On the Phase Sequences and Permutation Functions in the SLM Scheme for OFDM-IM Systems

In orthogonal frequency division multiplexing with index modulation (OFDM-IM), the active subcarriers can convey information bits by modulated symbols as well as their indices. OFDM-IM has attracted a great deal of attention from researchers by virtue of high energy efficiency. Unfortunately, OFDM-IM has inherited large peak-to-average power ratio (PAPR) of the classical OFDM signal, but there are few works dealing with it. Selected mapping (SLM) is a promising PAPR reduction technique that is distortionless, has good PAPR reducing capability, and can be easily adapted in OFDM-IM systems. In SLM for OFDM-IM systems, the phase sequences rotate the OFDM-IM block in the frequency domain before the inverse fast Fourier transform (IFFT) is applied. Also, in the recent work, permutation procedure in the frequency domain has been proposed before multiplication of phase sequences in SLM, which is newly introduced in OFDM-IM regime. Unfortunately, the theoretical investigation on the efficiency of the permutation in SLM is insufficient. In this paper, we investigate the phase sequence set (PSS) and permutation functions in the SLM scheme for OFDM-IM systems; First, the efficiency of the permutation is analyzed as a function of the number of active subcarriers. Second, the optimal conditions for the PSS and permutation functions are derived, which is verified through simulations.

SECTION I. Introduction

Index modulation (IM) is an emerging technique that exploits indices of active resources (such as spatial, time, and frequency resources) to convey information in addition to conventional $M$ -ary modulation symbols. The IM concept is applied to the orthogonal frequency division multiplexing (OFDM) [1], [2], which results in a novel scheme termed as OFDM with index modulation (OFDM-IM). Specifically, OFDM-IM activates only a subset of sub-carriers to carry data bits via both $M$ -ary complex data symbols and active sub-carrier indices. Thus, OFDM-IM achieves higher energy efficiency and reliability than the classical OFDM especially when it operates at a low bit rate. Therefore, OFDM-IM has attracted a great deal of attention from researchers as shown in recent surveys [3]–​[7]. In [5], two enhanced OFDM-IM schemes are proposed, where I- and Q- dimensions are jointly exploited for IM and linear constellation precoding is used for higher spectral efficiency and diversity gain. In [6], multiple-mode OFDM-IM is proposed to improve the spectral efficiency by conveying information through multiple distinguishable modes (signal constellations). In [7], the active subcarriers are selected in a layer-wise manner, which increases the length of IM bits and spectral efficiency.

As pointed out in [8], OFDM-IM inherited the high peak-to-average power ratio (PAPR) problem from the classical OFDM. It is widely known that the high PAPR causes in-band distortion and out-of-band radiation when the OFDM-IM signal passes through high power amplifier (HPA). Lots of PAPR reduction methods have been researched in OFDM literatures [9]. To solve the high PAPR problem in OFDM-IM, to borrow the methods for OFDM may work. However, those methods are not efficient because they do not consider the unique characteristic of OFDM-IM structure such as the presence of the inactive subcarriers [10]. At this point, there are only few works dealing with the large PAPR problem in OFDM-IM systems. For example, in [10]–​[12], the dither signals are used in the inactive subcarriers for PAPR reduction, but they induce in-band distortion inevitably.

Selected mapping (SLM) is a promising PAPR reduction technique that is distortionless, has good PAPR reducing capability, and can be easily adapted in OFDM-IM systems. The conventional SLM scheme is proposed in [13]. The SLM scheme has a probabilistic approach, where total $U$ alternative OFDM signals are generated and the best signal with the minimum PAPR is transmitted. To generate $U$ alternative OFDM signals, $U$ distinct phase sequences known to both transmitter and receiver are used. We denote the $U$ phase sequences as the phase sequence set (PSS) throughout this paper. Since the SLM scheme has a disadvantage of high computational complexity, the many modified versions with low complexity have been proposed [14]–​[16].

It is obvious that the PAPR reduction performance of the SLM scheme depends on the property of the used PSS. The authors in [17]–​[20] investigated the optimal condition for PSS in the SLM scheme for the classical OFDM system. In summary, the element-wise multiplication of two phase sequences needs to be aperiodic. Therefore, the randomly generated PSS gives the sub-optimal PAPR reduction performance because the random sequence has aperiodicity with high probability. Also, the PSS using the rows of the cyclic Hadamard matrix constructed from an maximum length sequence (MLS) provides the better PAPR reduction performance than the randomly generated PSS case. This is because the circular autocorrelation of an MLS is a Kronecker delta function. Although the works in [17]–​[20] cannot boost the PAPR reduction performance of the SLM scheme, the theoretical investigation on the PSS in SLM is meaningful because they provide the reasonable criteria for good PSS. Also, using the deterministic PSS based on the criteria is necessary in practical systems because of the PAPR reduction performance guarantee.

The conventional SLM scheme in [13] can be easily applied to the OFDM-IM systems, but the analytical results for the desirable PSS in [17]–​[20] cannot be naively applied to the OFDM-IM system because the OFDM-IM system has inactive subcarriers. Also, in [21], permutation procedure in the frequency domain has been proposed in the SLM scheme for OFDM-IM systems in order to make the subcarrier activation pattern (SAP) in the frequency domain diverse. This is because the envelope distribution of the OFDM-IM signal in the time domain depends on the SAP [22]. However, there is no work to investigate the permutation procedure in SLM in depth, which includes the rigorous analysis on the efficiency of the permutation and the condition for desirable permutation functions. To the author’s best knowledge, this is the first work to analyze the efficiency of permutation and the desirable PSS and permutation functions in SLM for OFDM-IM systems, which has a theoretical meaning. In this paper, the SLM scheme with permutation is considered for OFDM-IM systems. Then, our contribution includes;

• The efficiency of the permutation procedure is analyzed. Especially, the efficiency of permutation is derived as a function of the number of active subcarriers.

• The optimal conditions for PSS and the permutation functions are derived.

The rest of the paper can be summarized as follows. In Section II, OFDM-IM, PAPR, and the conventional SLM scheme is reviewed. In Section III, we introduce the conventional SLM with permutation procedure for OFDM-IM systems and investigate the efficiency of the permutation procedure. The optimal conditions for PSS and permutation functions are derived in Section IV. In Section V, we remark our contributions. Then, computer simulation results are presented to support the contributions in Section VI. Finally, Section VII concludes the paper.

A. Notations

Vectors are denoted by boldface letters $\mathbf {X}$ and its $i$ -th element is denoted by $X(i)$ . $\mathbf {X}_{1}\circ \mathbf {X}_{2}$ denotes element-wise multiplication of two vectors $\mathbf {X}_{1}$ and $\mathbf {X}_{2}$ . Also, ${}_{n}\mathrm {C}_{k}$ means $n$ combination $k$ , and $E[\cdot]$ denotes the statistical expectation. The complement of the set $\mathcal {I}$ is denoted by $\bar {\mathcal {I}}$ , and the cardinality of the set $\mathcal {I}$ is denoted by $|\mathcal {I}|$ .

SECTION II. OFDM-IM, PAPR, and SLM

A. OFDM-IM and PAPR

We consider an OFDM-IM system [2], where $N$ subcarriers are divided into $G$ groups. Each group has $n$ subcarriers and $N=nG$ . Unlike the conventional OFDM, not all the subcarriers are used for transmission in OFDM-IM. That is, only $k$ out of $n$ subcarriers are active in each group and clearly $k < n$ . Then $p_{1}$ bits are transmitted by the SAP of each group. The number of possible SAPs in a group is ${}_{n}\mathrm {C}_{k}$ and $p_{1} = \lfloor \log _{2} {}_{n}\mathrm {C}_{k}\rfloor$ . Once the SAP is selected, $p_{2} = k \log _{2} M$ bits can be conveyed by $M$ -ary modulation on the $k$ active subcarriers. Totally, $p = p_{1} +p_{2}$ bits are transmitted per group. Specifically, the $g$ -th group is formed as

$$\begin{equation*} \mathbf {X}^{g} = [X^{g}(0)X^{g}(1) \cdots X^{g}(n-1)]\end{equation*}$$ for $g=0,\cdots,G-1$ , where only the $k$ elements in $\mathbf {X}^{g}$ are nonzero values from $\mathcal {S}$ , used signal constellation, and the others are set to zero.

After the formation of the $g$ -th OFDM-IM group with length $n$ for all $g$ ’s, they are concatenated in interleaving pattern to maximize the diversity gain [23]. Throughout this paper, the interleaving pattern partitioning is assumed. Then the OFDM-IM block in the frequency domain is

$$\begin{align*} \mathbf {X}=&[X(0) X(1) \cdots X(N-1)]\\=&[X^{0}(0)\cdots X^{G-1}(0)\cdots X^{0}(n-1)\cdots X^{G-1}(n-1)]\end{align*}$$ or equivalently $$\begin{equation*} X(r\cdot G + g) = X^{g}(r)\end{equation*}$$ for $r=0,\cdots,n-1$ and $g=0,\cdots,G-1$ . Note that the possible indices in $\mathbf {X}$ which can be occupied by the elements of the $g$ -th group are $\{g,G+g,\cdots,(n-1)G+g\}$ . We assume the average power of used constellation $\mathcal {S}$ is normalized and thus $E[|X(i)|^{2}]=k/n, i=0,\cdots,N-1$ .

Fig. 1 shows an example of an OFDM-IM block when $N=8$ , $n=4$ , and $k=2$ are used, where the OFDM-IM block $\mathbf {X}$ is constructed by concatenating $\mathbf {X}^{0}$ and $\mathbf {X}^{1}$ in interleaved pattern. We define the SAP in the entire OFDM-IM block $\mathbf {X}$ as $\mathcal {I}$ and $|\mathcal {I}| = kG = K$ . It is clear that $X(i) \in \mathcal {S}$ for $i \in \mathcal {I}$ and $X(i) =0$ for $i \in \bar {\mathcal {I}}$ . For example, in Fig. 1, it is clear that $\mathcal {I}=\{1,2,3,6\}$ and $\bar {\mathcal {I}}=\{0,4,5,7\}$ . Also, we define the $g$ -th subset of $\mathcal {I}$ as $\mathcal {I}^{g}$ , which is the set of indices related to the $g$ -th group. That is,

$$\begin{equation*} \mathcal {I}^{g} = \{g,G+g,\cdots,(n-1)G+g\}\cap \mathcal {I}.\end{equation*}$$ In Fig. 1, $\mathcal {I}^{0}=\{2,6\}$ and $\mathcal {I}^{1}=\{1,3\}$ . Clearly, $\mathcal {I} = \bigcup _{g=0,\cdots,G-1}\mathcal {I}^{g}$ .

FIGURE 1.

An example of an OFDM-IM block generation when $N=8$ , $n=4$ , $G=2$ , and $k=2$ are used. Note that $\mathcal {I}=\{1,2,3,6\}$ , $\mathcal {I}^{0}=\{2,6\}$ , and $\mathcal {I}^{1}=\{1,3\}$ .

After generation of the OFDM-IM block $\mathbf {X}$ in frequency domain, it is transformed into the OFDM-IM signal $\mathbf {x}$ in time domain by unitary inverse fast Fourier transform (IFFT) as

$$\begin{equation*} \mathbf {x} = \text {IFFT}\{\mathbf {X}\}\end{equation*}$$ or $$\begin{equation*} x(m) = \frac {1}{\sqrt {N}}\sum _{i=0}^{N-1}X(i)e^{\frac {j2\pi im}{N}}\end{equation*}$$ for $m=0,\cdots,N-1$ .

In the classical OFDM case, for a large $N$ , elements of the OFDM signal are independent because of independent inputs to IFFT and approximately complex Gaussian distributed due to the central limit theorem. However, in OFDM-IM case, $x(0),\cdots,x(N-1)$ are approximately complex Gaussian distributed with zero mean, but they no longer have independency due to inactive subcarriers. Also, the PAPR of $\mathbf {x}$ is defined as

$$\begin{equation*} \text {PAPR}(\mathbf {x}) = \frac {\max _{m=0,\cdots,N-1}|x(m)|^{2}}{E[|x(m)|^{2}]}\end{equation*}$$ where $E[|x(m)|^{2}]=k/n$ .

B. The Conventional SLM Scheme

The conventional SLM scheme for OFDM systems is proposed in [13] and the conventional SLM for OFDM-IM systems is straightforward, which is depicted in Fig. 2. To generate $U$ alternative OFDM-IM signals, $U$ distinct phase sequences ${\mathbf P}_{u},u=1,\cdots,U$ known to both transmitter and receiver are used, where ${\mathbf P}_{u}=[P_{u}(0)\cdots P_{u}(N-1)]$ with $P_{u}(i)=e^{j\phi _{u}(i)}$ and $\phi _{u}(i)\in [0,2\pi)$ . Specifically, the $u$ -th alternative OFDM-IM signal is given by ${\mathbf x}_{u}= \text {IFFT}\{ \mathbf {P}_{u} \circ \mathbf {X}\}$ for $u=1,\cdots,U$ . In this paper, the set $\{\mathbf {P}_{1},\cdots,\mathbf {P}_{U}\}$ is referred to as PSS. Then, the OFDM-IM signal with the minimum PAPR $\mathbf {x}_{\tilde {u}}$ is transmitted as

$$\begin{equation*} \tilde {u}=\underset {u=1,\cdots,U}{\arg \min }{~\text {PAPR}}(\mathbf {x}_{u}).\end{equation*}$$ Many modified versions of SLM have been proposed [14]–​[16]. They mainly focus on lowering the computational complexity because of high complexity of the SLM scheme, but the original methodology remains the same.

FIGURE 2.

The conventional SLM scheme for OFDM-IM systems.

SECTION III. The SLM Scheme With Permutation for OFDM-IM Systems

Note that the SAP of an OFDM-IM block affects the envelope distribution of the corresponding OFDM-IM signal in the time domain and thus its PAPR [22]. Therefore, in OFDM-IM systems, the permutation procedure followed by the multiplication of the phase sequence has been newly introduced in [21]. That is, the SLM with permutation is depicted in Fig. 3. Note that the permutation procedure can be implemented with a negligible computational complexity because it does not need multiplications or additions.

FIGURE 3.

The SLM scheme with permutation for OFDM-IM systems.

FIGURE 4.

The PAPR reduction performances of the SLM scheme with $U=4$ . The efficiency of permutation is valid if $k$ is small compared to $n$ .

In particular, total $pG$ -bits generate the OFDM-IM block ${\mathbf X}$ , whose SAP is $\mathcal {I}$ . Then, the OFDM-IM block ${\mathbf X}$ is permuted by the $u$ -th permutation function $d_{u}(i),u=1,\cdots,U$ , where the relation between ${\mathbf X}$ and its permuted version ${\mathbf X}_{u}=[X_{u}(0)\cdots X_{u}(N-1)]$ is

$$\begin{equation*} X_{u}(d_{u}(i)) = X(i)\tag{1}\end{equation*}$$ for $i=0,\cdots,N-1$ . It is clear that the permutation should be done in each group individually. This is because the elements in a group have to be distributed evenly to maximize the frequency diversity gain, even after the permutation. Therefore, the possible indices which can be occupied by the elements of the $g$ -th group are still $\{g,G+g,\cdots,(n-1)G+g\}$ after permutation. It is obvious that the permutation procedure results in the change of the SAP of $\mathbf {X}$ , $\mathcal {I}$ , and we denote the SAP of ${\mathbf X}_{u}$ as $\mathcal {I}_{u},u=1,\cdots,U$ .

From now on, the remaining procedures are the same as the conventional SLM in subSection II-B. Specifically, the $u$ -th permuted version $\mathbf {X}_{u}$ is multiplied by the $u$ -th phase sequence ${\mathbf P}_{u}$ . Then, these $U$ alternative OFDM-IM blocks $\mathbf {P}_{u} \circ \mathbf {X}_{u},u=1,\cdots,U$ in frequency domain are IFFTed to generate $U$ alternative OFDM-IM signals as

$$\begin{equation*} {\mathbf x}_{u}= \text {IFFT}\{ \mathbf {P}_{u} \circ \mathbf {X}_{u} \}\end{equation*}$$ for $u=1,\cdots,U$ . Then, the PAPR values of them are calculated. Finally, the alternative OFDM-IM signal ${\mathbf x}_{\tilde u}$ having the minimum PAPR is selected for transmission.

Information about the selected phase sequence and permutation, $\tilde {u}$ , should be transmitted to the receiver for decoding as side information. It is clear that the amount of side information of the SLM scheme with permutation is the same as that of the original SLM scheme without permutation. Also, if the side information is perfectly known at the receiver, the bit error rate (BER) performance of OFDM-IM does not degraded by SLM with permutation.

A. Efficiency of Permutation in the SLM Scheme for OFDM-IM Systems

Now we investigate whether the permutation procedure in the SLM scheme in Fig. 3 helps to improve the PAPR reduction performance or not. First, in each group, the number of possible SAPs is ${}_{n}\mathrm {C}_{k}$ . Since ${}_{n}\mathrm {C}_{k} \geq 2^{p_{1}}$ , redundancy ${}_{n}\mathrm {C}_{k}-2^{p_{1}}$ SAPs are not used in a group. However, in this paper, we assume that all possible SAPs in a group are used with equal probability for simplicity. Clearly, the number of possible realizations of $\mathcal {I}$ becomes $({}_{n}\mathrm {C}_{k})^{G}$ . Then, we treat $\mathcal {I}$ as a random variable and its $({}_{n}\mathrm {C}_{k})^{G}$ realizations are denoted by $I_{\omega },\omega =1,\cdots,({}_{n}\mathrm {C}_{k})^{G}$ . We can also treat $\mathcal {I}_{u},u=1,\cdots,U$ as random variables and their respective realizations “corresponding to $\mathcal {I}=I_\omega$ ” are denoted by $I_{\omega,u},u=1,\cdots,U$ .

It is widely known that the $U$ alternative OFDM-IM signals have to be mutually independent to guarantee the PAPR reduction performance of SLM [19]. The method to lower the dependency between the alternative OFDM-IM signals will be discussed in subSections IV-A and IV-B. Thus we assume that the well-designed PSS (with mutual independency) is used. Then, the complementary cumulative distribution function (CCDF) of PAPR of the transmitted OFDM-IM signal $\mathbf {x}_{\tilde {u}}$ in Fig. 3 is

$$\begin{align*}&\hspace {-2pc}Pr\{\text {PAPR}(\mathbf {x}_{\tilde {u}}) >\gamma \} \\[3pt]=&\sum _{\omega =1}^{({}_{n}\mathrm {C}_{k})^{G}} Pr\{\text {PAPR}(\mathbf {x}_{\tilde {u}})>\gamma |\mathcal {I}=I_\omega \}Pr\{\mathcal {I}=I_\omega \} \\[3pt]=&\frac {1}{({}_{n}\mathrm {C}_{k})^{G}}\sum _{\omega =1}^{({}_{n}\mathrm {C}_{k})^{G}}Pr\{\text {PAPR}(\mathbf {x}_{\tilde {u}})>\gamma |\mathcal {I}=I_\omega \} \\[3pt]=&\frac {1}{({}_{n}\mathrm {C}_{k})^{G}}\sum _{\omega =1}^{({}_{n}\mathrm {C}_{k})^{G}}Pr\left \{{\bigcap _{u=1}^{U}\text {PAPR}(\mathbf {x}_{u})>\gamma |\mathcal {I}=I_\omega }\right \} \\[3pt]=&\frac {1}{({}_{n}\mathrm {C}_{k})^{G}}\sum _{\omega =1}^{({}_{n}\mathrm {C}_{k})^{G}}\prod _{u=1}^{U} Pr\{\text {PAPR}(\mathbf {x}_{u})>\gamma |\mathcal {I}_{u}=I_{\omega,u}\} \\[3pt]=&\frac {1}{({}_{n}\mathrm {C}_{k})^{G}}\sum _{\omega =1}^{({}_{n}\mathrm {C}_{k})^{G}}\prod _{u=1}^{U} F(I_{\omega,u},\gamma)\tag{2}\end{align*}$$ where $Pr\{\text {PAPR}(\mathbf {x}_{u})>\gamma |\mathcal {I}_{u}=I_{\omega,u}\}$ is a function of $\gamma$ and $I_{\omega,u}$ and we rewrite it as $F(I_{\omega,u},\gamma)$ .

1) Without Permutation

Without permutation, the $U$ SAPs of $U$ alternative OFDM-IM signals are identical. That is, $I_\omega = I_{\omega,1} = \cdots = I_{\omega,U}$ . Then, (2) becomes

$$\begin{align*} Pr\{\text {PAPR}(\mathbf {x}_{\tilde {u}})>\gamma \}_{\text {woP}}=&\frac {1}{({}_{n}\mathrm {C}_{k})^{G}}\sum _{\omega =1}^{({}_{n}\mathrm {C}_{k})^{G}}\left ({F(I_{\omega },\gamma)}\right)^{U} \\=&E\left [{\left ({F(\mathcal {I},\gamma)}\right)^{U}}\right].\tag{3}\end{align*}$$

2) With Permutation

With permutation, (2) becomes

$$\begin{align*}&\hspace {-2pc}Pr\{\text {PAPR}(\mathbf {x}_{\tilde {u}})>\gamma \}_{\text {wP}} \\=&\frac {1}{({}_{n}\mathrm {C}_{k})^{G}}\sum \limits _{\omega =1}^{({}_{n}\mathrm {C}_{k})^{G}}\prod _{u=1}^{U} F(I_{\omega,u},\gamma) \\=&E\left [{\prod _{u=1}^{U} F(\mathcal {I}_{u},\gamma)}\right].\tag{4}\end{align*}$$ If we choose the permutation functions to have independency between the random variables, $F(\mathcal {I}_{u},\gamma)$ ’s for $1\leq u \leq U$ , (4) becomes $$\begin{align*} Pr\{\text {PAPR}(\mathbf {x}_{\tilde {u}})>\gamma \}_{\text {wP}}\!\simeq \!\prod _{u=1}^{U}E\left [{F(\mathcal {I}_{u},\gamma)}\right] =\left ({E\left [{F(\mathcal {I},\gamma)}\right]}\right)^{U}\!\!\!\!\! \\\tag{5}\end{align*}$$ where the last equation holds because $\mathcal {I}$ and $\mathcal {I}_{u},u=1,\cdots,U$ are identically distributed regardless of permutation functions. The condition for low dependency between $F(\mathcal {I}_{u},\gamma)$ ’s will be discussed in subSection IV-C.

By Jensen’s inequality with (3) and (5), we have

$$\begin{equation*} \underbrace {Pr\{\text {PAPR}(\mathbf {x}_{\tilde {u}})>\gamma \}_{\text {woP}}}_{E\left [{\left ({F(\mathcal {I},\gamma)}\right)^{U}}\right]}\geq \underbrace {Pr\{\text {PAPR}(\mathbf {x}_{\tilde {u}})>\gamma \}_{\text {wP}}}_{\left ({E\left [{F(\mathcal {I},\gamma)}\right]}\right)^{U}}\tag{6}\end{equation*}$$ which means introducing the permutation procedure in SLM enhances the PAPR reduction performance. Note that the difference between the left-hand side (LHS) term and the right-hand side (RHS) term in (6) becomes large if the random variable $F(\mathcal {I},\gamma)$ has large fluctuation, which leads to the benefit of permutation in the SLM scheme.

B. Quantifying the Efficiency of Permutation

In this subsection, we quantify the efficiency of permutation in SLM. Since the closed form of $F(\mathcal {I},\gamma)$ in (6) or the envelope distribution of $\mathbf {x}$ is too complicated, instead we observe the correlation coefficient between the elements of the OFDM-IM signal $\mathbf {x}$ . This is because the elements of $\mathbf {x}$ are correlated (circularly symmetric) complex Gaussian with zero mean and their joint distribution are fully characterized by the correlation coefficient between the elements [24].

The correlation coefficient between $x(l)$ and $x(l+m)$ is

$$\begin{align*}&\hspace {-2pc}\rho _{\mathcal {I}}(l,l+m) \\=&\frac {E[x(l) x^{*}(l+m)]-E[x(l)]E[x^{*}(l+m)] }{k/n} \\=&\frac {1}{N}\frac {E\left [{\left ({\sum _{i_{1}=0}^{N-1}X(i_{1})e^{\frac {j2\pi i_{1} l}{N}}}\right)\left ({\sum _{i_{2}=0}^{N-1}X(i_{2})e^{\frac {j2\pi i_{2} (l+m)}{N}}}\right)^{*}}\right]}{k/n}\!\!\!\!\!\!\! \\=&\frac {1}{Gk}\sum _{i\in \mathcal {I}}e^{-\frac {j2\pi i m}{N}} \tag{7}\end{align*}$$ where we use $E[X(i_{1})X(i_{2})^{*}]=1$ for $i_{1}= i_{2}\in \mathcal {I}$ and otherwise $E[X(i_{1})X(i_{2})^{*}]=0$ . It is noted that $\rho _{\mathcal {I}}(l,l+m)$ only depends on the index difference $m$ and can be rewritten as $\rho _{\mathcal {I}}(m)$ . Now, we will observe the random variable $\rho _{\mathcal {I}}(m)$ instead of the random variable $F(\mathcal {I},\gamma)$ , where both two variables are the functions of the random variable $\mathcal {I}$ .

The correlation coefficient $\rho _{\mathcal {I}}(m)$ in (7) can be rewritten as

$$\begin{align*} \rho _{\mathcal {I}}(m)=&\frac {1}{Gk}\sum _{i\in \mathcal {I}}e^{-\frac {j2\pi i m}{N}} \\=&\frac {1}{Gk}\left ({\sum _{i_{0}\in \mathcal {I}^{0}}e^{-\frac {j2\pi i_{0} m}{N}}+\cdots +\sum _{i_{G-1}\in \mathcal {I}^{G-1}}e^{-\frac {j2\pi i_{G-1} m}{N}}}\right) \\=&\frac {1}{Gk}\left ({A_{0} + e^{-\frac {j2\pi m}{N}}A_{1} + \cdots +e^{-\frac {j2\pi (G-1)m}{N}}A_{G-1}}\right)\end{align*}$$ where $$\begin{equation*} A_{g} = \alpha _{g}+\alpha _{g+G}e^{-\frac {j2\pi Gm}{N}}+\cdots +\alpha _{g+(n-1)G}e^{-\frac {j2\pi (n-1)Gm}{N}}\tag{8}\end{equation*}$$ and $\alpha _{i}$ is the random variable indicating the subcarrier activation as $$\begin{align*} \alpha _{i} = \begin{cases} 1, & \text {if } X(i) \in \mathcal {S} \\ 0, & \text {otherwise} \end{cases}\tag{9}\end{align*}$$ for $i=0,\cdots,N-1$ . It is clear that $\alpha _{g}+\alpha _{g+G}+\cdots +\alpha _{g+(n-1)G} = k$ for $g=0,\cdots,G-1$ because there are $k$ active subcarriers per group.

As we explained, if $\rho _{\mathcal {I}}(m)$ has large fluctuation, the efficiency of permutation increases from (6). Therefore, we can measure the efficiency of permutation by the variance of $\rho _{\mathcal {I}}(m)$ . Since $A_{0},\cdots,A_{G-1}$ are i.i.d., the variance of $\rho _{\mathcal {I}}(m)$ is

$$\begin{align*} \text {var}(\rho _{\mathcal {I}}(m))=&\frac {1}{G^{2}k^{2}}\left ({\text {var}(A_{0})+\cdots +\text {var}(A_{G-1})}\right) \\=&\frac {1}{Gk^{2}}\text {var}(A_{0}). \tag{10}\end{align*}$$ Now we investigate the variance of $A_{0}$ .

1) $m = 0$ mod $n$

From (8), we have

$$\begin{equation*} A_{0} = \alpha _{0}+\alpha _{G}+\cdots +\alpha _{(n-1)G} =k\end{equation*}$$ and $$\begin{equation*} \text {var}(A_{0})=0.\tag{11}\end{equation*}$$

2) $m \neq 0$ mod $n$

Since $E[\alpha _{i}] = \frac {k}{n}$ for all $i$ ’s, from (8) we have

$$\begin{align*} E[A_{0}]=&E[\alpha _{0}]+E[\alpha _{G}]e^{-\frac {j2\pi Gm}{N}} \\&+\,\cdots +E[\alpha _{(n-1)G}]e^{-\frac {j2\pi (n-1)Gm}{N}}=0 \tag{12}\end{align*}$$ and thus $|E[A_{0}]|^{2}=0$ . Also, from (8), we have $$\begin{align*}&\hspace {-2pc}E[|A_{0}|^{2}] \\=&E\left [{\left ({\alpha _{0}+\alpha _{G}e^{\frac {j2\pi Gm}{N}}+\cdots +\alpha _{(n-1)G}e^{\frac {j2\pi (n-1)Gm}{N}}}\right)}\right. \\&\cdot \left.{\left ({\alpha _{0}+\alpha _{G}e^{-\frac {j2\pi Gm}{N}}+\cdots +\alpha _{(n-1)G}e^{-\frac {j2\pi (n-1)Gm}{N}}}\right)}\right] \\=&E\left [{\sum _{i_{0}=0,G,\cdots,(n-1)G}\alpha _{i_{0}}^{2}e^{\frac {j2\pi i_{0}m}{N}}e^{-\frac {j2\pi i_{0}m}{N}}}\right. \\&\left.{+\,\sum _{i_{1}\neq i_{2}}\alpha _{i_{1}}\alpha _{i_{2}}e^{\frac {j2\pi i_{1}m}{N}}e^{-\frac {j2\pi i_{2}m}{N}} }\right].\tag{13}\end{align*}$$ Clearly, we have $$\begin{equation*} E[\alpha _{i_{0}}^{2}] = \frac {k}{n}\tag{14}\end{equation*}$$ for $i_{0}=0,\cdots,N-1$ and $$\begin{equation*} E[\alpha _{i_{1}}\alpha _{i_{2}}] = \frac {_{n-2}\mathrm {C}_{k-2}}{_{n}\mathrm {C}_{k}} = \frac {k(k-1)}{n(n-1)}\tag{15}\end{equation*}$$ for $i_{1} \neq i_{2}$ and $i_{1},i_{2}\in \{0,G,\cdots,(n-1)G\}$ . The equation (15) can be derived by the fact $\alpha _{i_{1}}\alpha _{i_{2}}=1$ only if $\alpha _{i_{1}}=\alpha _{i_{2}}=1$ . Using (14) and (15), the equation (13) can be rewritten by $$\begin{align*} E[|A_{0}|^{2}]=&\frac {k}{n}\sum _{i_{0}=0,G,\cdots,(n-1)G}e^{\frac {j2\pi i_{0}m}{N}}e^{-\frac {j2\pi i_{0}m}{N}} \\&+\,\frac {k(k-1)}{n(n-1)}\sum _{i_{1}\neq i_{2}}e^{\frac {j2\pi i_{1}m}{N}}e^{-\frac {j2\pi i_{2}m}{N}}.\tag{16}\end{align*}$$

Then, we can rewrite (16) as

$$\begin{align*}&\hspace {-2pc}E[|A_{0}|^{2}] \\=&\left ({\frac {k}{n}-\frac {k(k-1)}{n(n-1)}}\right)\sum _{i_{0}=0,G,\cdots,(n-1)G}e^{\frac {j2\pi i_{0}m}{N}}e^{-\frac {j2\pi i_{0}m}{N}} \\&+\,\frac {k(k-1)}{n(n-1)}\left ({\sum _{i_{0}=0,G,\cdots,(n-1)G}e^{\frac {j2\pi i_{0}m}{N}}e^{-\frac {j2\pi i_{0}m}{N}}}\right. \\&\left.{+\,\sum _{i_{1}\neq i_{2}}e^{\frac {j2\pi i_{1}m}{N}}e^{-\frac {j2\pi i_{2}m}{N}}}\right) \\=&\frac {k(n-k)}{n-1} \\&+\,\frac {k(k-1)}{n(n-1)}\left ({e^{\frac {j2\pi 0m}{N}}+e^{\frac {j2\pi Gm}{N}}+\cdots +e^{\frac {j2\pi (n-1)Gm}{N}}}\right) \\&\cdot \,\left ({e^{-\frac {j2\pi 0m}{N}}+e^{-\frac {j2\pi Gm}{N}}+\cdots +e^{-\frac {j2\pi (n-1)Gm}{N}}}\right) \tag{17}\\=&\frac {k(n-k)}{n-1} \tag{18}\end{align*}$$ where the second term in (17) becomes zero because $m\neq 0\mod n$ . Then, from (12) and (18), we have $$\begin{equation*} \text {var}(A_{0})=\frac {k(n-k)}{n-1}.\tag{19}\end{equation*}$$

In conclusion, plugging (11) and (19) into (10) gives

$$\begin{align*} \text {var}(\rho _{\mathcal {I}}(m))=\begin{cases} \dfrac {1}{N}\dfrac {n}{n-1}\left ({\dfrac {n}{k}-1}\right), & m\neq 0\!\mod n \\ 0, & m = 0\!\mod n \end{cases}\tag{20}\end{align*}$$ where the zero variance means that the value of $\rho _{\mathcal {I}}(m)$ is a constant value when $m = 0\mod n$ . Note that as $k$ decreases, the variance of $\rho _{\mathcal {I}}(m)$ increases, which means the gap between the LHS and RHS terms in (6) becomes larger as $k$ decreases. In conclusion, the efficiency of the permutation increases as $k$ decreases.

SECTION IV. The Optimal Condition for PSS and Permutation Functions

In this section, we will derive the optimal condition for phase sequences and permutation functions in the SLM scheme for OFDM-IM systems. For simplicity, we will investigate the conditions for two alternative OFDM-IM signals $\mathbf {x}_{1}$ and $\mathbf {x}_{2}$ . First, we consider the optimal condition for PSS. As seen in (2), the phase sequences should make these two alternative OFDM-IM signals independent. That is, two sets $\{x_{1}(l)\}_{l=0}^{N-1}$ and $\{x_{2}(m)\}_{m=0}^{N-1}$ should be mutually independent.

In [24], it is known that $[x_{1}(0)\cdots x_{1}(N-1)]$ and $[x_{2}(0)\cdots x_{2}(N-1)]$ are respective complex Gaussian random vectors with zero mean due to the central limit theorem. Since two random vectors are linearly related, $[x_{1} (l)~x_{2}(m)]$ for any $l$ and $m$ is also complex Gaussian random vector and circularly symmetric. Then, uncorrelatedness or zero covariance between $x_{1} (l)$ and $x_{2}(m)$ induces the independency. The covariance between $x_{1} (l)$ and $x_{2}(m)$ is

$$\begin{align*}&\hspace {-1.7pc}\text {cov}(x_{1} (l) x_{2} (m)) \\[5pt]=&E[x_{1} (l) x_{2}^{*} (m)]-E[x_{1} (l)]E[x_{2}^{*} (m)] \\[5pt]=&\frac {1}{N}E\left [{\left ({\sum _{i_{1}=0}^{N-1}P_{1}(i_{1})X_{1}(i_{1})e^{\frac {j2\pi i_{1} l}{N}}}\right)}\right. \\[5pt]&\cdot \,\left.{\left ({\sum _{i_{2}=0}^{N-1}P_{2}(i_{2})X_{2}(i_{2})e^{\frac {j2\pi i_{2} m}{N}}}\right)^{*}}\right] \\[5pt]=&\frac {1}{N}E\left [{\sum _{i_{1}=0}^{N-1}\sum _{i_{2}=0}^{N-1}P_{1}(i_{1})P_{2}^{*}(i_{2})X_{1}(i_{1})X_{2}^{*}(i_{2})e^{\frac {j2\pi i_{1} l}{N}}e^{-\frac {j2\pi i_{2} m}{N}}}\right]\!\!\!\!\end{align*}$$ for $0\leq l,m \leq N-1$ . From (1), we have $$\begin{align*}&\hspace {-2pc}E[X_{1}(i_{1})X_{2}^{*}(i_{2})]\\=&\begin{cases} 1, & \text {if} ~i_{1} = d_{1}(i)~ {\mathrm {and}}~ {i_{2} = d_{2}(i)}~{\mathrm {for~all}}~ {i\in \mathcal {I}}\\ 0, & \text {otherwise}. \end{cases}\end{align*}$$ Then, $|\text {cov}(x_{1} (l) x_{2} (m))|$ becomes $$\begin{align*}&\hspace {-0.5pc}|\text {cov}(x_{1} (l) x_{2} (m))| \\&\quad {{\displaystyle {= \frac {1}{N}\left |{\sum _{i\in \mathcal {I}}P_{1}(d_{1}(i))P_{2}^{*}(d_{2}(i))e^{\frac {j2\pi d_{1}(i) l}{N}}e^{-\frac {j2\pi d_{2}(i)m}{N}}}\right |.} }}\tag{21}\end{align*}$$

A. Optimal Condition for PSS Without Permutation

As we analyzed, the efficiency of permutation is small if $k$ is large, which means the permutation may not used in the SLM scheme for large $k$ compared to $n$ . Therefore, we first consider the case of no permutation is used in the SLM scheme. In this case, clearly we have $d_{1}(i)=d_{2}(i)=i$ and thus (21) becomes

$$\begin{align*} |\text {cov}(x_{1} (l) x_{2} (m))|=&\frac {1}{N}\left |{\sum _{i\in \mathcal {I}}P_{1}(i)P_{2}^{*}(i)e^{\frac {j2\pi i (l-m)}{N}}}\right | \\=&\frac {1}{\sqrt {N}}\left |{\underset {i\in \mathcal {I}}{\text {IDFT}}\{P_{1}(i)P_{2}^{*}(i)\}}\right | \tag{22}\end{align*}$$ where $\mathop{\text {IDFT}}\limits_{i\in \mathcal {I}}\{\cdot \}$ denotes the unitary inverse discrete Fourier transform (IDFT) whose participating indices of input are the elements in $\mathcal {I}$ only. To boost the PAPR reduction performance of the SLM scheme, the sequence (22) should be as low as possible. In other words, the “punctured” version of the sequence $\mathbf {P}_{1}\circ \mathbf {P}_{2}^{*}$ has to be aperiodic.

To satisfy the condition, one of the sub-optimal solutions is to use the randomly generated PSS because random sequence has aperiodicity with high probability. Also, using the rows of the cyclic Hadamard matrix constructed from an MLS for $\mathbf {P}_{u}$ ’s for $u=1,\cdots,U$ is the sub-optimal and deterministic solution. It provides the better PAPR reduction performance than the randomly generated PSS case, which will be presented in simulation results. The reason is as follows;

Suppose that we use the PSS satisfying

$$\begin{equation*} \frac {1}{N}\left |{\sum _{i=0}^{N-1}P_{1}(i)P_{2}^{*}(i)e^{\frac {j2\pi i m}{N}}}\right |\leq c\end{equation*}$$ where $c$ is a constant. Since $|\bar {\mathcal {I}}|=N-K$ , we also have $$\begin{equation*} \frac {1}{N}\left |{\sum _{i\in \bar {\mathcal {I}}}P_{1}(i)P_{2}^{*}(i)e^{\frac {j2\pi i m}{N}}}\right |\leq \frac {N-K}{N}.\end{equation*}$$ Using the triangular inequality for (22), we have $$\begin{align*} |\text {cov}(x_{1} (l) x_{2} (m))|\leq&\frac {1}{N}\left |{\sum _{i=0}^{N-1}P_{1}(i)P_{2}^{*}(i)e^{\frac {j2\pi i m}{N}}}\right | \\&+ \,\frac {1}{N}\left |{\sum _{i\in \bar {\mathcal {I}}}P_{1}(i)P_{2}^{*}(i)e^{\frac {j2\pi i m}{N}}}\right | \\\leq&c+\frac {N-K}{N} = c + 1-\frac {k}{n}. \tag{23}\end{align*}$$ Therefore, without permutation, using the rows of the cyclic Hadamard matrix for $\mathbf {P}_{u}$ provides the small $c$ and thus small upper bound in (23).

B. Optimal Condition for PSS With Permutation

Now we consider the SLM scheme with permutation procedure as in Fig. 3. Using the substitution $i'=d_{2}(i)$ and $i = d_{2}^{-1}(i')$ , (21) is rewritten as

$$\begin{align*}&\hspace {-1.8pc}|\text {cov}(x_{1} (l) x_{2} (m))| \\=&\frac {1}{N}\left |{\sum _{i'\in \mathcal {I}'}P_{1}(d_{1}(d_{2}^{-1}(i')))P_{2}^{*}(i')e^{\frac {j2\pi d_{1}(d_{2}^{-1}(i')) l}{N}}e^{-\frac {j2\pi i'm}{N}}}\right | \\=&\frac {1}{\sqrt {N}}\left |{\underset {i'\in \mathcal {I}'}{\text {DFT}}\left \{{P_{1}(d_{1}(d_{2}^{-1}(i')))P_{2}^{*}(i')e^{\frac {j2\pi d_{1}(d_{2}^{-1}(i')) l}{N}}}\right \}}\right |\tag{24}\end{align*}$$ where $\mathcal {I}'=\{d_{2}(i)|i\in \mathcal {I} \}$ and $\mathop{\text {DFT}}\limits_{i'\in \mathcal {I}'}\{\cdot \}$ denotes the unitary discrete Fourier transform (DFT) whose participating indices of the input are the elements in $\mathcal {I}'$ only. Therefore, the optimal condition for the PSS is as follows; For all $l$ ’s, the input sequence of DFT in (24) should be aperiodic. Since the permutation functions affect that sequence and there are totally $N$ sequences to be considered because of $l=0,\cdots,N-1$ , it is hard to find the systematic PSS satisfying this property. Therefore, we defer it as a future work. Instead, we may use randomly generated PSS as a sub-optimal solution. Then, the input of DFT in (24) for all $l$ ’s become random sequences regardless of the permutation functions, and the random sequence is aperiodic with high probability.

C. Optimal Condition for Permutation Functions $D_{U}(I)$

In this subsection, we will investigate the optimal condition for permutation functions in the SLM scheme in Fig. 3. As we mentioned in subsubSection III-A2, permutation functions should be designed to lower the dependency between $F(\mathcal {I}_{u},\gamma)$ ’s for $1\leq u \leq U$ . Also, as in subSection III-B, the metric $\rho _{\mathcal {I}_{u}}(m)$ can be used instead of $F(\mathcal {I}_{u},\gamma)$ . Therefore, we investigate the covariance between $\rho _{\mathcal {I}_{1}}(l)$ and $\rho _{\mathcal {I}_{2}}(m)$ , where $\mathcal {I}_{1}$ and $\mathcal {I}_{2}$ are the SAPs of the first and second alternative OFDM-IM signals, respectively. From (7) we have

$$\begin{align*}&\hspace {-1.8pc}G^{2}k^{2}\text {cov}(\rho _{\mathcal {I}_{1}}(l)\rho _{\mathcal {I}_{2}}(m)) \\=&E\left [{\left ({\sum _{i_{1}\in \mathcal {I}_{1}}e^{-\frac {j2\pi i_{1} l}{N}}}\right)\left ({\sum _{i_{2}\in \mathcal {I}_{2}}e^{-\frac {j2\pi i_{2} m}{N}}}\right)^{*}}\right] \\&-\,E\left [{\left ({\sum _{i_{1}\in \mathcal {I}_{1}}e^{-\frac {j2\pi i_{1} l}{N}}}\right)}\right]E\left [{\left ({\sum _{i_{2}\in \mathcal {I}_{2}}e^{-\frac {j2\pi i_{2} m}{N}}}\right)^{*}}\right] \!\tag{25}\end{align*}$$ where the second term in (25) is zero except when $l=m=0$ and can be negligible because both $\rho _{\mathcal {I}_{1}}(l)$ and $\rho _{\mathcal {I}_{2}}(m)$ become constants when $l=m=0$ from (20). Then, the equation in (25) is rewritten as $$\begin{align*}&\hspace {-2pc}G^{2}k^{2}\text {cov}(\rho _{\mathcal {I}_{1}}(l)\rho _{\mathcal {I}_{2}}(m)) \\=&E\left [{\left ({\sum _{i_{1,0}\in \mathcal {I}_{1}^{0}}e^{-\frac {j2\pi i_{1,0} l}{N}}+\cdots +\sum _{i_{1,G-1}\in \mathcal {I}_{1}^{G-1}}e^{-\frac {j2\pi i_{1,G-1} l}{N}}}\right)}\right.\!\!\!\!\!\!\!\! \\&\cdot \,\left.{\left ({\sum _{i_{2,0}\in \mathcal {I}_{2}^{0}}e^{\frac {j2\pi i_{2,0} m}{N}}+\cdots +\sum _{i_{2,G-1}\in \mathcal {I}_{2}^{G-1}}e^{\frac {j2\pi i_{2,G-1} m}{N}}}\right)}\right] \!\!\!\!\! \\\tag{26}\end{align*}$$ where $\mathcal {I}_{u}^{g},u=1,2,g=0,\cdots,G-1$ is the subset of $\mathcal {I}_{u}$ , which is the set of indices related to the $g$ -th group only, i.e., $\mathcal {I}_{u}^{g} = \{g,G+g,\cdots,(n-1)G+g\}\cap \mathcal {I}_{u}$ .

Since two distinct groups are independent, the cross terms in (26) is

$$\begin{align*}&\hspace {-3pc}E\left [{\sum _{i_{1,g_{1}}\in \mathcal {I}_{1}^{g_{1}}}e^{-\frac {j2\pi i_{1,g_{1}}l}{N}}}\right]E\left [{\sum _{i_{2,g_{2}}\in \mathcal {I}_{2}^{g_{2}}}e^{\frac {j2\pi i_{2,g_{2}}m}{N}}}\right]\\=&\frac {k^{2}}{n^{2}}\sum _{i_{1,g_{1}}=g_{1},G+g_{1},\cdots,(n-1)G+g_{1}}e^{-\frac {j2\pi i_{1,g_{1}}l}{N}}\\&\cdot \,\sum _{i_{2,g_{2}}=g_{2},G+g_{2},\cdots,(n-1)G+g_{2}}e^{\frac {j2\pi i_{2,g_{2}}m}{N}}\end{align*}$$ for $0\leq g_{1}\neq g_{2} \leq G-1$ , where this term is zero except when $l=0\mod n$ and $m=0 \mod n$ . Then the cross terms in (26) is negligible because of (20). Then, we can rewrite (26) as $$\begin{align*}&\hspace {-0.5pc}G^{2}k^{2}\text {cov}(\rho _{\mathcal {I}_{1}}(l)\rho _{\mathcal {I}_{2}}(m)) \\&\qquad {{\displaystyle {= \sum _{g=0}^{G-1}\underbrace {E\left [{\sum _{i_{1,g}\in \mathcal {I}_{1}^{g}}e^{-\frac {j2\pi i_{1,g} l}{N}}\sum _{i_{2,g}\in \mathcal {I}_{2}^{g}}e^{\frac {j2\pi i_{2,g} m}{N}}}\right]}_{B_{g}}. } }}\tag{27}\end{align*}$$ Using (1) and (9), $B_{g}$ can be rewritten as $$\begin{align*} B_{g}=&E\left [{\left ({\alpha _{g} e^{-\frac {j2\pi d_{1}(g) l}{N}}+\alpha _{G+g} e^{-\frac {j2\pi d_{1}(G+g) l}{N}}}\right.}\right.\\&+\,\cdots +\left.{\alpha _{(n-1)G+g} e^{-\frac {j2\pi d_{1}((n-1)G+g) l}{N}}}\right) \\&\cdot \,\left ({\alpha _{g} e^{\frac {j2\pi d_{2}(g) m}{N}}+\alpha _{G+g} e^{\frac {j2\pi d_{2}(G+g) m}{N}}}\right.\\&+\,\cdots +\left.{\left.{\alpha _{(n-1)G+g} e^{\frac {j2\pi d_{2}((n-1)G+g) m}{N}}}\right)}\right].\end{align*}$$

Since the permutation is done in each group individually, we have

$$\begin{align*}&\hspace {-0.5pc}\{d_{u}(g),d_{u}(G+g),\cdots,d_{u}((n-1)G+g)\} \\&\quad\quad\qquad\qquad\qquad {{\displaystyle {=\{g,G+g,\cdots,(n-1)G+g\}.} }}\tag{28}\end{align*}$$ Using (14), (15), and (28), $B_{g}$ becomes $$\begin{align*}&\hspace {-1.7pc}B_{g} \\=&\frac {k(k-1)}{n(n-1)}\left ({e^{-\frac {j2\pi gl}{N}}+e^{-\frac {j2\pi (G+g)l}{N}} +\cdots +e^{-\frac {j2\pi ((n-1)G +g)l}{N}}}\right)\!\!\!\!\!\!\!\!\!\!\! \\&\cdot \,\left ({e^{\frac {j2\pi gm}{N}}+e^{\frac {j2\pi (G+g) m}{N}}+\cdots +e^{\frac {j2\pi ((n-1)G+g)m}{N}}}\right) \\&+\,\underbrace {\left ({\frac {k}{n}-\frac {k(k-1)}{n(n-1)}}\right)\sum _{i=0,G,\cdots,(n-1)G}e^{\frac {j2\pi (d_{2}(i+g)m-d_{1}(i+g)l)}{N}}}_{D_{g}} \\=&C_{g} + D_{g}. \tag{29}\end{align*}$$ The term $C_{g}$ in (29) becomes $$\begin{align*} C_{g} = \begin{cases} \dfrac {k(k-1)}{n(n-1)}n^{2}, & \text {if } l = 0 \mod n {~\text {and }} m = 0 \mod n\\ 0, & \text {otherwise}. \end{cases}\!\!\!\!\! \\\tag{30}\end{align*}$$ In (30), $C_{g}$ can be negligible because both $\rho _{\mathcal {I}_{1}}(l)$ and $\rho _{\mathcal {I}_{2}}(m)$ become constants when $l = 0 \mod n {\,\,\text {and }} m = 0 \mod n$ from (20). Therefore, we only investigate $D_{g}$ in (29). Then, (27) becomes $$\begin{align*}&\hspace {-2pc}G^{2}k^{2}\text {cov}(\rho _{\mathcal {I}_{1}}(l)\rho _{\mathcal {I}_{2}}(m)) \\=&\sum _{g=0}^{G-1}D_{g} \\=&\frac {k(n-k)}{n(n-1)}\sum _{g=0}^{G-1}\sum _{i=0,G,\cdots,(n-1)G}e^{\frac {j2\pi (d_{2}(i+g)m-d_{1}(i+g)l)}{N}} \\=&\frac {k(n-k)}{n(n-1)}\sum _{i=0}^{N-1}e^{\frac {j2\pi (d_{2}(i)m-d_{1}(i)l)}{N}} \\=&\frac {k(n-k)}{n(n-1)}\sum _{i'=0}^{N-1}e^{\frac {j2\pi (i'm-d_{1}(d_{2}^{-1}(i'))l)}{N}} \\=&\frac {k(n-k)}{n(n-1)}\sqrt {N}\underset {i'=0,\cdots,N-1}{\text {IDFT}}\left \{{e^{-\frac {j2\pi d_{1}(d_{2}^{-1}(i'))l}{N}} }\right \} \tag{31}\end{align*}$$ where the substitution $i'=d_{2}(i)$ and $d_{2}^{-1}(i')=i$ are used.

In conclusion, the optimal condition for the permutation functions is that (31) should have low envelope fluctuation or variance for all $l$ ’s. In other words, the input sequence of IDFT in (31) should be aperiodic for all $l$ ’s. Clearly, random permutation could be the sub-optimal solution.

SECTION V. Remarks

We remark the contributions of our work before simulation results are presented. For the conventional SLM scheme with permutation in OFDM-IM systems;

• We derived the efficiency of permutation and found that the efficiency of the permutation increases as $k$ decreases. (subSection III-B)

• If $k$ is large compared to $n$ , we may not use permutation. Without permutation, the optimal condition for the PSS is derived. Also, the deterministic PSS using the rows of the cyclic Hadamard matrix provides the sub-optimal PAPR reduction performance. (subSection IV-A)

• With permutation, the optimal condition for the PSS is also derived, and the randomly generated PSS is the sub-optimal solution. (subSection IV-B)

• The optimal condition for permutation functions is derived, where using the randomly generated permutation functions is the sub-optimal solution. (subSection IV-C)

• The derivation of the optimal conditions for PSS and permutation functions is meaningful because in practical systems using the deterministic PSS and permutation functions based on the derived criteria is necessary because of memory saving and PAPR reduction performance guarantee.

In an attempt to further enhance the PAPR reduction performance in the SLM scheme, future research directions will involve the systematic construction of PSS and permutation functions satisfying the derived optimal conditions by using sequence theory. However, we defer it as a future work because it is out of scope of our goal.

SECTION VI. Simulation Results

In this section, simulation results are presented to support the remarks in the previous section. Here, the following simulation parameters are used in all plots; $N=64$ (except for Fig. 5), $M=4$ (quadrature phase shift keying (QPSK) is used), $n=16$ , and $G=4$ , which is reasonable because the OFDM-IM system has an advantage over the classical OFDM when they operate at low bit rates [3]. CCDF is used to quantify the PAPR reduction performance. We performed $5\times 10^{7}$ trials by generating independent OFDM-IM blocks to plot each curve. In the legend, “perm.” denotes the permutation.

FIGURE 5.

The PAPR reduction performances of the SLM scheme with $U=4$ . $k=2$ and randomly generated PSS are used.

Fig. 4 depicts the CCDFs of OFDM-IM systems using the SLM scheme with $U=4$ and the randomly generated PSS is used. Two curves of original OFDM-IM without SLM for $k=2$ and $k=14$ are shown as benchmark. Although the number of trials is large enough, the original OFDM-IM case of $k=2$ has a rough appearance and there is no curve over PAPR $= 9$ dB. This is because the number of active subcarriers in an OFDM-IM block is only $K = 8$ in this case. In Fig. 4, for small $k$ compared to $n$ , such as $k=2$ , the permutation can improve the PAPR reduction performance over no permutation case while the gain of permutation is negligible for large $k$ , analyzed in subSection III-B. Due to lack of space, the plots of the other $k$ ’s are not depicted but they show the identical tendency.

Fig. 5 depicts the CCDFs of OFDM-IM systems using the SLM scheme with $U=4$ . Randomly generated PSS and $k=2$ are used for all curves. As depicted, the efficiency of permutation becomes large as $N$ decreases. Since OFDM-IM is adopted for communication systems requiring row bit rates such as narrow band internet of things (NB-IoT) [25], it is meaningful that the permutation has a benefit for small values of $N$ .

Fig. 6 depicts the CCDFs of OFDM-IM systems using the SLM scheme with $U=4$ and permutation procedure is not used. As seen from Fig. 6, without permutation, the PSS using the rows of the cyclic Hadamard matrix obviously provides the superior performance over the case using the random PSS as analyzed in subSection IV-A.

FIGURE 6.

The PAPR reduction performances of the SLM scheme with $U=4$ . Without permutation, the PSS constructed from the cyclic Hadamard matrix provides a better PAPR reduction performance over the random PSS case.

Fig. 7 depicts the CCDFs of OFDM-IM systems using the SLM scheme with $U=4$ and two original OFDM-IM systems without SLM. In Fig. 7, we can conclude that the permutation is not necessary if $k$ is large compared to $n$ . There are two reasons as we already mentioned; First, in the SLM scheme without permutation, the PSS can be well designed using the cyclic Hadamard matrix. Second, the efficiency of permutation is negligible when $k$ is large.

FIGURE 7.

The PAPR reduction performances of the SLM scheme with $U=4$ .

Fig. 8 depicts the CCDFs of OFDM-IM systems with $k=2$ using the SLM scheme with $U=2$ . The randomly generated PSS is used for all curves. We denote the variance of the concatenated version of the sequence (31) for all $l$ ’s as $\mu$ and generate the permutation functions having various $\mu$ ’s. That is, we measure the fluctuations of (31) by its variance $\mu$ . Through massive simulations, the empirically best permutation functions are obtained with $\mu =18.03$ . Note that no permutation case has the largest value of $\mu =63.01$ .

FIGURE 8.

The PAPR reduction performances of the SLM scheme with $U=2$ in OFDM-IM systems with $k=2$ . The permutation functions are manually generated according to various $\mu$ ’s.

Fig. 8 shows that the permutation functions with small $\mu$ are more appropriate for PAPR reduction. Thus, we confirm the validness of the optimal condition for the permutation functions derived in subSection IV-C. As we expected, the randomly generated permutation functions provide almost the same PAPR reduction performance as the empirically best case of $\mu =18.03$ . This is because the randomly generated permutation functions can lower the value of $\mu$ with high probability. However, using the deterministic permutation functions according to the derived optimal condition is more appropriate for practical situations from the view point of memory saving and PAPR reduction performance guarantee.

SECTION VII. Conclusions

OFDM-IM has become a promising technique whereby the specific activation of the frequency domain subcarriers is used for implicitly conveying extra information. Unfortunately, OFDM-IM has inherited a large PAPR problem from the classical OFDM system. To solve the large PAPR problem, SLM scheme is one of the promising schemes, but there are no works to analyze the optimal condition for phase sequences and permutation functions in the SLM scheme for OFDM-IM systems.

In this paper, useful aspects of the SLM scheme using permutation for OFDM-IM systems are described. First, there is a benefit of introducing the permutation procedure in the SLM scheme only if $k$ is small compared to $n$ . Therefore, if $k$ is large, the SLM scheme without permutation procedure could be a good choice with the PSS using the rows of the cyclic Hadamard matrix. Also, the optimal conditions for PSS and permutation functions in the SLM scheme are derived. These optimal conditions can guarantee the PAPR reduction performance of the SLM schemes, which is meaningful in practical systems. In order to support the analytical results, simulation results are presented.