Novel Relations between the Ergodic Capacity and the Average Bit Error Rate

0.14 MB·English

by  Ferkan Yilmaz

Novel Relations between the Ergodic Capacity and
⋆
the Average Bit Error Rate
Ferkan Yilmaz 1 and Mohamed-Slim Alouini 2
Electrical Engineering Program, Division of Physical Sciences and Engineering,
King Abdullah University of Science and Technology (KAUST),
Thuwal, Mekkah Province, Saudi Arabia.
1,2{ferkan.yilmaz,slim.alouini}@kaust.edu.sa
Abstract—Ergodic capacity and average bit error rate have The Shannon capacity of a signaling with bandwidth W
2 been widely used to compare the performance of different wire- over additive white Gaussian noise (AWGN) channels is
1 less communication systems. As such recent scientific research given in general by C(γ ) , W log (1 + γ ), where
0 and studies revealed strong impact of designing and imple- γ is the overall instanetnadneous condit2ional SNenRd (that is,
menting wireless technologies based on these two performance end
2
indicators.However andtothebestofourknowledge,thedirect it is the instantaneous SNR at the output of the overall

n links between these two performance indicators have not been system) and log () is the binary logarithm (i.e, it is the
a explicitlyproposedintheliteraturesofar.Inthispaper,wepro- logarithm to the2b·ase 2). Then, the ergodic capacity, i.e.,
5 J pboitseernroovrelraretelaotifonasnboevtewreaellnctohmeemrugondiciactcioanpascyistyteamndusthinegabveinraagrye Cavg ≡ E[W log2(1+γend)], where E[·] the expectation
operator, is defined by averaging the instantaneous capacity
modulation schemes for signaling with a limited bandwidth and
] operatingovergeneralizedfadingchannels.Morespecifically,we C(γend) over the probability density function (PDF) of the
T show that these two performance measures can be represented overall instantaneous SNR γ [2, Eq.(15.21)],i.e.,
end
I in terms of each other, without the need to know the exact end- ∞
.
s to-endstatisticalcharacterizationofthecommunicationchannel. C =W log (1+γ)p (γ)dγ, (1)
c Wevalidatethecorrectnessandaccuracy ofournewlyproposed avg Z 2 γend
[ relations and illustrated their usefulness by considering some 0
1 classical examples. wherepγend(γ)istheprobabilitydensity(PDF)oftheoverall
instantaneousSNRγ .Rememberingagainthatthechannel
v Index Terms—Ergodic capacity, bit error rate, binary modu- end
capacity is defined as the error-free maximum rate achievable
8 lation schemes, generalized fading channels.
7 in the signaling channel,where the definition of error-free re-
2 liesonthefactthattheremaybeaspecificsignallingtechnique
I. INTRODUCTION
1 (for example, it may be the combination of modulation, error
. Modern digital communications theory has brought the
1 correcting codes and linear/non-linear detection techniques)
changesto theradioengineering,oneofwhichis theneedfor
0 that enableserror-freesignaling providedthat the information
2 end-to-end performance measurements. The measure of that
rate is below the capacity of the corresponding channel. On
1 performanceisgenerallythebit-errorrate(BER),whichquan-
the other hand, errors occur even at very low transmission
: tifies the reliability of a communication system based on the
v rates. Indeed, noise is ubiquitous, which suggests that some
i employed detection scheme, signal-to-noise ratio (SNR), and
X errors are inevitable at any transmission rate. More precisely,
diversity technique. On the other hand, defining the Shannon
assuming that the instantaneous BER over generalized fad-
r capacity [1] of a signaling channel as a maximum error-free
a ing channels is compactly expressed by E(γ ). Then, the
rate is the cornerstone problem of modern communications end
average BER E E[E(γ )] is defined by averaging
theory.Notto mention,indeed,thatthe Shannoncapacityis a avg ≡ end
the instantaneous BER E(γ ) over the PDF of the overall
moreabstractideathantheBERasaperformancemeasureand end
instantaneous SNR γ [2, Section 1.1.3], i.e.,
itleadstootherimportanttheoreticalconclusionsthatunderlie end
∞
key concepts of information, entropy, and bandwidth.
E = E(γ )p (γ)dγ. (2)
avg Z end γend
⋆This work was supported by King Abdullah University of Science and 0
Technology (KAUST). When the signalling channel is fully loaded with the in-
⋆This work has been presented by Ferkan Yilmaz in IEEE International formation at the level of channel capacity, the transmission
Symposium on Wireless Communication Systems (ISWCS 2011), Aachen,
through the channel experiences errors. Regarding this fact
Germany, 6th-9th November, 2011. This work is presented to ensure a date
andtimestampfromarXivcentralauthority“http://arxiv.org/”onthepurpose and referring to both (1) and (2), we set out to see if there
of that this material is proven to be authors’ scholarly and technical work. are direct relationships between the ergodic capacity C
This work is moreover presented to share our newly obtained results with avg
and the average BER E . As the ergodic capacity C
the readership in advance of publication under the condition that copyright avg avg
and all rights therein are retained by authors or by other copyright holders. increases, the average BER E decreases as it seems that
avg
Allpersons copying thisworkandaccessing theinformation insideofitare they are inversely proportional. Moreover and to the best of
strictlyexpectedtoadheretotheterms,conditionsandconstraintsinvokedby
our knowledge, the following two important questions have
each author’s copyright. Inmostcases, this material cannot notbereposted,
reannounced, reuploaded and reused without the explicit permission of the not been answered so far: i)
authors orothercopyright holders.
⋆For more information about the related publications, the readers and 1) What is the ergodic capacity Cavg of a signalling
researchers arereferred to“https://sites.google.com/site/ferkanyilmaz/home”. channel if we have only the average BER Eavg of a
given specific modulation technique in the correspond- instantaneous SNR as mentioned before. When for a specific
ing channel? value γ , the instantaneous BER E(γ ) = 1/2, then the
end end
2) Whatisthe averageBERE ofa givenspecific mod- signalling channel becomes a fully dissipated channel, which
avg
ulation technique if we have only the ergodic capacity means that, referring to the channel certainty, the transferred
C of the corresponding channel? entropy through signalling channel becomes zero. From this
avg
Viewed from a somewhat broader perspective, each question point of view, one can conclude that there exists a kind of
aboveisadmittedlyopeningasetofnewideasandtechniques certainty measure, that is
in information theory, communications theory, or coding the-
Φ(γ )=1 2E(γ ), (3)
ory, although all of these questions have been defined with end − end
different scopes as well. which possesses the information about the channel certainty,
Inthispaper,weinvestigatethesetwoquestionsgivenabove and it has a close relation with the channelcapacity since the
and we provide two new simple analytical relations linking channel capacity is the maximum transmission rate without a
the ergodic capacity Cavg of a signalling channel and the transmission error as mentioned before. In more details, the
average BER Eavg of a specific modulation scheme in the signalingchannelisfullydissipated/lossy withΦ(γend)=0
corresponding channel. Additionally, three possible examples oritiserror-freewithΦ(γ )=1.Itisimportanttonotethat,
end
oftheseanalyticaltoolsaredemonstratedinthispaperinorder specifically from a practical point of view, the instantaneous
to show that these two techniques are promising technique BER E(γ ) has always a distribution between zero and
end
regardingtheperformancemeasuresofdigitalcommunication half for all binary modulation schemes (such as binary phase
systems. More precisely, without the need to know the statis- shift keying (BPSK), differentially encoded BPSK (BDPSK),
tical characterizations (such as PDF, cumulative distribution binaryfrequencyshiftkeying(BFSK)andnon-coherentBFSK
function (CDF), moment generating function (MGF), mo- (NC-FSK)binaryschemes).Asaresult,thecertaintymeasure
ments, Laguerre moments, etc.) for the overall instantaneous Φ(γ ) is distributed between zero and one as expected.
end
SNR ofthe communicationsystem., onecan readilyobtainthe
With the aforementioneddeduction conditioned on that the
ergodic capacity Cavg inn terms of the empirically obtained knowledgeoftheinstantaneousSNRγ ispreciselyknown,
end
average BER Eavg, and vice versa the average BER Eavg in the certainty measure Φ(γ ) can be readily defined as
end
terms of the ergodic capacity Cavg. Φ(γ )= E [θ(γ φ)], where E [] is the expectation
end φ end ψ
The remainder of this paper is organized as follows. Sec- − ·
operatorwithrespecttotherandomvariableψ,andθ()isthe
tionIIisdevotedtothemathematicaldevelopmentofconcepts ·
Heaviside’s theta (step) function [3, Eq.(1.8.3)], and where φ
related to the two questions mentioned above. After defining
is the instantaneous certainty measure having a non-negative
the certainty measure, the relation between the average BER, distributionfollowingthePDFp (ϕ)= E [δ(ϕ φ)],where
φ φ
averagecertaintyandtheoutageprobabilityareunderlinedby −
δ()istheDirac’sdeltafunction[3,Eq.(1.8.1)],thatisreadily
the PDF of the certainty measure. In particular, we show in ·
defined as
Section III that, for communication systems signalling with ∂Φ(ϕ)
p (ϕ)= , ϕ R+, (4)
binary modulation schemes over generalized fading channels, φ ∂ϕ ∈
the ergodic capacity C can be obtained in terms of the
avg with theintuitiveobservationthatΦ(γ )= γendp (ϕ)dϕ.
averageBEREavg.InSectionIV,weshowontheotherhand end 0 φ
Accordingly,both (3) and (4) suggestthat theRsignalling error
thattheaverageBERE canbecomputedfromtheergodic
avg
occurredthroughend-to-endtransmissioncanbe,withoutloss
capacityC withouttheneedtoknowtheexactthestatistical
avg
of generality,modeled as non-negativerandom variable (RV).
characterization of the overall end-to-end instantaneous SNR
Then, referring to (2), the average BER E = E[E(γ )]
γ . Itshouldbe mentionedthatseveralproofsof ourresults avg end
end of an arbitrary communication system signalling over gene-
haveintentionallybeenomittedduetospacelimitationbutare
ralized fading channels can be attained from the outage
available in the extended version of this paper. In addition,
probability P (ϕ) as shown in the following theorem.
all these results have been checked numerically for their γend
correctness and accuracy. Theorem 1. The average BER E = E[E(γ )] over ge-
avg end
neralized fading channels is alternatively given by
II. BACKGROUND RELATIONS
∞
1 1
Thepresentsectionisdevotedtothemathematicaldevelop- E (γ¯ )= P (ϕ)p (ϕ)dϕ, (5)
mentofthebasicandfundamentalconceptsrelatedtothetwo avg end 2 − 2Z0 γend φ
b
key questionsmentionedin the previoussection. We shall de- where γ¯ is the average SNR defined as γ¯ = E[γ ],
velopour main definitionsand theoremsin fullgenerality,for and P end(γ) E[θ(γ γ)] = ∞p end(u)du isentdhe
anarbitrarycommunicationsystemsignallingovergeneralized compleγmenedntary≡outagepreonbda−bilitythatRthγe inγsetnadntaneousSNR
fading channels, and we shall use the language of probability γ bexceeds a certain specified threshold γ.
end
theory throughout this paper to facilitate the mathematical
Proof: The proof is omitted due to space limitation.
exposition of the relation between the ergodic capacity C
avg
ofasignallingchannelandtheaverageBERE ofaspecific It may be here useful to define the average certainty
avg
modulation scheme in the corresponding channel. measure Φ E[Φ(γ )]. Referring to both (2) and
avg end
≡
Let us consider the instantaneous BER E(γ ) of a (3), the averaged certainty measure Φ can obtained as
end avg
specific communication system, where γ R+ is the Φ = 1 2E . On the other hand, utilizing Theorem 1,
end ∈ avg − avg
the average certainty Φavg can be alternatively given in the III. ERGODIC CAPACITY OBTAINED THROUGH THE
following corollary. BIT-ERROR RATEOF BINARY MODULATIONS
Corollary1. Theaveragecertainty Φ = E[Φ(γ )] over We show in this section that, for wireless communication
avg end
generalized fading channels is alternatively given by systemssignallingwithbinarymodulationschemesovergene-
∞ ralized fading channels, the ergodic capacity Cavg can be
Φavg(γ¯end)=Z0 Pγend(ϕ)pφ(ϕ)dϕ. (6) itmheplriecsitullytsobgtiaviennedinin(t8e)rmasndof(t1h0e)a.vTerhaigserBelEatRioEnaivsgibnydeuesdinga
b
Proof: The proof is obvious. promising technique for quality / performance assessment of
a wireless communication systems due to the fact that the
As an example, a unified instantaneous BER expression
average BER E is more measurable and controllable than
for binary modulation schemes is known to be given in [4, avg
the ergodic capacity C .
Eq.(13)] by Wojnar as avg
Γ(b,aγ ) Theorem 2. Consider a communication system signalling
E(γend), end , (7) with bandwidth W over generalized fading channels. Then,
2Γ(b)
withouthavingthe exact the statisticalcharacterizationofthe
where a depends on type of modulation scheme (1 for overall instantaneous SNR γ , the ergodic capacity C
2 end avg
orthogonal FSK and 1 for antipodal PSK), b depends on canbeobtaineddirectlyfromtheaverageBERE forbinary
avg
type of detection technique (1 for coherent and 1 for non- modulation schemes as
2
coherent), and Γ() and Γ(, ) are the Gamma function [5, ∞
· · ·
Eq. (6.1.1)]and the complementaryincomplete Gamma func- C (γ¯ )= (u) 1 2E (uγ¯ ) du, (11)
tion [5, Eq. (6.5.3)], respectively. Particularly in connection avg end Z0 Za,b h − avg end i
with (7), applying [6, Eqs. (8.4.16/1) and (8.4.16/1)] on (7), where the information certainty content (u) is given by
a,b
Z
we are now able to write the certainty measure Φ(γ ) for WΓ(b) 0
end (u)= G1,1 au , (12)
binary modulation schemes as Za,b log(2)u 1,2(cid:20) (cid:12)0,1 b(cid:21)
(cid:12) −
(cid:12)
Φ(γ )= 1 G2,0 aγ 1 , (8) where a and b are the modulationspeci(cid:12)fic parametersdefined
end Γ(b) 1,2(cid:20) end(cid:12)b,0(cid:21) before, and where log() is the natural logarithm.
(cid:12) ·
(cid:12)
where Gm,n[] is the Meijer’s G functio(cid:12)n [6, Eq. (8.3.22)]. Proof: The proof is omitted due to space limitation.
p,q ·
Accordingly using (4), (7) and (8), the PDF of the instanta-
ItisworthytomentionthatTheorem2isquiteusefulforthe
neous certainty measure φ, i.e., p (ϕ) for binary modulation
φ analysis of wireless communication systems. Specifically, in
schemes is obtained as
ordertofindtheergodiccapacityC ,theproposedtechnique
avg
(b/a)b in Theorem 2 eliminates the necessity to have the knowledge
p (ϕ)= ϕb−1e−(b/a)ϕ, ϕ R+, (9)
φ of the communication medium. More precisely, a key indica-
Γ(b) ∈
tionofTheorem2isthatthereisnoneedtoknowthedetailsof
which is the Gamma distribution with the parameters E[φ]=
the transmission medium and the statistical characterizations
a and E φ2 /E[φ]2 = (b+1)/b. In particular, substituting
oftheoverallinstantaneousSNRγ .Onejustneedstoknow
end
b = 1 in(cid:2)(9(cid:3)) results in the exponential PDF, i.e., pφ(ϕ) = i) the binary modulation scheme employed for signalling, ii)
(1/a)exp( ϕ/a). In addition, substituting b = 1/2 reduces
the signalling bandwidth W and iii) the average BER E
− avg
(9) to the power distribution of the half-Gaussian RV, i.e.,
curveovertheaverageSNR0<γ¯ < ,inordertoobtain
p (ϕ) = exp( ϕ/(2a))/√2πaϕ. By means of using (2), it end ∞
φ the ergodic capacity.
−
is at this pointconvenientto give the averagedcertainty Φ
avg
for binary modulation schemes as Corollary2. TheinformationcertaintycontentΦa,b(u)given
by (12) can also be represented as
(b/a)b ∞
Φavg(γ¯end)= Γ(b) Z Pγend(ϕ)ϕb−1e−(b/a)ϕdϕ. (10) (u)= W(b−1)γ(b−1,−au) , (13)
0 Za,b −log(2)ab−1ubexp(au+iπb)
b
It is useful to mention that we have described in this
section the relations between the average BER E (γ¯ ), wherei=√ 1istheunitimaginarynumber,andγ(, )isthe
avg end − · ·
the average certainty Φ (γ¯ ) and the complementary complementary incomplete Gamma function [5, Eq. (6.5.2)].
avg end
outageprobabilityPγend(γ), whichareentirelyeasy-to-obtain Proof: The proof is omitted due to space limitation.
(i.e., without need to have the exact PDF p (γ) of the
b γend Note that even if the novel integral relation given by
overall instantaneous SNR γ which is harder to obtain
end Theorem2,definingthe relationbetweenthe ergodiccapacity
experimentallythanthattheoutageprobability)andobviously
C and the average BER E , is simple to use, we
easy-to-use. avg avg
can specifically get its finite (N-terms) sum approximation
In the following sections, in addition to better illustrate the
converging rapidly and steadily and requiring few terms for
usefulness of the relations mentioned above, we shall obtain
an accurate result, as shown in the following corollary.
direct relations between the ergodic capacity C and the
avg
averageBERE forcommunicationsystemssignallingwith Corollary3. The novelrelationbetweenthe ergodiccapacity
avg
binary modulation schemes over generalized fading channels. C andtheaverageBERE ,i.e.,Theorem2canbegiven
avg avg
inafinite(N-terms)sumapproximationforbinarymodulation Then,assumethattheaverageBERE ofthecorresponding
avg
schemes as system for these SNR values are experimentally measured
N as E1, E2, ..., EN. Eventually, using available interpolation
Cavg(γ¯end)=nX=1wnZa,b(sn)h1−2Eavg(snγ¯end)i, (14) caanndebxetrraepaodlialytioanndteacchcnuiqrauteesly[1a0p]p,r[o1x1i]m,tahteedavaesrageBEREavg
where both wn and sn are defined in [7, Eqs. (22) and (23)], E (γ¯ ) γ¯ ; γ¯ N, E N (18)
respectively. avg end ≈Eint end { n}1 { n}1
(cid:0) (cid:1)
wheretheinterpolation/extrapolationtechnique (; , )are
int
Proof:Bychangingthevariableoftheintegrationin(11) E · · ·
implemented as a built-in function in standard mathematical
asu=tan(s)andthenusingtheGauss-Chebyshevquadrature
software packages such as MATHEMATICA®. As a result
(GCQ) formula [5, Eq.(25.4.39)],the proof is obvious.
and using (18) in (11), the ergodic capacity C can be
avg
Although the novel relation presented in Theorem 2 is approximated as
simple and straightforward to use, let us consider some of
its example applications in order to have a broad conceptual 10γ¯dB/10
C (γ¯ ) lim (u)
understanding of the relation and then to double check its avg end ≈γ¯dB→∞Z10−γ¯dB/10Za,b ×
analytical correctness.
1 2 uγ¯ ; γ¯ N, E N du. (19)
Example 1 (Average BER to Ergodic Capacity in Closed– h − Eint(cid:0) end { n}1 { n}1 (cid:1)i
Form).Let us consider the performance measures of binary As an illustration of this example, consider the BPSK per-
modulation over Rayleigh fading channels as a simple exam- formance curves of an L-branch diversity receiver signalling
ple,whosewell-knownunifiedaverageBERE isgivenfor overassumably an unknowngeneralizedfadingchannelswith
avg
all binary modulation schemes in [8, Eq. (16)], as bandwidthW are depicted in Fig. 1 for differentvalues of an
implementationparameter(i.e.,itistheparameterthatdirectly
b
1 1 aγ¯
E (γ¯ )= end , (15) orobliquelyeffectsonthedistributionofoverallinstantaneous
avg end 2 − 2(cid:18)1+aγ¯end(cid:19) SNRγend).Withouthavinginformationaboutneitherthetype
where a and b are the modulation parameters as explained ofthe L-branchdiversityreceivernorthe statistical properties
before.Substitutingboth(13)and(15)into(11)andthenusing of the wireless fadingchannelthat the communicationsystem
[5, Eq. (6.5.36)], we have issubjectedto,thecorrespondingergodiccapacityCavg canbe
readilycomputedbymeansof(19), yieldingresultsinperfect
∞
C (γ¯ )= W 1G1,1 au 0 agreement with the correct analytical results as illustrated in
avg end log(2)Z u 1,2(cid:20) (cid:12)0,1 b(cid:21)× details in Fig. 2 at the top of the next page. (cid:3)
0 (cid:12) −
(cid:12) 1
G1,1 a(cid:12)uγ¯ du. (16)
1,1(cid:20) end(cid:12)b(cid:21) IV. BIT-ERROR RATEOF THEBINARY MODULATION
(cid:12)
(cid:12) SCHEMES OBTAINED THROUGH THE ERGODICCAPACITY
Performing some simple algebraic manipulati(cid:12)ons using [6,
Eqs. (2.24.2/2) and (8.4.16/14)] and then utilizing [5, Theerrorcausedbythetransmissionofinformationthrough
Eq.(6.5.15)],theergodiccapacityC oftheRayleighfading a communication channel is clearly connected to the uncer-
avg
channels is readily obtained as tainty of the corresponding communication system. We show
in this subsection that, for communication systems signalling
W 1 1
C (γ¯ )= exp E , (17) withbinarymodulationschemesovergeneralizedfadingchan-
avg end log(2) (cid:18)γ¯end(cid:19) 1(cid:18)γ¯end(cid:19) nels, the average BER Eavg can also be implicitly given in
whichisin perfectagreementwith [9,Eq.(34)],whereEn() terms of the ergodic capacity Cavg using the results given in
is the exponential integral [5, Eq.(5.1.4)]. (cid:3)· (8) and (10).
In short,thisexampleshowsthatthe ergodiccapacityCavg Theorem3. Consideracommunicationsystemsignallingwith
ofacommunicationsystemscanbeobtainedfromtheaverage bandwidthW overgeneralizedfadingchannels.Then,without
BEREavg ofaspecificbinarymodulationschemeinthesame the need to have the exact statistical characterization of the
channel. In contrast, the next example is associated with the overall instantaneous SNR γ , the average BER E for
end avg
implementation and design of a communication system. binary modulation schemes can be obtained directly from the
Example 2 (AverageBERtoErgodicCapacityUsingNumer- ergodic capacity Cavg as follows
ical Techniques).At the implementationand design stage, the ∞
1
aavsepreacgieficBEbRinaEryavmgocdaunlaotfitoennsbfeorexanpeerinmouengthalnlyummbeearsuorfedSNfoRr Eavg(γ¯end)= 2 −Z0 Zba,b(u)ℑnCavg(−uγ¯end)odu(,20)
values.ItisassumedthatwehaveasetofdistinctSNRvalues where gives the imaginary part of its argument and
sfourchatlhlant0≤γ¯1≤1γ¯,22≤,....,.N≤γ¯.NF,wurhthereermγ¯morien,<γ¯10logh1a0s(γ¯ton)<beγ¯meaxx- where tℑhe{·i}nformation certainty content Za,b(u) is given by
∈ { } dB log(2) a b+1 b a
perimentally chosen as small as possible since the averaged (u)= exp , (21)
a,b
certaintymeasureΦ goesto zerowith the lowSNR values Z 2πaWΓ(b)(cid:16)u(cid:17) (cid:16)−u(cid:17)
avg
b
(i.e., the average BER E performance at the low-SNR where a and b are the parameters of the binary modulation
avg
region determines the ergodic capacity Φ of the channel). scheme as explained before.
avg
100
Exact Ergodic Capacity
14 Mapped Ergodic Capacity
10−1
12
10−2 Hz]10
s/
R bp (a),(b),(c),(d),(e)
E [8
erageB10−3 Capacity6
Av10−4 (a),(b),(c),(d),(e) dic
go4
Er
10−5
2
Experimental Average BER
Interpolated Average BER
10−6
−60 −50 −40 −30 −20 −10 0 10 20 30 0 5 10 15 20 25 30
SNR[dB] SNR[dB]
Fig. 1. Average BER Eavg of the L-branch diversity receiver over Fig. 2. Ergodic Capacity Cavg of the L-branch diversity receiver over
assumablyunknowngeneralized fadingchannels. assumablyunknowngeneralized fading channels.
Proof: The proof is omitted due to space limitation. V. CONCLUSION
It is obvious that the numerical results and applications In this paper, we define the channel certainty measure and
illustrated by Example 1 and Example 2 can also be readily then propose novel relations between the ergodic capacity
usedwithTheorem3.SoletusconsidernowtheNakagami-m (Shannon capacity) and the average BER performance of
fadingchannelsasadifferentexample,inordertovalidatethe an overall communication system using binary modulation
correctness and numerical accuracy of Theorem 3. schemes for signalling with bandwidth W and operating over
Example 3 (Ergodic Capacity to Average BER in Closed– generalizedfadingchannels. More specifically,these relations
Form).The ergodic capacity C with bandwidth W over suggest that, even withoutthe need to know the exact end-to-
avg
Nakagami-m fading channels is given in [12, Eq. (33)] by end statistical characterization of the overall communication
system, the ergodic capacity can be obtained through the
W m 0,1
C (γ¯ )= G3,1 , (22) medium of its corresponding average BER, and the average
avg end log(2)Γ(m) 2,3(cid:20)γ¯ (cid:12)m,0,0(cid:21)
end(cid:12) BER can also be explicitly obtained through the ergodic
(cid:12)
wheretheparametermdenotesthefadingfi(cid:12)gure(i.e.,diversity capacity. We finally validated the correctness and accuracy of
order) which ranges from 1/2 to the infinity [2, Section our newly proposed relations and illustrated their usefulness
2.2.1.4]. After performing some algebraic manipulations, we by considering a few classical examples.
have
πWΓ(m,m/γ¯ )
ℑnCavg(−γ¯end)o= log(2)Γ(m)end . (23) REFERENCES
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