Ergodic Capacity of Discrete- and Continuous-Time, Frequency-Selective Rayleigh Fading Channels with Correlated Scattering

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by  Martin Mittelbach

Ergodic Capacity of Discrete- and Continuous-Time, Frequency-Selective
Rayleigh Fading Channels with Correlated Scattering
Martin Mittelbach∗, Christian Mu¨ller∗, Konrad Schubert†, and Adolf Finger∗
∗Department of Electrical Engineering and Information Technology / Communications Laboratory
†Department of Mathematics
Dresden University of Technology, 01062 Dresden, Germany
E-mail: mittelbach, muellerc @ifn.et.tu-dresden.de, [email protected]
{ }
which can be adapted to serve as model for a single-
8 Abstract— We study the ergodic capacity of a fre- antenna, frequency-selective fading channel. However, in
0 quency-selective Rayleigh fading channel with corre- eithercase uncorrelatedscatteringis assumed,whichdoes
0 lated scattering, which finds application in the area of
not necessarily apply to UWB channels, as repeatedly
2 UWB.Underanaveragepowerconstraint,weconsider
a single-user, single-antenna transmission. Coherent documented in the literature [8], [9], [10]. This is one
n
reception is assumed with full CSI at the receiver and basic difference betweenconventionalandUWB channels.
a
J no CSI at the transmitter. We distinguish between Inthispaper,westudytheergodiccapacityoffrequency-
a continuous- and a discrete-time channel, modeled selectivefadingchannelswithcorrelatedscattering.Tothe
3
eitherasrandomprocessorrandomvectorwithgeneric
best of our knowledge this has not previously been exa-
covariance. As a practically relevant example, we exa-
T] mine an exponentially attenuated Ornstein-Uhlenbeck mined in the literature. We consider models appropriate
process in detail. Finally, we give numerical results, tocharacterizesmall-scalefadingeffects,i.e.,fadingdueto
I
. discuss the relation between the continuous- and the constructiveanddestructiveinterferenceofmultiplesignal
s
c discrete-time channel model and show the significant paths. We assume the small-scale fading to be Rayleigh
impact of correlated scattering.
[ distributed,whichisnotstandardinUWB channelmode-

ling [9]. However, detailed statistical evaluation of mea-
4 I. Introduction
v surements in [10] support this assumption also for UWB.
0 Due to the increasing role of wireless communication Furthermore, in [11] it is shown that the capacity of the
9 it is important to determine the maximum achievable in- Rayleigh fading channel provides a tight approximation,
0
formationratesovermultipathfading channels.Assuming even if other fading statistics are employed.
1
0 anergodicfadingprocessandsufficientlyrelaxeddecoding We distinguish between a continuous- and a discrete-
7 constraints,such that fluctuations of the channelstrength time channel, where the channel impulse response (CIR)
0 canbeaveragedout,thentheergodiccapacity isasuitable iseithermodeledasrandomprocessorrandomvectorwith
s/ performance measure. It basically represents the average genericcovariance.Since both models areimportant,they
c over all instantaneous channel capacities [1], [2]. are treated in parallel. The former is more general and
:
v In this paper, we examine the ergodic capacity of a more suitable to analysis, whereas the latter is more ade-
Xi single-user, single-antenna channel with full channel state quate for computer simulations and parameter estimation
information (CSI) at the receiver. The assumption of from measured data. Note, with the discrete approachwe
r
a coherent reception is reasonable if the fading is slow in modelequidistantsamplesoftheCIRratherthanvariable-
the sense that the receiver is able to track the channel distant physical paths as in [12]. Modeling the sampled
variations. The transmitter has no CSI but knows the impulseresponsebetterdescribestheeffectivechanneland
statistical properties of the fading. Further, we imply isconsideredmorerobustsinceonlyaggregatephysicalef-
an average power constraint on the channel input and fectsneedtobe reflected[10].Thisisparticularlyrelevant
Rayleigh-distributed fading. where frequency-selective propagation phenomena occur.
Under the above constraints, the ergodic capacity was For the continuous-time Rayleigh fading channel, we
investigated for the case of flat Rayleigh fading, e.g., in examine a detailed example with special covariance. We
[3], [4], [5]. The recent interest in ultra-wideband (UWB) utilize an exponentially attenuated Ornstein-Uhlenbeck
technologies makes it important to examine the capacity process being mathematically tractable and capturing
also of frequency-selective fading channels. As related an exponential power decay, which is common in UWB
information-theoretic work, consider for instance [6],[7], channel modeling [9], [12]. Additionally, it incorporates
where in [6] a model with two scattered components was exponentially correlated scattering as measured in [8].
studied. In [7] a multi-antenna system was considered, This paper is organized as follows: Section II specifies
thechannelmodels,andinSectionIIIwederiverespective
expressionsfortheergodiccapacity.InSectionIVweana-
This research was supported by the Deutsche Forschungsgemein-
schaft(DFG)undergrantFI470/7-1. lyzetheexampleofanexponentiallyattenuatedOrnstein-
Uhlenbeck process,givenumericalresults,anddiscuss the are real i.i.d. Gaussian processes with zero mean and co-
relations between the continuous- and the discrete-time variance function
model. Section V finally concludes the paper.
The following notation is used. The operator E[] de- R(τ,τ′)=R˜(τ −τ′)g(τ)g(τ′), τ,τ′ ∈R. (2)
·
notes expectation, j is the imaginary unit, and X is the The indicator 1 is introduced since only delays τ 0
A ≥
complex conjugate of X. We use the abbreviations R- are meaningful and u is some suitable function modeling
integral for Riemann integral and i.i.d. for independent power decay over τ. Thus, (X ), (Y ) are attenuated,
τ τ
and identically distributed. A wide-sense stationary ran- non-stationary versions of (X˜ ), (Y˜ ) with X =Y =0
τ τ τ τ
domprocessisreferredtoasstationaryprocess.Wedefine for τ<0. Finally, the continuous-time Rayleigh fading
the sets Z := 0,...,K 1 , K N, and W:= –W,W , channel model is defined as (H ):=(X +jY ), τ R.
W > 0. FuKrthe{r, X = (X−k),}k∈Z∈K, represents(cid:0)a c2olu2m(cid:1)n Note, by now nothing is saτid abouτt banτd- or∈time-
vector of size K with components Xk. Matrix notation is limitation of the channel. Combining a stationary process
equivalent using two indices. with a decaying function allows us to take advantage of
well-investigated random processes as described, e.g., in
II. Channel models
[14, Ch. 3.5/3.7] while capturing the decaying nature of
As general fading multipath channel model we consider measuredCIRs.Fornormalizationwewillusetheconstant
a linear time-varying system with equivalent lowpass im- c:=2 ∞ R(τ,τ)dτ, which represents the mean energy
pulse response (Hτ,t) being a complex random process in contaiRn−e∞d in (Hτ). It is finite if g2 is R-integrable. This
the time variable t R and the delay parameter τ R. is obviously a reasonable assumption from a practical
Then a realization ∈h(τ,t) := H (ω) is the channe∈l re- viewpoint. Further conditions of R˜ being continuous and
τ,t
sponse at time t due to an impulse at time t τ [1], [13]. g being R-integrable we will motivate later.
Next, for fixed τ we assume the channel to −be invariant Uncorrelatedscatteringcanalsobe includedinthe con-
within coherence intervals of fixed length. Hence, we may tinuous modelutilizing a covariancefunction R˜ informof
considerthe randomprocess(H ), τ R, inthe discrete aDiracdeltadistribution,i.e.,R˜(τ˜)=c˜δ(τ˜)forsomec˜>0.
τ,n
time variable n Z. Further, we imply∈(H ) to be sta- However, to rigorously derive the ergodic capacity in this
τ,n
tionaryandinde∈pendent for fixedτ, whichcorrespondsto case we need to extend the mathematical tools applied in
the block fading model. As a consequence of this major this work, as will be briefly discussed in Section III.
simplification we are able to drop the time index n and
B. Discrete-Time Rayleigh Fading Channel Model
model the channel as random process (H ) in the delay
τ
variable τ R. Another widely-used assumptionof uncor- Forthediscrete-timemodelweassumethechanneltobe
related sca∈ttering, i.e., E[HτHτ′] = 0 for τ,τ′∈R with τba=ndl-l,iml itZed, ttoo Wobt.aTinhetnhe(Hcoτm)pclaenxbraensdaommplperdocaetsdse(lHay)s
τ 6= τ′, does not necessarily hold [8], [9], [10]. Therefore, in thWe di∈screte delay variable l Z. Note, we have infinitle
weassumecorrelatedscattering,whichisasubstantial dif- ∈
expansion in delay due to band-limitation and H =0 for
ference to previous work. l
l<0 due to H =0for τ<0.In the followingwe will refer
InadditiontofadingweassumeadditivewhiteGaussian τ
to H as the l-thchanneltap. Next, we approximate(H ),
noise (AWGN) at the receiver. Below, we distinguish bet- l l
l Z, by a random vector H:=(H ,...,H ) of size
ween a continuous- and a discrete-time channel. Thus the ∈ 0 L−1
L:= WT +1.Here, L models the number ofsignificant
noiseiseithermodeledascomplexwhiteGaussianprocess ⌊ d⌋
(Z ),t R,withi.i.d.realandimaginarypart,eachofzero channeltapswithTd beingthe channeldelayspread.This
t
mean a∈nd power spectral density N0 or as complex white practically feasible approximation is well-founded since
Gaussian process (Z ),n Z, with2 i.i.d. real and imagi- CIRs are sufficiently close to zero for delays τ>Td, which
n
∈ is mathematically captured by u in (1). Note, for flat
nary part, each of zero mean and variance N0.
2 fading we have L=1, for frequency-selective fading we
Next, we will specify stochastic properties of (H ) to
τ have L>1, and for UWB we clearly have L 1. Finally,
obtainaRayleighfadingchannelmodelforcontinuousand ≫
we denote the L-dimensional complex random vector H
discrete time.
asH=X+jY,whereX,Y arereali.i.d.Gaussianvectors
with zero mean and covariance matrix
A. Continuous-Time Rayleigh Fading Channel Model
Let (X˜τ), (Y˜τ), τ∈R, be real stationaryi.i.d. Gaussian Γ:=(γik), γik :=̺ikσiσk, i,k ∈ZL. (3)
processeswith zero meanand continuous covariancefunc-
Therein, 2σ2 is the mean power of the l-th channel tap
tionR˜.LetgbeanR-integrablefunction,i.e., ∞ g(τ)dτ H with σ2l:=E[X2]=E[Y2]=E[H 2]/2. If we define
exists as improper R-integral. We define R−∞ p l :=p :=l (σ2), ll Z , thlen p:=|pl|+p is the mean
re im l ∈ L re im
g(τ):=u(τ)1 (τ), τ R, (1) power delay profile,which is relatedto u in (1). The coef-
[0,∞) ∈ ficients ̺ik represent the normalized correlation between
with 1 the indicator function being 1 if τ A and 0 oth- tap H and H . Uncorrelated scattering is included as a
erwise.AThen (X ) := (X˜ g(τ)), (Y ) := (Y˜∈g(τ)), τ R, speciali case, wkhere the covariance satisfies Γ=diag(p/2).
τ τ τ τ
∈
For normalization we set the mean power of H to 1, i.e., B. Continuous-Time Rayleigh Fading Channel
lL=−01σl2=12. Subsequently, we refer to the above defined Theorem 1: The ergodiccapacity(4)ofthe continuous-
Pmodel as the discrete-time Rayleigh fading channel. time Rayleigh fading channel is given by
III. Calculation of Ergodic Capacity W/2
C = 1 exp 1 Ei 1 df, (6)
We now calculate the ergodic capacity for the defined ln(2)Z W/2 (cid:16)2ασˆ2(f)(cid:17) 1(cid:16)2ασˆ2(f)(cid:17)
−
continuous-anddiscrete-timeRayleighfadingchannelun-
where
der the general conditions specified in the beginning of
theintroduction.Unlessstatedotherwise,capacityexpres- σˆ2(f)= ∞ ∞R(τ,τ )cos(2π(τ–τ )f)dτdτ , f R, (7)
′ ′ ′
Z Z ∈
sions are given in [bits/s] for the continuous model and in 0 0
[bits/s/Hz] for the discrete model. with R as defined in (2) and Eim(z):= 1∞e−tzt−mdt. In
particular, Ei1(z)=–Ei(–z) with Ei theRexponential inte-
A. Capacity Formulae gral [16, 5.1]. If R˜ in (2) satisfies R˜(τ˜)=c˜δ(τ˜), we have
In the continuous case the ergodic capacity within the uncorrelated scattering and obtain
frequency band W is calculated by [6], [1]
C = W exp 1 Ei 1 , (8)
ln(2) cα 1 cα
W/2
(cid:0) (cid:1) (cid:0) (cid:1)
C =E(cid:20)Z log2 1+α|Hˆf|2 df(cid:21), (4) wherec=2c˜ ∞ g2(τ)dτ istheonlyparameteroftheCIR
−W/2 (cid:0) (cid:1) thathasaninRfl−u∞enceonthecapacity.Expression(8)iswell
where (Hˆf):=( ∞ e−j2πfτHτdτ),f R, is the Fourier known and was already derived in [6].
transform of theR−p∞rocess (H ) and α∈:= P defines the Proof: Here we just provide the main parts of the
τ N0W proofthatallowtounderstandtheunderlyingmethods.A
average signal-to-noise ratio (SNR). The type of integral
tocalculateC andHˆ isastochasticR-integralasdefined, complete proof containing omitted details is given in [17].
f
Relevant properties of stochastic R-integrals and of mean
e.g., in [15, Ch. 3.4]. Clearly, its existence is necessary for
square calculus can also be found in [18, Ch. 2.1-2/8.1-2],
these quantities to make sense. Expression (4) is valid
[15, Ch. 3.3/3.4/3.6],[19, Ch. 8-4.].
assuminganinformationcarryingcomplexenvelope input
signal band-limited to W with constant power spectral (i) Stochastic R-integral: Let (Uτ), τ R, be a complex
∈
density P and mean power constraint P. random process with E[Uτ 2]< . Then it can be shown
For thWe discrete-time channel model (4) is also appli- thatthestochasticR-inte|gra|lIU∞:= ∞ Uτdτ existsifand
cable if the discrete-time Fourier transform (DTFT), i.e., only if E[UτUτ′] is R-integrable, i.eR.−,∞
sHˆpfec:=truPmlL=−o0f1tHhlee−chj2aπnfnl/eWl v,efct∈orWH,.isHuosweedvetro,cwailtchultahteedthise- ∞ ∞ E[UτUτ′]dτdτ′ (9)
Z Z
crete approach we aim at a numerically easy-to-compute −∞ −∞
model,preferablydiscreteinthefrequencydomainaswell. exists as improper R-integral. Then we obtain
Therefore,weapproximatetheDTFTbyanN-pointDFT,
∞
E[I ]= E[U ]dτ. (10)
i.e., evaluating the spectrum at N points. Thus we calcu- U τ
Z
rlaatnedoHˆmn:v=ecHtˆofr|fHˆ=n:=W/(NHˆ, ,n.∈..Z,HNˆ, an)d. Tobhtisaiancttuhaellcyommepalenxs If we have another compl−ex∞process (Vτ), τ R, with
0 N 1 ∈
we are dividing the spectrum in−to N flat, parallel sub- E[Vτ 2]< for whichIV := ∞ Vτ dτ exists,then it can
| | ∞
channelswhichcorrespondstoanOFDM-basedsystemap- further be shown that R−∞
proach with N sub-carriers [2, Ch. 5.3.3/5.4.7]. The er- ∞ ∞
godic capacity is then given by E[IUIV]=Z Z E[UτVτ′]dτdτ′. (11)
−∞ −∞
CN =E(cid:20)N1 N−1log2(1+α|Hˆn|2)(cid:21), (5) Fo(uiri)ierExtirsatnesnfcoermof((HˆHˆff))=: W( e∞usee−j(29π)fτtHoτpdrτo)v,efthaRt,thoef
nX=0 the CIR(H ), τ R,exists.RT−∞hus we setU :=e j∈2πfτH
τ τ − τ
where α again means average SNR, now α:=NP0. Con- and obtain (9) w∈ith E[UτUτ′]=2R(τ,τ′)e−j2πfτej2πfτ′,
sidering the parallel channels in frequency, the AWGN where R is the covariance given in (2). If R in (2) is such
process at the receiver translates into a complex zero that R˜ is continuous and g is R-integrable then (9) exists
mean Gaussian random vector with independent real and for all f R and hence (Hˆ ). Note, for (Hˆ ) to exist, we
f f
∈
imaginary parts with N independent components each can alternatively require R to be R-integrable.
having variance N20. The average power constraint of P (iii) Distribution of (Hˆf): It can be shown that any
on each discrete-time channel input symbol converts to linear transformation of a Gaussian process is a Gaussian
NP onthe setofsub-channels(per OFDMsymbol).Note process. Thus (Hˆ ) is a complex Gaussian process com-
f
that WCN C as N . posedoftworealGaussianprocesses(Xˆ ):=(Re[Hˆ ])and
In thNe fo→llowing we→eva∞luate (4) and (5). (Yˆ ):=(Im[Hˆ ]). The process (Xˆ ) hasfzero mean,fwhich
f f f
follows from (10) with U :=Re[e j2πfτH ] and from and obtain
τ − τ
E[H ]=0.We calculate the covarianceof (Xˆ ) using (11)
withτU as before and V :=Re[e j2πf′τH ]fresulting in CN = ln1(2)exp α1 Ei1 α1 =:Cus, (14)
fRE,e[fX[ˆ′]f∈rXeˆRτpfl′a]a=cneddR0R∞byRaI0s∞mi[nR]((tτ2h,)aτ.τt′C)tohcorerspe(sr2poπ−oc(enτsdfsin(−Ygˆlτy)′,fτhw′a)e)sdszhτeordowτm′weifatohnr i(onσfd02te,hp.e.e.nfl,daσteL2nR−t1ao)y.fleELigx,hpNrfea,sdsaiinnodng(cid:0)(ct1ah4s(cid:1)e)emiassei(cid:0)dadneenr(cid:1)tipvioceawdleitrnodt[3ehl]ea,yc[4a]pp.raocfiitlye
f
· ·
and identical covariance. Again we use (11) with U := Proof: Theorem 2 is similarly proved as Theorem 1 but
τ
Re[e j2πfτH ] and V :=Im[e j2πf′τH ] to determine with less effort. We may write the N-point DFT Hˆ of the
− τ τ − τ
the cross-covariance between (Xˆf) and (Yˆf). We obtain channel vector H using matrix notation. Let Φ:=(ϕnl)
fE,[fXˆ′fYˆRf′]a=ndR0∞ER[X0ˆ∞fRYˆf(′τ],=τ′)sEin[Xˆ(2fπ′Yˆ(fτ]f.−Thτu′fs′)t)hdeτpdrτo′cessfoers wexiitshteϕnncel:=ofe−Hˆj2nisl/Nno,wn∈evZidNe,ntl.∈FZuLrt,htehre,ntrHaˆn=sfoΦrHm.inTghae
(Xˆ )∈,(Yˆ )areidenticallyd−istributedbutnotindependent. complex Gaussianvector this way yields a complex Gaus-
f f
(iv) Distribution of Hˆ 2: From (iii) it follows for any sian vector [18, 7.5-2] with Xˆ:=Re[Hˆ], Yˆ :=Im[Hˆ] being
f
f R that Xˆ ,Yˆ are| i.i.|d. Gaussian random variables realGaussianvectors.Wecalculatemeansandcovariances
f f
ha∈ving zero mean and variance E[Xˆ2]=σˆ2(f) with σˆ2(f) using the linearity of E[] and the independence of X=
as in (7). Thus Hˆ 2=Xˆ2+Yˆ2 isfan exponentially dis- Re[H]andY =Im[H].We· obtainthatXˆ,Yˆ areidentically
tributedrandom|vafr|iablewfithprfobabilitydensityfunction distributed with zero mean and covariance E[XˆnXˆn′]=
qf((zˆv))=Ex(i2sσˆte2n(fce))a−n1decxaplc[(u2lσˆat2i(ofn))o−f1Czˆ:],Tzˆh≥ec0r.iterion(9)fora PγikiL=−a0s1iPnkL(=3−)01.γTikhecocsr(o2sπs-(cionv−arikann′c)e/Nis)gifvoernnb,ynE′∈[XZˆnNYˆna′]n=d
asastenrotedcUaholfasnso:l=tyivclaiofRlgiEd-2i[(nfU1otfe+rgUfiαrfna′|li]Htˆteiosfi|Rne2x)t-eiitsnghttreeaangtnirdIoaUnbthl:be=eoouRpv−nrWeodWr/pa2/Wer2rieUt2si.ef.sDTdf(uh1eue0xs)t,iosi(ftt1swh1iee)f EYaPˆHnˆn[iLXd=ˆa−n20vr1Ye=aˆPnriiX′.aˆ]kLi.n==2−dc0.1−e+γGEEiYkaˆ[[XusX2ˆˆisnnsn2i′is(]aYˆ2=nnπa]σgˆ(.rian2aInitnnw−dfooiatlkmlnhonw′σvˆ)es/axn2Nrpfoiaoa)rsnbafelioelnnlsrtni(wan1∈l,il3tynZ)h′.N∈dzTieZsthrthoeNrairbtemwufXoetˆiraetnehnd,,
finite integration boundaries it is sufficient to show that | n| n n
random variable with parameter (2σˆ2) 1. Now we ex-
E[UfUf′] is continuous. This is equivalent to E[Uf2] being change summation and expectation inn (−5) and evaluate
continuous. The continuity of E[U2] can either be shown
f E[log (1+α Hˆ 2)] to obtain (12). Finally, if we have
directly or equivalently by showing that the process (Uf) uncor2related|scnat|tering, i.e., Γ in (3) is diagonal, then
is continuous in mean square.Now we can apply (10) and
obtainC=E[I ]= W/2 E[U ]df.WecalculateE[U ]by σˆn2= lL=−01σl2=12 for n∈ZN, and (12) implies (14). (cid:4)
evaluating theUintegR−raWl/E2[Uf]f= 0∞log2(1 + αzˆ)qf(fzˆ)dzˆ PIV. ExOarmnpsltee:inE-xUpholneennbteicakllpyraotcteesnsuated
with qf as in (iv). Finally, this yRields (6).
(vi) Uncorrelated scattering: The covariance R in (2) Now, we consider an example for the continuous-time
becomes R(τ,τ )=c˜δ(τ τ )g(τ)g(τ ) for R˜(τ˜)=c˜δ(τ˜). Rayleigh fading channel with special covariance. We use
′ ′ ′
−
Plugging R into (7) and using the fundamental property an exponentially attenuated Ornstein-Uhlenbeck process
of the Dirac distribution of ∞ h(τ)δ(τ τ′)dτ=h(τ′) capturing an exponential power decay. This is a common
−
yields σˆ2(f)=c˜ ∞ g2(τ)dτR,−∞independent of f. Then assumption in UWB channel modeling and was justified
(6) immediatelyRi−m∞plies (8). Note, this is only a formal by measurements [9], [12]. In addition, it incorporates ex-
derivation. The introduced mathematical tools are not ponentially correlatedscattering. Based on UWB channel
applicabletorandomprocesseswithDirac-typecovariance measurements,asimilarcorrelationmodelwasusedin[8].
functions, which do not even exist in the classical sense. Finally, we discuss relations to the discrete-time model.
A rigorous mathematical treatment involves stochastic
A. Definition
differential equations. (cid:4)
AstationaryOrnstein-UhlenbeckprocessisarealGaus-
C. Discrete-Time Rayleigh Fading Channel sian processes with zero mean and covariance function
Theorem 2: The ergodic capacity (5) of the discrete- R˜(τ − τ′):=2dae−a|τ−τ′| for τ,τ′∈R with parameters
time Rayleigh fading channel is given by a,d>0 [14, Ch. 3.7.2/3.7.3]. Using the notation of Sec-
tion II-A we set (X˜ ), (Y˜ ) to be normalized Ornstein-
N 1 τ τ
C = 1 − exp 1 Ei 1 (12) Uhlenbeck processes with d:=2a and specify the attenu-
N Nln(2) nX=0 (cid:16)2ασˆn2(cid:17) 1(cid:16)2ασˆn2(cid:17) ation function u in (1) as u(τ):=√bce−bτ for τ∈R with
parametersb,c>0,wherec isthe constantdefinedinSec-
where
tion II-A. We obtain the attenuated Ornstein-Uhlenbeck
L 1L 1
σˆ2 = − − γ cos(2π(i k)n/N), n Z , (13) processes (Xτ)=(X˜τg(τ)), (Yτ)=(Y˜τg(τ)), τ R, repre-
n ik − ∈ N sentingindependentrealandimaginarypartof∈(H ),each
Xi=0 Xk=0 τ
with covariance function
withγ asin(3).If (3)satisfiesΓ=diag(σ2,...,σ2 ),we
have uinkcorrelated channel taps (uncorrel0ated scaLt−te1ring) R(τ,τ′)=ce−a|τ−τ′|be−b(τ+τ′)1[0, )(τ)1[0, )(τ′), (15)
∞ ∞
for τ,τ R. vectors with unit mean energy as stated in Section II-B.
′
∈
Inaddition,wechooseεtobecloseto1,i.e.,truncationin
B. Analytical Calculations
timeisnegligible.Note,thisdiscreteversionofthechannel
The expressions given in this subsection are derived in
resembles the behavior of the continuous channel in time
the Appendix in condensed form. domain.However,thespectrumisdifferentduetoaliasing
Calculating (7) using (15) we get
since the Ornstein-Uhlenbeck process is not band-limited
c(a+b) becauseof (16).This,inturn,isnegligibleifεˆiscloseto1.
σˆ2(f)= (16)
(a+b)2+(2πf)2 Toobtainequalityinfrequencydomain,wehavetosample
alowpass-filteredversionofthecontinuouschannel,which
byelementaryintegration.Onerepresentationoftheclosed
of course has different behavior in time. Since we aim at
form solution of (6) is given by
equivalence in time, we consider the former approach.
n Finally, for the discrete and the continuous model to be
C = lnW(2) ∞ n+11 k+12Lk21(β1W2/4)Ln−21k(β1β2), (17) comparable we have to set c:=W/εˆ.
nX=0 Xk=0 − As a numerical example, we set a:=b:=1, and ε:=
2
whereLµisthegeneralizedLaguerrepolynomialoforderk 0.998 resulting in T =6.2146, when normalized to sec-
k d
[20,8.970],β :=2π2/(αc(a+b)),andβ :=(a+b)2/(4π2). onds. In Example 1 (Fig. 1), we set εˆ:=0.998 leading
1 2
Upper and lower bounds for (17) are given by to W =101.32, when normalized to [Hz], c=101.52, and
L=630. In Example 2 (Fig. 2), we set εˆ:=0.800 leading
Cθ =W+l4o√g2β2(cid:0)+1e+−θ/2βe1−aθαrcσˆt2a(nW/2)(cid:1)W−/24ln√(β2)2 ar,ctan(cid:16)W√β/(221(cid:17)8) tsoionWs a=re0.9c7o9n6si,decr=ed1.a2s24f5u,nactnidonLo=f 7α. aCnadpaacrietygievxepnreisn-
ln(2) (cid:18)√β2+e−θ/β1(cid:19) [bits/s/Hz], where normalization to the respective band-
width W is performed if required. The values of C/W
wherewehaveanupperboundforθ=0andalowerbound
are obtained by numerically evaluating (6) and C is
N
for θ=γ with γ being the Euler constant.
computed with N=6300. As reference curves the AWGN
In case the effect of the channel outside the band W
channel capacity C =log (1+α) is plotted. Another
is negligible, i.e., W is sufficiently large (depending on awgn 2
referenceisthe capacityC foruncorrelatedchanneltaps
us
the parameters a,b,c,α), then (17) can be closely ap-
(14), which is an upper bound for the correlated case due
proximated by integrating (6) with (16) over entire R
to Jensen’s inequality.
yielding
WeobservedifferencesbetweenC andC/W whichare
N
C = π exp(β β )Γ(1,β β ), (19) duetotheappliedapproximations,i.e.,truncationintime,
≈ ln(2)√β1 1 2 2 1 2 aliasing, and discretization of spectrum. They are prima-
where Γ(µ,z):= z∞e−ttµ−1dt is the incomplete gamma rilyminor,particularlyinExample2,butincreasewithα.
function [20, 8.35R0.2]. The distance to C is considerable in Fig. 1 but small in
us
Note, C in (19) is just an approximate expression for Fig.2.Thisdepends onthe concentrationofthespectrum
≈
proper parameter ranges but not the capacity for infinite within the considered band, which in turn is controlled
bandwidth, i.e., not limW C since α and thus β1 de- by the degree of correlation (parameter a) and the de-
→∞
pend on W as well. lay spread (parameter b). The tightness of the bounds is
parameter-dependent.EspeciallythelowerboundinFig.2
C. Numerical Results and Relation to the Discrete-Time
isverytightforα>15dB.Finally,C is wellapproximated
Rayleigh Fading Channel
by C in Example 1 for α<15dB but is inappropriate in
Here,we givenumericalexamples forthe previouslyde- ≈
Example 2, since εˆis not close enough to 1.
rivedexpressionsandshowthe connectiontothe discrete-
time channel. To do so, we first define two constants for V. Conclusion
the continuous-time channel. Let εˆ (0,1) be the portion
of the channel within the frequency∈band W in terms of In this paper, we determined the ergodic capacity of
a frequency-selective Rayleigh fading channel with cor-
meanenergy,i.e.,εˆ:=2 W/2 σˆ2(f)df,wherec is defined
c W/2 related scattering. We considered a continuous- and a
inSectionII-A.IfwefixεˆR,−thenwegetW =a+btan(πεˆ)by discrete-time channel and examined a detailed exam-
π 2
elementaryintegration.Furtherletε (0,1)betheportion ple incorporating exponential power decay and exponen-
∈
of the channel within the time interval [0,Td] in terms of tially correlated scattering. Analytical, approximate, and
meanenergy,i.e.,ε:=2 TdR(s,s)ds.Ifwe fix ε,then we boundingexpressionsaswellasnumericalresultswerepre-
c 0
get Td=−2b1ln(1−ε) byRelementary integration. sented and the relation between continuous- and discrete-
Consider the continuous-time channel within the fre- time models were discussed. The results illustrate sig-
quencybandW.Torepresentthischannelbythediscrete- nificant differences between the capacities for correlated
time model we sample the covariance function R of (15) and uncorrelated scattering. Future work includes for
over the range [0,T ] [0,T ] by 1 -spacing to get the example:consideringotherstationaryprocesses,assuming
d × d W
covariancematrixΓof(3).WenormalizeΓtohavechannel non-perfect CSI, further elaborating (17), analyzing the
4 again substituting s=β f2.
3.5 CCCau/swWgn (ii) With 0∞ t1e−tµ Ei1(t+ν)d1t= πν(µs−in1()/µ2πe)−ν/2Wµ−1,µ(ν)
3 CCCCNθθ==/γ0W//WW [92.12,362.R1.5,.38..1345]9.a3n],dwWhe−r14e,41W(y) =is yt14heey2ΓW(h12i,tyt2)ak[e22r0’s,
[bits/s/Hz]2.52 ≈ W-function [2A0c,k9n.2o2w0]l,ewdegfimκn,eaλnllty get (19).
Capacity1.5 TheauthorswishtothankLotharPartzsch,Department
of Mathematics at Dresden University of Technology, for
1
valuable comments and discussions.
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R