Power Loading in Parallel Channels with Channel

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Power Loading in Parallel Channels with Channel Distribution Information at the Transmitter

arXiv:1003.5097v1 [cs.IT] 26 Mar 2010

Justin P. Coon, Senior Member, IEEE, and Rafael Cepeda, Senior Member, IEEE

Abstract—In this paper, the ergodic capacity of parallel channels is investigated with a view to designing the power allocation strategies that optimise this performance measure under the assumption that perfect, instantaneous channel knowledge is available at the receiver whereas the transmitter only has knowledge of the fading distributions and/or the first and second moments of the subchannel gains. Upper and lower bounds on the capacity are derived for general fading distributions, and low and high SNR optimal power allocation strategies for three parallel fading channels are given. The specific channels of interest are a singleinput single-output Rayleigh fading channel, a selection channel, and a receive diversity channel with maximum ratio combining (MRC). Moreover, it is shown that, for the selection and MRC channels, as the number of diversity branches M → ∞, the upper and lower bounds are tight, and the power loading that maximises mutual information for all fading distributions follows a waterfilling principle, but where the mean channel gains are used to calculate the power values instead of the instantaneous channel gains. Finally, practical examples using new measured ultrawideband channel data are provided. Index Terms—Capacity, parallel channel, OFDM, power loading, fading, channel sounding.

I. I NTRODUCTION Parallel channels are frequently encountered in modern communication systems. For example, orthogonal frequency division multiplexed (OFDM) transmissions operate in a parallel channel where each subchannel is a different frequency bin, or subcarrier, that is ideally orthogonal to all other subchannels [1], [2]. Also, when full, instantaneous channel state information is available at the transmitter (CSIT), so-called singular value decomposition (SVD) beamforming can be exploited to convert a multiple-input multiple-output (MIMO) channel into a parallel channel, where each subchannel is an eigenmode of the original channel [3]. Such channels arise in multiantenna communication systems. Other examples of systems that operate in parallel channels include time-division multiplexed (TDM) transmissions and frequency-hop spreadspectrum (FH-SS). In recent years, the design of subchannel power allocation strategies has become a popular research topic for systems operating in parallel channels. Power loading is typically performed such that some objective function is optimised. For example, power loading schemes have been developed to maximise spectral efficiency [4] and system throughput [5]– [7], as well as to minimise bit-error rate (BER) [8]–[10] and transmit power [8], [11]–[13]. Much of the work to date has J. P. Coon and R. Cepeda are with Toshiba Research Europe Ltd., 32 Queen Square, Bristol, BS1 4ND, UK; tel: +44 (0)117 906 0700, fax: +44 (0)117 906 0701, email: {justin, rafael}@toshiba-trel.com.

focused on the case where full CSIT is available (see, e.g., [3], [14]–[16]). However, if the channel changes frequently over time, or comprises a large number of subchannels, the aquisition of full CSIT may require a prohibitively large amount of feedback or processing. To address this issue, power loading schemes based on imperfect (e.g., outdated) channel knowledge were developed in [17], [18]. Further schemes that rely on quantised channel information and limited feedback of channel information were presented in [10], [19], [20]. An alternative approach to designing power allocation schemes for parallel channels, which does not rely on instantaneous/perfect channel feedback, is to utilise the statistical information about the channel when assigning power to each subchannel. In [21], the authors developed a power loading strategy that minimises the average BER for Rayleigh fading subchannels by using only knowledge of the first moments (means) of the channel gains. Furthermore, in [22], [23], the authors developed novel power and bit allocation algorithms to maximise the spectral efficiency and minimise the power consumption using statistical knowledge of the channel. In this article, we take a different approach to previous work encountered in the literature by studying the ergodic capacity of a parallel channel when channel distribution information is available at the transmitter (CDIT). This information may include the probability distribution that models the random channel gains, or simply the low-order moments of the channel gains. Specifically, we consider a parallel channel with N fading subchannels, where the input-output relationship of the nth such subchannel can be modelled by yn = hn xn + zn

(1)

where xn is the transmitted signal, hn is the channel transfer coefficient, zn is additive Gaussian noise, and yn is the received signal. Our goal is to calculate the power allocation strategy that achieves capacity (in the ergodic sense) for the channel defined in (1) using only CDIT. If the subchannel statistics are identical over all subchannels, the optimal power allocation scheme reduces to a balanced power allocation across the subchannels. However, in many cases of interest, subchannel statistics differ dramatically. One motivating example can be found in OFDM systems operating in channels with extremely wide bandwidths, where the average gain of the channel response decays with frequency according to a power law [24], [25]. In addition, channels with correlated channel impulse response taps experience variations in the average power profile of their respective channel frequency responses [26]. The novel contributions presented in this article are:

COON AND CEPEDA: POWER LOADING IN PARALLEL CHANNELS WITH CDI AT THE TRANSMITTER

the derivation of bounds on the ergodic capacity for parallel channels with general fading distributions when power loading is applied using knowledge of the mean subchannel gains; • optimal power allocation strategies at low SNR (to second order) and high SNR for three types of parallel fading channels, each being based on Rayleigh fading subchannels: a single-input single-output (SISO), a selection channel, and a receive diversity channel with maximum ratio combining (MRC); • the derivation of a power loading principle that is optimal at low and high SNR for any fading channel, as well as for any SNR in the selection and receive diversity channels discussed above when the number of diversity branches grows large; • a high SNR analysis of SISO, selection, and MRC parallel channels through the characterisation of the excess rate – as defined in [27]; • the introduction of the concept of relative gain – which is the ratio of the capacity of a system with full CSIT to the capacity of a system with only CDIT at asymptotically low SNR – with an aim to analyse the three parallel channels mentioned above; • the application of the new techniques and analysis outlined above to measured ultrawideband (UWB) channel data, which was obtained using a state-of-the-art timedomain UWB channel sounder [28]. The rest of the paper is organised as follows. In Section II, we provide bounds on the capacity for general fading distributions. Following this analysis, we investigate the capacity of different types of parallel channels for various SNR regions in Section III. In Section IV, we utilise measured channel data to evalulate the theoretical expressions given in the first part of the paper. Finally, conclusions are drawn in Section V. •

II. C APACITY

FOR

G ENERAL FADING D ISTRIBUTIONS

Let us denote the power transmitted on the nth subchannel by Pn , and let N0 be the variance of the zero-mean, additive white Gaussian noise zn . Note that we assume without loss of generality that the power spectral densities of the noise processes on all subchannels are identical. It follows that the instantaneous capacity, expressed as a function of the powers {Pn }, of a parallel channel with subchannel gains {γn } is given by [29]   N X Pn (2) γn C (P1 , . . . , PN ) = log 1 + N0 n=1 where the units are in nats. The ergodic capacity is simply    N X Pn CE = E {C} = E log 1 + (3) γn N0 n=1

i.e., the expectation of C, taken with respect to each channel gain γn . In practical communication systems, it is common to place a constraint on the power that is transmitted across the subchannels. This restriction can take several forms. For example,

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one may choose to allow the transmit power to vary across the subchannels, fixing the power on each subchannel for the duration of a transmitted frame and maintaining a constant average transmit power across all subchannels. We refer to this approach as a short-term power constraint. Alternatively, the transmit power per subchannel may be allowed to change over time as long as the sum of the average (with respect to time) powers over all subchannels adheres to some constraint, known as the long-term power constraint (cf. [15]). When considering ergodic capacity with CDIT only, allowing the power per subchannel to vary with time does not provide any advantage since there is insufficient information available in the system at any given moment to determine whether an increase or a decrease in power on a given subchannel would improve the capacity. Thus, we restrict our attention to the short-term power constraint, i.e., we allow the power to vary across the subchannels, but we do not allow the power on a given subchannel to change with time. Now, the goal is to choose the set (

Pn : Pn ≥ 0,

N X

Pn = P

n=1

)

(4)

such that CE is maximised, where P is the short-term power constraint. With this objective in mind, we define the poweroptimal ergodic capacity as follows. Definition 1: The power-optimal ergodic capacity of a parallel channel where CDI is available at the transmitter and full CSI is available at the receiver is defined as ⋆ CE

(

= sup CE (P1 , . . . , PN ) : Pn ≥ 0, N X

n=1

)

Pn = P, {Pn } are deterministic . (5)

The set of optimal powers {Pn⋆ } is the set that yields the supremum given above. The constraint that the set {Pn⋆ } must be deterministic precludes the solution from being dependent upon the instantaneous channel gains {γn }. This is the key assumption that leads to the novel results presented in this paper. In order to maximise the functional1 CE [f1 , . . . , fN ] =

N Z X

n=1

  Pn γn dγn (6) fn (γn ) log 1 + N0

the density functions {fn } of the channel gains must be known. In many practical scenarios, however, the estimated or assumed fading distributions may not be exact. Thus, it is important to obtain bounds on capacity that are applicable to any fading distribution. We provide such bounds below, reserving the treatment of the case where the assumed fading distributions are accurate for the next section. 1 We employ the convention of square brackets to denote dependence upon functions, which is often used to study the calculus of variations.

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reciprocal of mean subchannel SNR

A. Upper Bounds We can derive reasonably tight bounds on the power-optimal ⋆ ergodic capacity CE . Let Pnw denote the power loading strategy based on the traditional waterfilling scheme [30], but where the means of the channel gains, denoted by {m1,n }, are used in place of the instantaneous channel gains {γn }, i.e.,  + N0 Pnw = ν − (7) m1,n

power allocated to subchannel (shaded area)

water level

+

where (x) = max {0, x} and ν is chosen to satisfy + N  X N0 = P. ν− m1,n n=1

(8)

We refer to (7) as statistical waterfilling in accordance with [26, cf. §VI] (also, see [31]). An illustration of this power loading technique is given in Fig. 1. Now, we have the ⋆ following upper bound on CE :    N X Pn⋆ ⋆ γn CE = E log 1 + N0 n=1   N X P⋆ ≤ log 1 + n m1,n N0 n=1   N X Pnw (9) m1,n ≤ log 1 + N0 n=1 where the first inequality follows from Jensen’s inequality and the second inequality results from the standard waterfilling optimisation. At high SNR, the upper bound given by (9) can be made tighter by considering the case where full CSIT is available. In particular, it is clear that    ⋆ N X Pw ⋆ CE ≤ E log 1 + n γn (10) N0 n=1

where



Pnw =

 + N0 ξ− γn

B. Lower Bounds We can clearly state the following trivial lower bound:    N X Pnw ⋆ . (14) γn CE ≥ E log 1 + N0 n=1

⋆ From (9) and (14), we see that CE lies between w w CE (P1 , . . . , PN ) and its upper bound given by invoking Jensen’s inequality. Moreover, due to the fact that log (1 + x) ≃ x for small x, it follows that statistical waterfilling is in fact optimal to first order at low SNR. This optimality holds true at high SNR as long as the distribution of the nth subchannel gain satisfies a finite logarithmic moment condition. To see this, we first recall that Pnw → P/N at high SNR. It follows that if |E {log γn }| < ∞, we have the following asymptotic relations:

log (1 + SNR · m1,n ) ∼ log SNR + log m1,n

(15)

E {log (1 + SNR · γn )} ∼ log SNR + E {log γn }

(16)

and

At asymptotically high SNR, (10) is at least as tight as (9). This result follows from an application of Jensen’s inequality and the relation2       ⋆ Pw Pw ∼ E log 1 + n γn (13) E log 1 + n γn N0 N0 ⋆

which results from the fact that Pnw → P/N and Pnw → P/N as P → ∞. In fact, in many practical examples, such as for subchannels experiencing Rayleigh fading, (10) is tighter than (9) at high SNR. use the symbol ∼ to denote asymptotic equivalence.

Fig. 1. Illustration of statistical waterfilling concept. The total shaded area corresponds to the total transmit power.

(11)

is the traditional waterfilling power allocation given full CSIT, with ξ satisfying + N  X N0 ξ− = P. (12) γn n=1

2 We

Subchannel

where

P . (17) N N0 Thus, the ratio of the upper and lower bounds given by (9) and (14) tends to one as SNR → ∞. It is important to note that |E {log γn }| < ∞ for many practical fading distributions, including Rayleigh fading. While the calculation of (9) is straightforward for a given set {m1,n }, computing (14) may require a Monte Carlo approach for complicated fading distributions for which the expectation cannot be manipulated to yield a simple form. In this case, we may use Markov’s inequality to derive a lower bound that can sometimes be more easily computed:     N X N 0 an ⋆ (18) (e − 1) CE ≥ sup an 1 − Fn Pnw a n=1 n SNR =

COON AND CEPEDA: POWER LOADING IN PARALLEL CHANNELS WITH CDI AT THE TRANSMITTER

where Fn (x) = Pr (γn ≤ x) is the cumulative distribution function (CDF) of the nth subchannel gain. In particular, (18) is a useful bound when analysing the capacity of some diversity channels, which will be discussed in more detail in the next section.

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where CCSIT is the capacity when full CSIT is available. The excess rate is defined in a manner similar to that presented in [27]: ⋆ G , lim (CE − N log (SNR)) (23) SNR→∞

where As discussed above, statistical waterfilling becomes the optimal power loading strategy as SNR → 0. This result is corroborated by calculating the low-SNR optimal power loading strategies for a parallel fading channel. In particular, this can be done easily by replacing the logarithm in the capacity expression with the corresponding truncated Taylor polynomial of order q, then solving the resulting constrained maximisation problem using Lagrange multipliers. For q = 1, 2, the first and second-order optimal power allocation strategies follow  P, if n = arg max m1,n (19) Pn(1) = 0, otherwise and Pn(2) =

N02 m2,n



+ m1,n −φ , N0

∀n

(20)

respectively, where mi,n is the ith moment of the channel gain for the nth subchannel and φ is chosen to satisfy the power P (2) constraint n Pn = P . The first-order optimal allocation can also be concluded from the bounds (9) and (14), which are asymptotically equal as P → 0 due to the linearity of log (1 + x) at x ≃ 0. Clearly, the second-order optimal allocation follows a waterfilling principle. Furthermore, it is well-known that at very low SNR, waterfilling dictates that power is only allocated to the channel with the maximum gain. This is precisely the statement given in (19). The corresponding low-SNR approximation of the second-order, power-optimal ergodic capacity can be written as ! N (2) (2) X m2,n Pn Pn ⋆ m1,n − CE ≃ N0 2N0 n=1 =

1 2

X

n:φ≤m1,n /N0

m21,n − φ2 N02 . m2,n

(21)

Finally, we note that at high SNR, the capacity-optimal powers satisfy Pn⋆ = P/N for all n and for any fading distribution. This unsurprising result follows directly from the asymptotic relation given by (13). D. Relative Gain and Excess Rate In the next section, we analyse the capacity and optimal power allocation strategies for three channels where the fading distributions are known a priori. In particular, we make use of the concepts of relative gain and excess rate to study the behaviour of these channels at low and high SNR, respectively. We define the relative gain as CCSIT R , lim ⋆ SNR→0 CE

P . N N0

SNR =

C. Power Loading at Low and High SNR

(22)

(24)

These two performance measures will become particularly useful when we study measured UWB channel data in Section IV. III. C APACITY

FOR

K NOWN FADING D ISTRIBUTIONS

In this section, we focus on the calculation of the capacity and optimal power allocation strategy for three specific parallel fading channels, each of which is defined by the input-output characteristics of its subchannels. The first channel that is considered has SISO subchannels. For the second channel, a selection process defines each subchannel. Finally, the third channel has single-input multiple-output (SIMO) subchannels, where MRC is employed at the output. For each of these channels, we focus on the low and high SNR regimes in particular. All channels that are considered arise from cases where the subchannel fading distributions are derived from Rayleigh distributed fading processes. A. SISO Fading Channels Consider a system operating in a parallel channel with N possibly dependent subchannels, each of which experiences Rayleigh fading. The channel gain for the nth subchannel is denoted by γn . This gain is modelled as an exponentially distributed random variable (r.v.) with mean µn ; thus, the density function of γn is given by fn (γ) =

1 − µγ e n, µn

γ ≥ 0.

(25)

The ergodic capacity of this parallel channel can be expressed as (see, e.g., [16])   N X N0 N0 P µ n n e CE = (26) E1 Pn µn n=1 where E1 (x) =

Z



x

e−t dt. t

(27)

The maximisation of (26) over {Pn } must, in general, be performed numerically. However, we can derive an alternative ⋆ expression for CE , which we encompass in the following proposition. Proposition 1: The power-optimal ergodic capacity of a parallel channel where CDI is available at the transmitter, and with exponentially distributed channel gains with positive, bounded means {µn }, can be expressed as ⋆ CE =

N X µn ⋆ Pn (1 − λPn⋆ ) N 0 n=1

(28)

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where λ is a constant chosen to satisfy the Kuhn-KarushTucker (KKT) conditions related to the optimality of {Pn⋆ }. ⋆ Moreover, the function CE is concave in {Pn⋆ }. Proof: See Appendix A. It is important to note that the expression for the poweroptimal ergodic capacity given above is only valid for the optimal powers {Pn⋆ }, for which there exists no closed-form representation in general. Note, however, that (28) is an N dimensional, elliptic paraboloid, and the concavity of this simple expression may be used to derive efficient numerical techniques to compute the optimal values {Pn⋆ } [32]. At low SNR, we can apply (20) and (21) with m1,n = µn

(29)

m2,n = 2µ2n

(30)

Proof: We can write RSISO

=

= =

where µ ¯ denotes the sample average of {µn } and µmax is the maximum mean channel gain. If the subchannels are independent, then the relative gain can be calculated exactly to yield RSISO =

1 µmax

N |A| X 1 X (−1) 2 µ i=1 i A∈2Si (αi (A))

(32)

where Si = {1, . . . , N } r i, the set 2Si denotes the power set of Si , and X 1 1 αi (A) = + . (33) µi µj j∈A

Proof: See Appendix B. Equation (31) in Proposition 2 can be simplified further to yield p RSISO ≤ 1 + 2 (N − 1). (34) Thus, we have that

RSISO = O

√  N

(35)

and the advantage obtained by power loading based on full, instantaneous knowledge of the channel increases fairly slowly with the number of subcarriers. For the case where the subchannels are independent and identically distributed (i.i.d.), µ1 = · · · = µN and (32) leads to the corollary given below, which shows that the increase in the relative gain is considerably slow in this scenario. Corollary 1: The relative gain of a parallel channel with i.i.d Rayleigh fading subchannels, each with a mean gain of µ, satisfies RSISO = log N + O (1) . (36)

N X X

|A|

(−1)

i=1 A∈2Si (1 N N −1  X X i=1 n=0 N   X k=1

2

+ |A|)  n N − 1 (−1) 2 n (n + 1) k+1

N (−1) k k

N X 1 = k

and to calculate the capacity. Moreover, we can invoke (19) to calculate the relative gain, which we give in the following proposition. Proposition 2: The relative gain of a parallel channel with possibly dependent, Rayleigh fading subchannels with mean gains {µn } is bounded by ! 12 √ N µ ¯ N −1 2 X 2 2 RSISO ≤ + µ −µ ¯ (31) µmax µmax N n=1 n

N |A| (−1) 1X1 X =  2 P µ i=1 µ 1 1 + A∈2Si j∈A µ µ

k=1

= log N + C + O N −1



(37)

log µn

(39)

where C ≈ 0.577 is Euler’s constant, the next to last equality follows from [33, 0.155 4], and the final equality follows from an application of Euler’s summation formula. At high SNR, the capacity-optimal powers satisfy Pn⋆ = P/N for all n, as stated above. Using this power loading strategy along with (26) and the series representation of E1 (cf. [34, 5.1.11]), we arrive at the following standard highSNR approximate expression for the power-optimal ergodic capacity: ⋆ CE ≃ N log (SNR) + GSISO (38) where GSISO = −N C +

N X

n=1

is the excess rate. We will see in the next two sections that the asymptotic expression for capacity is similar for different channels based on Rayleigh fading, where the difference arises in the excess rate term. B. Selection Channels As mentioned in Section I (also, see [24], [25]), subchannel statistics differ dramatically in OFDM systems operating in channels with extremely wide bandwidths. One example of such a system can be found in ultrawideband (UWB) communications [35], [36]. Recent research in the area of UWB communications has shown that performing transmit antenna selection on a per-subcarrier basis in OFDM systems operating over very large bandwidths subject to equivalent isotropic radiated power (EIRP) constraints maximises the input-output mutual information in systems with multiple transmit antennas [37]. In such systems, antenna selection is performed by choosing to transmit using the antenna that corresponds to the channel with the maximum gain for each subcarrier. Thus, different antennas will transmit information on different frequencies, but only one antenna will be active on any given subcarrier. This approach can obviously be generalised for use with other parallel channels, in which case we denote each subchannel resulting from the selection process as a selection channel.

COON AND CEPEDA: POWER LOADING IN PARALLEL CHANNELS WITH CDI AT THE TRANSMITTER

We analyse the capacity of a parallel selection channel in which selection is performed over M independent, identically distributed (i.i.d.) channels for each subchannel. Suppose the parallel channel contains N such subchannels. Each of these subchannels experiences fading that is distributed according to the maximum of M independent Rayleigh distributed r.v.’s. Equivalently, the distribution of the power of a given subchannel is derived from the maximum of M exponentially distributed r.v.’s. Denoting the gain on the nth subchannel by γn , we have that γn =

max

i∈{1,...,M}

γn,i

(40)

where the density function of γn,i is given by (25). It follows that the density function of γn is given by [38] M−1 γ γ M  fn (γ) = 1 − e− µn e− µn . (41) µn

By expanding the binomial in (41), we can carry out the integration in (3) to yield the following expression for the ergodic capacity of this parallel channel (see, e.g., [16]):    N X M  X M iN0 i+1 PiNµ0 . (42) (−1) e n n E1 CE = Pn µn i n=1 i=1 At low SNR, we can apply (20) and (21) with m1,n = µn (ψ0 (M + 1) + C)

(43)

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all power to the subchannel that has the maximum gain (out of M N choices), while statistical waterfilling selects the subchannel where the mean gain m1,n is maximised. It follows from Proposition 3 and the relation M X  1 = log M + O M −1 ψ0 (M + 1) = −C + k

that

Rsel = O

! √ MN . log M

(49)

Consequently, the relative gain increases slowly with both the number of subcarriers and the number of selection branches. For the case where the subchannels are i.i.d., we can derive an approximation for Rsel in a similar manner as was presented in the proof of Corollary 1: Corollary 2: The relative gain of a parallel selection channel derived from i.i.d Rayleigh fading subchannels, each with a mean gain of µ, can be bounded by Rsel ≃ 1 +

log N log M + C

(50)

when N and M are large. At high SNR, we can use standard series representations of the E1 function to arrive at the following asymptotic expression for the power-optimal ergodic capacity: ⋆ CE ∼ N log (SNR) + Gsel

and   π2 m2,n = µ2n (ψ0 (M + 1) + C)2 − ψ1 (M + 1) + 6 (44) to obtain an approximation for the ergodic capacity, where ψk (x) is the polygamma function of order k [34]. Moreover, we can invoke (19) to calculate the relative gain, which we give in the following proposition. Proposition 3: The relative gain of a parallel selection channel with possibly dependent subchannels with mean gains given by (43) is bounded by µ ¯ µmax (ψ0 (M + 1) + C) ! 21 √ N 2 X 2 MN − 1 µ −µ ¯2 + . (45) µmax (ψ0 (M + 1) + C) N n=1 n

Rsel ≤

If the subchannels are independent, then the relative gain can be calculated exactly to yield

MN X 1 X (−1)|A| 1 2 µmax (ψ0 (M + 1) + C) i=1 µi (αi (A)) A∈2Si (46) where Si = {1, . . . , M N } r i, the set 2Si denotes the power set of Si , and X 1 1 + . (47) αi (A) = µi µj

Rsel =

j∈A

Proof: The proof is similar to that given in Appendix B, but noting that at low SNR, the optimal power allocation when full CSIT is available for all subchannels is to allocate

(48)

k=1

(51)

where Gsel = −N C +

N X

log µn + N

n=1 M  X

= GSISO + N

i=2

M i

 M  X M i=1



i

(−1)i log i.

i

(−1) log i (52)

It is interesting and insightful to study the power-optimal capacity as the number of diversity branches M of the selection channel grows large. In fact, such an analysis leads to an expression for the optimal power loading strategy for large M . This strategy is encompassed in the following proposition. Proposition 4: Consider a parallel selection channel distributed according to (41). As M → ∞, the optimal power loading strategy tends to statistical waterfilling, which is given by  + N0 w Pn = ν − (53) µn (ψ0 (M + 1) + C) P w where ν satisfies P = N n=1 Pn . Proof: See Appendix C. This result can be understood intuitively by noting that the mean channel gains increase monotonically with the number of diversity branches M (cf. (43)). Thus, the effective SNR on a given subchannel grows without bound as M → ∞, suggesting that an equal power allocation strategy is optimal, a condition that is satisfied by statistical waterfilling. In fact, Proposition 4 is stronger than it may first appear since it implies that for every ε > 0, there exists a number of diversity branches M such that the suboptimality of the statistical

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waterfilling capacity (measured appropriately) is less than ε. In other words, one may theoretically determine the number of diversity branches such that, when statistical waterfilling is used for power loading, the power-optimal ergodic capacity can be approached to an arbitrarily close degree. C. MRC Channels The third parallel fading channel that we analyse is the parallel MRC channel. As the name suggests, each subchannel of this parallel channel is viewed as the composition of several (say, M ) channels that are combined according to the MRC rule, i.e., the received signal on the ith branch is processed with a matched filter, and the outputs of the M branches are added together. For this analysis, we again assume that each constituent channel experiences Rayleigh fading that is identically distributed on a given subchannel and independent across subchannels. Thus, the composite channel gain γn for the nth subchannel is distributed according to a scaled χ22M r.v. The density function of γn is given by γ

γ M−1 e− µn fn (γ) = M , µn (M − 1)!

γ ≥ 0.

(54)

It can be shown that by carrying out the integration in (3) and using various relationships between the incomplete gamma function Γ (a, x) and the En -function Z ∞ −t e dt (55) En (x) = xn−1 tn x

one can obtain the following expression for the ergodic capacity of the MRC parallel channel:   N M−1 X N0 X N0 . (56) e Pn µn Ei+1 CE = Pn µn n=1 i=0

Note that the expression for CE presented here has a slightly different form than reported in [16]. Specifically, the form given here was chosen due to its similarity to the expressions for CE given above for SISO and selection channels. At low SNR, we can apply (20) and (21) with m1,n = µn M

(57)

m2,n = µ2n M (M + 1)

(58)

and to obtain an approximation for the power-optimal ergodic capacity. We can also invoke (19) to approximate the relative gain, which we give in the following proposition. Proposition 5: The relative gain of a parallel MRC channel with possibly dependent subchannels with mean gains {µn } is approximated by ! 21 √ N µ ¯ N −1 M +1 X 2 + µ −µ ¯2 . (59) RMRC ≤ µmax µmax M N n=1 n

Proof: The calculation is similar to that detailed in Appendix B. From Proposition 5, we see that r M +1 (N − 1) (60) RMRC ≤ 1 + M

and thus, as with the SISO and selection channels, the relative gain increases at most with the square root of the number of subchannels. Furthermore, the bound on the relative gain decreases with an increasing number of diversitypbranches, albeit slowly, eventually converging to RMRC ≤ 1+ (N − 1). At high SNR, the optimal power allocation strategy is Pn⋆ = P/N for all n. By employing the series expansion for En (cf. [34, 5.1.12]), we can calculate the following high-SNR approximation of the power-optimal ergodic capacity of the parallel MRC channel: ⋆ CE ∼ N log (SNR) + GMRC

(61)

GMRC = GSISO + N (C + ψ0 (M )) .

(62)

where As with the parallel selection channel, we can study the power-optimal capacity as the number of diversity branches M of the MRC channel grows large. Such an analysis leads to the following proposition, which is analogous to that presented above for the parallel selection channel. Proposition 6: Consider a parallel MRC channel distributed according to (54). As M → ∞, the optimal power loading strategy tends to moment-based waterfilling, which is given by +  N0 w Pn = ν − (63) µn M PN where ν satisfies P = n=1 Pnw . Proof: See Appendix D. IV. M EASURED C HANNELS As discussed above, one example of a parallel channel with subchannels that experience unequal mean fading gains can be found in OFDM-based UWB systems. In order to better understand these channels, channel measurement campaigns were conducted using a state-of-the-art time-domain multiantenna UWB channel sounder. A. Description of Sounding Equipment The sounder, manufactured by MEDAV, interrogates the propagation channel by using trains of pseudo noise (PN) sequences of 4,095 pulses or chips [28]. These are generated in baseband at a clock rate of 6.95 GHz and later up-converted, using the same clock, to cover the bandwidth from approximately 3.5 to 10.5 GHz. In turn, the receiver down-converts captured signals, does a periodic sub-sampling of them, and uses a phase shifter to allow the sampling of complex channel impulse responses (CIRs) in the time domain. Fig. 2 shows the configuration of the sounding equipment for measurements. Test signals are generated and bandpass filtered to avoid out-of-band emissions. The test signals are then amplified, transferred to a biconical antenna for radiation and, after travelling through the propagation environment, received by a similar antenna connected to a bandpass filter [39], [40]. After correcting the gain of the incoming signals with an automatic gain control (AGC) unit, the receiver uses an analogue-to-digital converter (ADC) and performs a matched filtering of the data, with the known PN sequence, in a digital

COON AND CEPEDA: POWER LOADING IN PARALLEL CHANNELS WITH CDI AT THE TRANSMITTER

DSP / control unit

MEDAV sounder

0 dBm +30 dB

Test signal generator AGC / ADC / DSP

3.5 – 10.5 GHz

Agilent 83020A

107

10 m long sucoflex cable Tx biconical antenna

T1

5 m long sucoflex cable 3.5 – 10.5 GHz

2 1

Rx biconical antenna

2

1

T2

Fig. 2.

Interconnection of sounding equipment.

signal processing (DSP) unit. Before sounding and after a warming-up period, the system response, phase imbalance and crosstalk are characterised using cabled or open connections. These measured parameters are then used for calibration to leave only the combined response of the antennas and the environment on the data recordings.

Concrete column Rx

2 1 4 3 21

4 3 21

Transmit array (T1, T2) Receive array (Rx)

B. Sounded Environment and Experimental Method UWB (2×4) MIMO channel measurements were conducted in an open-plan modern office environment in the central area of Bristol, UK. The sounding environment had typical scattering objects such as personal computers, liquid crystal displays (LCDs), non-metallic cubicle partitions, desks and metallic cabinets. The ceiling of the office was made of perforated metallic tiles and the floor supported by a metallic structure covered with non-metallic material. The dimensions of the sounded office are: 12.74 m wide, 30.84 m long and 2.39 m high. For this work, each recorded CIR results from averaging 256 captured CIRs in hardware. Under these conditions, the observation time of the system for a recorded CIR is 155 ms. The antennas were mounted on fibreglass masts at 1.3 m from the floor. The transmit antenna mast was attached to the movable part of an x − y automated positioning system [41]. The positioners and the sounder were remotely controlled, via Ethernet connection, to prevent human intervention in the area of measurements. In parallel, a spectrum analyser, connected to a biconical antenna and a low noise amplifier, periodically scanned the spectrum of interest. This information was sent via Ethernet to a remote computer to check the “health” of the test signal and the presence of interfering signals. Fig. 3 shows a diagram of the sounded environment. Two specific locations were selected to capture line-of-sight (LOS) and non-line-of-sight (NLOS) data. These locations, T1 (LOS) and T2 (NLOS), are grids of x−y points in which two transmit antennas, 25 cm apart, were displaced at distances of 3 cm. The receive antenna mast was positioned at a fixed location (Rx), and a four-element linear antenna array was formed by securing the first antenna and the following ones at 3, 6 and 12 cm from it. Each measurement grid had 441 (21 × 21) points, so a total of 3,528 (441 × 8) CIRs were recorded for each location T. In general, the distance between transmit and receive antennas ranges from approximately 7.02 m to 6.51 m for T1 and from 6.41 m to 5.77 m for T2. Note that NLOS conditions were achieved by locating the transmit antenna in

Fig. 3.

Illustration of sounded environment and antenna locations.

such a way that a concrete column was always shadowing it from the receive one. C. Post-processing and results We now analyse the measurements obtained in these campaigns to evaluate the theoretical analysis detailed in the preceding sections. For SISO analysis, we take the CIRs measured from the first transmit antenna to the first receive antenna. Selection channels are constructed by taking the CIRs from the two transmit antennas to the first receive antenna. Finally, MRC channels are formed by extracting the CIRs measured from the first transmit antenna to the receive antennas spaced apart by 12 cm distance. Each of these CIR is converted into the frequency domain, by using a discrete Fourier transform (DFT), and we retain the channel frequency response characteristics for the 5-to-6 GHz band. With a frequency spacing of roughly 1.7 MHz, this amounts to 588 frequency samples, i.e., channel frequency response coefficients. A power plot of three consecutive NLOS measurements in the 5.3-to-5.5 GHz band is illustrated in Fig. 4. From this figure, we can see that the measurements are fairly independent of one another. The channel measurements were normalised such that the average mean channel gain (in frequency) is one. Fig. 5 depicts the mean channel gain (averaged over the 441 available snapshots) for the LOS and NLOS channels. It is certainly clear from this figure that the mean fading gains vary considerably with frequency. Thus, one would expect an unequal power loading strategy based on the statistics of these channels to perform better than a balanced power allocation scheme. In Fig. 6, capacity results are illustrated for the measured LOS SISO, selection, and MRC channels. The corresponding results for the measured NLOS channels are shown in Fig.

SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY 10 Lower bound (Jensen) Upper bound (Jensen) Balanced power allocation

2.6 0 2.4 −10

2.2 MRC

2

−20 C/CAWGN

Power of channel frequency response (dB)

108

−30

−40 5.3

5.35

5.4 Frequency (GHz)

5.45

1.8 1.6

Selection

1.4

5.5

1.2

Fig. 4. Plot of the power of three consecutive channel frequency response measurements in the 5.3-to-5.5 GHz band.

1 SISO

0.8 0.6 −20

LOS

−15

−10

−5

0

Mean channel gain (dB)

4 2

Fig. 6.

5 SNR (dB)

10

15

20

25

30

Normalised capacity and bounds for measured LOS channels.

0 3

−2 −4

Lower bound (Jensen) Upper bound (Jensen) Balanced power allocation

5

5.1

5.2

5.3

5.4 5.5 5.6 Frequency (GHz)

5.7

5.8

5.9

6 2.5

NLOS MRC 2

2

C/CAWGN

Mean channel gain (dB)

4

0

Selection 1.5

−2 −4

5

5.1

5.2

5.3

5.4 5.5 5.6 Frequency (GHz)

5.7

5.8

5.9

6

1 SISO

Fig. 5. Mean channel gains for LOS and NLOS measured channels in the 5-to-6 GHz band.

0.5 −20

Fig. 7.

7. In these figures, the upper and lower bounds on capacity defined by (9) and (14) are depicted along with the capacity when a balanced power allocation is employed. The lower bound and the balanced power capacity are both averaged over the 441 snapshots of measured data to emulate the expectation in the ergodic capacity expression. Since a large range of SNR values are considered in these graphs, it is beneficial for ease of comparison to normalise these capacity results with respect to the capacity of a parallel additive white Gaussian noise (AWGN) channel with N = 588 subchannels, which is given by CAW GN = N log (1 + SNR) . (64) It is evident from Fig. 6 and Fig. 7 that statistical waterfilling is capable of providing significant gains at low SNR, which exemplifies typical operating conditions in wideband systems with a strict power budget, such as UWB. In this example, the optimal powers could not be computed for two reasons: 1) the exact fading distribution is not known, although it appears to closely resemble a Rayleigh distribution; 2) numerical inaccuracies arise, particularly at low SNR, when executing numerical optimisation algorithms and N is reasonably large. It should be noted, however, that small systems were studied (e.g., N = 8), and it was found that statistical waterfilling very closely resembles the optimal power distribution in the

−15

−10

−5

0

5 SNR (dB)

10

15

20

25

30

Normalised capacity and bounds for measured NLOS channels.

examined cases (results not shown). Moreover, the capacity resulting from the application of statistical waterfilling was very close to the power-optimal ergodic capacity in these examples. It is also useful to examine the maximum percent error (MPE) of the ergodic capacity, which is defined as MPE =

CUB − CLB × 100% CLB

(65)

where CUB and CLB are the upper and lower bounds given by (9) and (14), respectively. This metric quantifies the deviation of the statistical waterfilling power allocation strategy from the optimal power allocation since the power-optimal ergodic capacity lies between CLB and CUB . The MPE is illustrated for the LOS/NLOS SISO/MRC channels in Fig. 8. (The MPE for the selection channel is similar to the MRC channel, and is thus not shown.) The results shown in this figure suggest that the difference in the two bounds does, indeed, tend toward zero at high and low SNR. Moreover, this supports Proposition 6 since the MPE for the MRC channel is significantly lower than the SISO channel. Estimates of the relative gains for the measured LOS channels are depicted in Fig. 9. The results corresponding to

COON AND CEPEDA: POWER LOADING IN PARALLEL CHANNELS WITH CDI AT THE TRANSMITTER 20

109

300

LOS NLOS

18

200

SISO 16

100 Excess rate

14

MPE

12 10

0

−100

8 6

−200

MRC

SISO Selection MRC

4 −300

2 −20

Fig. 8.

−15

−10

−5

0

5 SNR (dB)

10

15

20

25

0

50

100

150

30

200 250 300 Number of subcarriers

350

400

450

500

Fig. 10. Estimates of the excess rate for the measured LOS SISO, selection, and MRC channels.

MPE for measured LOS and NLOS channels.

300

3.5

200

3

Excess rate

Relative gain

100

2.5

0

−100

2

−200

SISO Selection MRC 1.5

0

50

100

150

200 250 300 Number of subcarriers

350

400

450

SISO Selection MRC −300

0

50

100

150

200 250 300 Number of subcarriers

350

400

450

500

500

Fig. 9. Estimates of the relative gains for the measured LOS SISO, selection, and MRC channels.

the NLOS channels are very similar, and are thus not shown here. Fig. 9 illustrates the advantage that having knowledge of the full CSIT gives relative to having only CDIT. This advantage decreases as the number of diversity branches in the channel increases, with the MRC channel providing a lower relative gain than the selection channel. Moreover, Fig. 9 also supports the analysis given in Section III related to the growth of the relative gain with the number of subchannels N . In particular, it√is easy to see from this figure that growth is slower than N , which is the result given in (35), (49), and (60). For completeness, we provide results for the excess rate of the LOS and NLOS channels discussed above in Fig. 10 and Fig. 11, respectively. These figures show the improvement in capacity that is achieved with increasing N when diversity transmission/reception is employed. In contrast, SISO channels exhibit a degradation in the excess rate with increasing numbers of subchannels. It should be noted that although these results correspond to measured channels, it was found that theoretical channels based on Rayleigh fading subchannels yield very similar curves. One can conclude that

Fig. 11. Estimates of the excess rate for the measured NLOS SISO, selection, and MRC channels.

the distributions of the measured channels are very close to the theoretical distributions discussed in Section III. V. C ONCLUSIONS In this paper, we investigated the ergodic capacity of parallel channels and analysed the subchannel power allocation strategies that maximise this quantity under the assumption that perfect, instantaneous channel knowledge is available at the receiver whereas the transmitter only has knowledge of the distributions of the subchannels. We derived bounds on the ergodic capacity for general fading distributions, and showed that a power loading strategy based on statistical waterfilling is optimal in the low and high SNR regimes. We also studied three types of parallel fading channels: a single-input single-output channel, a selection channel, and a receive diversity channel with maximum ratio combining, all of which possessed Rayleigh fading constituent subchannels. We proved that, for the diversity channels, as the number of diversity branches M → ∞, the upper and lower bounds on capacity become tight, and the power loading that maximises mutual information follows the statistical waterfilling principle.

110

SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY

Finally, we utilised newly measured UWB channel data – which was obtained through the use of a state-of-the-art, time-domain, multiantenna UWB channel sounder – to provide practical results for systems employing statistical waterfilling and balanced power allocations. These results corroborated our theoretical analysis, and demonstrated that power loading using statistical waterfilling is effective in a range of practical scenarios. ACKNOWLEDGMENT The authors would like to thank the directors of the Telecommunications Research Laboratory, Toshiba Research Europe Ltd for their continued support. A PPENDIX A P ROOF OF P ROPOSITION 1 The proof of Proposition 1 is as follows. From (26), we can formulate the KKT conditions for optimality: N X

Pn⋆ ≥ 0, N0 2 (Pn⋆ )

µn

e

Pn⋆ = P,

νn ≥ 0,

n=1

N0 ⋆ µn Pn

E1



N0 Pn⋆ µn





νn Pn⋆ = 0

1 − νn + λ = 0 Pn⋆

(66)

for n = 1, . . . , N where νn is a Lagrange multiplier for the inequality constrainsP{Pn⋆ ≥ 0} and λ is a multiplier for the N ⋆ equality constraint n=1 Pn = P . Treating νn as a slack variable, we have   N0 N0 1 N0 ⋆ µn Pn , n = 1, . . . , N λ≥ ⋆ − e E1 2 Pn Pn⋆ µn (Pn⋆ ) µn (67) and !   N0 N N0 1 0 ⋆ ⋆ e Pn µn E1 Pn − ⋆ + λ = 0. (68) 2 Pn⋆ µn Pn (Pn⋆ ) µn It follows that the optimal power distribution satisfies the following system of equations !   N0 N N0 1 0 ⋆ ⋆ e Pn µn E1 Pn − ⋆ + λ = 0, 2 Pn⋆ µn Pn (Pn⋆ ) µn n = 1, . . . , N N −1 X

(69)

Pn⋆ = P.

Multiplying both sides of this equation by (Pn⋆ )2 µn /N0 and summing over {n : Pn⋆ > 0} yields the expression given for ⋆ in the theorem. CE ⋆ To see that CE is concave in {Pn⋆ }, first note that if λ = 0, ⋆ then CE is linear, and thus concave, in {Pn⋆ }: ⋆ CE =

that λ≥

µn >0 N0

n:Pn >0

since lim e

Pn⋆ →0

N0 ⋆ µn Pn

E1



N0 Pn⋆ µn



= 0.

Moreover, for Pn⋆ > 0, the KKT conditions state that   N0 N0 N0 1 ⋆ µn Pn e E − ⋆ + λ = 0. 1 2 ⋆µ ⋆ P P n (Pn ) µn n n

(72)

(73)



µn > 0.

(77)

For the general case where Pn⋆ 6= 0, the KKT conditions state that    N0 N0 1 N0 λ = ⋆ 1 − ⋆ e Pn⋆ µn E1 . (78) Pn Pn µn Pn⋆ µn Thus, it is sufficient to show that the following relation holds f (x) = xex E1 (x) ≤ 1,

0 < x < ∞.

(79)

To prove this inequality, we use the continued fraction (cf. [34, 5.1.22])   1 1 1 2 2 ··· (80) E1 (x) = e−x x+ 1+ x+ 1+ x+ from which it follows that   1 1 1 2 2 x f (x) = x ··· = x+ 1+ x+ 1+ x+ x + F (x)

(81)

where 0 < F (x) < 1 for 0 < x < ∞. Thus, f (x) ≤ 1. P ROOF

The multiplier λ is chosen such that the constraint is satisfied. Note that   X N0 N0 ⋆ µn ⋆ ⋆ ⋆ Pn E1 CE (P1 , . . . , PN ) = e (71) Pn⋆ µn ⋆

(74)

Thus, it suffices to consider the case where λ 6= 0. In this ⋆ case, the second partial derivative of CE with respect to Pn⋆ is negative for all n if λ is positive. From the KKT conditions, we have    N0 N0 1 N0 λ ≥ ⋆ 1 − ⋆ e Pn⋆ µn E1 , n = 1, . . . , N. Pn Pn µn Pn⋆ µn (75) If Pn⋆ → 0 (at the edge of the domain of the right-hand side of the inequality above), then it can be shown by using the following asymptotic expansion of E1 (cf. [34, 5.1.51])   6 2 1 e−x (76) 1 − + 2 − 3 + ··· E1 (x) ∼ x x x x

(70)

n=0

N 1 X µn Pn∗ . N0 n=1

A PPENDIX B OF P ROPOSITION 2

We begin by noting that as SNR → 0, the optimal power allocation strategy for the case where full CSIT is available is to allocate all transmit power to the channel exhibiting the highest instantaneous gain. Moreover, (19) implies that the optimal power allocation strategy for the case where only CDI is available at the transmitter is to allocate all transmit power to the channel with the highest mean gain. Thus, we can invoke the Taylor expansion of log (1 + x) to rewrite RSISO as  SNR·N · E [γmax ] + O SNR2 E [γmax ]  = RSISO = lim SNR→0 SNR·N · µmax + O SNR2 µmax (82)

COON AND CEPEDA: POWER LOADING IN PARALLEL CHANNELS WITH CDI AT THE TRANSMITTER

where γmax denotes the largest channel gain and µmax denotes the largest mean channel gain. Thus, we must calculate, or estimate, the expectation E [γmax ]. In the most general case, where the channel gains {γn } may be dependendent upon one another, it is difficult to calculate E [γmax ] directly. However, we can make use of the following theorem to obtain an upper bound. Theorem 1 ( [38, Th. 5.5.2 (Arnold and Groeneveld)]): Let Xn , for n = 1, . . . , N , be possibly dependent variates with E [Xn ] = µn and V [Xn ] = σn2 . Then for any constants {cn } " # N X  ¯ ≤ cn X(n) − µ E n=1 )1 (N N   2 X X 2 2 2 (µn − µ ¯) + σn (cn − c¯) (83) n=1

111

Now, we can write Z N  X x 1 ∞ − µx Y  1 − e− µn dx xe i µ i=1 i 0 n∈Si Z ∞ N X 1 X |A| = (−1) xe−αi (A)x dx µ i 0 S i=1

E [γmax ] =

A∈2

=

N X i=1

|A| 1 X (−1) 2 µi (αi (A)) A∈2Si

n=1

√ =µ ¯+ N −1

N 2 X 2 µ −µ ¯2 N n=1 n

! 21

(84)

which when combined with (82) leads to the result stated in (31). For the case where the subchannels are independent, the channel gains are independent, non-identically distributed (i.n.d.) exponential r.v.’s, the density and cumulative distribution functions of which are given by γ 1 − µγ fn (γ) = e n ⇔ Fn (γ) = 1 − e− µn . (85) µn The distribution function of the maximum of N i.n.d. r.v.’s is given by N Y F˜ (γ) = Fn (γ) (86)

αi (A) =

f˜ (γ) =

N X i=1

fi (γ)

Y

Fn (γ)

(87)

n∈Si

where Si = {1, . . . , N } r i. For our example of exponentially distributed r.v.’s with different means, we have N  X γ 1 − µγ Y  e i (88) 1 − e− µn . f˜ (γ) = µ i=1 i n∈Si

X 1 1 + . µi µj

(90)

j∈A

It follows that RSISO =

1 µmax

P ROOF

N X 1 X (−1)|A| 2. µ i=1 i A∈2Si (αi (A))

(91)

A PPENDIX C OF P ROPOSITION 4

We prove the proposition by showing that the upper and lower bounds given by (9) and (18) are asymptotically equivalent (as M → ∞). We first consider the upper and lower bounds on the power-optimal capacity of the nth subchannel. We assume the channel quality is good enough on this subchannel such that a nonzero power is allocated for transmission; this is a valid assumption since if the converse were true, the upper and lower bounds on capacity would, of course, both be zero. The cumulative distribution function of the maximum of M independent, exponentially distributed r.v.’s, each with mean µn , is M  . F˜ (x) = 1 − e−x/µn Now, letting βn = Pnw µn /N0 , where Pnw is dependent upon M , we can write (using (9) and (18))

n=1

which, upon taking the derivative, yields the following density of the maximum channel gain:

(89)

where 2Si is the power set of Si and

n=1

where µ ¯ and c¯ denote the arithmetic means of {µn } and {cn }, respectively, and X(n) is the nth ordered variate such that X(1) ≤ · · · ≤ X(n) ≤ · · · ≤ X(N ) . Due to the assumption that each subchannel experiences Rayleigh fading, we have that E [γn ] = µn and V [γn ] = σn2 = µ2n . Now, letting cn = δN −n where δi is the Kronecker delta function, we can invoke Theorem 1 to obtain the inequality 2 ! 12  N −1 N −1 + E [γmax ] ≤ µ ¯+ N2 N !1 N   2 X 2 (µn − µ ¯) + µ2n ×

i

1 ≥ lim

M→∞

 M   −1 an −βn (e −1) an 1 − 1 − e log (1 + βn (ψ0 (M + 1) + C))

.

(92)

For every M ∈ N, we can calculate βn for all n, and thus we can choose a constant c such that 0 < c < βn < ∞. For M large enough, we can define an = log (c log M ) > 0.

(93)

Following on from above, and utilising properties of continuity

112

SUBMITTED TO THE IEEE TRANSACTIONS ON INFORMATION THEORY

where 0 < α < 1. Following on from above, we can write

and limits, we can write

1 ≥ lim

 M   −1 −βn (c log M−1) log (c log M ) 1 − 1 − e

log (1 + βn αM ) Γ (M, αM ) · M→∞ log (1 + βn M ) Γ (M ) Γ (M, αM ) = lim M→∞ Γ (M ) M−1 X (αM )k = lim e−αM M→∞ k!

1 ≥ lim

log (1 + βn (ψ0 (M + 1) + C)) log log M + O (1) = lim M→∞ log (1 + βn (log M + O (1)))   −1 −1 M log (c log M ) 1 − M −cβn eβn − lim M→∞ log (1 + βn (log M + O (1))) !M −1 eβn = 1 − lim 1 − −1 M→∞ M cβn M→∞

=1

k=0

= lim e−αM eM−1 (αM ) M→∞ " = lim e

−αM

M→∞

= 1 − lim e−αM M→∞

(95)

and the last equality follows from the fact that  α M =0 1− γ M→∞ M lim

(96)

for any finite α > 0 and 0 < γ < 1. Since this relation holds for any n, it follows that the upper and lower bounds given by (9) and (18) are asymptotically equivalent, which concludes the proof.

(αM ) M!

We prove the proposition by showing that the upper and lower bounds given by (9) and (18) are asymptotically equivalent (as M → ∞). We first consider the upper and lower bounds on the power-optimal capacity of the nth subchannel. We assume the channel quality is good enough on this subchannel such that a nonzero power is allocated for transmission; this is a valid assumption since if the converse were true, the upper and lower bounds on capacity would, of course, both be zero. The cumulative distribution function of the MRC subchannel is   Γ M, µxn . (97) F˜ (x) = 1 − Γ (M ) Now, letting βn = Pnw µn /N0 , where Pnw is dependent upon M , we can write (using (9) and (18))  Γ M, βn−1 (ean − 1) an · . (98) 1 ≥ lim M→∞ log (1 + βn M ) Γ (M ) For every M ∈ N, we can calculate βn for all n. Thus, we can choose to define an = log (1 + βn αM ) > 0

(99)

(100)

where the first equality follows from properties of continuity, the second follows from [33, 8.352 2], the third follows from [34, 6.5.11] and the last equality follows from [34, 6.5.34]. Now, let M (αM ) uM = αM (101) e M! and consider the infinite series ∞ X

uM .

(102)

M=1

This series is convergent only if lim uM = 0.

M→∞

A PPENDIX D P ROOF OF P ROPOSITION 6

#

M

(94)

where we have used the asymptotic relation   1 ψ0 (M + 1) = log M + O M

M

(αM ) eM (αM ) − M!

(103)

Thus, if we can prove convergence of the series, then we can prove that the final limit in (100) is zero, and consequently that the upper and lower bounds for the capacity of the nth MRC subchannel are asymptotically equivalent. Using the ratio test, we have (α (M + 1))M+1 eαM M ! uM+1 = lim M→∞ (αM )M eα(M+1) (M + 1)! M→∞ uM M  1 −α = αe lim 1 + M→∞ M 1−α = αe < 1. (104) lim

Thus, the series converges, and the stated result follows. Since this relation holds for any n, we have shown that the upper and lower bounds given by (9) and (18) are asymptotically equivalent. R EFERENCES [1] S. B. Weinstein and P. M. Ebert, “Data transmission by frequencydivision multiplexing using the discrete Fourier transform,” IEEE Trans. Commun., vol. COM-19, no. 5, pp. 628–634, Oct. 1971. [2] R. van Nee and R. Prasad, OFDM for Wireless Multimedia Communications, 1st ed. Boston: Artech House, 2000. [3] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Europ. Trans. Telecommun., vol. 10, no. 6, pp. 585–595, Nov. 1999. [4] M. Lei, P. Zhang, H. Harada, and H. Wakana, “An adaptive power distribution algorithm for improving spectral efficiency in OFDM,” IEEE Trans. Broadcast., vol. 50, no. 3, pp. 347–351, Sep. 2004.

COON AND CEPEDA: POWER LOADING IN PARALLEL CHANNELS WITH CDI AT THE TRANSMITTER

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