) of the kernel in E q. (16) for the spe c tral plane-wav e representation of nonstatio nary mo de-stirred fields, for selected v alues of τ at ω / (2 π ) = 1 GHz: (a) mag nitude; (b) phas e. 17 R x x 1/ τ 1/ T C R Re(z) Im(z) Figure 3: Contour of in tegr ation for Eq . (25) in the c o mplex z -plane ( τ > T ). 18 References [1] P .- S. Kildal, Micr ow. Opt. T e chn. L ett, vol. 3 2, p. 112, 2 002. [2] A. Sor ren tino, G. F erra ra and M. Migliacc io, IEEE Antennas and Wir eless Pr op ag. L et t ., vol. 56, no. 6, pp. 1825– 1830, 2008. [3] G. 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Elektr on., vol. 1 1, pp. 1812– 1821, 1963 . 20 App endix: Instan taneous energy densit y Here, we deriv e the general expression for th e in stan taneous energy of a n onstati onary field. Subs ti- tution of the general expression Eq. (11) for Y ( t ) in to Eq. (29) yields w e ( t ′ ) = − ǫ | Y ( t 0 ) | 2 τ exp  − 2( t ′ − t 0 ) τ  − ǫY ∗ ( t 0 ) τ exp  − t ′ − t 0 τ  Z + ∞ −∞ exp  j ω t ′  1 − exp h − (1 + j ω τ ) t ′ − t 0 τ i 1 + j ω τ d Z X ( ω ) + ǫY ( t 0 ) τ exp  − t ′ − t 0 τ  Z + ∞ −∞ exp  − j ω t ′  n − j ω τ + exp h − (1 − j ω τ ) t ′ − t 0 τ io 1 − j ω τ d Z ∗ X ( ω ) + ǫ τ Z + ∞ −∞ Z + ∞ −∞ exp  j ( ω 1 − ω 2 ) t ′  1 − exp h − (1 + j ω 1 τ ) t ′ − t 0 τ i 1 + j ω 1 τ × n − j ω 2 τ + exp h − ( 1 − j ω 2 τ ) t ′ − t 0 τ io 1 − j ω 2 τ d Z X ( ω 1 )d Z ∗ X ( ω 2 ) . (50) In tegration of w e ( t ′ ) yields the instan taneous W e ( t ) as W e ( t ) = Z t t 0 w e ( t ′ )d t ′ = W e 1 ( t ) + W e 2 ( t ) + W e 3 ( t ) + W e 4 ( t ) (51) with W e 1 ( t ) = − ǫ | Y ( t 0 ) | 2 2  1 − exp  − 2( t − t 0 ) τ  (52) W e 2 ( t ) = ǫY ∗ ( t 0 ) Z + ∞ −∞ exp (j ω t 0 ) 1 + ω 2 τ 2  exp  − (1 − j ω τ ) t − t 0 τ  − exp  − 2( t − t 0 ) τ  d Z X ( ω ) (53) W e 3 ( t ) = ǫY ( t 0 ) 2  1 − exp  − 2( t − t 0 ) τ  Z + ∞ −∞ exp ( j ω t 0 ) 1 − j ω τ d Z ∗ X ( ω ) − ǫY ( t 0 ) Z + ∞ −∞ j ω τ exp ( − j ω t 0 ) 1 + ω 2 τ 2  1 − exp  − (1 + j ω τ ) t − t 0 τ  d Z ∗ X ( ω ) (54) W e 4 ( t ) = ǫ Z + ∞ −∞ Z + ∞ −∞ I 1 ( t ; ω 1 , ω 2 ) + I 2 ( t ; ω 1 , ω 2 ) + I 3 ( t ; ω 1 , ω 2 ) + I 4 ( t ; ω 1 , ω 2 ) (1 + j ω 1 τ )(1 − j ω 2 τ ) d Z X ( ω 1 )d Z ∗ X ( ω 2 ) (55) and I 1 ( t ; ω 1 , ω 2 ) = ω 2 ω 1 − ω 2 exp [j( ω 1 − ω 2 ) t 0 ] { 1 − exp [j( ω 1 − ω 2 )( t − t 0 )] } (56) I 2 ( t ; ω 1 , ω 2 ) = 1 1 − j ω 1 τ exp  − 2 t 0 τ  exp [j( ω 1 − ω 2 ) t 0 ]  1 − exp  − (1 − j ω 1 τ ) t − t 0 τ  (57) I 3 ( t ; ω 1 , ω 2 ) = j ω 2 τ 1 + j ω 2 τ exp [j( ω 1 − ω 2 ) t 0 ]  1 − exp  − ( 1 + j ω 2 τ ) t − t 0 τ  (58) I 4 ( t ; ω 1 , ω 2 ) = − 1 2 exp  − 2 t 0 τ  exp [j( ω 1 − ω 2 ) t 0 ]  1 − exp  − 2( t − t 0 ) τ  . (59) 21 F or ω 1 = ω 2 ∆ = ω , I 1 ( t ; ω ) = − j ω ( t − t 0 ) (60) I 2 ( t ; ω ) = 1 1 − j ω τ exp  − 2 t 0 τ   1 − exp  − (1 − j ω τ ) t − t 0 τ  (61) I 3 ( t ; ω ) = j ω τ 1 + j ω τ  1 − exp  − ( 1 + j ω τ ) t − t 0 τ  (62) I 4 ( t ; ω ) = − 1 2 exp  − 2 t 0 τ   1 − exp  − 2( t − t 0 ) τ  . (63) Considering that h d Z ( ω ) i = 0 and h d Z ( ω 1 )d Z ∗ ( ω 2 ) i = δ ( ω 1 − ω 2 )d F X ( ω ), we h a ve that h W e 2 i = h W e 3 i = 0 (64) h W e 4 i = ǫ Z + ∞ −∞ I 1 ( t ; ω ) + I 2 ( t ; ω ) + I 3 ( t ; ω ) + I 4 ( t ; ω ) 1 + ω 2 τ 2 d F X ( ω ) . (65) 22