As a background for understanding the rationale behind ECM, this section provides a brief overview of the application of SLM to a basic pilot-assisted OFDM system. Figure 1 shows a block diagram representation of a pilot-assisted SLM-OFDM transceiver. A description of the transmitter and receiver of conventional SLM-OFDM can be summarised as follows.
Transmitter
Consider an OFDM symbol block X of size N
v
. For 0≤k≤N
v
−1 where k represent the subcarrier index, X can be represented as
$$\begin{array}{*{20}l} \boldsymbol{X} &= \left[ \boldsymbol{X}[\!0] ~ \boldsymbol{X}[\!1] ~ \boldsymbol{X}[k] ~ \dots ~ \boldsymbol{X}\left[N_{v}-1\right] \right]. \end{array} $$
((1))
By letting X be a pilot-assisted OFDM symbol, let L represent the pilot spacing, i.e. the number of subcarriers between two consecutive pilots, then for an equi-spaced pilot structure, the total number of pilots, N
p
=N
v
/L. Hence, the number of data subcarriers, N
d
is N
v
−N
p
.
For an equi-spaced pilot arrangement, a contiguous set of subcarriers within X can be partitioned into a number of clusters as depicted in Fig. 2. Thus, as an example, X can be considered as an aggregate of N
c
clusters (of the same size) i.e.
$$\begin{array}{*{20}l} \boldsymbol{X} = \left\{ \boldsymbol{X}_{1} ~ \boldsymbol{X}_{2} ~ \boldsymbol{X}_{c} ~ \dots ~ \boldsymbol{X}_{N_{c}} \right\} \end{array} $$
((2))
where c for 1≤c≤N
c
represents the cluster index and each cluster is denoted by X
c
.
Using, for example, the cluster structure in Fig. 2, N
c
=N
p
/2, since each cluster consist of two pilots. Let W=2L be the cluster size, then for 0≤w≤W−1 where w is the cluster subcarrier index, each subcarrier in a given cluster X
c
is denoted by X
c
[ w]. Similar to [16], X
c
[ w] is expressed as
$$\begin{array}{*{20}l} \boldsymbol{X}_{c}[\!w] &= \boldsymbol{X}[\!cW + w] = \boldsymbol{X}[\!k] \\ &=\left\{ \begin{array}{l} \boldsymbol{X}_{c}[\!w_{e}] = \boldsymbol{X}[\!cW + w_{e}]\\ \boldsymbol{X}_{c}[\!w_{o}] = \boldsymbol{X}[\!cW + w_{o}]\\ \boldsymbol{X}_{c}[\!w_{d}] = \boldsymbol{X}[\!cW + w_{d}]\\ \end{array}\right. \end{array} $$
((3))
where w
e
and w
o
are the w-indices of the first and second pilots in each cluster, respectively (see Fig. 2). Similarly, w
d
is associated with data in each cluster. From the expression in (3), it can be seen that both unclustered, X[ k] and clustered, X
c
[ w] representations can be used interchangeably.
In polar coordinate form, X
c
[ w] may be expressed as
$$\begin{array}{*{20}l} \boldsymbol{X}_{c}[\!w] &= \boldsymbol{A}_{c}[\!w]\exp\left(j\theta_{c}[\!w]\right) \end{array} $$
((4))
where A
c
[ w] and θ
c
[ w] are respectively the amplitude and phase components of X
c
[ w].
Using an N-point inverse fast Fourier transform (IFFT) where N>N
v
, a time-domain signal x of size N is obtained from X. For 0≤n≤N−1, each time-domain signal sample, x[ n] is expressed through [17]
$$\begin{array}{*{20}l} \boldsymbol{x}[\!n] &= \frac{1}{N} \sum\limits_{k = 0}^{N_{v}-1} \boldsymbol{X}[\!k]\exp(j2\pi nk/N) \\ &= \text{IFFT}\{ \boldsymbol{X} \}, \end{array} $$
((5))
where IFFT {·} denotes the IFFT function. Finally, the length of the OFDM signal x is further extended by a cyclic prefix (CP) to mitigate channel fading, reduce inter-symbol interference (ISI) and facilitate the use of frequency-domain equalisation [18]. The PAPR of x is calculated from [19]
$$\begin{array}{*{20}l} \text{PAPR}\{\boldsymbol{x}\} &= \frac{\max \left\{ \vert \boldsymbol{x}\vert^{2} \right\}}{E\left\{ \vert\boldsymbol{x}\vert^{2}\right\}} \end{array} $$
((6))
where E{·} denotes expectation function. Note that the use of CP has no noticeable influence on the PAPR evaluations [20].
SLM
A detailed description of the well-known SLM PAPR reduction technique can be found in [7]. Using U different sequence vectors B
u[ k]= exp(j
α
u[ k]) for 1≤u≤U where α
u[ k]∈(0,π] represent positive valued phase sequence values, SLM generates U alternative OFDM signals and selects (for transmission) the modified signal with the lowest PAPR.
Let \(\bar {u}\) represent the u-index of the phase sequence vector that produced the lowest PAPR signal, \(\boldsymbol {x}^{\bar {u}}[\!n]\) is defined by [13]
$$\begin{array}{*{20}l} \boldsymbol{x}^{\bar{u}}[\!n] &= \text{IFFT}\left\{ \boldsymbol{X}[\!k] \boldsymbol{B}^{\bar{u}}[\!k] \right\}. \end{array} $$
((7))
From the expression in (7), it can be noted that in order to achieve successful data recovery, the value of \(\boldsymbol {B}^{\bar {u}}[\!k]\) or its u-index \(\bar {u}\) must be correctly known or determined at the receiver [21].
Receiver
As in a standard baseband OFDM receiver, all CP samples are first removed from the received signal before transforming the remaining signal samples into the frequency domain through a fast Fourier transform (FFT) to produce \(\boldsymbol {\bar {Y}}[\!k]\), which is given by [13]
$$\begin{array}{*{20}l} \boldsymbol{\bar{Y}}[\!k] = \boldsymbol{H}[\!k] \boldsymbol{X}[\!k] \boldsymbol{B}^{\bar{u}}[\!k] + \boldsymbol{V}[\!k]. \end{array} $$
((8))
The terms H[ k] and V[ k] respectively represent the channel gain and the independent and additive white Gaussian noise (AWGN) at the kth subcarrier. In a similar manner to X[ k] in (3), H[ k] and V[ k] can also be respectively represented in clustered forms as H
c
[ w] and V
c
[ w]. Thus, the expression for \(\boldsymbol {\bar {Y}}[\!k]\) can be re-written as
$$\begin{array}{*{20}l} \boldsymbol{\bar{Y}}_{c}[\!w] = \boldsymbol{H}_{c}[\!w]\boldsymbol{X}_{c}[\!w]\boldsymbol{B}^{\bar{u}}_{c}[\!w] + \boldsymbol{V}_{c}[\!w]. \end{array} $$
((9))
After the FFT, the next stage involves SLM de-mapping of \(\boldsymbol {\bar {Y}}[\!k]\). Normally, at this point, the value of SI must be determined.
FDC SI estimation
Let \(\hat {u}\) represent an estimate of the SI. Using the FDC-based SI estimation technique described in [13] and assuming all the U candidate SLM sequences B
u are known at the receiver, \(\hat {u}\) can be computed from
$$\begin{array}{*{20}l} \hat{u} &= \underset{u}{\arg\max~}Re \left\{ \boldsymbol{R}^{u} \right\} \end{array} $$
((10))
where R
u is the FDC function, computed from [13]
$$\begin{array}{*{20}l} \boldsymbol{R}^{u} &= \frac{1}{N_{p}-1} \sum_{p=1}^{N_{p}-1} \boldsymbol{\bar{H}}^{u}\left[p\right] \cdot \boldsymbol{\bar{H}}^{u}\left[p-1\right]^{*} \end{array} $$
((11))
where
$$ \boldsymbol{\bar{H}}^{u}\left[p\right] =\left. \left(\boldsymbol{\bar{Y}}\left[p\right] \boldsymbol{B}^{u}\left[p\right]^{*}\right) \right/ \boldsymbol{X}\left[p\right] $$
((12))
and where ∗ is the complex conjugation operator and p for 0≤p≤N
p
−1 represents the pilot indices. From the study in [16], the FDC method was shown to require a total of 2 U
N
p
−U complex multiplications (CMs) and U(N
p
−2) complex additions (CAs).
SLM de-mapping
Using the SI estimate \(\hat {u}\) and if \(\hat {u} = \bar {u}\), a variable denoted by Y
c
[ w] (free of the SLM term \(\boldsymbol {B}^{\bar {u}}_{c}[\!w]\)) is obtained from
$$\begin{array}{*{20}l} \boldsymbol{Y}_{c}[\!w] &= \boldsymbol{\bar{Y}}_{c}[\!w] \boldsymbol{B}^{\hat{u}}_{c}[\!w]^{*} \\ &=\left\{ \begin{array}{l} \boldsymbol{Y}_{c}[\!w_{e}] = \boldsymbol{H}_{c}[\!w_{e}]\boldsymbol{X}_{c}[\!w_{e}] + \boldsymbol{V^{'}}_{c}[\!w_{e}], \\ \boldsymbol{Y}_{c}[\!w_{o}] = \boldsymbol{H}_{c}[\!w_{o}]\boldsymbol{X}_{c}[\!w_{o}] + \boldsymbol{V^{'}}_{c}[\!w_{o}], \\ \boldsymbol{Y}_{c}[\!w_{d}] = \boldsymbol{H}_{c}[\!w_{d}]\boldsymbol{X}_{c}[\!w_{d}] + \boldsymbol{V^{'}}_{c}[\!w_{d}], \end{array}\right. \end{array} $$
((13))
where \(\boldsymbol {V^{'}}_{c}[\!w] = \boldsymbol {V}_{c}[\!w] \boldsymbol {B}^{\hat {u}}_{c}[\!w]^{*}\). After SLM de-mapping, the next stage involves channel estimation and channel equalisation.
Channel estimation and equalisation
In practical systems, it is usually assumed that the transmitted pilots are known at the receiver. Using, for example, a least squares (LS) channel estimation method, sub-channel estimates \(\boldsymbol {\hat {H}}_{c}[\!w_{e}]\) and \(\boldsymbol {\hat {H}}_{c}[\!w_{o}]\) can be computed through [22]
$$\begin{array}{*{20}l} \boldsymbol{\hat{H}}_{c}[\!w_{e}] &= \boldsymbol{Y}_{c}[\!w_{e}] \Big/ \boldsymbol{X}_{c}[\!w_{e}] \\ &= \boldsymbol{H}_{c}[\!w_{e}] + \Big(\boldsymbol{V^{'}}_{c}[\!w_{e}] \Big/ \boldsymbol{X}_{c}[\!w_{e}]\Big), \end{array} $$
((14))
$$\begin{array}{*{20}l} \boldsymbol{\hat{H}}_{c}[\!w_{o}] &= \boldsymbol{Y}_{c}[\!w_{o}] \Big/ \boldsymbol{X}_{c}[\!w_{o}] \\ &= \boldsymbol{H}_{c}[\!w_{o}] + \Big(\boldsymbol{V^{'}}_{c}[\!w_{o}] \Big/ \boldsymbol{X}_{c}[\!w_{o}]\Big). \end{array} $$
((15))
At high signal-to-noise ratio (SNR), the additive noise terms in (15) become negligible and as a result, the expressions for \(\boldsymbol {\hat {H}}_{c}[\!w_{e}]\) and \(\boldsymbol {\hat {H}}_{c}[\!w_{o}]\) are reduced to
$$\begin{array}{*{20}l} \boldsymbol{\hat{H}}_{c}[\!w_{e}] &\approx \boldsymbol{H}_{c}[\!w_{e}] \text{~~and~~} \boldsymbol{\hat{H}}_{c}[\!w_{o}] \approx \boldsymbol{H}_{c}[\!w_{o}]. \end{array} $$
((16))
Through, for example, a linear interpolation between \(\boldsymbol {\hat {H}}_{c}[\!w_{e}]\) and \(\boldsymbol {\hat {H}}_{c}[\!w_{o}]\), data sub-channel estimate \(\boldsymbol {\hat {H}}_{c}[\!w_{d}]\) is obtained. Using the data sub-channel estimate, a channel equalised data term \(\boldsymbol {\hat {Y}}_{c}[\!w_{d}]\) is obtained from
$$\begin{array}{*{20}l} \boldsymbol{\hat{Y}}_{c}[\!w_{d}] &= \boldsymbol{Y}_{c}[\!w_{d}] \Big/ \boldsymbol{\hat{H}}_{c}[\!w_{d}] \\ &\approx \boldsymbol{X}_{c}[\!w_{d}] \frac{\boldsymbol{H}_{c}[\!w_{d}]}{\boldsymbol{\hat{H}}_{c}[\!w_{d}]} + \Big(\boldsymbol{V^{'}}_{c}[\!w_{d}] \Big/ \boldsymbol{\hat{H}}_{c}[\!w_{d}]\Big). \end{array} $$
((17))
From the expression in (17), it can be noted at high SNR and in the absence of noise enhancement due to the term \(\left (\boldsymbol {V^{'}}_{c}[\!w_{d}] \Big / \boldsymbol {\hat {H}}_{c}[\!w_{d}]\right)\), the use of a standard quadrature amplitude modulation (QAM) demodulation (based on minimum Euclidean distance) produces an estimate of the transmitted data using [16]
$$\begin{array}{*{20}l} \boldsymbol{\hat{X}}_{c}[\!w_{d}] &= \underset{\mathcal{C}_{q} ~\in~\mathbb{Q}}{\min} \left|\boldsymbol{\hat{Y}}_{c}[\!w_{d}] - \mathcal{C}_{q}\right|^{2} \end{array} $$
((18))
where \(\mathbb {Q}\) is a set of Q constellation points \(\mathcal {C}_{q}\) for 1≤q≤Q such that \(\boldsymbol {\hat {X}}_{c}[\!w_{d}] \in \mathbb {Q}\).