Modeling Multipath Fading Responses Using Multitone Probing Signals and Polynomial ‘Approximation By LJ. GREENSTEIN and B. A. CZEKAJ tonuscrit vested August 7, 1980) We show in quite u general way that highly accurate modeling of muitipath fading responses is possible using low-order complex poly- hnomiats. This appliee to all trretril radio systems in the chan: relized common carrier bande below 15 GH, where channel widths fare 40 MHz or lo. The context of the study is a new multipath tesperiment being conducted in Net Jersty over @ 23-mile path at IL Gite, The transmitted signal consists of upto nine tones ina 40-MIT2 ‘bandwidth, Theve tones are coherently processed, sampled, and dig itied in the revelver and recorded. during fading evente, for later Offtne reductions, Sinple routines eam be used to determine pol homiat coelcients from these recorded! data. This paper describes the signal processing and data reduction methods and analyzes them to assees the accuracy of pobmonial fitting. The analysis use a mean-square error measure and assumes a representative form for The underlying response function. Our results predict Dat the vast ‘ajaits of multipath fading responses can be accurately approxi Inte over bandtwid hs of 40 (2) SLIz using frst. (Second) order ‘Complex polynomial | wrrooucrion [Moulkipadh fading (hereafter abbreviated oP) on tarestrial micro wave paths can be a major ease of cutage in digital radio systems Numerous efforts have been aimed at understanding, analyzing, and correcting this source of disruption, and some have led to new sats: tical modele for MPP responses” "The patticular model that ingptes the presont work approximates the seer responce by a low-oeer complox pelynomial in frequency * For a particular 2-mile path in Georgi, fea shown that a Bost order pulyimnil suffices to characterize th fading exponse in &25- 193 MHz band centered near 6 GHz ‘The joint probability distribution for the polynomial coefficients was derived for that path, thus permitting 1 complete statistical description of the MPP response. [Now another experiment is being instrumented this time for & 23- rile path in Nev Jereey operiting inthe 1-GH baal. The wim of he row experiment is to add to, and i several reapecta improve upon, the ‘ta rs acl to quantify che earier polynomial model. "The improve >menta include higher measurement signal to noise ratios (sus, higher sampling rates (39 measurementa per second rather than 8), coherent processing to obtain phese information (previously absent, and a ‘Wider measurement bandwidth (0 MHz rather than 26 M2). (Given the highly variable nalury of multipath fing each improved smasauremente for anew path in u different frequoney band and loale Should add inporantly to our nowledge of this paenounenn The basic design of the experiment can be simply stated: Aa many 1s nino coherendy-phased tones within « 40-MPz bandwidth are transmitted from Murray Hill and coherently demodulated in a re iver at Crawford Fl; che demodulated tones are sampled, digi, tnd wereene by desktop computer/eontoller and the digo dat, if deemed interesting, are recorded on magnetic tape for later off Hine processing "The recorded data willbe in a form that facilitates gelynonial approximation using simple, ecient computor routines. The data will bbe quice goneral in form, however, ie, amenable to modeling via any mathematieal approximation considered promising. "The present scudy evaluates the accuracy of polynomial sppron mation, relating eta the experiment parameters and to he meas of signal procesing and data reduction Section I describes the signal proceasingin the transmitter and receiver, and derives signal and nose Zelationships used in the subsequent error analvsea. Section III de teribes the methods of polynomial iting lo be considered, and defines ‘the moan-square error measures Unat wil be used to wvaluate them. ‘Section TV aaalyees the errors in the polynomial ting caused by ‘measuremenc noice, and Section V analyzes the errors caused by finite sampling of the frequency response. In gonoral, the erors increase ‘with the bandwidth over which the fitting is done. In analyzing the ‘errors caused by finite sampling, we assume a general form for the >t response function that has beca applied successfullyin other da fitting sttien® and assume cither worst-ase or fypenl values forthe anetion parameters "The mear-aquare error calculations permit prediction of the mis ‘mum bandwidths for which polynomial fitting is valid. Section VI summarizes the results computed, under ather stringent meanvasunre fervor requirements, for polynomial orders of one, fo, four, sx, and wight 104 THE BELL SYSTEM TECHNICAL JOURNAL, FEBRUARY 1981 Ih SIGNAL PROCESSING ANALYSIS FOR THE MPF EXPERIMENT 121 Propasaton pat and rselo channels ‘Maulipath fading responses are to he measured on » 23-nile path Doeowen Murray Hill and Crawford Hil, similar to the one wed by CCraword and Jskex in their earlier experiments." The transmitting fantenna tt Murray Hil is 656 feot above sea level, and the receiving fantonna atop Crawford Hill i425 fee aboveses level. An experimental Tieenge hus heen obtained to operate over this path in three 40-Mz channels within the 11-GHe common carrer band. These channela are fencered at 11465, 11,45, and 1.625 GH The initil measurements ‘ill be inthe band centered st 11548 GH ‘We desribe here the signal processing relationships that undertie the experiment design, ‘The dlalls of circuitry, components, and ‘yuipment wl be reported separately by those who have developed (he srr measurement stem. 22 Transmitted signal "The transmitod signal is created by the two-stage upronventon of ‘baseband signal having the fom ote) =a + de cosindat +0), 0 ‘where is even and the oler parametors willbe discussed. The up Conversion places the signal in an RP chanecl centered. at radian frequency er = 2a, (fo 11M GH}, ence the transmitted signal win ¥ Tp, coslact + not + 4), a whore py ix the power of the nth (ranamlted tone and a toxal of N=] tones are transmitted, From (1), se aee that pis proportional lo d and that p, (m0) ears the ame proportionality ti 74 “The variation ofp, wich mis clearly symmetrical about n= 0 because it derives from amplitude settings of the baseband tones. Nonuniform ‘arieions of 7 ean early be compensa for via baseband adjust {nents in thw receive. In Section TV we consider nonuniform variations fer which receiver noise effects are minimized "The frequency apacing belween tranzmitted tones, Af, may be 5, 19, oF 2) MHz, Since the trameniesion i confined to a channel of 40 Mis width, and occupies a bandwidth Af, we havo the constraints te 28 tones whan 3f~ 8 Mil (N+ 11-35 tones when 37 10 Mic anu (41) 223 toncs when 3/20 Mila We will consider the four patsicular combinations N= 2, Af = 20 MME N= 4, 8/ = 10 Mig N'~ 6,8/=5 MHz snd N= 8, Af= 5 Mil Finally, we mention that the N/2 baseband tones are dorivel from MODELING MPF RESPONSES 195 ‘2 common 5-Mlle reference, and so can be relatively phased in any tanner desired. For purposes of analysis, we wil assume all 3 to be sero here since any phasing in the transmitter are eaaly undone in the receiver, no generality is leet. Ono erterion for choosing the actual (ye is minimization of the peak factor of the RF signal (2). The Thasehand phase adjustments that accomplish chs have been derived for N= 2, 4,6, and 8 We will une the resulting minimized peak factors in making noise calculations later 2.9 Response ofthe propsgston meatum ‘We denote the complex signal gin of the propagation medium by i “The quantity gy can be compute from familiar nlio path equations Note that ui meseared from che center of Une channel. During ‘onfading periods, [F(u) | = 1 throughout the channel bandwidth, during multipath fading, Fs) varieg with «in a randomly Line "The function F(«) contains two phase factors of no interest to us. ‘One is exp(jo), where gy is the phase shift through the medium at '= 0; the ther i exp(jat,), where fis the nominal propagation time ‘long the path, (For a 26-mile path fy = 0.13 ma) The investigation of ‘multipath fading can be simplified, with no loss of information, by removing these wo factors. Thus the function of interest to uss Ha) = Flo) exp -¥e-+ af ‘The sim of our modeling ofort ato find suitable functions for approx mating (4), and to statistically characlorize tho parameters of those fenetona ‘We will soc in Section 25 chat the response function aetully sampled by the measurement system in cw -roml(ereg) ‘where ¥ and 0 are the (possibly) random oF unknown phases of Treucnty references in the receiver. To obtain samples of che desied Teton [2 (a) | fom samples of the messured function [G()] il ‘therefore require performing the operation ‘ Hw B]) 196. THE BELL SYSTEM TECHNICAL JOURNAL, FEBRUARY 1963, at onch ofthe sampling frequancins [a= 0, #0... (N/D80] We ‘vill show later how to nccomplish this in the data processing 24 Decompositions of Fs), Gta, and Ml) ‘We demonstrate here « useful decomposition for complex response functions such as #4), Gla, and #0). We wil trat only Hc), roting that the same mathomaties and notation apply to Mc) and Glor Since w in measured frou an azbitrary microwave frequency (297) ‘hore sno physical Teton ¢owstume complex conjugate symmetry for ‘Hoh. In ix most general form, 7) can be expressed as He) = Halu) + jlo), o ‘where Hla) ar Halu) axe euch functions having complex conjugate symmetry. Aocordingly, we can write Hof) ~ Hols) + jHales), Hola) = Hla) + jbale. 8) {By transmitting and coherently receiving N + 1 tones spuecl by 4, one could in theory abtain messurements of the two even fanetions alio=0, du, ++, (2/2) Su; and of the two odd funetions ub w= Au, (28/2) Bo. [he total number of samples, 2. + 2), is eonsistot ‘with meamaring the amplitudes apd phases of une N+ 1 received tones) In reality, the receiver obtains thexe amples for the corre ponding st of G funecions, which differ from the H fonctions f Vand Gare not both 0, Obtaining H samples from @ samples is discueed in Section 26 “Another dkyartute ofthe receiver witputs from the desired samples is the presence of measurement ee. We will defer th introduction ‘of noive wall Section 27 25 Signa! processing inthe receiver "The neceiver input at mri Val =e Ym |Fleonlat+ndet+ ga, ‘where | Fo] anil are the magnitude and phase of Find; isthe ‘orm (nanading} path gain, and we have used (2) with all thse en be ar "The signal goes through o two stage down conversion which amounts to quadrature demodulation. That is, to baceband oulpuls are obe tained which correspond to mixing Val) with 2 cas(ed +) and with Sy'snlort +). A nonzero valve of signifies that che Hr and W MODELING MPF RESPONSES 197 :forones inthe reiver are nat in phase aynehronicm with thore in the transmiter, Buch of the baseband siguls consists of a de component plus sinaeoide nt = Oe, «=, (N/2}4ux The mth sinaaoid in each of these surly rough quadrature demerdlation, via the local references cos[n(Aut + 8)] and —sinfridor + 8), to produce two more de ‘outputs. These references are al derived from a 5-MIl2 source ip the receiver, ond nonzero 4 significs that this source is not in phase tyohroniam sith the one in the tranemitar Using (5) and ordinary celgonamecric identities, the following state- meni ens he pe Se aca Sainte (seis) To) « (Towne) , (6) =(emecs) w (ie) The de outputs produced by demodulation via 2eoatent + J) Beinbost + ad conlidet +O ae T= Th) « (condo) ti th) ® (apetctns) > Ui) The de out produced by demotion va (Gan?) sod ant + ae U- Qe). (VEpaGdtnde) (on) «Hees» Pen paierp tention Pemeare eon ememety inn oarry cpt ah eM iy il ten 28 seein LN. (ay Ni. 12) Due Capo. + Hv Pocard 198 THEBELL SYSTEM TECHNICAL JOURNAL, FEBRUARY 1981 [Any set of samples that seems intervtling, or is par of a sequence that ‘outs interesting, is revorded on magnetic tape for subsequent of line processing 2.8 Relating the Gand H functions By wing 46), the Hfunetions (te), Has, ee) can be easily contained from the sorresponding G inetions once 4@ and AT are Specified, Defining » new function G(o; A) A Cfo) exp(j84), we frst ‘befor the matrix operation ootur sn] forse inde |[G.06] ay [areisay}=[sinae coed |] Goieh Z An identical equation relates Gaus A) and Gulu A) to Gas) and {Gia}. The H functions ae then obeined! from Usexe new G functions, for apecifed 87, ns follows: Heieh) _|vmwdT sin w8T][ Gnu; 40) Heid |*[sinaar oon wat |Lexte: aor ‘An identical cqustion relates Hes) and Huu) to Gifu; 80) and lo; 89), “The operation indicated by (18) loads to the result He) ~ 0, ‘the phase reeponge ofthe function co bo analysed is forced to zero ai U2 '0. [To an th, combine (6), (12) and (18) for w = 0.) This is « “folcome rimplifasion in the data snd entail no Tos of uaefl infor. mation, Fortunately, sin 4® and cos 4b are readily obtained from the ‘heamured G samples, oF os x10) G10) sin ad = — ——, condo = os Yeo IO + TLIO) ‘To geo this use (8) ab 9 = and recall hac A = Y— 40.1 “Hy way of contrast, the vale of 47 Lo ue in (14 is not so read spectied or dewornined. Yet, W yet the full benefit of polynomial todclng (accurate fitting using low-order functions), AT must be farefully chowen,” Wr have arrived at x criterion for choosing 47 bused upon the following data reduetion procure: For a givon AT, (1s applied and the revlling #! samples are Bted by a nite-order ‘complex polynornil in jo. We consider that value of 4 to be optimal for which the polynomial Gtting is best, in some least-squares sense nso gmt AE Cs ca ean Mae eS Berhad PSP Mee RSE ime pee ae rae MODELING MPF RESPONSES 199 etind later. In Section 64, wo wil identify daca-derived measure that accurately prodicts the optimal AT: We havo shown that the de receiver outputs axe proportional Jrequency samaples of he function Glo), ad tha Uke unwanted phase factors that distinguish Go) froma 77(a) ean be removed in the data processing, Not to readily removed are the oles associated with the Aigtized outputs. These consist of hoth additive Gaussian nce fom the input anv componenis of the reesiver and quantiring noise from the [C-bit analog-<o-digital conversions, ‘Receiver noiee produoss an eddtve random component for each de output defined by (10) to (12). These 2.N + 1) noises are zero-mesn and mutually independent. All have the game variance except (hate ‘srcocinted with (10), Le, n ~0, for which the variance i 3 dB higher. ‘These findings flow from the receiver procesing described in Section 25. ‘We shall now asmume that each of the de outpute in (10) to (12) is adjusted by n flor 1/(ab V2pa) m= 0, N/2, before being digitized. ‘Accordingly, the variance ofthe Gna noise waociated with a given, ‘output came i po [ATINeiDuth, OFS ATEN Bohs m= 1, NP ‘Table I defines the quantities in (18) and gives values for each. Using those data and assuring uniform Lone powers, we obtain the following us) ‘Table System parameters usod in noiee analyaie 7 eS oS ar rharmal tl tat aso opt coe lh of oer proce 4h, Nat ges ctor ones acer ae “Ieben estan 200 THE BELL SYSTEM TECHNICAL JOURNAL, FEBRUARY 198% result: fin AB, form 0, les the eange ~T9 AB 2 AB, where the preci value depends on form = 0 08 dB higher. “Assuring che de ovtputs are amplitude-adjusted as indicat, the input w every Lebit A/D conversion ix precsly 2 sample of & fanetion. During normal propagation, the sample ie within = 1 [sce and G)]_ To provide some room for excom gain, we assume ‘quantizer amplitude Limits tet at += L60 (4-4R margin). Axa result, the ‘Tuantiaing or foreach digitied sample can be characterized as an ‘tditive olne uniformly distributed on [—8/2, 8/2], where A = 2 x ri O96 x 10" The quastizing noike variance is chen 954i), every sample. (17) Comparing this with of above, we find jstifencon fr ignoring quan- tivation effects, Alternately, chey can be accounted for using an ap- proximate correction factor given in Section IV. "To simplify matters further, we intrudes the notation Hag Heltbe), Hien d Hednbud, ote. (18) ‘These are the quantities produced by (13) and (18) for any speifet combination of &@ and AT. We nocount forthe noisiness of the HE ‘samples vn the notation| Flare 8 Horn + farm Hina 8 Hon Sine ty 18) wher Banas Sou ht ae ce mando noige samples. Since the {sare Dprodueel by phase rotations of the noises asqocialed will the Semples, they are all Gaus zero-menn, and mutually independent, Sa Uke the onginel noses. Moreover, their variances ar denticl 1 ‘tha given by (16), We thas have an accurate, simple decription for the noisines of the data to be proceed, ob = ati 38 x 10 Ii, POLYNOMIAL FITTING AND ERROR MEASURES £21 Fitting polynomial o the Ht sarales "The implicit asmumption ofthe polynomial Sitting approach is that, over some finite bandwidth 2 centered on f= 0, the response function Hic} ean be aecurataly appronimated by a low-order complex poy- omin ie, Ha) = Hie) 2 SABLE, folsom) ‘sing (7) and (8), we can brea this eepresentation down as fllwss nia Lawar'= $ aust + Eatin en Twi Peele MODELING MPF RESPONSES. 201 toy =F wacion = Banta + Fwd ey ‘The A's and By's are slowly varying random coefficients; in any given ‘mesiurament interval, they collectively characterize the shor(term frequency seeponce ofthe propagation medium. "The flings indicated above ean be done, fr every 60-ms measure- ment interval, by uxing the 21N +1) HT samples obtained in that interval ‘The way che fing is done depends onthe valucs of Mend IN- We now consider three possible cases 800 1M = Nth Nm 2,4, 6,078 ‘Theft samples obtained using N + 1 tones can bo fitted precisely sing an Nthorder complex pelynomial. Thus, when M =X, fting ‘comms of matehing each summation in (21) and (22) to he appropri fle H samples at the sample frequencies. The resulting equations for the Ave are a9 follows (identical equations apply to the Bye, with Fiyaand Hy'sveplacing the Hrs and Hai): (Bow a0, [aye B Pindtlce Hrs} K-24 oo 8, Act E Dibesne fan REM a1 28) where Din and Dim are the (mith elomenta of the N/2 x N/2 matrices [D"] and [D"), respectively; [D'] and [D°] ae the inverse ofthe macrices [4] and [a], respectively; and the (m, 2th elements of a") and [d"} ore a= (1m en and aay (Dm 5) ‘The matrees [D"] and (D*] for = 2,4 6 nd 8 are given in Table TE Note for future reference that the derived Av'sand By'sare weighted sums of the H samples ‘Since N’ean he as high as eight, this method of Ating suggests the posuibility ofeighth-oder polynomial mexteling. Earlier stadia, ho rugget that this orde is unnecessarily high for bandwidths of 1 to-40 Milz** Reductions using M ~ N= 8 may therefore involv cacessive demands on data storage and analysis, and unduly complicate 202 THE BELL SYSTEM TECHNICAL JOURNAL, FEBRUARY 1981