On the stationarity and ergodicity of fading channel ... - IEEE Xplore

The 14" IEEE 2003 International Symposium on Persona1,lndoor and Mobile Radio Communication Proceedings
On the Stationarity and Ergodicity of Fading
Channel Simulators Basing on Rice's
Sum-of-Sinusoids
Matthias Patzold
Department of Iqforniation aiid Commuiiication Technology
Faculty of Eugineering and Science. Agder University Collegc
Grooseveien 36. N-4876 Grinistad, Norway
E-mail: niattliias.paetzoldOliia.no
Abstract - Channel simulators basing on Rice's sumof-sinusoids
are playing an important role in fading
channel modelling. However, the parameters of the
sum-of-sinusoids have to be determined carefully to
fully exploit the potential that this powerful procedure
has to offer. This paper presents general conditions
under which the sum-of-sinusoids procedure results in
a stationary and ergodic channel simulator. Moreover,
with the help of the introduced conditions, several established
parameter computations methods will be investigated
with respect to their usability to design stationary
and ergodic fading channel generators. It turns
out t,hat. if and only if the gains and frequencies are constant.
quantities and the phases are random variables,
t,hen the sun-of-sinusoids defines a stationary and ergodic
process.
l

. the 14" IEEE 2003 International Symposium on Persona1,lndoor and Mobile Radio Communication Proceedings
I\,
//,it) = liiii
v. - 1
s
It=]
c ,.,,

(16)
The 14'" IEEE 2003 International Symposium on Personai.lndoor and Mobile Radio COmmuniMtion Proceedings
4. If t = t,, is fired iind Q,.,, is a randoni variable. then
,ij(fiI) = (fi,(t) I t = to] is ii random variable.
'1. If I,otli f = t.,l aiid H,.,, iilo fixed: t,lien fi,(to) is a real
lilllllller alld rqual to i;,(t[]).
Fiji r,hc perioriuiuice e\duat,ion uf the stochastic simulation
iiioclcl: it, is important to know the conditions on which
the stocIi?istic: process fit(t) is stationary and ergodic. A
i:leilr iiiiilerstanding of statimary and ergodic processes is
iiiip)rt,ant, for tlie intentioil of the papcr. We will therefore
I-c\-ii-n. these t,wu twins I,rirfly.
.4. 5tntroiioi?j
IV. PERF0RMAWC.E CRITERIA
;I;(+) is said to be stect-sense station,.
id jL,(f + c) have the same statistics for
;illy ,' E R. A sss prcx litis the following properties:
(i) TIP brst-orcler c1eiisit.y of [i;(t) is independent, of time:
I.

(11)]
The 1 4Ih IEEE 2003 International Symposium on Persona1,lndoor and Mobile Radio Communication Proceedings
.A wiiip;wison of (IN) ancl (18) slioms that fii(t) is autocor- out in Fig. 1. Therefore; all together 2" - 1 = 7 classes
rdiitioii crg

The 14" IEEE 2003 lntemalional Symposium on Personal.lndoor and Mobile Radio Communication Proceedings
irhk til replace the dctermiuistic phases' by random
~iliascs fl,.,,. In this mse. the .lakes niethod results in a
Class I i:liitniiei siniiilator, which is SSS; mean-ergodic, and
iiiitoi:i,rrrlatii)ii-n.godic. However. this is in contrast to the
i~iiirlysis of .lakes' simulator in [12]; where it has been assii~~d
t,liirt tlrc gaiiis c?.,~ are rnndo'in variables. But this
iissiiiiipt.ioii is not in t,he sense of the origiiial Jakes method
iiiiil leads not to the desired resnlts.
Piwiiuier,cr computat,iun mctlrods
TAHIX I1
ovEwiI;\v oi? P.~H.W~JTER COMPUTATION
\I t:~I1iO1)23 ,AX13 THE SI'ifllSTICAi. PROPERTIES OF
'THE ItESUL'I'INC: CHARNEL SIMULAI'ORS
SSS
VII. CONCLUSION
Mean- Auto-
WSS ergodic carrel..
ergodic
111 this paper. tlie stationiiry and ergodic properties of
Raj-lcigli fading ch21nnel simulators using the sum-ofsiiiiisoitls
priiiciple have been andysed. Depending on
n-lint,licr t,lie model parameters are random variables or
uuistnnt, giiiuitities, altogether seven classes of stochastic
rkiiiiinel siinulat~ors hiwe heen defined. It turned out that if
r~iid oiily if the phases are random variables and the other
iiiodid parameters are constitiit, then the channel simulator
is Ijotli stationary and ergodic. If the frequencies
arc ridom variablesl then tlie stochastic channel simulat,(ir
is stationary but non-autocorrelation-ergodic. The
however. is given when the ga,ins are randoni
wiiiililrs aiirl t.he other paranieters are constant. Theii, a
uiiii-stiitioliaiy stocliast,ic clia~i~iel simulator is obtained.
rcstilts presented here are of fuiidairieiital importance
fur illi: perforinancr evaluation of existing design methods.
hlureover. tlie results give strategic: guidelines to engineers
for tlie developmeiit of new design methods.
REFERENCES
S. 0. Rice, "Mathematical analysis of random iioisc..' Bell Syd
Tech. J.. vol. 23, pp. 282-332. .Jul. 1944.
S. 0. Rice, "Mathematical analysis of random mise:' Bell Sy,~l.
Tech. J., vol. 24. pp. 46-156. Jan lY45.
W. C. Jakes. Ed.. Micmulaur: Mobile Comrnunzcations. Piscataway,
NJ: IEEE Press. 1993.
hl. Pitzold. U. Killat. F. Lauc. aid Y. 1,i. ..On the statistical
properties of detelminislic siniiilatioii ~iiodels for iriobile fading
chaoncis." IEEE Rms. Veh. Tecirnol.. voi~ 47. 110. 1. pi). 4i?-
209. i'eb. IWX.
P. Heher. "A statistical discrete-time model for the 7%
multipath channel." IEEE Rans. Veh. Teehnol.. vol. 41.
pp. 461-488. Nov. 1892.
hl. Pitzold. Mobzle Fading Chanmls. Chichester: .John Wile?
82 Sons, 2002.
R.1. Pitzold and N. Yousscf. -%lodeliing and siiiiiilati~ii of
directioii-selective and frecliiency~selective niohile radio cliiiiinels."
lnt. J. Electr, and Comnli~n.. vol. AE;U-55. 110 ti. pp
433-1142, Nov. 2001. '
hl. Pitzold. "System Functions and Characteristic Quantities of
of Spatial Deterministic Gaussian Uricorrelated Scattcring Processes,"
in Proc. IEEE 57th Veh. Technol. Conf.. VTCS003-
Spnsg, Jeju, Korea, accepted for publication, Apr. 20113.
H. Schulze, "Stochastische hlodelle und digitale Simulation
voii hlobilfunkkanilei~." in U.R.S.l/lTG Conf. zn, Kleziiheubach
1988. Germany (FR). Proc. Kleinheubaclrer Reports oftlre Gcrinan
PTT. Darmstadt. Germany. 1980. vol. 32. pp. 173-483
h.1. Pitzold, IJ. Killat. and I:. Lonc. "A deterininistic digit,ai sindation
model for Siiauki processes wifh applicatiw to it sha&
owed Rayleigh laird mobile radio clrunnel.'' IEEE 7i.m~. Veh.
Techmi.. vol. 46. no. 2. pp 318-331. May 1996.
Y. R. Zheng and C. Xiao, "improved nrodels for the $er)
toll of milltipie oncol-re~ated nayyleigii fadilrbr waverorms.:' I
Communicallons Letters. vol. ti. no. 6. pp. %ti-258. J,m 1002
RI. F. Pop and N. C. Beaulieu. -'Limitatiaua oisuiii-of~sinosoids
fading chairiiel simulators;" IEEE 7Sans. 011 ConwnzLnzcafio7ir.
49, no. 4, pp. 699-708. ~pr. 2001
P. hl. Crespo and J. Jinrhez. "Computer simulation of radio
channels using a liariiionic decomposition technique." IXEE
Xns. Veh. Teehnol., vol. 14. no. 3, pp. 414-410. Aug. lY95.
R. H. Clarke. "A statistical theory of mobile-radio reception.''
Bell Syst. Tech. Journal. vol. 47. pp. 957-1000. Jul. /Aug. l96X.
A. Papoulis and S. U. Pillai. Probabilitg. Random Variables .
and Stochastic Processes. New York: h?lcGmw-tIill. 4th edition.
2002.
W. R. Bennett. "Distribution of the sum of raiidonrly phased
components," Quart. Appl. A4ath..' vol. 5. pp. 385-39% hlq
1948.
hl. Pitzold.'U. Killat. F. Law. and Y. LI. "On the problems
uf Monte Carlo method based sinidation models for nwhile ritdio
chnnnels." in Proc. IEEF 4th Int. Synzp. on Spread SJEC.
trum Techniqzies k Applzcntions. ISSSTA '96. hkyerre. Ccrmany,
Sept. 1996, pp. 1214-1230.
M. Piitzold and F. Lam, "Statistical properties of .lakes. fading
channel simulator." in Proc. IEEE 48th Veh. Technol. ConJ.,
VTC'98. Ottawa. Ontario. Canada. May 1998, pp. 712-718.
'11. aliould be noted that. the phases Oz.,, in Jakes' channel simuiator
are equal to 0 lor ali i = 1. 2 and Ti=
1, 2, . . . , N, [le].
1525