A C Van Bochove

P T De Boer

R O Taniman

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Bitloading with reduced computational complexity order
for a multicarrier multimode PON
R.O. Taniman1, A.C. van Bochove, and P.T. de Boer
1 University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands
r.o.taniman@ewi.utwente.nl
Nowadays, multicarrier transmission technique, e.g., optical OFDM, receives quite
intense attention in the optical communications and networking field due to its potential
to provide dispersion compensation. Adaptive modulation level is commonly and
naturally employed in relation to multicarrier transmission, provided (sub)channel state
information is available. The modulation level adaptations usually done by a bitloading
mechanism. In our work, given some modulation formats that are o be used, we attempt
to reduce the computational complexity order of the bitloading algorithm from the one
that has been presented in the literature. The bitloading alorithm is applied to a
multicarrier multimode PON.
Introduction
Multicarrier transmission technique offers a potential topr vide dispersion compensation
from a rather different perspective where signal distortion is mitigated by way of parallel
narrowband transmissions. This technique is beneficial fora multimode PON which
has to offer a system capacity in the order of Gbps and cover few kilometers distances.
Especially in the case of optical OFDM where some inherent digital signal processing
is used, it’s quite straightforward to use an adaptive modulation scheme: temporally
adapting the modulation level of each subcarrier to optimize the system capacity, subject
to a certain prescribed probability of bit error, based on the (sub)channel state information.
In our multicarrier multimode PON, the modulation level adaptation, as a part of the
dynamic capacity allocation [1], is determined by the OLT where it is assumed that the
(sub)channel state information is available to the OLT.
Choosing the modulation level of each subcarrier is usually ca led bitloading. In [2],
an optimal algorithmic bitloading was presented. The complexity of this bitloading,
assuming the use of linear search, isO
(
N2K
)
, whereN is the total number of subcarriers
andK is the maximum allowable number of bits per symbol. The bitloading is usually
used as a part of a heuristic dynamic capacity allocation algorithm. In this case, a low-
order algorithmic complexity is required as the algorithm is to be used in a real-time
system (e.g., adaptive to (sub)channel condition and/or traffic load). Indeed some paper
authors, e.g., [3, 4] became reluctant to employ an adaptiveoptimal bitloading scheme
due to its algorithmic complexity, hence preventing them tofurther optimize the system
capacity. In this paper, we describe our attempt to reduce the complexity of the bitloading
algorithm, especially as a part of our dynamic capacity alloc ti n algorithm as described
in [1].
Proceedings Symposium IEEE/LEOS Benelux Chapter, 2007, Brussels
171
Adapting the bitloading algorithm
The optimal bitloading algorithm (Algorithm MinCost) given in [2] (see Algorithm
1) is of the cost minimization type (given a certain total number of bitsbfixed to be loaded).
TheAlgorithm MinCost in fact works by using a greedy approach, i.e. loadingbfixed
bits one by one, based on the least additional cost∆c j,k ≡ c j,k+1− c j,k at each step (c j,k
is the cost to loadk bits onto subcarrierj andc j,0 ≡ 0 ∀ j, meaningc j,K = ∑K−1k=0 ∆c j,k).
Note thatAlgorithm MinCost requires∆c j,k be strictly increasing ink. For our
multicarrier multimode PON we instead would like to maximize the number of loaded
bits while not exceeding a certain costcfixed (Problem MaxBit). Nevertheless, it is
possible to employAlgorithm MinCost in order to find a solution toProblem
MaxBit, givencfixed, by increasing thebfixed one by one while the resultingc∗ ≤ cfixed
until the state after which increasing thebfixed one bit more would mean that the resulting
c∗ > cfixed. We then haveb∗ = bfixed of this state.
Algorithm 1 Algorithm MinCost
Require: ∆c j (b j) = [c j (b j +1)−c j (b j)]; c = 0 and for all j, let b j = 0
1: while ∑ j b j < bfixed do
2: j∗← argmin
j
∆c j (b j)
3: c← c+∆c j∗ (b j∗)
4: b j∗ ← b j∗ +1
5: end while
Is the solution found using (repeated applications of) theAlgorithm MinCost for
theProblem MaxBit optimal? We establish the optimality of this solution as follows:
Suppose there is anotherAlgorithm alt that can loadbalt > b∗ with cbalt ≤ cfixed.
UsingAlgorithm MinCost with balt bits to be loaded would result inc∗balt > cfixed.
However, the optimality ofAlgorithm MinCost dictates that givenbfixed = balt, no
other algorithms can satisfy / result inc < c∗balt, meaning,c≥ c∗balt must hold which then
implies thatc > cfixed. BecauseAlgorithm alt is among these other algorithms, it
implies thatcbalt > cfixed too: a contradiction. Hence, it can be established that repeat d
applications ofAlgorithm MinCost results in an optimal solution toProblem
MaxBit.
Algorithm 2 Algorithm MaxBit
Require: ∆c j (b j) = [c j (b j +1)−c j (b j)]; c = 0 and for all j, let b j = 0
1: while truedo
2: j∗← argmin
j
∆c j (b j)
3: c← c+∆c j∗ (b j∗)
4: if c > cfixed then
5: break{finish the while loop}
6: else
7: b j∗ ← b j∗ +1
8: end if
9: end while
Exploiting the greedy approach ofAlgorithm MinCost, modifying Algorithm
MinCost in order to find an optimal solution toProblem MaxBit can be quite
straightforward, namely, by testing whether (c∗b ≤ cfixed and) c∗b+1 > cfixed for b =
0,1,2, . . . increasingly (wherec∗b is the minimum cost to loadb bits). The value ofb
that makes the test return true is then the value ofb∗, i.e. the maximum total number
Bitloading with reduced computational complexity order for a multicarrier multimode . . .
172
of bits that can be loaded given a fixed cost ofcfixed. We call the resulting algorithm:
Algorithm MaxBit (see Algorithm 2).
Reducing the complexity order of the bitloading algorithm
Due to line 7 (loading additional bit) in Algorithm 2, the (linear) search for the minimum
additional cost at line 2 must be repeated every time a bit is loaded. This repetition renders
the complexity order rather high, namely,O
(
N2K
)
. in order to reduce the complexity
order, we proposeAlgorithm MaxBit2 (see Algorithm 3) which employs a sorting
in ascending order of the additional costs∆c j,k that needs to be performed only once. To
enable the sorting, the∆c j,k should be precalculated for allj andk. Subsequent necessary
operations for the bitloading are given at line 2-10.
Algorithm 3 Algorithm MaxBit2
Require: arraysA j (containing data pairs:subcarrier andcost); i = 1; c = 0 and for all j, let b j = 0
1: construct the sorted arrayA usingcost fields as the keys
2: while truedo
3: j∗← A[i].subcarrier
4: c← c+A[i].cost
5: if c > cfixed then
6: break{finish the while loop}
7: else
8: b j∗ ← b j∗ +1
9: i← i +1
10: end if
11: end while
Algorithm MaxBit2 for the bitloading works correctly if∆c j,k is strictly increas-
ing with k. This requirement comes fromAlgorithm MinCost as the basis
for Algorithm MaxBit2. By using the modulation formats in Table 1 for the
adaptive modulation scheme and following the common practice to use square signal
constellations, one can satisfy this requirement (note that the bits are consequently loaded
per 2 bits and∆c j,k ≡ c j,k+2− c j,k with k is even). From the table, ˆak is the signal
amplitude of an outermost point in the signal constellations which is proportional to the
average energy of this constellation point. The last row of the table is a trick to ensure
that no more than 8 bits can be loaded onto a subcarrier. For our specific system, we set
c j,k = g j · âk, whereg j is the overall gain of subchannelj, in order to avoid clipping.
TABLE 1: The modulation formats and their associ-
atedâk.
k bits/symbol modulation format âk
0 - 0
2 QPSK 1
4 16-QAM 3
6 64-QAM 7
8 256-QAM 15
10 - ∞
0 2 4 6
0
2
4
6
8
k
∆c
j,
k
FIGURE 1: ∆c j,k with g j = 1 andK = 8
Because arrayA consists ofNK elements, using an ordinary efficient sorting results
in O (NK lg(NK)) complexity order forAlgorithm MaxBit2. This is already an
improvement fromO
(
N2K
)
(whenN is large enough andK is small - this is a typical
Proceedings Symposium IEEE/LEOS Benelux Chapter, 2007, Brussels
173
system configuration). But by exploiting the fact that∆c j,k is strictly increasing withk,
we can further reduce the resulting complexity order. First, we put∆c j,k into subarrays
A j for each j. Then, by using the merging process of the well-known merge sort ( ee
e.g., [5]), we construct the sorted arrayA. The trick is to start the merging process from
arraysA j rather than from individual elements because eachA j is already naturally sorted.
As the merging process can be illustrated as a binary tree (seFigure 2 as an example),
the number of merging stages can be found to bedlg(N)e. In each stage, at mostNK
comparisons need to be done. Now, it becomes clear that the resulting complexity
order of this peculiar sorting isO (NK lg(N)) which becomes the complexity order of
Algorithm MaxBit2 as it is dominated by the complexity order of the sorting part
(at line 1).
1 1 2 2 2 3 4 4 4 6 8 12
A
2 4 81 2 4
A1 A2
1 2 4 3 6 12
A3 A4
1 2 2 4 4 8 1 2 3 4 6 12
A12 A34
FIGURE 2: A sorting example withN = 4 andK = 3
Conclusion
We have shown that through some modifications, we were able tor duce the compu-
tational complexity order of the bitloading algorithm fromO
(
N2K
)
as listed in the
literature toO (NK lg(N)). This complexity reduction can lend support to the use of the
bitloading algorithm to further increase the system capacity of a multicarrier multimode
PON.
Acknowledgements
This work is supported by the Dutch Ministry of Economic Affairs in the IOP GenCom
project ”Full Service Access Networks using Multimode Fibre”. The authors would also
like to thank Dr. Q. Yu for very useful discussions.
References
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Berlin, Germany, Sep. 2007, vol. 2, pp. 55-56.
[2] C.Y. Wong, R.S. Cheng, K.B. Letaief and R. Murch, ”Multiuser OFDM with adaptive subcarrier,
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[3] D. Kivanc, G. Li and H. Liu. ”Computationally efficient bandwidth allocation and power control for
OFDMA,” IEEE Trans. Wireless Communications, vol. 2, no. 6, pp. 1150-1158, Nov. 2003.
[4] J. Gross, H. Geerdes, H. Karl and A. Wolisz, ”PerformanceAnalysis of Dynamic OFDMA Systems
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[5] D.E. Knuth, The Art of Computer Programming vol. 3, 2nd ed., Reading, Massachusetts: Addison-
Wesley, 1998, ch. 5, pp. 158-168.
Bitloading with reduced computational complexity order for a multicarrier multimode . . .
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Bitloading with reduced computational complexity order for a multicarrier multimode PON

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Bitloading with reduced computational complexity order for a multicarrier multimode PON

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