delay-limited transmission in ofdm systems

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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

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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.




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.





−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





−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.





−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.





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





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.





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





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.

delay-limited transmission in ofdm systems

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