Spread Spectrum OFDM-IM With Efficient Precoded Matrix and Low Complexity Detector

This paper introduces a novel scheme that employs efficient precoded matrices for index modulation orthogonal frequency division multiplexing namely (EP-OFDM-IM). This technique implements Discrete Fourier Transform (DFT) and Fast Walsh Hadamard Transform (FWHT) precoded matrices for spreading the subset of active M-ary data symbols and their indices to transmit additional information. This aims to enhance the system’s transmit diversity (TD) by exploiting multipath and index diversities. Moreover, a low complex minimum means square error greedy detector (MMSE-GD) is proposed that can achieve optimal performance for perfect channel state information (CSI) and superior performance compared to maximum likelihood (ML) under imperfect CSI. The theoretical analysis based on pair-wise error probability is derived to estimate the bit error probability (BEP) of the proposed scheme for the performance analysis. Simulation results demonstrate that our proposed scheme EP-OFDM-IM significantly outperforms various existing competitive benchmark IM schemes. Specifically, at a spectral efficiency (SE) of 1.5 bps/Hz, and BEP of 10−3, our proposed scheme achieves addition of 15dB gain.

SECTION I. Introduction

In wireless communication, orthogonal frequency division multiplexing (OFDM) is a prominent multicarrier transmission technique due to its robustness to intercarrier interference. However, for the future generation wireless system, OFDM is not suitable due to the lack of flexibility and inadequate confinement of the spectrum [1]. Furthermore, the problem with the frequency shifts because of Doppler effects in high mobility applications reduced the spectral efficiency (SE) due to cyclic prefix to handle interference. Therefore, there is a need for a flexible waveform to encounter these challenges. OFDM with index modulation (OFDM-IM) proposed in [2] is a growing multicarrier modulation scheme that selectively activates a subset of M-ary data symbols, setting the remaining ones to zero. This approach helps to transmit classical M-ary symbols with additional information on the indices over these active sub-carriers.

OFDM-IM has gained significant attention from researchers to enhance the error performance and the data rate. Dual mode OFDM with IM (DM-OFDM-IM) proposed in [3], modulates silent subcarriers alongside active ones to improve system throughput. In zero-padded tri-mode OFDM with IM (ZTM-OFDM-IM) [4], only a subset of available subcarriers are used for modulation within each OFDM subblock, employing two distinct constellations to achieve high SE. Multi-mode IM-aided (MM-OFDM-IM) was introduced in [5] to carry extra bits by utilizing inactive subcarriers. In [6], an approximate mathematical expression on the BEP of the maximum likelihood (ML) detector is proposed, while [7] investigates the achievable rate. The integration of multiple input multiple output (MIMO) with OFDM-IM is proposed in [8]. In the OFDM-IM with in-phase and quadrature (I/Q) scheme [9], index modulation is applied independently to both I/Q components, effectively doubling the number of index bits. This approach significantly enhances the SE. Based on the OFDM-IM framework, several advanced techniques have been proposed in the literature to enhance the transmit diversity (TD) of the system. The implementation of multiple constellations for the identical information transmitted across the inactive and active sub-carriers was performed by combining the TD technique with index modulation OFDM (OFDM-IM-TD) in [10]. Interleaving is introduced into the OFDM-IM system by increasing the Euclidean distance between data symbols to achieve frequency diversity in [11]. To obtain the highest diversity gain, the combination of space-time block codes with coordinates interleaving (CI) and index modulation OFDM was introduced in [12]. In coordinate interleaving index modulation, the real and imaginary parts of each complex symbol are transferred independently onto the sets of subcarriers to improve the frequency diversity of the system against frequency-selective fading channels. A compressed sensing approach integrated with index modulation is introduced in [13], enhancing energy efficiency and diversity order at the expense of increased complexity. The integration of the spread spectrum (SS) with index modulation (IM-OFDM-SS) was introduced in [14], utilizing the spreading codes to transmit the information bits and their indices thereby increasing the diversity gain. Conversely, the grouped linear constellation precoder (LCP) was introduced in [15] to maximize multipath diversity gain for traditional OFDM.

In this paper, a novel efficient precoded (EP) OFDM with index modulation (EP-OFDM-IM) scheme is introduced which is designed to enhance the TD of the system. To the best of our knowledge, the existing literature has not addressed the potential advantages of using precoded OFDM with index modulation for index diversity and multipath diversity. The main contributions are outlined as follows:

• The novel scheme EP-OFDM-IM is presented which implements Fast Walsh Hadamard Transform (FWHT) and Discrete Fourier Transform (DFT) precoded matrices to spread the subset of active sub-carriers and their indices. This approach significantly reduced the baseband processing complexities, increased the SE, and TD of the system.

• An approximate mathematical expression on the bit error probability (BEP) of EP-OFDM-IM is derived to provide insights into diversity and coding gain. This particular shed light on the effects of different spreading matrices on the run-time computational complexity.

• A low complexity minimum means square error greedy detector (MMSE-GD) is proposed, which offers optimal performance compared to ML in perfect channel state information (CSI) and superior performance under imperfect CSI.

• Extensive simulations are provided to demonstrate that EP-OFDM-IM outperforms the existing competitive benchmark schemes including OFDM, OFDM-IM, CI-OFDM-IM, and IM-OFDM-SS.

The subsequent sections of this paper are organized as follows: In Section II, the transmitter side of the EP-OFDM-IM system model with precoding matrices FWHT and DFT with the comparison of the computational complexity of various spreading matrices is presented. Section III provides the derivation of the theoretical upper bound of BEP.

In Section IV, the receiver side along with a low-complexity detector is introduced. The detailed simulation results are provided in Section V, Finally, Section VI concludes the paper.

A. Notation

The lower-case bold letters represent vectors, while upper-case bold letters denote matrices. The superscripts $(.)^{T}$ , $(.)^{H}$ , and $(.)^{*}$ denote the transpose, hermitian, and conjugate respectively. $C(,)$ is binomial coefficient, $\lfloor. \rfloor$ represents the floor function and $Q(.)$ is Guassian tail probability. Kronecker product is denoted by $\otimes$ , while Frobenius norm is presented by $\lVert.\rVert$ .

SECTION II. System Model

We propose an efficient precoding OFDM-IM scheme named EP-OFDM-IM. The transmitter design of the novel scheme is depicted in Fig. 1. For each OFDM block, t information bits are transmitted. These bits are divided into G clusters, each consisting of $p=p_{1}+p_{2}$ bits. These bits are then utilized to construct OFDM sub-blocks of length ${N} = N_{t}/G$ , where $N_{t}$ represents the size of the fast fourier transform (FFT). The selection of indices has been done in two different mapping techniques. First, a straightforward look-up table is used to provide active indices corresponding to specific bits for the mapping operation as shown in Table 1. This method is very efficient for the smaller values. For the higher values of N and K, it is very difficult to design the look-up table. The second method is based on combinatorial theory an effective technique to map the information bits on active subcarrier indices. For each OFDM sub-block g (where $g=1,2,3\ldots., G$ ), the index selector activates on K sub-carriers out of N based on the corresponding $p_{1}= \lfloor \log _{2}(C(N,K)) \rfloor$ , while the rest of the $N-K$ sub-carriers are zero-padded. The activated sub-carriers for each sub-block g is given by $a_{g}= \{ i_{1}, i_{2} \ldots i_{K}\}$ , where $i_{K} \in \{1,2,\ldots \ldots N\}$ for $k = \{1,2,\ldots.K\}$ . For smaller values of N and K, the selection of subcarrier indices can be conducted using either a lookup table or by using a one-to-one mapper based on combinatorial methods [2]. In order to determine the data symbols, the remaining $p_{2}= K \log _{2}(M)$ from the total p bits, are mapped onto the M-ary amplitude phase modulation (APM) constellation and denoted by ${s}_{g}=[s_{1}, s_{2} \ldots.. s_{K}]$ with $s_{K} \in E$ . where E is M-ary APM constellation. The data rate R is calculated as follows:

$$\begin{equation*}R =\frac {[\log _{2}(C(N,K))]+ K\log _{2}(M)}{N} \tag {1}\end{equation*}$$

TABLE 1 Look-up Table for $N = 4$ , $K = 2$

FIGURE 1.

Transmitter design of EP-OFDM-IM.

OFDM block creator uses $a_{g}$ and ${s}_{g}$ to generate the M-ary symbol vector $m =[m_{1},m_{2},\ldots m_{N}]^{T}$ , where, $m_{i}k = s_{K}$ for the $i_{K} \in a_{g}$ and $m_{i}=0$ for the $i \notin a_{g}$ . Before applying the inverse fast fourier transform (IFFT), m undergoes multiplication with the precoded matrices D. It’s important to note that D is a square matrix with size $N \times N$ , which can be meticulously generated to effectively spread $a_{g}$ and ${s}_{g}$ across M subcarriers, resulting in substantial diversity gains. The details of designing the D will be presented later. The spread OFDM-IM precoded vector denoted by v = Dm is transmitted over the N flat fading channel. The channel impulse response matrix in the frequency domain is denoted by H $= diag\{h_{1},\ldots h_{N}\}$ , where, $h_{i}$ elements are independently and uniformly distributed with $h_{i} \sim \mathcal {CN}(0, \sigma ^{2})$ .

The received signal is represented by

$$\begin{equation*} \mathbf {y}=\mathbf {H}{\mathbf {D}} \mathbf {m}+ \mathbf {w} \tag {2}\end{equation*}$$

The noise vector is denoted by ${w}=[w_{1},w_{2},\ldots \ldots w_{N}]^{T}$ with the distribution $\mathcal {CN}(0, N_{o})$ . As a result, the signal-to-noise ratio (SNR) is specified as $\overline {\gamma }= \sigma ^{2} / N_{o}$ .

For the measurement of the received signal $\widehat {\mathbf {m}}$ , the optimal ML detector [16] is used and is estimated by (3)

$$\begin{equation*} \widehat {\mathbf {m}}=\arg \min _{ \mathbf {m}}\left \Vert {{\mathbf {y}-\mathbf {H}\mathbf {D}\mathbf {m} }}\right \Vert ^{2} \tag {3}\end{equation*}$$

A. Precoded Matrices for OFDM-IM

In this paper, we implement two less computational complexity precoded matrices namely DFT and FWHT. To the best of our knowledge, DFT has been discussed in [17] named DFT-s-OFDM but implementation of DFT on OFDM-IM is missing in the literature. The Walsh Hadamard (WH) is a highly popular [18] spreading matrix with high computational complexity and poor error performance in binary phase shift keying (BPSK). Therefore, we introduced a highly efficient precoded matrix FWHT with less computational complexity. The result shows that the bit error rate performance is highly improved. The Walsh Hadamard is the most common spreading matrix which is defined recursively as follows

$$\begin{align*} \mathbf {D}_{1}=\frac {1}{\sqrt {2}}\begin{bmatrix}1 & 1 \\ 1 & -1 \end{bmatrix}, \mathbf {D}_{k}=\mathbf {D}_{k-1}\otimes \mathbf {D}_{1}. \tag {4}\end{align*}$$

It can be observed that the WH size must be $N=2^{k}$ , which makes it particularly suitable for low-complexity implementations because all elements are real-valued. In contrast, FWHT is an efficient algorithm [19] that uses a butterfly structure to compute the WH matrix as shown in Fig. 2. FWHT repeatedly splits the input vector with size n, into two smaller WH of size $n/2$ . This recursive breakdown makes the algorithm faster.

FIGURE 2.

Example of fast walsh-hadamard transform estimation for input vector (1,0,1,1,0,0,1,0).

The Discrete Fourier Transform (DFT) [20] of the N sub-carrier is represented by

$$\begin{equation*} \mathbf {I}_{k} = \sum _{n=0}^{N-1}\mathbf {m_{n}} \cdot \mathbf {D}_{N}^{kn}. \tag {5} \end{equation*}$$

In equation (5), $\mathbf {m} = [m_{1},m_{2},\ldots.,m_{N}]^{T}$ is M-ary modulated symbols and $\mathbf {D}_{n}k$ is the DFT matrix which acts as a precoder. The DFT matrix is calculated as follows

$$\begin{align*} \mathbf {D}_{N} & = e^{\frac {-j2\pi }{N}} = Cos\left ({{\frac {2\pi }{N}}}\right ) - jSin\left ({{\frac {2\pi }{N}}}\right ) \tag {6} \\ \mathbf {D} & = \frac {1}{\sqrt {N}} \begin{bmatrix} 1 & ~1 & ~1 & ~\: \cdots & ~1 \\ 1 & ~\: w & ~\: w^{2} & ~ \cdots & ~\: w^{N-1} \\ \vdots & ~\: \vdots & ~\: \vdots & ~\: \ddots & ~\: \vdots \\ 1 & ~\: w^{(N-1)} & ~\: w^{2(N-1)} & ~\: \cdots & ~\: w^{(N-1)(N-1)} \end{bmatrix} \tag {7}\end{align*}$$

The information bits can be retrieved using the Inverse Discrete Fourier Transform (IDFT) mentioned in (8).

$$\begin{equation*} x_{k} = \frac {1}{N}\sum _{k=0}^{N-1} I_{k} \cdot D_{N}^{-kn}. \tag {8} \end{equation*}$$

B. Precoded Matrices Computational Complexity

The computational complexity comparison of precoded matrices WH, FWHT, and DFT has been conducted using two methods: first, by analyzing the floating-point operations (FLOP) operational complexity, and second, by evaluating the run-time complexity. WH transform performs linear operation by multiplying with the Hadamard matrix by $\mathbf {H}_{n}$ , where $n = 2^{k}$ . If the input vector is x, the output of WH transform is $\mathbf {y}=\mathbf {H}_{n}\cdot \mathbf {x}$ where $\mathbf {H}_{n}$ is $n \times n$ . Since each entry in y depends on n matrix-vector multiplication, therefore the FLOP computational complexity of WH is $O(n^{2})$ . FWHT is a divide-and-conquer algorithm that optimizes the WH by repeatedly dividing the input vector into smaller WH of size $n/2$ as shown in Fig. 2. FWHT performs $n/2$ additions and subtraction, resulting in total n operations per stage. Since there are $\log _{2} (n)$ operations, the FLOP computational complexity of FWHT is $O(n\log (n))$ .The standard multiplication of DFT has FLOP computational complexity $O(n^{2})$ [21], while the Fast Fourier Transform (FFT) provides less computational complexity of $O(n\log (n))$ [22]. It is evident that FWHT computational complexity is less compared to WH and DFT precoded matrices. Let’s take a look at the second method, the run-time computational complexities of precoded matrices using Matlab run on 12th generation core i7 with a speed of 1.8 GHz have been compared. Table 2 represents the run time performance of each precoding matrix. The computational run-time complexity varies based on the selection of active sub-carriers. When the number of active subcarriers is less, the run time computational complexity is less. These results further demonstrate that the computational complexity of FWHT is less than that of WH and DFT. This efficiency advantage translates to optimized resource utilization, minimizing waste, and contributing to greener, more environmentally friendly technologies which are crucial in ensuring sustainable and responsible technological development in existing and future networks [23].

TABLE 2 Run Time Computational Complexity Comparison of Different Precoding Matrices

SECTION III. BEP Performance Analysis of EP-OFDM-IM

Let m be the transmitted vector and $\hat {\mathbf {m}}$ be the estimated signal at the receiver side. If H is the channel response then pairwise error probability (PEP) [24] is represented by (9)

$$\begin{equation*}P\left ({{\mathbf {m}\rightarrow \hat {\mathbf {m}} | \mathbf {H}}}\right )=Q\left ({{\sqrt {\zeta /\left ({{ 2N_{0}}}\right ) } }}\right ) \tag {9}\end{equation*}$$ where $\zeta = \left \Vert {{\mathbf {H} \mathbf {D}(\mathbf {m}-\hat {\mathbf {m}})}}\right \Vert ^{2}$ , which can also be written as $\zeta =\sum _{i=1}^{N}\Theta _{i}\left |{{h_{i}}}\right |^{2}$ , where $\Theta _{i}=\left \vert {{ d_{i}(\mathbf {m}-\hat {\mathbf {m}}) }}\right \vert ^{2}$ , $d_{i}$ is the ith row of precoding matrix D. The unconditional (PEP) can be estimated by taking the average of $P\left ({{\mathbf {m}\rightarrow \hat {\mathbf {m}}|\mathbf {H}}}\right)=\frac {1}{\pi }\int _{0}^{\pi /2}\exp \left ({{-\frac {\zeta }{4N_{0}\sin ^{2}\phi }}}\right)d\phi$ and written by using the alternative form of Q function [24] in (10) $$\begin{equation*}P\left ({{\mathbf {m}\rightarrow \hat {\mathbf {m}} | \mathbf {H}}}\right )=\frac {1}{\pi }\int \limits _{0}^{\pi /2} {\mathcal {M}}_{\zeta }\left ({{-\frac {1}{4\sin ^{2}\phi } }}\right ) d\phi. \tag {10}\end{equation*}$$

In (10), the Momentum Generating Function (MGF) is represented by ${\mathcal {M}}_{\zeta }$ , The MGF of $\mu _{i}=\Theta _{i} \left \vert {{ h_{i} }}\right \vert ^{2}$ in the presence of channel model is estimated as ${\mathcal {M}}_{\mu _{i}}(t) = (1 - \Theta _{i}\lambda ^{2}t)^{-1}$ and finally expression is written as ${\mathcal {M}}_{\zeta }(t)=\prod _{ i=1}^{N} (1 - \Theta _{i}\lambda ^{2}t)^{-1}$ . By replacing MGF value with (10), we obtain

$$\begin{equation*} P\left ({{\mathbf {m}\rightarrow \hat {\mathbf {m}}}}\right )=\frac {1}{\pi }\int _{0}^{\pi /2}\prod _{i=1}^{N}\left ({{\frac {\sin ^{2}\phi }{\sin ^{2}\phi +\frac {\Theta _{i}\lambda ^{2}}{4N_{0}}}}}\right )d\phi. \tag {11} \end{equation*}$$

Let us define the set ${\mathcal {D}}_{\mathbf {m},\hat {\mathbf {m}}} = \{i | \Theta _{i}\cancel {=} 0\}$ for the given precoding matrix D, and the cardinality of the matrix is represented by $\Omega _{\mathbf {m},\hat {\mathbf {m}}} = \left \vert {{{\mathcal {D}}_{\mathbf {m},\hat {\mathbf {m}}} }}\right \vert$ . The $\sin ^{2}\phi \left ({{\sin ^{2}\phi +\frac {\Theta _{i}\lambda ^{2}}{4N_{0}}}}\right) \leq \left ({{1 +\frac {\Theta _{i}\lambda ^{2}}{4N_{0}}}}\right)^{-1} = \frac {4}{\Theta {i}\bar {\gamma }}$ for $0 \leq \sin ^{2}\phi \leq 1$ at high signal to noise ratio (SNR) for $i \in {\mathcal {D}}_{\mathbf {m},\hat {\mathbf {m}}}$ . Based on these facts, the (11) can be approximated as

$$\begin{equation*} P\left ({{\mathbf {m}\rightarrow \hat {\mathbf {m}}}}\right )\approx \frac {\left ({{\bar {\gamma }/4}}\right )^{-\Omega _{\mathbf {m},\hat {\mathbf {m}}}}}{2\prod _{i\in {\mathcal {D}}_{\mathbf {m},\hat {\mathbf {m}}}}\Theta _{i}}. \tag {12} \end{equation*}$$

The coding and diversity gain is estimated from (12) as follows

$$\begin{align*} D_{c}& =\min _{\mathbf {m}\ne \hat {\mathbf {m}},\,\Omega _{\mathbf {m},\hat {\mathbf {m}}}=D_{d}}\left ({{\prod _{i\in {\mathcal {D}}_{\mathbf {m},\hat {\mathbf {m}}}}\Theta _{i}}}\right )^{\frac {1}{D_{d}}}. \tag {13} \\ D _{d}& =\min _{\mathbf {m}\ne \hat {\mathbf {m}}}\Omega _{\mathbf {m},\hat {\mathbf {m}}}, \tag {14} \end{align*}$$

Finally, the upper bound on the BEP using union bound theory on $P\left ({{\mathbf {m}\rightarrow \hat {\mathbf {m}}}}\right)$ as follows

$$\begin{equation*} P_{b}\leq \frac {1}{pCM^{K}}\sum _{\mathbf {m}}\sum _{\hat {\mathbf {m}}}\frac {S\left ({{\mathbf {m},\hat {\mathbf {m}}}}\right )\left ({{\bar {\gamma }/4}}\right )^{-\Omega _{\mathbf {m},\hat {\mathbf {m}}}}}{2\prod _{i\in {\mathcal {D}}_{\mathbf {m},\hat {\mathbf {m}}}}\Theta _{i}}, \tag {15}\end{equation*}$$ where $S(m,\hat {m})$ is the difference bits between m and $\hat {\mathbf {m}}$ . The precoded matrix D can be designed for the given parameters N, K, M to maximize the coding and diversity gain as shown in (13) and (14). When $N=4$ , $M=4$ ,and $K=1$ , the EP-OFM-IM scheme using the FWHT precoded matrix achieve $D_{c}=1$ and $D_{d}=2$ which is double compared to OFDM-IM.

SECTION IV. Receiver Side of EP-OFDM-IM Scheme

The receiver side of the EP-OFDM-IM system is shown in the Fig. 3. To implement this novel technique practically, we introduced a low-complexity MMSE-GD to improve the error performance and reduce the system complexity. The comparison between MMSE-GD and the optimal detector ML has been conducted to demonstrate the superiority of MMSE-GD.

FIGURE 3.

Receiver design of EP-OFDM-IM.

A. Maximum Likelihood (ML) Detector

The ML detector searches all the possible combinations and jointly determines the K active indices i of spreading code and M-ary constellation symbols [16]. The data symbols and active indices are denoted by $\hat {i}$ and $\hat {{m}}$ hat respectively. The ML detector makes decisions based on the

$$\begin{equation*} (\hat {i},\hat {s}) = \mathop {argmin}\limits _{i,s} {\left \|{{{y} - \mathbf {H} \mathbf {D} \mathbf {m} }}\right \|^{2}}. \tag {16}\end{equation*}$$

The ML detector performance is good when the active index K and modulation scheme M are low. Since the computation complexity of ML detection is $O(M^{K})$ , and it grows exponentially, which make this detector impractical when K and M increase. After looking into this problem we decided to design a low complexity detector that can overcome this computational complexity problem.

B. MMSE-Greedy Detector (GD)

The structure of this detector at the receiver side is shown in the Fig. 3. After performing the FFT, the signal is applied to the MMSE equalizer which multiplies the signal with an equalization matrix denoted by ${E}=diag \{e_{1},e_{2},\ldots,e_{N}\}$ , where

$$\begin{equation*} e_{i}=\frac {h_{i}^{*}}{\left |{{h_{i}}}\right |^{2}+\bar {\gamma }^{-1}} \tag {17}\end{equation*}$$ The new signal is then multiplied with dispreading matrices $D^{-1}$ to retrieve the sending data symbols $$\begin{equation*} \overset {\sim }{\mathbf {m}} = \mathbf {D^{H} Q} y \tag {18}\end{equation*}$$

In (16), FWHT matrices D perform the search in a less computational manner which helps to reduce the complexity of the detector. The GD detector receives the despreading signal $\mathop {\mathbf {m}}\limits ^{\sim }$ and calculates the transmitted symbols $\hat {\mathbf {m}}$ . The estimation of the detector is based on two parts. In the first part, GD identifies the K active indices i.e $\hat {i}=\{\hat {\alpha _{1}}, \hat {\alpha _{2}}\ldots.\hat {\alpha _{K}}\}$ which corresponds to highest energy values $\left \vert {{\mathbf {y}(\alpha) }}\right \vert ^{2}$ . In the second part, the data symbols are detected by using the ML decision

$$\begin{equation*} \hat {\mathbf {m}} \hat {\alpha }= \mathop {argmin}limits_{\mathbf {m}(\hat {\alpha })\in S} \left \|{{{y} (\hat {\alpha }) - \hat {\alpha } {D} \mathbf {m} (\hat {\alpha })}}\right \|^{2}. \tag {19}\end{equation*}$$

SECTION V. Simulation Results of EP-OFDM-IM

In this section, the simulation results of EP-OFDM-IM is presented and compared with the competitive existing bench-mark schemes including IM-OFDM-SS, CI-OFDM-IM, OFDM, and OFDM-IM. The Matlab software is used for the simulation. In the first part, the implementation of various precoding matrices WH, DFT, and FWHT in OFDM-IM has been demonstrated in Fig. 4. For the given system parameters, $N=4$ is the total number of subcarriers, $K=2$ is the active subcarrier and $M=4$ is the modulation scheme. It is observed that FWHT outperforms both WH and DFT. However, DFT provides optimal performance compared to WH. The choice of transform depends on the specific system requirements and trade-offs between BEP performance and computational complexity.

FIGURE 4.

BEP analysis of EP-OFDM-IM using WH, DFT, and FWHT, for (N, K, M) = (4,2,4) and ML detector.

The BEP has been estimated at different configurations $N=4$ , $k=2$ , and $M=2$ , The implementation of DFT shows the optimal performance in the case of 4-phase shift Keying (PSK) while better performance in the case of binary PSK in comparison with the traditional WH precoding matrix. On the other hand, the FWHT spreading matrix demonstrates superior BEP performance as compared to DFT and WH. At 10⁻⁴ BEP, FWHT achieves 15 dB more gain as shown in Fig. 5. The FWHT consistently outperforms both WH and DFT in terms of error correction, achieving the lowest BEP values across all SNR levels. This suggests that FWHT is a promising choice for optimizing error correction in EP-OFDM-IM systems.

FIGURE 5.

BEP performance of EP-OFDM-IM using WH,DFT and FWHT, when (N,K, M) = (4,2,2) and ML detector employed.

Fig. 6 shows the theoretical upper bound on the BEP of EP-OFDM-IM has been compared when various configurations of subcarrier, active subcarrier, and modulations scheme employed. The theoretical bounds become very tight across all precoding matrix cases as the SNR increases. The comparison indicates that the theoretical upper bounds for all three transforms WH, DFT, and FWHT are generally tighter than the simulation results, particularly at higher SNR levels. This suggests that there is room for further optimization of the system to achieve BEP performance closer to the theoretical limits. Additionally, the relative performance of the different transforms varies depending on the specific configuration, highlighting the importance of careful system design to select the most suitable transform for a given application.

FIGURE 6.

Comparison between the simulation results and theoretical upper bounds on the BEP of EP-OFDM-IM for various configurations of (N, K, M) and spreading matrices.

Fig. 7 shows the comparison between the EP-OFDM-IM and the existing competitive benchmark scheme to show the superiority of the proposed scheme. The following configuration has been used for the comparison: PSK modulation scheme, ML detector, 1.5 bit per subcarrier. The results demonstrate that the proposed scheme outperforms the existing benchmark scheme. at 10⁻³ BEP, our proposed scheme achieves 15dB more gain. This indicates that EP-OFDM-IM with FWHT is a promising approach for achieving low BEP rates while maintaining high SE, making it a potential candidate for future wireless communication systems.

FIGURE 7.

BEP performance comparison between EP-OFDM-IM and various existing competitive benchmarks scheme at 1.5 bps/Hz and ML detector employed.

In Fig. 8, the novel scheme EP-OFDM-IM is implemented with low complexity detector MMSE-GD and compared with the ML in the presence of perfect CSI, where the wireless channel experiences no impairments. The results demonstrate that the proposed MMSE-GD achieves optimal performance as compared to ML. It should be noted that the choice between ML and MMSE-GD involves a trade-off between performance and complexity, while the choice of transform depends on system requirements.

FIGURE 8.

BEP performance comparison of EP-OFDM-IM with ML and low-complexity detector MMSE-GD under perfect CSI.

In Fig. 9 the performance of EP-OFDM-IM with MMSE-GD is compared with ML in the presence of imperfect CSI where multiple impairments such as noise, interference, and fading are considered. In this situation, fading effects are not fully compensated for, leading to more distorted signals at the receiver, which increase the BEP more than 10⁻². The results show that even under the worst channel condition our low complexity MMSE-GD with both the efficient matrices DFT and FWHT performs better as compared to ML. However, the FWHT precoded matrix implementation of OFDM-IM with MMSE-GD demonstrates superior performance than the DFT matrix.

FIGURE 9.

BEP performance comparison of EP-OFDM-IM with ML and low-complexity detector MMSE-GD under imPerfect CSI.

In summary, it can be seen that the implementation of OFDM-IM with an efficient precoding matrix FWHT and DFT significantly improves the BEP compared to the current competitive benchmarks scheme. Additionally, the proposed low-complexity detector MMSE-GD makes it possible for OFDM-IM to be practically implemented in the real scenario in highly reliable and low-complexity 6G applications [25]. In EP-OFDM-IM, the selection of subcarriers is based on the application requirements, however, it utilizes fewer subcarriers as compared to traditional OFDM. This technique provides improved SE with less energy, which makes EP-OFDM-IM suitable for Internet of Things (IoT) applications. The high SE and TD of EP-OFDM-IM make it a promising candidate waveform for the next-generation wireless networks where efficiency, flexibility, and performance are critical [26].

SECTION VI. Conclusion

In this paper, a novel scheme called EP-OFDM-IM is proposed, that effectively employs low-complexity, efficient precoding matrices, namely FWHT and DFT, to spread M-ary data symbols across active subcarriers. Additionally, it uses active subcarrier indices to transmit extra bits. Taking into account channel estimation errors, we have derived both an upper bound and an approximate mathematical expression for the BEP. The BEP performance of EP-OFDM-IM and other existing competitive benchmark schemes have been simulated. The results demonstrate that EP-OFDM-IM outperforms these benchmarks. Additionally, a low-complexity detector, MMSE-GD, has been proposed, which outperforms the ML detector under imperfect CSI. This advancement makes the practical implementation of EP-OFDM-IM feasible. This proposed technique will significantly enhance the reliability and efficiency of machine-type communication.