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*=2*L* 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 *k*th 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 \(\boldsymbol {\hat {X}}_{c}[\!w_{d}] \in \mathbb {Q}\).