Table Of ContentModeling 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
