We consider a multiuser OFDM system with one transmitter, *K* users and *N* subcarriers. Perfect channel knowledge is assumed at the transmitter, where resource allocation is executed. A resource allocation scheme is effective for *L* OFDM symbols, while *L* is determined by temporal channel variation and the time duration of one OFDM symbol. In a fast time-varying environment *L* cannot be large. For example, one frame in the worldwide interoperability for microwave access (WiMAX) [13] is composed of 48 OFDM symbols. The resource allocation scheme is updated for every *L* OFDM symbols via the signalling overhead. There are 2^{
M
}data rates that may be allocated to each subcarrier. It means that *M* bits are necessary to identify one discrete rate. According to [2], if the water filling strategy is used, *NM* bits are required for expressing one resource allocation scheme and sent at first. Thereafter, data symbols follow. Note that *N*⌈ log_{2}(*K*)⌉ bits are always needed to notify each receiver of which subcarriers are assigned to it. This amount is constant in this study.

We aim at maximizing the weighted sum rate subject to limited transmission power and individual minimum rate required by users. When the water filling strategy is adopted, it is stated as

$\begin{array}{ll}\text{maximize}& R=\sum _{k=1}^{K}{w}_{k}\sum _{n\in {\mathcal{S}}_{k}}{r}_{k,n}\\ \text{subject to}& {r}_{k,n}={\text{log}}_{2}(1+{p}_{k,n}{g}_{k,n})+\frac{M}{L},\\ k=1,\dots ,& K,n=1,\dots ,N\\ \sum _{n\in {\mathcal{S}}_{k}}{r}_{k,n}\ge {R}_{k},k=1,\dots ,K\\ \sum _{k=1}^{K}\sum _{n\in {\mathcal{S}}_{k}}{p}_{k,n}\le P\\ {\mathcal{S}}_{k}\cap {\mathcal{S}}_{l}=\varnothing ,k,l=1,\dots ,K,k\ne l\\ {p}_{k,n}\ge 0,k=1,\dots ,K,n=1,\dots ,\mathrm{N.}\end{array}$

(1)

The non-negative power and rate allocated to subcarrier *n* for user *k* is denoted by *p*_{k,n} and *r*_{k,n}, respectively. They are related by the equality constraint, where *g*_{k,n}is the CNR of subcarrier *n* of user *k*. Once subcarrier *n* is assigned to a user, *M* bits must be sent at first for every *L* OFDM symbols. The average rate for signalling over each subcarrier is *M*/*L*. The achieved rate for user *k* is weighted by positive *w*_{
k
}, which is given by the system for the purpose of fairness control among users. Each user requires a minimum rate *R*_{
k
}, expressed by the second constraint. The transmission power is limited to *P*, illustrated by the third constraint. The set of users is referred to as $\mathcal{K}$. The set ${\mathcal{S}}_{k}$ contains the subcarriers assigned to user *k*. We denote the cardinality of ${\mathcal{S}}_{k}$ by *s*_{
k
}. One subcarrier is assigned to at most one user at any specific time to avoid interference to each other, illustrated by the last constraint in (1). The variables of problem (1) are *r*_{k,n}, *p*_{k,n} and ${\mathcal{S}}_{k},k=1,\dots ,K,n=1,\dots ,\mathrm{N.}$

The dual optimum of (1) can be obtained by the dual method [10]. The Karush-Kuhn-Tucker conditions [14] are applied to (1), where *K* + 1 dual variables appear. Then, the ellipsoid method is used to let these dual variables iteratively converge. The number of iterations is related to $\mathcal{O}\left({(K+1)}^{2}\right)$[8, 15]. In each iteration (*K* + 1)*N* equations must be calculated. Hence, the complexity for determining the dual optimum of (1) is $\mathcal{O}\left(N{K}^{3}\right)$. The concrete solution is available in [11]. This can be viewed as the extension from single-user water filling to multiuser water filling, where different rates and powers are allocated to subcarriers to meet transmission constraints.

To notify receivers of the employed resource allocation scheme with a smaller signalling overhead, the same rate

*r*_{
k
} may be allocated to the subcarriers assigned to user

*k*=1,…,

*K*. Then, only

*KM* bits are sufficient to distinguish data rates for all subcarriers. The signalling overhead is reduced by (

*N*−

*K*)

*M* bits. The resource allocation problem (1) can be rewritten as

$\begin{array}{ll}\text{maximize}& \stackrel{\u0304}{R}=\sum _{k=1}^{K}{w}_{k}{s}_{k}{r}_{k}\\ \text{subject to}& {r}_{k}={\text{log}}_{2}(1+{p}_{k,n}{g}_{k,n}),\\ k=1,\dots ,& K,n=1,\dots ,N\\ {s}_{k}{r}_{k}\ge {R}_{k}+\frac{M}{L},k=1,\dots ,K\\ \sum _{k=1}^{K}\sum _{n\in {\mathcal{S}}_{k}}{p}_{k,n}\le P\\ {\mathcal{S}}_{k}\cap {\mathcal{S}}_{l}=\varnothing ,k,l=1,\dots ,K,k\ne l\\ {p}_{k,n}\ge 0,k=1,\dots ,K,n=1,\dots ,\mathrm{N.}\end{array}$

(2)

In (1), *M* bits must be sent for each *subcarrier* to identify the employed rate in the current *L* OFDM symbols. Thus, on average *M*/*L* bits/OFDM symbols must be additionally achieved over each *subcarrier*. In (2), only *M* bits of signaling overhead is necessary for each *user*. Hence, on average *M*/*L* bits/OFDM symbols must be additionally achieved for each *user*. Thus, each minimum required rate is increased by *M*/*L* for signalling. In the following, we first extract two classical single-user resource allocation problems to quantify the instantaneous per-symbol performance loss of the proposed strategy. Thereafter, a heuristic method is designed to solve (2). The variables of problem (1) are *r*_{
k
},*p*_{k,n} and ${\mathcal{S}}_{k},k=1,\dots ,K,n=1,\dots ,\mathrm{N.}$