Better than Rician: Modelling millimetre wave channels as Two-Wave with Diffuse Power

Better than Rician: Mo delling millimetre w a v e c hannels as Tw o-W a v e with Diffuse P o w er Eric h Z¨ oc hmann ∗ , Sebastian Caban, Christoph F. Mecklen br¨ auk er, Stefan Pratschner, Martin Lerc h, Stefan Sc h warz, Markus Rupp No vem ber 6, 2021 Abstract This contribution provides exp erimen tal evidence for the tw o-wa ve with diffuse p o w er (TWDP) fading mo del. W e ha v e conducted tw o indo or millimetre w av e measurement campaigns with directive horn antennas at b oth link ends. One horn antenna is moun ted in a corner of our lab oratory , while the other is steerable and scans azimuth and elev ation. Our first measuremen t campaign is based on scalar netw ork analysis with 7 GHz of bandwidth. Our second measurement campaign obtains magnitude and phase information, additionally sampled directionally at several p ositions in space. W e apply Ak aike’s information criterion to decide whether Rician fading sufficiently explains the data or the generalized TWDP fading mo del is necessary . Our results indicate that the TWDP fading hypothesis is fav oured o ver Rician fading in situations where the steerable antenna is pointing to wards reflecting ob jects or is slightly misaligned at line-of- sigh t. W e demonstrate TWDP fading in several different domains, namely , frequency , space, and time. 1 In tro duction Accurate mo delling of wireless propagation effects is a fundamental prerequisite for a prop er communication system design. After the introduction of the double- directional radio channel mo del [1], wireless propagation researc h ( < 6 GHz) started to mo del the wireless c hannel agnostic to the antennas used. More than a decade later, propagation research fo cusses now on millimetre w av e bands to unlo ck the large bandwidths av ailable in this regime [2 – 5]. A t millimetre w av es ( mmW av e s), omnidirectional an tennas hav e small effective an tenna areas, resulting in a high path-loss [6 – 10]. T o ov ercome this high path-loss, researchers ha ve proposed to apply highly directive an tennas on b oth link ends [11 – 14]. Most researc hers aim to achiev e high directivity with antenna arrays [15 – 20] and a ∗ Christian Doppler Lab oratory for Dep endable Wireless Connectivity for the So ciety in Motion, Institute of T elecommunications, TU Wien 1 few with dielectric lenses [21 – 23]. When the link-quality dep ends so muc h on the ac hieved b eam-forming gain, an tennas must b e considered as part of the wireless channel again. Small-scale fading is then influenced by the antenna. According to Durgin [24, p. 137], “The use of directive antennas or arrays at a receiver, for example, amplifies several of the strongest multipath wa ves that arriv e in one particular direction while attenuating the remaining wa ves. This effectively increases the ratio of sp ecular to nonsp ecular received p o wer, turning a Rayleigh or Rician fading channel in to a TWDP fading channel.” The men tioned tw o-w av e with diffuse p o wer ( TWDP ) fading channel describ es this spatial filtering effect b y tw o non-fluctuating receive signals together with man y smaller diffuse comp onen ts. 1.1 Related work The authors of [25] in vestigated a simple wall scattering scenario and analysed ho w fading scales with v arious antenna directivities and different bandwidths. Increasing directivity [25], as well as increasing bandwidth [25, 26], results in an increased Rician K-factor. The authors of [27] analysed fading at 28 GHz with high gain horn antennas on b oth link ends. They observe high Rician K-factors ev en at non-line-of-sight ( NLOS ). This effect is explained by spatial filtering of directiv e antennas, as they suppress many multipath comp onen ts [25]. Outdoor measuremen ts in [28, 29], show a graphical agreemen t with the Rice fit, but esp ecially Fig. 10 in [29] might b e b etter explained as TWDP fading. TWDP fading has already successfully b een applied to describ e 60 GHz near b ody shadowing [30]. F urthermore, as quoted ab o ve, TWDP m ust b e considered for arrays, as they act as spatial filters [24, 31]. While theoretical work on TWDP fading is already adv anced [32 – 38], e xperimental evidence, esp ecially at millimetre wa ves, is still limited. F or enclosed structures, such as aircraft cabins and buses, the applicability of the TWDP mo del is demonstrated by F rolik [39 – 43]. A deterministic tw o ray behaviour in ra y tracing data of mmW av e train-to-infrastructure communications is shown in [44]. A further extension of the TWDP -fading mo del, the so called fluctuating tw o-ray fading mo del, was also successfully applied to fit mmW av e measurement data [36]. Our group has conducted t wo measuremen t campaigns [45, 46] to directionally analyse receiv e p o wer and small-scale fading parameters for mmW av e s. This con tribution is based on [45, 46]. 1.2 Outline and con tributions With this contribution, we aim to bring scientific rigour to the small-scale fading analysis of millimetre wa ve indo or channels. W e show in Section 2 – b y means of an information-theoretic approac h [47] and null hypothesis testing [48] – that the TWDP mo del has evidence in mmW av e communications. W e hav e conducted t wo measurement campaigns within the same lab oratory with different channel sounding concepts. Our measurements are carried out in the V-band; the applied center frequency is 60 GHz. F or b oth measurement 2 campaigns, 20 dBi horn antennas are used at the transmitter and at the receiv er. The first measurement campaign ( MC1 ) samples the channel in azimuth ( ϕ ) and elev ation ( θ ), keeping the antenna’s (apparent) phase center [49, pp. 799] at a fixed ( x, y ) – co ordinate. The transmitter is moun ted in a corner of our laboratory . The sounded environmen t as well as the mec hanical set-ups are explained in Section 3. F or MC1 , we sounded the channel in the frequency-domain by aid of scalar netw ork analysis, describ ed in Section 4. These channel measurements span ov er 7 GHz bandwidth, supp orting us to analyse fading in the fr e quency domain . F or the second measurement campaign ( MC2 ), describ ed in Section 5, we impro ved the set-up mechanically and radio frequency ( RF ) – wise. By adding another linear guide along the z -axis, w e k eep the an tenna’s phase cen ter constan t in ( x, y , z ) – coordinate, irresp ective of the antenna’s elev ation. F urthermore, we c hanged the sounding concept to time-domain channel sounding. This approach allo ws us to utilise the time domain and to sho w channel impulse resp onses in Section 7. Additionally , by adjusting ( x, y , z , ϕ, θ ), we sample the c hannel in the sp atial domain at all directions ( ϕ, θ ). These improv ements enable us to show spatial correlations in Section 6, a further analysis to ol to supp ort the claims from MC1. In summary , we demonstr ate TWDP fading for directional mmW av e indo or c hannels in the fr e quency-domain , in the sp atial-domain , and in the time-domain . 2 Metho dology - F ading mo del iden tification TWDP fading captures the effect of interference of tw o non-fluctuating radio signals and many smaller so called diffuse signals [31]. The TWDP distribution degenerates to Rice if one of the t wo non-fluctuating radio signals v anishes. This is analogous to the well known Rice degeneration to the Rayleigh distribution with decreasing K factor. In the framework of mo del selection, TWDP fading, Rician fading, and Rayleigh fading are hence nested hypotheses [47]. Therefore, it is also obvious that among these alternatives, TWDP alw ays allows the b est p ossible fit of measurement data. Occam’s razor [50] asks to select, among comp eting hypothetical distributions, the hypothesis that makes the fewest assumptions. Different distribution functions are often compared via a go odness- of-fit test [51]. Nev ertheless, the authors of [52] argue that Ak aik e’s information criterion ( AIC ) [47, 53 – 55] is b etter suited for the purp ose of choosing among fading distributions. Later on, the AIC w as also used in [56 – 60]. The AIC can b e seen as a form of Occam’s razor as it p enalizes the num b er of estimable parameters in the approximating mo del [47] and hence aims for parsimony . 2.1 Mathematical description of TWDP fading An early form of TWDP w as analysed in [32]. Durgin et al. [31] introduced a random phase sup erposition formalism. Later, [35] achiev ed a ma jor break- through and found a description of TWDP fading as conditional Rician fading. 3 F or the b enefit of the reader, we will briefly rep eat some imp ortant steps of [35]. The TWDP fading mo del in the complex-v alued baseband is given as r complex = V 1 e j φ 1 + V 2 e j φ 2 + X + j Y , (1) where V 1 ≥ 0 and V 2 ≥ 0 are the deterministic amplitudes of the non-fluctuating sp ecular comp onen ts. The phases φ 1 and φ 2 are indep enden t and uniformly distributed in (0 , 2 π ). The diffuse comp onen ts are mo delled via the law of large n umbers as X + j Y , where X, Y ∼ N (0 , σ 2 ). The K -factor is the p o wer ratio of the sp ecular comp onents to the diffuse comp onen ts K = V 2 1 + V 2 2 2 σ 2 . (2) The parameter ∆ describ es the amplitude relationship among the sp ecular comp onen ts ∆ = 2 V 1 V 2 V 2 1 + V 2 2 . (3) The ∆-parameter is b ounded b et ween 0 and 1 and equals 1 iff b oth amplitudes are equal. The second moment of the env elop e r = | r complex | of TWDP fading is given as E  r 2  = Ω = V 2 1 + V 2 2 + 2 σ 2 . (4) Exp ectation is denoted by E . F or b ounded amplitudes V 1 and V 2 , a clever choice of σ 2 normalises Ω, that is Ω ≡ 1. Starting from (4), by using (2) we arriv e at σ 2 = 1 2(1 + K ) . (5) Giv en the K and ∆ parameter (Ω ≡ 1), the authors of [61] provide a formula for the amplitudes of b oth sp ecular comp onen ts V 1 , 2 = 1 2 r K K + 1  √ 1 + ∆ ± √ 1 − ∆  . (6) Real-w orld measurement data hav e Ω 6 = 1. T o work with the formalism in tro duced ab ov e, we normalise the measurement data through estimating ˆ Ω b y the metho d of moments. The second moment Ω of Rician fading and TWDP fading is merely a scale factor [62, 63]. Notably , we are more concerned with a prop er fit of K and ∆. Generally , estimation errors on Ω propagate to K and ∆ estimates. How ev er, [62] achiev ed an almost asymptotically efficient estimator with a moment-based estimation of Ω. Our en velope measurements are partitioned into 2 sets. W e take the first set ( r 1 , . . . , r n , . . . r N ) for parameter estimation of the tuple ( K, ∆) as describ ed in Section 2.2, and hypothesis testing as describ ed in Section 2.3. The first set is carefully c hosen to obtain env elop e samples that are approximately indep enden t and iden tically distributed. The second set ( r 1 , . . . , r m , . . . r M ) is the complemen t 4 of the first set. W e use the elements of the second set to estimate the second momen t via ˆ Ω = 1 M M X m =1 r 2 m , (7) where m is the sample index and M is the size of the second set. Partitioning is necessary to av oid biases through noise correlations of ˆ Ω and ( ˆ K , ˆ ∆) [64]. By considering the estimate (7) as true parameter Ω, all distributions are parametrised by the tuple ( K, ∆), solely . Example distributions are sho wn in Fig. 1. The cumulativ e distribution function ( CDF ) of the env elop e of (1) is giv en in [35] as F TWDP ( r ; K , ∆) = (8) 1 − 1 2 π 2 π Z 0 Q 1  p 2 K [1 + ∆ cos ( α )] , r σ  d α . The Marcum Q-function is denoted by Q 1 ( · , · ). F or ∆ → 0, Equation (8) reduces to the well known Rice CDF F Rice ( r ; K ) = 1 − Q 1  √ 2 K , r σ  . (9) It might sound tempting to ha ve a second strong radio signal presen t; in fact, how ever, tw o wa ves can either sup erpose constructively or destructively and ev entually lead to fading that is more severe than Rayleigh [39 – 43]. W e observ e the highest probability for deep fades for TWDP fading in Fig. 1. 0 0.5 1 1.5 2 2.5 0 1 2 3 p ( r/ √ Ω ) PDFs 0 0.5 1 1.5 2 2.5 norm. amplitude r / √ Ω 0 0.5 1 p ( r/ √ Ω < abscissa ) CDFs Rice: K = 20 , ∆ = 0 Ra yleigh: K = 0 , ∆ = 0 TWDP: K = 20 , ∆ = 1 Figure 1: Comparison of Rayleigh, Rician, and TWDP fading. The TWDP distribution with ∆ = 1 deviates from the Rice distribution. TWDP fading’s probability for deep fades is higher than for a Rayleigh distribution. 5 2.2 P arameter estimation and mo del selection Note that our mo del of TWDP fading (1) do es (obviously) not contain noise. Ov er our wide frequency range (in MC1 w e hav e 7 GHz bandwidth) the receive noise p ow er sp ectral density is not equal. A statistical noise description that is v alid o ver our wide frequency range is frequency-dep enden t. T o av oid the burden of frequency-dep enden t noise mo delling, w e only take measurement samples whic h lie at least 10 dB ab o v e the noise p o wer and ignore noise in our estimation. Ha ving the env elop e measuremen t data set ( r 1 , . . . , r n , . . . r N ) at hand, we are seeking a distribution of whic h the observed realisations r n app ear most lik ely . T o do so, we estimate the parameter tuple ( K ,∆) via the maximum lik eliho o d pro cedure ( ˆ K , ˆ ∆) = arg max K, ∆ N X n =1 ln ∂ F TWDP ( r n ; K, ∆) ∂ r = arg max K, ∆ N X n =1 ln f TWDP ( r n ; K, ∆) = arg max K, ∆ N X n =1 ln L ( K, ∆ | r n ) . (10) W e denote the probability densit y function ( PDF ) by f ( · ), n denotes the sample index, and N the size of the set. T o solve (10), we first discretise K and ∆ in steps of 0 . 05. Next, we calculate ∂ F TWDP ( r ; K , ∆) ∂ r for all parameters via numerical differen tiation. Within this family of distributions, we search for the parameter v ector maximizing the log-likelihoo d function (10). F or the optimal Rice fit, the maxim um is searched within the parameter slice ( K, ∆ = 0). An exemplary fit of Rician and TWDP fading is shown in Fig. 2. As a reference, Ra yleigh fading ( K = 0 , ∆ = 0) is shown as well. 6 0.5 1 1.5 2 r/ √ Ω 0 0.2 0.4 0.6 0.8 1 p ( r/ √ Ω < abscissa) Ra yleigh: K =0 , ∆ =0 Rice: ˆ K =1.8 , ∆ =0 measured TWDP: ˆ K =16 , ˆ ∆ =0.9 cell edge Figure 2: CDF: Distribution fitting for exemplary frequency domain measuremen t data. Illustration of the maxim um likelihoo d fitted Rice dis- tribution and the maximum likelihoo d fitted TWDP fading distribution. The Rician K -factor and the TWDP K -factor deviate significantly . Rayleigh fading is plotted as reference. T o select b etw een Rician fading and TWDP fading, we employ Ak aike’s information criterion (AIC). The AIC is a rigorous wa y to estimate the Kull- bac k–Leibler divergence, that is, the relativ e entrop y based on the maximum- lik eliho o d estimate [47]. Giv en the maximum-lik eliho od fitted parameter tuple ( ˆ K , ˆ ∆ ) of TWDP fading and Rician fading, we calculate the sample size corrected AIC [47, p. 66] for Rician fading (AIC R ) or TWDP fading (AIC T ) AIC R/T = (11) − 2 ln L R/T ( ˆ K , ˆ ∆ | r ) + 2 U R/T + 2 U R/T ( U R/T + 1) N − U R/T − 1 , where U is the mo del order. F or Rician fading the mo del order is U R = 1, since w e estimate the K -factor, only . F or TWDP fading U T = 2, as ∆ is estimated additionally . The second moment Ω (estimated already with a differen t data set b efore the parameter estimation) is not part of the ML estimation (10) and therefore not accoun ted in the mo del order U . W e choose b et ween Rician fading and TWDP fading based on the low er AIC. 2.3 V alidation of the chosen mo del Based on (11), one of the tw o distributions, Rice or TWDP , will alwa ys yield a b etter fit. T o v alidate whether the chosen distributions really explains the data, 7 w e state the following statistical hypothesis testing problem: H 0 : ( F Rice ( r ; ˆ K ) , if AIC R ≤ AIC T F TWDP ( r ; ˆ K , ˆ ∆) , else H 1 : ( ¬ F Rice ( r ; ˆ K ) , if AIC R ≤ AIC T ¬ F TWDP ( r ; ˆ K , ˆ ∆) , else (12) The Bo olean negation is denoted by ¬ . Our statistical to ol is the g-test [65, 66] 1 . A t a significance level α , a null hypothesis is rejected if G = 2 m X i =1 O i ln  O i E i  ? > χ 2 (1 − α,m − e ) , (13) where O i is the observed bin count in cell i and E i is the exp ected bin count in cell i under the null hypothesis H 0 . The cell edges are illustrated with vertical lines in Fig. 2. The cell edges are chosen, suc h that 10 observ ed bin coun ts fall in to one cell. The estimated parameters of the mo del are denoted b y e . F or Rician fading we estimate e = 2 (Ω , K ) parameters and for TWDP fading w e estimate e = 3 (Ω , K, ∆) parameters in total. The (1 − α ) – quantile of the c hi-square distribution with m − e degrees of freedom is denoted by χ 2 (1 − α,m − e ) . The prescrib ed confidence level is 1 − α = 0 . 01 . 3 Flo or plan and set-ups for MC1 and MC2 Our measured environmen t is a mixed office and lab oratory ro om. There are office desks in the middle of the room and at the windo w side, there are lab oratory desks, see Fig 3. The main interacting ob jects in our channel are office desks, a metallic fridge, a wall, and the surface of the lab oratory desk. These ob jects are all marked in Fig. 3. 1 The w ell known c hi-squared test approximates the g-test via a local linearisation [67]. 8 RX/ TX TX/ RX office desks l abor atory desk fridge L O S wall Figure 3: Flo or plan of the measured en vironment. The flo orplan indicates the multipath comp onen ts that are visible in the measurement results. TX and RX switch roles for MC2. TX/RX in the right upp er corner of the ro om is alwa ys static. RX/TX in the middle of the ro om is steerable, indicated by the spider’s web. Our directional measurements are carried out by using the traditional ap- proac h of mechanically steered directional antennas [68, 69]. As directional an tennas, 20 dBi conical horn antennas with an 18 ◦ 3 dB op ening angle are used. Our p olarisation is determined by the LOS polarisation. When TX and RX are facing each other at LOS, the p olarisation is co-p olarised and the E-field is orthogonal to the flo or. In MC1 , the essential mechanical adaptation to the state-of-the-art directional channel sounding set-up [70, 71] is that the elev ation- o ver-azim uth p ositioner is mounted on an xy-p ositioning stage. Thereby , we comp ensate for all linear translations caused by rotations and keep the phase cen ter of the horn antenna alw ays at the same ( x, y ) co ordinate, see Fig. 4. The z co ordinate is roughly 70 cm ab o ve ground but v aries 13 cm for different elev ation angles. 9 TX an tenna steerable RX antenna y -axis x -axis θ -axis ϕ -axis Figure 4: Photograph of the mechanical set-up for MC1 from the re- ceiv er p oin t of view. The receive antenna, a conical 20 dBi horn, is mounted on a multi-axis p ositioning and rotating system. The azimuthal and elev ation angle are con trolled to scan the whole upp er hemisphere. The multi-axis system mo ves and rotates the horn antenna such that its phase center stays at the same ( x, y ) co ordinate during the directional scan. F or MC2 w e add another linear guide along the z -axis to comp ensate for all in tro duced offsets. The horn antenna’s phase center is thereby lifted upw ards b y one metre. Now we are able to fix the phase center of the horn antenna at a sp ecific ( x, y , z ) co ordinate in space. The whole mechanical set-up and the fixed phase center is illustrated in Fig. 8. 4 MC1: Scalar-v alued wideband measuremen ts A wireless channel is said to b e small-scale fading, if the receiver ( RX ) cannot distinguish b et w een different multipath comp onen ts ( MPC s). Depending on the 10 p osition of the transmitter ( TX ), the p osition of the RX and the p osition of the in teracting comp onen ts, the MPC s interfere constructively or destructively [72, pp. 27]. The fading concept only asks for a single carrier frequency , whose MPC s arrive with different phases at the RX . By spatial sampling a statistical description of the fading pro cess is found. In MC1 , the spatial ( x, y ) – co ordinate (of TX and RX ) is kept constant. Dif- feren t phases of the impinging MPC s are realised b y changing the TX frequency o ver a bandwidth of 7 GHz. Thereb y we implicitly rely on frequency translations to estimate the moments of the spatial fading pro cess. 4.1 Measuremen t set-up W e measure the forw ard transfer function with an Rohde and Sch warz R&S ZV A24 vector netw ork analyser ( VNA ). The VNA can measure directly up to 24 GHz. F or mmW av e up-con version and down-con version, we employ mo dules from Pasternac k [73]. They are based on radio frequency integrated circuits describ ed in [74]. The up-conv erter module and the do wn-conv erter module are op erating built-in synthesizer phase-lo c ked lo ops ( PLL s), where the lo cal oscillator (LO) frequency is calculated as f LO = 7 / 4 · s PLL · 285 . 714 MHz ≈ s PLL · 500 MHz . (14) The scaling factor of the synthesizer PLL coun ters is denoted by s PLL . F or f LO ≈ 60 GHz , the scaling factor is s PLL = 120. T o av oid crosstalk, we measure the transfer function via the conv ersion gain (mixer) measurement option of our VNA and op erate the transmitter and receiver at different baseband frequencies: 601 to 1100 MHz and 101 to 600 MHz. The set-up is shown in Fig. 5. 11 R&S ZV A24 601–1100 MHz 101–600 MHz PLL 57 . 0 − 63 . 5 GHz up-conv erter PLL 57 . 5 − 64 . 0 GHz down-con verter 285 . 714 MHz ∼ -3dB -10dB -10dB 57 . 6–64 . 6 GHz Figure 5: RF set-up for MC1. The combination of different PLL scaling factors allows for a measurement bandwidth of 7 GHz. The reference clo c k for the up-conv erter and the down-con verter is shared. The p o wer splitter has an isolation of 30 dB. T o av oid p ossible leak age on the clo c k distribution netw ork, atten uators additionally decouple b oth conv erters. The transfer function is measured applying the conv ersion gain (mixer) measuremen t option of the R&S VNA. 4.2 Receiv e p o wer and fading distributions In Fig. 6, we show the estimated received mean pow er of 7 GHz bandwidth, normalised to the maximum RX p o wer, that is P RX,norm. ( ϕ, θ ) = ˆ Ω( ϕ, θ ) max ϕ 0 ,θ 0 ( ˆ Ω( ϕ 0 , θ 0 )) . (15) As already mentioned in Section 2, we partition the frequency measurements in to tw o sets. The normalised receive p o wer is calculated according to (7), with frequency samples spaced by 2 . 5 MHz. Ev ery tenth sample is left out as these samples are used for fitting of ( K, ∆) and hypothesis testing. W e display the results via a stereographic pro jection from the south p ole and use tan ( θ / 2 ) as azim uthal pro jection. All samplings p oin ts, lying at least 10 dB ab o ve the noise lev el, are sub ject of our study . They are displa yed with red, white or blac k mark ers. Sampling p oints where we decided for TWDP fading, following the pro cedure describ ed in Section 2, are marked with red diamonds. White circles mark p oin ts for whic h AIC fa vours Rician fading. F our p oin ts are marked black. These points failed the n ull hypothesis test and w e neither argue for Rician fading nor for TWDP fading. TWDP fading o ccurs whenever the line-of-sight ( LOS )- link is not p erfectly aligned or if the interacting ob ject cannot b e describ ed by a pure reflection. 12 Figure 6: Estimated directional receive p o w er of MC1. There are four main interacting ob jects leading to stronger receive p o wer (marked in the figure). TWDP fading o ccurs whenever the LOS -link is not p erfectly aligned or the reflecting structure is not p erfectly plain. Red diamonds mark TWDP fading and white circles mark Rician fading. Black markers show p oin ts where b oth distributions are rejected by the hypothesis test. Directions less than 10 dB ab o v e the noise level are not ev aluated. In Fig. 7, the K -parameter of the selected hypothesis is illustrated. Figure 7 sho ws either the Rician K-factor or the TWDP K-factor, dep ending on the selected hypothesis. Note that their definitions are fully equiv alent. F or Rician fading, the amplitude V 2 in (1) is zero by definition. Whenever the RX p o w er is high, the K -factor is high. Below the K -estimate, the estimate of ∆ is sho wn. Here again, by definition, ∆ = 0 whenever we decide for Rician fading. F or in teracting ob jects, the parameter ∆ tends to be close to one. Note, that decisions based on AIC select TWDP fading mostly when ∆ is ab o v e 0 . 3. Smaller ∆ v alues do not c hange the distribution function sufficien tly to justify a higher mo del order. 13 Figure 7: Estimated K -factor and ∆ -parameter of MC1. W e plot the K -factor estimate of the selected hypothesis. The K -factor b ehav es analogously to the RX p o wer. At LOS, the K -factor is far ab o ve 20 dB. The desk reflection has a surprisingly high K -factor of ab out 15 dB. Other reflections hav e K -factors of approximately 10 dB. The ∆-parameter for reflections tends to b e close to 1. Mark ers hav e the same meaning as in Fig. 6. 14 5 MC2: V ector-v alued spatial measuremen ts x-axis y-axis z-axis ϕ -axis θ -axis phase cen ter RX antenna steerable TX antenna Figure 8: Photograph of the impro ved mec hanical set-up for MC2 from the receiver p oin t of view. Our mec hanical set-up consists no w of fiv e indep enden t axis to fully comp ensate all offsets introduced by rotation. A sc hematic sketc h is sup erimp osed. All five axis are necessary to rotate the horn an tenna around the phase center at a fixed ( x, y , z ) co ordinate. Notice that TX and RX switch roles as compared to Fig. 4. In contrast to MC1 , we no longer rely on frequency translations and are indeed sampling the channel in space. The fading results we presen t in Section 5.3 are ev aluated at a single frequency . F ading is hence determined by the obtained spatial samples, exclusively . 5.1 Measuremen t set-up A t the transmitter side, a 2 GHz wide wa veform is produced by an arbitrary w av eform generator ( A W G ). A multi-tone wa veform (OFDM) with Newman phases [75 – 77] is applied as sounding signal. The signal has 401 tones (sub- carriers) with a spacing of 5 MHz. This large spacing assures that our system 15 is not limited by phase noise [78]. The TX sequence is rep eated 2 000 times to obtain a coherent pro cessing gain of 33 dB for i.i.d. noise. The Pasternac k up-con verter (the same as in MC1 ) shifts the baseband sequence to 60 GHz. The 20 dBi conical horn antenna, together with the up-conv erter is moun ted on a fiv e axis p ositioner to directionally steer them. As receiver, a signal analyser ( SA ) (R&S FSW67) with a 2 GHz analysis bandwidth is used. The received in-phase and quadrature ( IQ ) baseband samples are obtained from the SA . The whole system is sketc hed in Fig. 9. In MC2 , for feasibility reasons, TX and RX switc h places compared to MC1 . The RX in form of the SA is put on to the lab oratory table. The RX 20 dBi conical horn antenna is directly mounted at the RF input of the SA . The SA is lo cated on a table close to a corner of the ro om; the RX an tenna is not steered. Similar to the set-ups of [79 – 82], proper triggering b etw een the arbitrary w av eform generator and the SA ensures a stable phase b et ween subsequent measuremen ts. arbitrary w av eform generator PLL up-con verter 285 . 714 MHz ∼ sp ectrum analyzer R&S FSW67 0–1 GHz 59–61 GHz 10 MHz sync. ref. Figure 9: RF set-up for MC2. The VNA from Fig. 5 is replaced with an A W G and an SA. This leads to a set-up where we obtain phase information as w ell. An option of the SA gives us direct access to the baseband IQ samples. 5.2 Receiv e p o wer F or the calculation of the RX p o w er, av eraged o ver 2 GHz bandwidth, we perform a sweep through azimuth and elev ation at a single co ordinate. The LOS and wall reflection from MC1 are still visible in Fig. 10. F ading is ev aluated at a single frequency in the subsection b elo w. Nevertheless, we already indicate fading distributions by markers in Fig. 10 in order to b etter orient ourselves later on. 16 Figure 10: Estimated directional receive p ow er of MC2. Due to the elev ated p osition of the steerable horn antenna, tw o interacting ob jects from Fig. 6, namely the desk and the fridge, are no longer visible. LOS and the wall reflection are still present. These regions are the only ones which are spatially sampled. Markers hav e the same meaning as in Fig. 6. As the steerable horn antenna is ab o ve the office desks and the fridge level, these in teracting ob jects do not b ecome apparen t. In case the steerable TX do es not hit the RX at LOS accurately , the table surface acts as reflector and a TWDP mo del explains the data. F or wall reflections, with non-ideal alignmen t, TWDP also explains the data b est. 5.3 F ading distributions T o obtain different spatial realisations, with the horn antenna p ointing into the same direction, the co ordinate of the apparent phase cen ter is mov ed to ( x, y , z ) – p ositions uniformly distributed within a cub e of side length 2 . 8 λ , see Fig. 11. W e realise a set of 9 × 9 × 9 = 729 directional measurements. This results in a spacing b et ween spatial samples of 0 . 35 λ in each direction. Although λ / 2 sampling is quite common [25, 27], we choose the sampling frequency to b e co-prime with the wa velength, to circumv ent p erio dic effects [83]. W e restrict our spatial extend to a void changes in large-scale fading. Only at directions with strong reception lev els do w e p erform spatial sampling 2 . Similar as in the previous section, w e partition the measurements into 2 sets. The partitioning is made according to a 3D chequerboard pattern. The first set is used for the estimation of the second moment ˆ Ω and the second set is used for the parameter 2 Spatial sampling for all directions takes more than three days. 17 tuple ( K , ∆). x y z 2 . 8 λ 2 . 8 λ 2 . 8 λ Figure 11: Spatial sampling grid. F or one sp ecific direction, w e dra w 9 × 9 × 9 = 729 samples uniformly from a cub e of side length 2 . 8 λ . The distance b et ween samples is 0 . 35 λ with a rep eat accuracy of ± 0 . 004 λ . The orien tation of the horn an tenna is indicated via the cone shap e at the sampling p oin ts. The b est fitting K -factors, in b oth regions with strong reception, are illus- trated in Fig. 12, top part. Belo w the ∆-parameters are provided. Remem b er, the RX in form of an SA is put on the lab oratory table. In case the TX is not p erfectly aligned, a reflection from the table surface yields a fading statistic captured by the TWDP mo del. The in teraction with the wall, similar to Fig. 7, has again regions b est mo delled via TWDP fading. 18 Figure 12: Estimated K -factor and ∆ -parameter of MC2. Due to the elev ated p osition, the wall reflection has a 6 dB increased K -factor as compared to MC1, see top part of Fig. 7. If the beam is not p erfectly aligned, Rician fading turns again in to TWDP fading. W all reflections describ ed by TWDP fading hav e a ∆-parameter of close to one. The table surface reflection leads to a significantly smaller reflected comp onen t (∆  1). The encircled sampling p oin ts are sub ject of further study in Sections 6 and 7. Markers hav e the same meaning as in Fig. 6. 19 6 MC2: Efficien t computation of the spatial cor- relation The wall reflection from the previous section is now sub ject to a more detailed study . Our spatial samples are used to show spatial correlations among the dra wn samples. Our three-dimensional sampling problem, see again Fig. 11, is treated via t wo-dimensional slicing. F or the calculation of the spatial (2D) auto correlation function, we apply the Wiener–Khin tchine–Einstein theorem, that relates the auto correlation function of a wide-sense-stationary random pro cess to its p o w er sp ectrum [84]. In tw o dimensions, this theorem reads [85, 86] F 2 D { C ( x, y ) } = S ( x 0 , y 0 ) , (16) where C is the 2D-auto correlation and S is the p o wer sp ectral densit y of a 2D signal. The op erator F 2 D denotes the 2D F ourier transform. W e calculate all 2D auto correlation functions C ( z ,f ) of one x − y slice at height z at a single frequency f through F 2 D  C ( z ,f ) ( x, y )  = F 2 D  <  H ( z ,f )  x 0 , y 0   conj  F 2 D  <  H ( z ,f )  x 0 , y 0  . (17) The sym b ols  denotes the Hadamard multiplication. The op erator conj {·} denotes complex conjugation. T o ensure a real-v alued auto correlation matrix (instead of a generally complex representation [86]), from the complex-v alued c hannel samples only the real parts <{·} are taken. The spatial auto correlation of the imaginary parts are identical. One could also analyse the magnitude and phase individually . While the correlation of the magnitude stays almost at 1, the phase correlation patterns are similar to those of the real part. The 2D F ourier transform F 2 D is realised via a 2D discrete F ourier transform ( DFT ). The 2D DFT is calculated via a m ultiplication with the DFT matrix D from the left and the righ t. T o mimic a linear conv olution with the DFT , zero padding is necessary . W e hence take the matrix f H ( z ,f ) f H ( z ,f ) =  <{ H ( z ,f ) } 0 0 0  . (18) F urthermore, the finite spatial extend of our samples acts as rectangular window. The rectangular window leads to a triangular env elop e of the the auto correlation function. This window ed spatial correlation is denoted by C ( z ,f ) window ed = D H   D f H ( z ,f ) D  (19)  conj  D f H ( z ,f ) D   D H . 20 T o comp ensate the windowing effect, we calculate the spatial correlation of the rectangular window, constructed in accordance to (18) S = D H  D  1 0 0 0  D   conj  D  1 0 0 0  D  D H . The matrix 1 denotes the all-ones matrix. Matrix S comp ensates the truncation effect of the auto correlation through elemen t-wise (Hadamard) division, denoted b y  . Finally , the efficient computation of the spatial correlation (17) reads C ( z ,f ) = C ( z ,f ) window ed  S . (20) A t a distance of 0 . 35 λ , the measurement data is still correlated, therefore w e are able to view our correlation results on the finer, interpolated grid. The in terp olation factor is 20. That means that we calculate our spatial correlations on a grid of 0 . 35 λ / 20 = 0 . 0175 λ distance. The very efficien t implemen tation of (20) is applied to all (parallel) 2D slices and to all frequencies. All realisations in z and f are av eraged ¯ C = 1 9 1 401 9 X z =1 401 X f =1 C ( z ,f ) . (21) F urthermore, we plot one-dimensional auto correlation functions, ev aluated along x and y , together with their tw o-dimensional representations. W e provide tw o spatial correlation plots ev aluated at an azimuth angle of ϕ = 340 ◦ and ϕ = 160 ◦ in Fig. 13, b oth at an elev ation angle of θ = 110 ◦ . The top part of Fig. 13 sho ws a correlation pattern dominated by a single wa v e. The spatial correlation b elo w sho ws an interference pattern, whic h is intuitiv ely explained by a sup erposition of tw o plane wa ves. The one dimensional correlations, ev aluated either at the x -axis or at the y -axis, show this oscillatory b eha viour as well. 21 W all: LOS: Figure 13: Spatial correlation plot at ϕ = 160 ◦ , 340 ◦ and θ = 110 ◦ . F or the w all reflection at ϕ = 160 ◦ , the pattern shows an interference of tw o plane w av es, supp orting the TWDP fading assumption. F or LOS at ϕ = 340 ◦ , we observe a spatial correlation pattern dominated by one wa ve. The white dashed lines illustrate plane wa v e phase fronts. 22 7 MC2: Time gated fading results T o confirm that our observ ations are not artefacts of our measurement set-up, for example back-lobes of the horn antenna, we now study the wireless channel in the time domain. Our 2 GHz wide measurements from MC2 allo w for a time resolution of approximately 0 . 5 ns. This corresp onds to a spatial resolution of 15 cm. W e plot the ch annel impulse resp onses (CIRs) as a function of distance, namely the LOS excess length ∆ s , that is h (∆ s ) = h  τ − τ LOS  c 0  . (22) The scatter-plot of the CIR s for ϕ = 160 ◦ is shown in Fig. 14. The LOS CIR at ϕ = 340 ◦ is display ed as reference as well. The steerable TX is p ositioned more than a metre apart from the w all. This amounts in an excess distance of appro ximately tw o to three metres. A t this excess distance, a cluster of m ultipath comp onents is present. Note, if the horn antenna points to wards the wall, the wa ve emitted by the back-lobe of the horn an tenna is receiv ed at zero excess distance. Still, the receive p o w er of the back-lobe is far b elo w the comp onen ts arriving from the wall reflection. F ading is hence determined by the w all scattering b eha viour. 23 W all: LOS: Figure 14: Scatter-plot of the CIRs. W e plot the CIRs as a function of spatial distance, where ∆ s = 0 corresp onds to the LOS distance. Our spatial resolution (a channel tap) is 15 cm. The spatial extend of our sampled cub e (729 samples) is 2 . 8 λ = 1 . 4 cm, a magnitude smaller than the spatial resolution. The scatter-plot is ev aluated at a wall reflection ( ϕ = 160 ◦ ) and at LOS ( ϕ = 340 ◦ ). The mean p o wer is plotted with a continous red line. W e observ e that the arriv al cluster centred at 2 . 5 m fades very deeply . The gray highlighted region around 2 . 5 m is further analysed in Fig. 15. 24 1 . 8 2 . 0 2 . 2 2 . 4 2 . 6 2 . 8 3 . 0 3 . 2 ∆ s in m 0 1 2 3 4 5 6 | h (∆ s ) | - observed values mean power Rice TWDP Figure 15: Violin-plot of the CIR time-gated for the w all reflecti on. This figure shows a zo om-in of the gray highlighted region in Fig. 14. In contrast to Fig. 14, the y-axis is in linear scale. Thereby the violin plot indicates the distribution at each tap. The marker shows the mean v alue. The marker st yle co des b est fitting distribution. The gray highligh ted region of Fig. 14 (top part) shows a reflection cluster that corresp onds to the excess distance of the wall reflection. The distributions of each channel tap are represented by a violin plot in Fig. 15. A violin plot illustrates the distribution estimated via Gaussian kernels [87]. Fig. 15 clearly demonstrates that the TWDP -decided distributions hav e multiple mo des. The AIC decisions are plotted as markers at the mean p o wer levels. W e ev aluated the fading statistic in space for ϕ = 160 ◦ at the channel tap corresp onding to approximately 2 . 5 m excess distance. This channel tap is mid in the cluster b elonging to the wall reflection. Fig. 16 clearly shows a TWDP fading b eha viour, confirmed by AIC. 25 0.5 1 1.5 2 norm. amplitude r / √ Ω 0 0.2 0.4 0.6 0.8 1 p ( r/ √ Ω < abscissa) Ra yleigh: K =0 , ∆ =0 Rice: ˆ K =1 , ∆ =0 measured TWDP: ˆ K =4.8 , ˆ ∆ =1 Figure 16: CDF: Distribution fitting for spatial measurement data, time gated by the channel tap at 2 . 5 m, ϕ = 160 ◦ , and θ = 110 ◦ . Note that, similar to the fitting result in Fig. 2, the estimated Rician K-factor is again m uch smaller than the TWDP K-factor. 8 Conclusion W e demonstrate, by means of model selection and h yp othesis testing, that TWDP fading explains observed indo or millimetre wa ve channels. Rician fits of rep orted studies must b e considered with caution. As t wo exemplary fits, in Figs. 2 and 16, sho w, Rician K-factors tend to be muc h smaller than their TWDP companions. There is more p o wer in the sp ecular comp onen ts than is predicted b y the Rician fit. The TWDP fading fit accoun ts for a possible cancellation of t wo sp ecular wa ves. Our results are verified through t wo indep enden t measurement campaigns. F or MC1 and MC2 w e even used different RF – hardw are. While MC1 w as limited to results in the frequency-domain, MC2 allo wed a careful study in the spatial-domain and the time-domain. Ha ving this strong evidence at hand, we claim that the TWDP fading mo del is more accurate to describ e mmW av e indo or channels. The flexibility of this mo del allo ws furthermore to obtain Rician fading (∆ = 0) and Rayleigh fading ( K = 0) results with the same c hannel mo del. In link-level simulations, TWDP fading with ∆ = 1 shows a worse bit error ratio ( BER ) than Rayleigh fading. T o demonstrate this known effect [39 – 43], we pro vide a BER plot in the App endix. Rayleigh fading can hence not b e used as w orst case b ound, esp ecially for mmW av e scenarios. 26 App endix - BER and capacit y loss for TWDP fading − 10 0 10 20 30 40 SNR in dB 10 − 3 10 − 2 10 − 1 10 0 BER Ra yleigh “bound” A WGN bound TWDP: K = 50, ∆ = 1 TWDP: K = 10, ∆ = 1 TWDP: K = 50, ∆ = 0.75 TWDP: K = 10, ∆ = 0.75 Rice: K = 50, ∆ = 0 Rice: K = 10, ∆ = 0 Figure 17: Sim ulation: Bit error ratio of 4-QAM with TWDP fading. The BER p erformance of TWDP fading p oten tially lies ab ov e the normally c hosen “Rayleigh b ound”, for example for ∆ = 1. W e simulate the BER of 4-QAM transmissions with Gra y-mapping and TWDP - fading. The channel is p erfectly known to the receiv er. The receiver employs a zero-forcing equalizer. The symbols are normalised to symbol p o wer of one, the SNR is the inv erse noise p o wer. Our simulations assume frequency flat fading and the only channel tap is generated according to a TWDP statistic. A fading c hannel tap, given K and ∆, is thus simulated as describ ed by Equation (1). W e simulate TWDP flat-fading with indep enden t c hannel realisations. Rayleigh fading and Rician fading are included as limiting cases and are sim ulated as references as w ell, see Fig. 17. As ∆ increases, the BER -p erformance gets worse and worse, confer the dashed line for ∆ = 0 . 5. Finally , Fig. 17 shows the w orse-than-Rayleigh regions [42] for ∆ = 1. A t this p oint, we would like to p oin t out that the unco ded BER, of course, has little significance. The maxim um capacity loss δ C (in bit/s/Hz) o ccurs as K → ∞ and is b ounded by [35] δ C ( K → ∞ , ∆) = 1 − log 2 (1 + p 1 − ∆ 2 ) ≤ 1 . (23) DFT discrete F ourier transform mmW a ve millimetre wa ve A WG arbitrary wa veform generator PLL phase-lo c k ed lo op 27 SNR signal-to-noise ratio SA signal analyser LO lo cal oscillator RF radio frequency RX receiv er TX transmitter CDF cum ulative distribution function AIC Ak aike’s information criterion TWDP t wo-w av e with diffuse p o w er LOS line-of-sigh t NLOS non-line-of-sigh t IQ in-phase and quadrature CDF cum ulative distribution function PDF probability density function BER bit error ratio MC1 first measurement campaign MC2 second measurement campaign VNA v ector netw ork analyser CIR channel impulse resp onse MPC m ultipath comp onen t References [1] Martin Steinbauer, Andreas F Molisch, and Ernst Bonek. The double- directional radio channel. IEEE Antennas and Pr op agation Magazine , 43(4):51–63, 2001. [2] Christopher R Anderson and Theo dore S Rappap ort. In-building wideband partition loss measurements at 2.5 and 60 GHz. 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