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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 7, JULY 2009 3747
Delay-Limited Transmission in OFDM Systems:Performance Bounds and Impact of
System Parameters
Gerhard Wunder, Member, IEEE , Thomas Michel, Student Member, IEEE ,and Chan Zhou, Student Member, IEEE
Abstract—Delay matters in future wireless communication. Anappropriate limit for rates achievable under delay constraintsis the delay limited capacity (DLC). In this work, the DLC of OFDM systems is investigated. Despite its complicated correla-tion structure the OFDM DLC is fully characterized for lowand high SNR. It is shown that (under weak assumptions) theOFDM DLC is almost independent of the fading distribution inthe low SNR region but strongly depends on the delay spread
thereby achieving a capacity gain over AWGN capacity. In thehigh SNR region the roles are exchanged. Here, the impactof delay spread is negligible while the impact of the fadingdistribution becomes dominant. The relevant quantities and theirasymptotic behaviour are derived without employing simplifyingassumptions on the OFDM correlation structure. Using a generalconvergence framework the analysis further shows that if thedelay spread becomes large even the predominant impact of thefading distribution vanishes and DLC capacity loss comparedto AWGN capacity approaches 0.58[nats/s/Hz]. The convergencespeed, the loss due to non-uniform power delay profile, and therelation to ergodic capacity is also analyzed and underlined withsimulations and application examples. The main conclusion hereis that OFDM fully takes advantage of the degrees of freedom of theunderlying fading channel in terms of delay spread and, regardless
of the fading distribution, delay sensitive capacity measures such as the DLC converge to the ergodic capacity. Finally, since universalbounds are obtained which apply to any fading distribution theresults can also be used for other classes of parallel channelsextending the range of applicability.
Index Terms—Delay limited capacity, orthogonal frequencydivision multiplexing (OFDM), power control, rate allocation,parallel Gaussian channels.
I. INTRODUCTION
MODERN wireless services are very sensitive to delay
and require a certain rate to be provided in each time
slot. This sensitivity can be translated directly to the centralquestion motivating our work: What is the maximum data rate
achievable under delay limitations? Assuming a block fading
process and capacity achieving codes, this question can be
made more precise: What is the maximum data rate achievablein each fading state under a long term power constraint, so that
the temporal structure of the fading process can not cause a
Manuscript received July 28, 2008; revised February 6, 2009 and February27, 2009; accepted March 5, 2009. The associate editor coordinating thereview of this paper and approving it for publication was S. Hanly.
The authors are with the Fraunhofer German-Sino Mobile CommunicationsLab, Heinrich-Hertz-Institut, Einstein-Ufer 37, D-10587 Berlin, Germany (e-mail: {wunder, michel, zhou}@hhi.fhg.de).
Digital Object Identifier 10.1109/TWC.2009.080991
failure of the provided service? An answer to this question
provides not only an appropriate performance limit for delay
sensitive services such as e.g. streaming services in LTE
systems (Long Term Evolution of 3GPP UMTS system). It
also gives structural insights into the general system behavior
yielding guidelines for engineering wireless communication
systems.
It is known that in general multiple degrees of freedom infading channels allow reliable communication in each fading
state under a long term power constraint. This is due tothe possibility of recovering the information from several
independently faded copies of the transmitted signal. The rateachievable in each fading state is called zero outage capacity
or alternatively delay limited capacity (DLC) [1]. Not only
multiple input multiple output (MIMO) channels but also
frequency selective multi-path channels offer multiple degrees
of freedom. This is in contrast to single antenna Rayleigh flat
fading channels, where a DLC does not exist.
This work investigates the DLC of frequency selective
multi-path channels using orthogonal frequency divisionmultiplexing (OFDM) to mitigate inter-symbol interference.OFDM can be considered as a special case of parallel fading
channels with correlated fading process. Pioneering work on
this topic was carried out in [2][3][4][5][6][7][8]. Unfortu-
nately, these results do not carry over to the OFDM case: since
the subcarriers are highly correlated due to oversampling of
the channel in the frequency domain the fading distribution
is commonly degenerated which significantly complicates the
analysis. This particularly affects the critical impact of the
delay spread and the number of subcarriers. Hence, even
though the information-theoretic foundations are established,
the characterization of the OFDM DLC remains an openquestion.
Our main contributions are as follows: We derive the OFDM
DLC in a general setting and analyze the impact of systemparameters such as delay spread, power delay profile (or the
multi-path intensity profile) and fading distribution along with
the study of suboptimal resource allocation strategies. We
focus on two cases in particular: the behaviour at low and
high signal to noise ratio (SNR). The OFDM DLC in the
low SNR regime is characterized by its first and second order
Taylor expansion, which are explicitly calculated in terms of
(large) delay spread for OFDM. It is shown that so-called rate
water filling using solely order statistics of subcarrier gains
1536-1276/09$25.00 c 2009 IEEE
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3748 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 7, JULY 2009
is the optimal resource allocation strategy. Similar analysis
is carried out in the high SNR regime where it is shown
that simple channel inversion achieves close-to-optimal per-formance also in the (degenerated) OFDM case where this
time the DLC depends on the fading distribution; the analysis
culminates in a general convergence theorem again in terms of
(large) delay spread for OFDM where even the impact of the
fading distribution vanishes showing a universal capacity loss
of 0.58[nats/s/Hz] compared to AWGN capacity. This showsthat OFDM fully takes advantage of the degrees of freedom
of the underlying fading channel in terms of delay spread
and, regardless of the fading distribution, short term capacity
measures such as the DLC converge to the ergodic capacity.
The remainder of this paper is organized as follows: Section
II presents the OFDM system model. In Section III the OFDM
DLC is introduced and suboptimal power allocation strategies
are discussed. In Section IV-A the behavior at low SNR is
studied while Section IV-B focuses on the high SNR regime.We conclude with some final remarks in Section VII.
A. Notations
All terms will be arranged in boldface vectors. Common
vector norms (such as ·1 for the l1-norm) will be employed.
The expression z ∼ CN (0, 1) means that the complex-valued
random variable z = x + jy is circular symmetric Gaussian
distributed, i.e. the real and imaginary parts are indepen-
dently Gaussian distributed with zero mean and variance 1/2:
x, y ∼ N (0, 1/2). A sequence of random variables is called
iid if 1.) any subset is an independent set and 2.) all randomvariables are circular symmetric. The expectation operator
(e.g. with respect to the fading process) will be denoted as
E (respectively Eh
or Eh̃
).Pr(A)
denotes the probability of
an event A. All logarithms are to the base e unless explicitly
defined in a different manner.
II. OFDM COMMUNICATION MODEL
Assuming familiarity with the general model consider a
standard OFDM communication system where a single user
uses K subcarriers for information transmission. The complex
channel gain on subcarrier k is by means of Fast Fourier
Transform (FFT) given by
h̃k =L
l=1
c̃l e−2πj(l−1) · (k−1)
K , k = 1,...,K, (1)
where L ≤ K is the delay spread, and c̃l are the complex
path gains that are modeled as independent, zero mean random
variables with variance σl > 0 for all l. The vector of variances
σ = [σ1,...,σL]T is called the power delay pro file (PDP)
and the channel energy is normalized, i.e. ||σ||1 = 1. We
say that the channel has a uniform PDP if σ1 = . . . = σLand a non-uniform PDP otherwise. Note that in practice the
PDP is typically non-uniform. The channel (path) gains are
defined as hk := |h̃k|2 (respectively cl := |c̃l|2) and the
distribution of the channel gains is called the (joint) fading
distribution. It is worth pointing out that we do not makeany assumptions on the fading distribution. Even the case of
point masses (i.e. discrete fading distributions) induced e.g. by
h̃K
x1
x2
xK
p1(h)
p2(h)
pK (h)
n1
n2
nK
y1
y2
yK
h̃1
h̃2
Fig. 1. General system model: data of K streams xk is sent over parallelfading channels with arbitrary fading distribution hk generated by eqn. (1)and received under AWGN with iid nk ∼ CN (0, 1).
some vector quantizer is covered in our analysis provided that
the rates defined below are achievable. The general model is
summarized in Fig.1.
Given the channel gains h = [h1,...,hK ]T the rate achiev-
able over all K parallel Gaussian channels with a certain
power allocation p = [ p1,...,pK ]T reads as
R(h,p) =1
K
Kk=1
rk(hk, pk) =1
K
Kk=1
log (1 + pkhk) , (2)
where rk(hk, pk) denotes the rate achievable on subcarrier
k. We further introduced the factor 1/K so that all rates
are normalized to spectral ef ficiency and given in [nats/s/Hz].
The small impact of the OFDM guard interval on the spectralef ficiency shall be neglected here.
Now, assume that the system is subjected to a long termpower constraint, i.e.
Eh
Kk=1
pk(h)
≤ P ∗, (3)
where we use pk(h) to denote that the power allocation may
depend on the current fading realization. This means that whilethere is no peak power constraint per fading state, in average
the power constraint P ∗ has to be met. Please note that due tonon-linear components in the transmitter path such ideal power
control scheme is dif ficult to implement in practice. Therefore,
the results should be seen as a limit for any practical power
control scheme.
III. A PERFORMANCE MEASURE FOR DELAY LIMITED
TRANSMISSION
A. Optimal rate allocation
We introduce the DLC C d (P ∗) for an OFDM system,
which is a special case of parallel fading channels.
De finition 1: The delay limited capacity C d (P ∗) of an
OFDM system under a long term power constraint P ∗ is given
by
C d (P ∗) = supp∈P∗
inf h∈H
R (h, pk(h)) (4)
where
H ⊆RK+ is the set of possible channel gains and
P ∗ comprises all power allocation policies advising a power
allocation pk(h) ∀k to every h ∈ H such that (3) holds.
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WUNDER et al.: DELAY-LIMITED TRANSMISSION IN OFDM SYSTEMS: PERFORMANCE BOUNDS AND IMPACT OF SYSTEM PARAMETERS 3749
In other words, C d (P ∗) is the maximum rate which can be
achieved for all possible channel gains without violating the
average power constraint P ∗.
Definition 1 implies that in order to achieve C d (P ∗) we
need to find the power allocation pk(h) ∀k that supports a
given rate C d with minimum power. For any h ∈ H this
optimization problem is equivalent to:
minp∈RK
Kk=1
pk
subj. to1
K
Kk=1
rk(hk, pk) ≥ C d
(5)
Using the relation between power and rate on subcarrier k in
eqn. (2) the problem can be easily solved and the resulting
optimal rate allocation is given byerk
hk− λ
−= 0, k = 1,...,K
1
K
Kk=1
rk = C d
λ > 0
⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭
(Rate Waterfilling)
where [·]− := min {·, 0} and λ ∈ R is a Lagrange multiplier.The rate allocation is called rate water filling (RW) because
substituting λ = log(λ̃) and hk = log(1/h̃k) yields the
classical waterfilling rule. Solving for λ and after some algebra
we obtain the single user OFDM delay limited capacity C dwith power constraint P ∗ (corresponds to Theorem 3.2 in [7])
P ∗ =Eh
⎛⎝
|D (C d,h)| exp
C dK|D(C d,h)|
K k∈D(C d,h)
h1/|D(C d,h)|
k
⎞⎠
− 1
K Eh
⎛⎝ k∈D(C d,h)
1
hk
⎞⎠ (6)
where the random variable D (C d,h) ⊆ {1,...,K } denotes
the set of active subcarriers and |D (C d,h)| its cardinality.Since the numerator in (6) can be bounded by a constant and
by applying arithmetic-geometric mean inequality to the last
term, the delay limited capacity C d is greater than zero if and
only if
RK+
1
k∈D(C d,h) h1/|D(C d,h)|k
dF h (h) <
∞. (7)
Here, F h denotes the joint fading distribution function. The
class of fading distributions for which (7) holds is called
regular in [3]. It will become apparent in the following that
the correlation structure of the channel gains in OFDM pro-
vides the main challenge in proving and analyzing regularityaccording to (7).
Let us now introduce two important suboptimal power
allocation strategies.
B. Suboptimal rate allocation
It is evident from the expression for the DLC that the
major dif ficulty is the rate waterfilling operation for all channel
gains. In order to avoid this complexity we introduce the
notion of rate water filling for expected ordered channel gains,
or so-called statistical rate water filling (SRW) as follows:for a given vector h of real elements let us introduce the
total ordering hk[K] ≥ hk[K−1] ≥ . . . ≥ hk[1], i.e. hk[1] is
the minimum value and hk[K] is the maximum value; the
distribution of hk[ p] is known to be the p-th order statistics of
a sample h. Based on the order information we can deduce a
fixed rate allocation on the subcarriers, avoiding optimal RW.The key idea is to allocate a fixed rate budget to the p-th
ordered subcarrier. Defining the terms
ζ p :=
+∞ 0
1
hdF hk[p] (h)
where F hk[p] is the marginal distribution of the p-th ordered
channel gain and using these factors in the optimization
problem (5) the SRW rate allocation is given by:
erk[p]
ζ −1 p −
λ−
= 0, k = 1,...,K
1
K
K p=1
rk[ p] = C d
λ > 0
⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭
(Statistical Rate Waterfilling)
The performance of SRW is illustrated in Fig.2 and it can be
observed that it does particularly well in the low SNR region.
This will be exploited in the low SNR analysis where it is
shown that it becomes optimal as SNR goes to zero.
There is an interesting second rate allocation termed chan-
nel inversion (CI) introduced in [3] where the powers assertedto the subcarriers are all the same. It is easy to see then that
the CI rate allocation according to
rk = log
1 +
eC dhkKk=1 h
1/Kk
, k = 1,...,K,
(Channel inversion)
always leads to a rate higher than the requested rate at
the expense of power consumption. Hence, this is also a
suboptimal solution. CI is illustrated in Fig.2. In contrast toSRW it performs well in the high SNR region. This will be
exploited in the high SNR analysis where it is shown that itbecomes optimal as SNR goes to infinity.
The relevant performance measures for SRW and CI play a
significant role in the forthcoming analysis. Next, we analyzeexistence of DLC in OFDM systems.
C. Existence of DLC
Denote the guaranteed rates achievable under SRW by
C SRW d (P ∗). By the suboptimality of SRW the DLC is clearly
non-zero if ζ K = Eh(h−1∞ ) < ∞, i.e.
RK+
1
h∞dF h (h) < ∞. (8)
and it is therefore of general interest when eqn. (8) holds. Thefollowing theorem states a strikingly weak suf ficient condition
on the existence of the DLC.
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−30 −20 −10 0 10 20 30 40 50 600
5
10
15
SNR [dB]
C
d
[ b p s / H z ]
equal power strategy
equal rate strategy
rate water−filling strategy
OFDM DLC
Fig. 2. RW, SRW, and CI allocation policies for L = K = 16independent subcarriers. In addition, the suboptimal strategy of equal ratebudget assertion is also depicted which is suboptimal independent of the SNRregime. Summarizing, simple suboptimal schemes approximate very well thedelay limited capacity curve over a large SNR range and thus yield useful
insights in the general behaviour of the OFDM delay limited capacity.
Theorem 1 (Non-zero DLC): Suppose there is a pair of
path gains that have a joint distribution with bounded den-
sity in some open neighborhood of zero. Then, C d (P ∗) ≥C SRW d (P ∗) > 0 for any P ∗ > 0.
Proof: We examine under which conditions
Eh(h−1∞ ) < ∞ holds. Applying the standard inequality
E (X ) ≤ ∞i=0 Pr (X ≥ i) for some non-negative random
variable X and using the inequality c1 = h1 /K ≤h∞, and c∞ ≤ c1 the expectation can be written as:
Eh
1
h∞
≤ 1 +
+∞i=1
Pr
Ll=1
cl ≤ 1
i
(9)
By assumption there are two channel gains say ci1 , ci2 with
joint distribution with bounded density (by some real constant
0 ≤ cde < ∞) in some open neighborhood of the zero say
[1/i0] × [1/i0]. Hence, we have for i ≥ i0
Pr
Ll=1
cl ≤ 1
i
≤
[1/i]×[1/i]
dF (ci1 , ci2)
≤ cde
[1/i]×[1/i]
dci1dci2
≤ cdei2
rendering the sum in (9) and hence the DLC finite which
proves the claim.
Theorem 1 connects time and frequency domain in OFDM
and shows that under mild assumptions two independent paths,
L = 2, are suf ficient for C d (P ∗) > 0. The theorem indeed
fails to hold for L = 1: even though a single path gain has
two real independent components (real and imaginary part),each component is chi-square distributed with one degree of
freedom of which the density is unbounded.
IV. THE IMPACT OF SYSTEM PARAMETERS
It is now of great interest to understand the impact of
the ergodic fading process and its parameters. So the delayspread L and the power delay profile σ as well as the fading
distribution itself obviously affect the OFDM delay limited
capacity. Since the expression in (6) is still very complicated,
we focus on the behavior in the low and the high SNR regime
and carry out a detailed analysis.
A. The low SNR regime
1) Low SNR rate control: In the following theorem we
characterize the first and second term in the Taylor expansion
of C d(P ∗) for small P ∗; both order terms were shown by
S. Verdu in [9] to characterize the system at very low SNR(i.e. low spectral ef ficiency). The theorem tells us that, albeit
generally suboptimal, SRW rate control becomes optimal atlow SNR. Note that the results can neither be obtained from
the approach in [9] since the capacity formula in eqn. (6) has
no simple differentiation expressions. For the ease of notation
we define h∞ := h∞.Theorem 2 (Low SNR optimality of SWF): Suppose that
Eh
h−1∞
< ∞.
i.) The first order limit is given by:
C d (0) := limP ∗→0
C d (P ∗)
P ∗=
1
Eh
h−1∞
(10)
Hence, SRW rate control is first order optimal in the low
SNR regime.
ii.) Define the sub-linear term as Δd (P ∗) := C d (0) P ∗ −C d (P ∗). Then, the second order limit is given by:
limP ∗→0
Δd (P ∗)(P ∗)
2 = K Eh χ−1h h−1∞ 2E3
h
h−1∞
(11)
Here, χh is the (random) multiplicity of subcarriers with
maximum channel gain.
iii.) Suppose that the joint fading distribution is absolute
continuous. Then, the following limit holds:
limP ∗→0
Δd (P ∗)
(P ∗)2 =
K Eh
h−1∞
2Hence, SRW rate control is first order and second order
optimal in the low SNR regime.
Proof: The proof is deferred to Appendix VIII-A.Consequently, by concavity of the capacity and some fixed
multiplicity χ ≥ 1 of subcarriers with maximum channel gain,
we have for any P ∗:
P ∗
Eh
h−1∞
− K (P ∗)2
2χEh
h−1∞
2 ≤ C d (P ∗) ≤ P ∗
Eh
h−1∞
(12)
The behaviour of the DLC and the first and second order
approximations from Theorem 2 for different numbers of taps
is illustrated in Fig.3; here, we depict C d over E b/N 0 (E b:energy per bit, N 0: noise spectral density) which itself is
related to the capacity expression via C dN 0 E b = P ∗, andletting P ∗ → 0 then yields the minimum transmitted energy
per bit. It can be observed that the approximations are very
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−3.475 −3.47 −3.465 −3.46 −3.455 −3.45 −3.445 −3.44 −3.435 −3.430
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5x 10
−3
Eb /N
0[dB]
C d
[ b p s / H z ]
Cd
1st order
2nd order
−6.5 −6.45 −6.4 −6.350
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5x 10
−3
Eb /N
0[dB]
C d
[ b p s / H z ]
Cd
1st order
2nd order
−8.2 −8 −7.8 −7.6 −7.4 −7.20
0.5
1
1.5
2
2.5
3
3.5
4
x 10−3
Eb /N
0[dB]
C d
[ b p s / H z ]
Cd
1st order
2nd order
−10 −9 −8 −7 −6 −50
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
Eb /N
0[dB]
C d
[ b p s / H z ]
Cd
1st order
2nd order
Fig. 3. OFDM DLC, 1st and 2nd order behaviour over E b/N 0 for L = K and L = 4 (upper left) L = 16 (upper right), L = 64 (lower left) andL = 512 (lower right).
CX
XX
h1
1
1
h2
p=1/3 p=1/3
p=1/3
A B
Fig. 4. Three fading states with equal probability.
tight for practical values of L while the range where the
approximation is useful degrades for very large L.
Rather interesting, the multiplicity of the maximum subcar-rier gain occur in the expressions; a scenario where this mat-
ters is discussed in the example in Fig.IV-A1 for one subcarrier
with three fading states occurring with equal probability. This
mimicks e.g. a mobile at the cell border employing handover.
Here, clearly Eh
h−1∞
= 1 but in fact according to Theorem 2
the DLC growth over energy per bit indicated by the sublinear
term will be Δd(P ∗)/P ∗P ∗→0→ 4/3.
2) An explicit formula for OFDM: Appealing to Theorem 2
the forthcoming analysis reduces to the study of the expectedmaximum of the channel gains. However, the expressions do
not show how the DLC depends on the system parameters
which we investigate by means of an asymptotic analysis, i.e.
for large L, K . This analysis turns out to be quite accurate
even for very small L.
We make use of the following result [10][11, Theorem
1]: under very mild assumptions on the fading distribution
we have that h∞ equals approximately log(L) with large
probability (recall that we set ||σ||1 = 1), i.e.
Pr(log(L) − 4log[log(L)] ≤ h∞
≤ log(L) + 4log [log (L)])= 1 − O
log−4 (L)
(13)
even for moderate L, K ≥ L when the complex path gains are
iid; moreover, the upper bound also holds when the PDP is
non-uniform [10]. We can apply this result to the DLC wherewe have to show that from the convergence in probability
given in eqn. (13) it follows convergence of the expected
maximum of the channel gains as well. Leaving out technical
details we can show the following:
Theorem 3: Suppose that the complex path gains are in-
dependent. Moreover, assume that the path gain distribu-tion allows for some bounded Lipschitz constant in an -
neighborhood of zero uniformly. Then the following result
holds:
limsupL→∞
limP ∗→0
C d (P ∗)
log(L) P ∗≤ 1 (14)
Equality in (14) holds for uniform PDP.
Proof: The theorem is a direct consequence
of the uniform integrability property proved
in [11, Lemma 1][10] showing essentially that
Eh h−1∞ = (log (L) + O (log [log (L)]))
−1for large L.
Note that the DLC compares favorably by the factor log(L)with the capacity of AWGN in the low SNR regime.
In order to make the theorem useful in practice we can write
limP ∗→0
C d (P ∗)
log(L) P ∗≤ 1 +
cLo g (L)
log L=: ψ (L) , any L (large),
(15)
where g (L) := log [log (L)] and cLo > 0 is a constantindependent of K (≥ L); consequently, the limit in eqn. (14)
regarding L is independent of how K, L scale. On the other
hand for small but non-zero P ∗ we have by Theorem 2 and
eqn. (12)
C d (P ∗)
log(L) P ∗≥ 1 − cLo
g (L) + Kχ−1P ∗ log(L)
log(L)
(16)
which now indeed depends on K . Hence, for fixed L and
growing K the lower bound (16) becomes arbitrarily small.
This effect can be typically alleviated by the fact that for K L the number of subcarriers having approximately the same
maximum channel gain h∞ is also increasing. A very good
estimate is hk ≥ hk∗ cosπLK
with hk∗ = h∞ and |k − k∗| <
K/(2L) [12] so that Kχ−1 = O (L) and the lower bound
becomes independent of K . Note that there is also an impact
of the PDP which is treated in Sec.VI.The unknown small constant cLo > 0 can be found numeri-
cally. The approach is demonstrated in Fig.5 where we depict
C d(P ∗) over E b/N 0 (in dB scale) and the approximations for
different L. It is seen that the minimum energy per bit, atwhich reliable transmission is possible, goes to minus infinity
with order − log[log(L)] as indicated by Theorem 3.
B. The high SNR regime
After the treatment of the low SNR regime we turn towards
high SNR. In contrast, here not only the delay spread but alsothe fading distribution is important. Furthermore CI instead of
SRW becomes the asymptotically optimal rate allocation.
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−12 −10 −8 −6 −4 −2 0 2
1
2
3
4
5
6
7
8
9
10x 10
−3
Eb /N
0[dB]
C
d
[ b p s / H z ]
L=2,...,1024
Fig. 5. The solid curves depict the DLC over E b/N 0 in an OFDM channelwith L = 2...1024 taps for Rayleigh fading and uniform PDP. The upperbound on the minimum energy per bit marked by the crosses is given byeqn. (15) with cLo ≈ 1.2; the constant is found by simple linefitting for thesmaller L’s while by virtue of Theorem 3 the formula will hold also for the
larger L’s. The gain over of the AWGN channel (−1.59[dB]) is marked bythe dashed curve.
1) High SNR rate control: Defining the quantity h :=Kk=1 h
−1/Kk we have by using the suboptimal CI rate control
law C d (P ∗) ≥ log
P ∗/Eh
h
provided that Eh
h
< ∞,
i.e. for regular fading distributions [3]. We can extend this
result to an upper bound without using any simplifying as-sumptions on the fading distribution; it is also remarkable
that it is tight for large K and large P ∗ regardless whether
the distribution is continuous or not. Further, note that the
quantity Eh h is not at all always meaningful; a simple
counterexample is given in the already discussed three fadingstate example in Sec. IV-A where the DLC is non-zero but
Eh
h
= ∞.
Theorem 4 (High SNR optimality of CI): Suppose that
Eh
h
< ∞.
i.) The following upper limit holds:
lim supP ∗→∞
[C d (P ∗) − log(P ∗)] ≤ logEh
h
+1
K .
(17)
ii.) The following lower limit holds:
liminf P ∗→∞
[C d (P ∗) − log(P ∗)] ≥ log Eh h . (18)
Both bounds coincide for large K (or the distribution is
continuous [3]) and, hence, CI rate control is optimal in
the high SNR regime then.
Proof: The proof is given in Appendix VIII-B.Theorem 4 states that as long as Eh
h
< ∞ the DLC
lies in some target corridor determined by Eh
h
. Hence, it
suf fices to evaluate the term Eh
h
in (17) in the high SNR
regime which characterizes the fixed capacity gap compared
to the log(P )-scaling of AWGN depending on the fadingdistribution. A simple explicit asympotic expression for this
gap is now provided.2) An explicit formula for OFDM: In order to get some
insight let us carry out again an asymptotic analysis. The
following theorem shows that while the fading distribution
10 15 20 25 30 35 40 452
4
6
8
10
12
14
SNR [dB]
C
d
[ b p s / H z ]
IncreasingL=2,4,32uniform PDP
L=4 linear decreasingPDP
log(P*)−0.58[nats]
Fig. 6. Scaling in the high SNR region: The black line indicates the scalingat high SNR given by log (P ∗) − 0.58 (for Rayleigh fading). The dashedlines give the OFDM DLC for L = K = {2, 4 (non-uniform), 4 (uniform),32}.
matters for small L the impact quickly vanishes in the limit
for large L. The more independent channel gains that can
be obtained, the faster will be the convergence, as stated inthe following theorem. Here, avoiding technicalities in the
proofs (such as L, K being prime numbers) L, K are generally
assumed to be dyadic numbers (or just divisible); this is not too
restrictive since K is dyadic in practice. Please note that the
following convergence holds for any performance measure thatfulfills the conditions stated in the proof, i.e. monotonicity and
uniform integrability (such as peak-to-average power ratio).
The proof technique improves on the approach taken by [13]
where weak convergence of the joint distribution of any finitesubset of subcarriers to a joint Gaussian distribution is shown,
which excludes performance measures such as Eh
h
definedon all subcarriers.
Theorem 5: Suppose that complex path gains c̃1,..., c̃L are
iid; further suppose that their marginals are circular symmetric
and have uniformly bounded and suf ficiently smooth densitieswith exponential tails for all L. Then, the following upper
limit holds:
limsupL,P ∗→∞
[C d (P ∗) − log(P ∗)] ≤∞
0log(h)exp(h) dh
:=H F ≈−0.58
(19)
Equality in (19) holds for uniform PDP.
The upper bound becomes tight in the presence of K o-th
order diversity with convergence rate K −1/3o (for constants see
eqn. (34)) where K o is the number of independent subcarriers.
Proof: see Appendix VIII-C.
Interestingly, there is always a loss in capacity compared to
AWGN under the assumptions of the theorem (e.g. Rayleigh-
, Nakagami-fading etc.). The capacity loss equals that of
AWGN capacity to ergodic capacity as we will show in the
next subsection. An illustration is shown in Fig.6 where alsothe non-uniform case is exemplarily incorporated showing
almost no impact of the PDP.
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V. CONVERGENCE TO ERGODIC CAPACITY
The results can be related to ergodic capacity. The ergodic
capacity is given by [3]
C e = Eh (max {log(ξh1) , 0}) (20)
where ξ is chosen such that
P ∗ = Ehmaxξ − h−11 , 0 . (21)
Note that the first order term of the DLC is not bounded withrespect to L. The same is true for the ergodic capacity [9] so
that we can say that DLC reflects the behavior of the ergodiccapacity in the low SNR regime. A similar statement can be
shown for the high SNR case.
Corollary 6: Under the assumptions of Theorem 5 with
uniform PDP the DLC converges to the ergodic capacity as
K → ∞.
Proof: We only have to show that C e scales as log(P ∗)+H F as P ∗ → ∞. A rigorous proof of this fact can be found
in [11, Lemma 1].The preceding results have some interesting implications.
They indicate that the DLC approaches the ergodic capacity
even if the channel gains are not independent. In the asymp-
totic regime coding over spectral degrees of freedom could
substitute the coding over fading blocks in case of ergodiccapacity.
VI . APPLICATIONS TO LTE SYSTEMS
In order to apply the results to practical scenarios we need
to find a way to tackle also non-uniform PDPs. The impact of
the PDP has been touched already in Theorem 3 proving the
”order-optimality” of uniform PDP. Using essentially Theorem
3, we can even incorporate non-uniform PDP σ1 ≥,..., ≥ σL(at least one inequality is strict) as follows: instead of directly
representing channel gains cl, ∀l, in (1) consider the followingtriangular structure
c̃L =√
γ Lc̃(L)L
c̃L−1 =√
γ Lc̃(L−1)L +
√γ L−1c̃
(L−1)L−1
...
c̃1 =√
γ Lc̃(1)L +
√γ L−1c̃
(1)L−1 + ... +
√γ 1c̃
(1)1
whereγ l := σl−σl+1
andσL+1 := 0
for some appropriate iid
sequence of random variables c̃(l)i , l = 1,...,L,i = l,....,L;
hence, we can write c̃l =Li=l
√γ ic̃
(l)i . Note that the actual
distribution of these random quantities which might be dif ficultto calculate will not be needed in the subsequent analysis.
Clearly, there are random phases ejϕi(h), i = 1,...,L, such
that:Li=1
il=1
√γ ic̃
(l)i e−
2πj(l−1)(k−1)K , k = 1,...,K
∞
=
L
i=1
ejϕi(h)
i
l=1
√γ ic̃
(l)i e−
2πj(l−1)(k−1)K , k = 1,...,K
∞
:=h(γi)∞
Applying triangle inequality and Jensen’s inequality to
E[1/(·)2] yields a strict upper bound1. The bound can be
improved by observing that the phases will be close to
independent. Since Eh(h(γi)∞ ) ≤ ψ(L)log(iγii) using eqn.
(15) we obtain the formula:
C d (P ∗) P ∗L
i=1
ψ (L)log
iγii
(22)
Note that this approach is tight only for uniform PDPs butit constitutes 1.) a rigorous upper bound and 2.) captures the
right behaviour, i.e. a more spread out PDP will have a higher
DLC (so-called Schur-concavity). In the following we discuss
an application example which is related to LTE performance
evaluation.
A. Bandwidth request for delay-sensitive services
Using formula (22) we can estimate the number of usersthat can be supported at the cell border which request a
fixed rate. We consider an OFDM system with the system
parameters given in [14]. The system has 1024 subcarriersand the bandwidth is 5 MHz. The multi-path fading channel
is modelled as Pedestrian A/B [15] with 4 and 29 taps non-
uniformly. Suppose the receive SNR at the cell border is -10
dB, the maximal number of users that can be supported with
different service requirement is given in Tab. I. It can be seen
that the capacity for high data rate services (e.g. Videophone)
is scarce if several users are at the cell border in the same
time. Hence, an increase of the bandwidth from 5 MHz to 20MHz as discussed in 3GPP LTE is advisable.
VII. CONCLUSION
In this work, we studied the delay limited capacity of
OFDM systems. It was shown that explicit expressions can
be found for the low and high SNR regime even for the
challenging correlation structure of OFDM. The presented
results are not restricted to OFDM but can be carried overother classes of parallel channels such as e.g. MIMO. Still
an open problem is the complete characterization of the
DLC for arbitrary SNR and arbitrary power delay profile.
Here, universal bounds seem very dif ficult to derive. It is an
interesting but unproven conjecture that the DLC is in generalSchur-concave with respect to the power delay profile which
implies that a uniform profile maximizes the DLC in all cases.
VIII. PROOFS OF THE THEOREMS
A. Proof of Theorem 2
Note that the proof of i.) can be easily devised using
the same technique as in ii.) and the proof of iii.) is then
straightforward so they are both omitted.
ii.) Our starting point is eqn. (6) where we use the expansion
of the exponential function up to the quadratic term, i.e.
exp(x) = 1 + x + 0.5x2 + o
x2
. Fix > 0 and set the
”virtual” channel gain of any subcarrier k with hk ≥ h∞ −
1
Eh(h−1∞ ) is bounded by
L
i=1 Eh
i
l=1√γ ic̃
(l)ie−
2πj(l−1)(k−1)K , k = 1,...,K
∞
−2
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TABLE IMAXIMUM NUMBER OF SUPP ORTABLE USERS AT THE CELL BORDER F OR 3GPP PEDESTRIAN A/ B CHANNEL, 3 KM / H. T HE VALUES IN ROW 3, 4 IS
OBTAINED BY THE SIMULATED DL C AND IN ROW 5, 6 IS THE OBTAINED BY UPPER BOUND IN (22).
Real-time Streaming Services Conversational voice High quality streaming audio VideophoneRate Requirement 13 kb/s 128 kb/s 384 kb/s
Nr. of users (Ped. A, DLC simulated) ≤ 34 ≤ 3 ≤ 1Nr. of users (Ped. B, DLC simulated) ≤ 67 ≤ 6 ≤ 2
Nr. of users (Ped. A, bound) ≤ 34 ≤ 3 ≤ 1Nr. of users (Ped. B, bound) ≤ 77 ≤ 7 ≤ 2
to h∞. Denote by χh () the multiplicity of the number of
subcarriers that are assigned the maximum channel gain by
this strategy for any channel realization h. Then, we obtain
for suf ficiently small P ∗
P ∗ ≥ Eh
C d + 0.5C 2dKχ−1
h ()
h∞
. (23)
The equation is an upward open parabola in C d where one zero
is negative and one is positive where the latter is increasing
in P ∗. Solving this equation for C d yields the inequality
C d (P ∗) ≤ − Eh
h−1∞
K Eh
χ−1h
() h−1∞
+
E2h
h−1∞
K 2E2
h
χ−1h () h−1
∞
+2P ∗
K Eh
χ−1h () h−1
∞
.
Expanding the square root function yields
C d (P ∗) ≤ 1
Eh
h−1∞
P ∗− K Eh
χ−1h () h−1
∞
2 (1 + )E3
h
h−1∞
(P ∗)2 . (24)
Subtracting the first order expression (10) from (24) we arrive
for some > 0 at
Δd (P ∗) ≥ K Ehχ−1
h () h−1∞
2 (1 + )E3h
h−1∞
(P ∗)2
≥ K Eh
χ−1h
h−1∞
−
2 (1 + )E3h
h−1∞
(P ∗)2 (25)
and thus have established a lower bound on Δd(P ∗) for any
, > 0. The last inequality (25) follows from the following
argument: observe that χh () ≥ 1 and, almost surely with
respect to the fading distribution, for any realization h we
have
χ−1h () h−1
∞ → χ−1h h−1
∞ , → 0,
and provided that Eh χ−1h h−1∞ ≤ Ehh−1∞ < ∞ we obtain
by dominated convergence [16]:
lim→0
Eh
χ−1h
() h−1∞
= Eh
χ−1h
h−1∞
Furthermore, it is easily established that
Δd (P ∗) ≤ (1 + ) K Eh
χ−1h h−1
∞
2E3
h
h−1∞
(P ∗)2
. (26)
Combining (25) and (26) leads to the desired result (11).
Since SRW will assert rates only to the subcarrier with the
best channel gain in the low SNR region this will generate the
same optimal first and second order terms provided that theprobability of multiple subcarrier allocation for the optimal
scheme is zero.
B. Proof of Theorem 4
We can use the following strategy for an upper bound C d:
fix > 0 and suppose that for any fading state h we set the
values that are below to . In other words we do not allow
”virtual” channel gains below .Now, define hk := max {hk, }. Then, using (6) we have
for suf ficiently large P ∗
P ∗ ≥Eh
eC dK
k=1
(hk)− 1K
− 1
K
K
k=1
Eh
1
hk
(27)
= Eh
eC d
Kk=1
(hk)− 1K
− Eh
1
h1
.
Obviously, the second term grows without bound as → 0for many fading distributions such as Rayleigh fading. Fur-
thermore the growth depends on P ∗. Let us bound this termas follows: we have
Eh
1
h1
=
∞0
1
h1dF h (h1)
≤ −1.
Clearly, the term is related to P ∗. Since the underlying
optimal rate control law is waterfilling and the minimumchannel gain is at least we have D(h) ≡ {1,...,K } if
λ ≥ −1. Thus the above equation (27) is certainly true if
P ∗ =
k∈D(h)
pk =
Kk=1
λ − 1
hk
+≥ K
which is a rough estimate. Hence, we obtain
Eh
1
h1
≤ P ∗
K
and finally for any > 0
C d ≤ log⎛⎝ P ∗ 1 + 1
K Eh
Kk=1 (hk)
− 1K
⎞⎠ .
Now observe thatKk=1 (hk)
− 1K ≤ h, and, hence, almost
surely we have
lim→0
Kk=1
(hk)−1K = h. (28)
Using (28) we can once again invoke the dominated conver-
gence theorem and obtain
EhK
k=1
(hk)− 1K →
Eh hprovided that Eh
h
< ∞ which is true by assumption.
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C. Proof of Theorem 5
Suppose that Ln, K n are dyadic numbers where for any
index n ≥ 1, h̃n1 , ...., h̃nKn is a family of complex-valuedrandom variables (so-called triangular array) generated by
the sequence c̃1, ...., c̃Ln via the DFT operation (1). Let
E nm be a finite subset of {1,...,K n} for some m ≥ 1 and
2m ≤ Ln ≤ K n defined by
E nm :=
k : k =(i − 1) K n
2m+ 1, i = 1,..., 2m
;
denote the corresponding subset of random variables by h̃Enm
generating a subfamily of complex-valued random variables
with cardinality |E nm|. Furthermore, we superimpose a naturalorder on the elements of E nm such that k1 < k2 < ... < k2m
for ki ∈ E nm. In order to establish convergence we need to
prove the following steps.
i.) (Monotonicity): We can frequently use the following
well-known inequality [17] for some a,b > 0 dividing K :
Eh
Kk=1
h− 1K
k
≤
ak1=1
RK+
bk2=1
h− 1b
k1+(k2−1)a dF h (h)
1a
(29)
By using the lower bound in (18) and the integral inequality
in (29) it is straightforward to see that (by the structure of
the DFT) if the fading distribution is generated by a complexpath gain distribution of which the marginal densities can be
written as f ̃ci (c̃i) , c̃i ∈ C, where f ̃ci is circular symmetric,
i.e. it is invariant under complex phase factors then the fading
distribution is invariant regarding k1 in (29); hence we arrive
at R2K
n
+
Knk=1
h̃k− 2Kn
dF ̃hn
h̃
≤ R2K
n
+
k∈Enm1
h̃k− 2
|Enm1 | F ̃hn
h̃
≤ R2K
n
+
k∈Enm2
h̃k− 2
|Enm2 | F ̃hn
h̃
uniformly in n for fixed m2 ≤ m1 and 2m1 ≤ K n.
ii.) (Convergence): Suppose convergence in distribution
(denoted as D→ [18]) of the random vector h̃Enm to a Gaussianrandom vector h̃mG ∼ 2m
n=1 CN (0, 1) for fixed m ≥ 1 is
required. By means of the Cramer-Wold device convergence
is equivalent to convergence of
ξ (αi, β i)
:=M i=1
αi
h̃nki
r
+ β i
h̃nki
i D→n→∞
N
0,M i=1
α2i
2+
β 2i2
(30)
for arbitrary real numbers αi, β i, and indices ki ∈ E nm, i =1,...,M,M ≤ 2m. Evaluating the variance of ξ (αi, β i) yields
for uniform PDP
Eh̃EnmG
(ξ (αi, β i))2
=M i1=1
M i2=1
αi1αi21
2Ln
Lnl=1
cos
2π (l − 1) (ki1 − ki2)
K n
+
M
i1=1
M
i2=1
β i1β i21
2Ln
Ln
l=1
cos2π (l − 1) (ki1 − ki2 )
K n
+M i1=1
M i2=1
αi1β i21
Ln
Lnl=1
sin
2π (l − 1) (ki1 − ki2)
K n
and by applying central limit theorem for triangular arrays
to the random variable ξ (αi, β i) (which is the sum of inde-pendent random variables by the iid assumption) proves (30)
for any set ki ∈ E nm, i = 1,...,M . Note that by the assumed
exponential decay of the marginals, Lindeberg’s condition for
triangular arrays will be clearly satisfied and convergence of
h̃EnmD→ h̃mG follows (see also [13] where a somewhat similar
convergence is proved).
iii.) (Uniform integrability): The following conditionholds:
limα→∞
supn≥1
Ehn
h I h ≥ α
= 0 (31)
The proof is sketched: Using (29) and properties of the FFT
in (1) the expectation can be upperbounded as:
Ehn
h ≤ Ehn
h̃1
−1 h̃K/2−1
= Ehn
⎛
⎜⎝
⎛
⎝L/2
l=1
c̃2l−1
⎞
⎠2
−⎛
⎝L/2
l=1
c̃2l
⎞
⎠2
−1⎞
⎟⎠Both sums are independent and converge in distribution to
a circular symmetric Gaussian distribution. By assumption
on the smoothness on the marginals this implies pointwise
convergence of densities and by Scheffé’s theorem [18] the
convergence of the expectations. Since the limit expectationis finite for two independent circular symmetric Gaussian
distributed complex path gains (refer to Theorem 1) this in
turn implies eqn. (31).
iv.) (Lower bound): Using (17) we set Ehn(h) =Ehn(exp[log(h)]). Then, by Jensen’s inequality and the con-
vexity of exp(·) we arrive at
Ehnelog(h) ≥ eEhn [log(h)]
= exp
− ∞0
log(h) dF hn1 (h)
= exp
− ∞0
log(h)exp(−h) dh
− (n)
(32)
where (n) → 0 as n → ∞. The first equality follows fromthe assumption that the marginal distribution of the complex
path gains is circular symmetric and the second equality isdue to the central limit theorem and uniform integrability.
Monotonicity, convergence of h̃EnmD
→h̃mG , and uniform
integrability now ensures that Eh̃E
nm (h) converges to E
h̃mG
(h)for any m ≥ 1 and uniform PDP. Finally it suf fices to show
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that Eh̃mG
(h) → exp(−H F ) as m → ∞, and to provide a
measure of convergence speed for independent subcarriers;note that by the assumed smoothness of the marginals the
lower bound convergence in eqn. (32) will happen much
faster (or it is even exact for Rayleigh fading) and we focus
on the convergence of Eh̃mG
(h) with K o = 2m independent
subcarriers.
We need the following technical result for K o > 1 indepen-
dent channel gains (not even necessarily OFDM subcarriers).
Proposition 1: Suppose that K o > 1 channel gains are iid
with marginal density f . Then, the following upper bound
holds: RKo+
h dF h (h)
≤
K oK o − 1
Ko
×⎛⎝⎡⎣i≥1
f
a−i− f
b−i⎤⎦ 1
K o
−i≥1 a
Ko−1Ko
i f b
+
i − f a
+
i ⎞⎠Ko
Here, 0 ≤ a−/+i < b
−/+i ≤ ∞ are interval boundaries such
that f (h) ≤ 0, h ∈ a−i , b−i
and f (h) ≥ 0, h ∈
a+i , b+i
.
Proof: This lemma is proved in [11, Proposition 8].
Now define the following random variable (i.e. partial
sums):
h(Ko) := − 1
K o
Kok=1
log(hk)
By independence, we have h(Ko)
→(
−H F ) in probability and
we now show Eh(Ko)(exp(h(Ko))) → exp(−H F ). Definingsome real co H F and splitting up the events in sets
{h(Ko) ≤ co} and {h(Ko) > co} we obtain in the first case
the inequality
Eh
eh
(Ko)
I
h(Ko) ≤ co
≤ Eh
min
eh
(Ko)
, eco
(33)using the set function I{·}. Defining further the event set A :={h(Ko) + H F
≤ } for some small > 0 the RHS of eqn.
(33) can be upperbounded as follows: since
Eh mineh(Ko)
, eco I {A} ≤ e−H F +
≤ e−H F + 2e−H F
for suf ficiently small and
Eh
min
eh
(Ko)
, ecoI {Ac}
≤ eco Pr (Ac)
we just need a bound on the probability Pr (Ac) depen-
dent on . The probability can be easily upperbounded by
Tschebyscheff’s inequality, i.e.
Pr (Ac) ≤ σoK o2
whereσo = Eh
[log (h1) − Eh (log(h1))]
2
and by choosing the suf ficiently slowly converging zero se-
quence Ko= O
1/K
1/3o
the third term can be upper-
bounded by:
Eh
eh
(Ko)
I
h(Ko) > co
=
RKo+
h I
h(Ko) > co
dF h (h)
≤
RKo+
h2
dF h (h)
12
RKo+
I
h(Ko) > co
2dF h (h)
12
≤
K oK o − 1
Ko2
Pr
h(Ko) > co
12
The first inequality follows from Cauchy’s inequality and the
bound for the first integral follows from the independence
of hk, k ∈ {1,...,K o} in combination with Prop. 1 with
monotonously decreasing density f . The probability can beagain tackled with Tschebyscheff’s inequality. It follows
Pr
h(Ko) > co
≤ σo
K o (co + H F )2
and putting terms together
C (P ∗)
≥ log(P ∗) + H F + log
1 + 2Ko
+coσo
e−H F K o2Ko
+
√eσo
e−H F √
K o (co + H F )
Ko large≥ log(P ∗) + H F + 2
3√
K o+ coσo
e−H F 3√
K o
+
√eσo√
K o (co + H F )(34)
concludes the proof.
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Gerhard Wunder received his graduate degreeof electrical engineering (Dipl.-Ing.) in 1999 andthe Ph.d degree (on the peak-to-aver power ratioproblem in OFDM) in electrical engineering in 2003from Technische Universität (TU) Berlin, Germany.
He is now with the Fraunhofer German-SinoLab for Mobile Communications, Heinrich-Hertz-Institut, leading several industry and researchprojects in the field of wireless communication sys-tems. He is also a lecturer for detection/ estimation
theory, stochastic processes and information theoryat the TU Berlin, department for mobile communications. Recently, he alsoreceived the habilitation and Privatdozent degree from the TU Berlin incommunication engineering. His general research interests include estimationand information theory as well as crosslayer design problems.
Thomas Michel received his graduate degree inBusiness Administration and Engineering (Dipl.-Wirtsch.-Ing.) from TU Dresden, Germany in 2003and his Ph.D. degree in Electrical Engineering fromTU Berlin, Germany in 2008.
Chan Zhou received the graduate degree (Dipl.-Ing) in technical computing engineering from Tech-nische Universität (TU) Berlin , Germany in 2004.He is currently pursuing the Ph.D. degree at TUBerlin. His research interests include radio resourcemanagement and scheduling, wireless propagationand channel modeling, feedback and control channeldesign.