05-03-2013, 01:00 PM
Codes, Correlations and Power Control in OFDM
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ABSTRACT
Practical communications engineering is continuously
producing problems in interest to the coding theory
community. A recent example is the power-control problem
in Orthogonal Frequency Division Multiplexing (OFDM).
We report recent work which gives a mathematical
framework for generating solutions to this notorious problem
that are suited to low-cost wireless applications. The key
result is a connection between Golay complementary
sequences and Reed-Muller codes. The former are almost
ideal for OFDM transmissions because they have a very low
peak-to-mean envelope power ratio (PMEPR), while the
latter have efficient encoding and decoding algorithms and
good error correction capability. This result is then
generalised in two ways. Firstly we study polyphase Golay
sequences, motivating the introduction of non-binary
generalisations of the Reed-Muller codes.
Introduction
Orthogonal frequency division multiplexing (OFDM) is a method of transmitting data simul-
taneously over multiple equally-spaced carrier frequencies, using Fourier transform processing
for modulation and demodulation [1]. The method has been proposed or adopted for many
types of radio systems such as wireless Local Area Networks [2] and digital audio and digi-
tal video broadcasting [3]. OFDM oers many well-documented advantages for multicarrier
transmission at high data rates, particularly in mobile applications.
The principal diculty with OFDM is that when the sinusoidal signals of the n carriers add
mostly constructively the peak envelope power is as much as n times the mean envelope power.
If the peak envelope power is subject to a design or regulatory limit then this has the eect
of reducing the mean envelope power allowed under OFDM relative to that allowed under
constant envelope modulation. This reduces the eective range of the OFDM transmissions
and is particularly acute in mobile applications where battery power is a constraint. Moreover,
to prevent signal distortions and spectral growth due to non-linearities inherent in electronic
components, power ampliers must be operated below their compression point where power is
converted most eciently. This results in more expensive and ineciently used components.
OFDM Codes
In this section we sketch how the theory developed above can be used to construct OFDM
codes. Full details can be found in [5, 6]. We concentrate on codes in which the alphabet
size 2h is equal to 2t for t = 1;2; 3. We also focus on codes with n = 16 or n = 32 carriers.
These parameter choices are the most important for low-cost applications of OFDM (such as
mobile wireless applications).
In contrast to classical coding theory, where the two parameters of fundamental impor-
tance are rate and (normalised) minimum distance, we have a third parameter, the PMEPR
of the code. We dene this to be the maximum of the PMEPRs of all the codewords in the
code. We also dene the rate of a length n OFDM code C over Z
2t to be log2 jCj=(nt). The
denominator here expresses the maximum number of bits that can be transmitted per OFDM
symbol using uncoded 2t-PSK modulation on n carriers, while the numerator is equal to the
number of information bits encoded by C.
Binary Coding Options
As a simple illustration of the kinds of coding options available, we consider the case of n = 16
carriers. By taking a single `Golay coset' identied by Corollary 6.1 in the case m = 4, we
get a binary, length 16 code with rate 0.31, minimum Hamming distance 8 and a PMEPR of
2. Using instead 8 of the 12 `Golay cosets', we obtain a code still having a PMEPR of 2, but
with an increased rate of 0.50 and decreased minimum Hamming distance of 4. A compromise
option can be obtained using four out of the six cosets identied by Corollary 6.1 that lie in
the Kerdock code of length 16 [7].
Decoding Algorithms
In this section we outline decoding algorithms for codes of the type described in Section 7.
One possible rst step in obtaining a decoding algorithm for such a code is to apply an
appropriate generalisation of the supercode decoding method [24]. This method applies to
codes that are the union of cosets of a binary code C; the basic idea is subtract in turn
each possible coset representative from the received codeword and to decode the result as a
codeword of C, the best decoding result over all cosets determining the coset representative.
Applying the supercode method reduces our decoding problem to that of nding an e-
cient decoding algorithm for RM2h(1;m). We report on two distinct approaches to this prob-
lem. The rst approach, reported in [25], is a natural generalisation of the fast Hadamard
transform (FHT) algorithm for decoding the binary rst-order Reed-Muller code RM(1;m).
It is a maximum-likelihood soft-decision algorithm that works in the Euclidean domain: it
operates on the complex vector y obtained by applying an inverse fast Fourier transform to
the sampled, received signal. This signal is in turn a noise-corrupted version of the transmit-
ted signal modelled by the real part of (1). The second approach, developed in [5], works in
the `coding' domain: it has as input a vector containing just the phase information in the
components of y. It results in a decoder for RM2t(1;m) with respect to both Hamming and
Lee distance and uses t real-number FHTs and some additional computation.