Preamble-Based Signal-to-Noise Ratio Estimation for Adaptive Modulation in Space–Time Block Coding-Assisted Multiple-Input Multiple-Output Orthogonal Frequency Division Multiplexing System

Abstract

This paper presents algorithms to estimate the signal-to-noise ratio (SNR) in the time domain and frequency domain that employ a modified Constant Amplitude Zero Autocorrelation (CAZAC) synchronization preamble, denoted as CAZAC-TD and CAZAC-FD SNR estimators, respectively. These SNR estimators are invoked in a space–time block coding (STBC)-assisted multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) system. These SNR estimators are compared to the benchmark frequency domain preamble-based SNR estimator referred to as the Milan-FD SNR estimator when used in a non-adaptive $2\times 2$ STBC-assisted MIMO-OFDM system. The performance of the CAZAC-TD and CAZAC-FD SNR estimators is further investigated in the non-adaptive $4\times 4$ STBC-assisted MIMO-OFDM system, which shows improved bit error rate (BER) and normalized mean square error (NMSE) performance. It is evident that the non-adaptive $2\times 2$ and $4\times 4$ STBC-assisted MIMO-OFDM systems that invoke the CAZAC-TD SNR estimator exhibit superior performance and approach closer to the normalized Cramer–Rao bound (NCRB). Subsequently, the CAZAC-TD SNR estimator is invoked in an adaptive modulation scheme for a $2\times 2$ STBC-assisted MIMO-OFDM system employing M-PSK, denoted as the AM-CAZAC-TD-MIMO system. The AM-CAZAC-TD-MIMO system outperformed the non-adaptive STBC-assisted MIMO-OFDM system using 8-PSK by about 2 dB at BER = ${10}^{-4}$. Moreover, the AM-CAZAC-TD-MIMO system demonstrated an SNR gain of about 4 dB when compared with an adaptive single-input single-output (SISO)-OFDM system with M-PSK. Therefore, it was shown that the spatial diversity of the MIMO-OFDM system is key for the AM-CAZAC-TD-MIMO system’s improved performance.

1. Introduction

Higher system throughput and greater system capacity are necessary for fifth-generation (5G) mobile communication systems to offer high connectivity for new applications and use cases. One of the key enabling techniques for 5G to improve throughput and capacity is multiple-input multiple-output (MIMO) [1,2] by performing spatial multiplexing of multiple signals. In order to support the growth of wireless communication, 5G New Radio (NR) standards were created by the 3rd Generation Partnership Project (3GPP) to enhance spectrum efficiency for mobile broadband [3]. One of the improvements that 5G NR introduces is the employment of orthogonal frequency division multiplexing (OFDM) as the signal waveform due to its beneficial properties such as high spectral efficiency and robustness to channel fading. More specifically, the multipath challenge in wide-bandwidth channels can be mitigated by employing OFDM that uses a series of narrow-band overlapping sub-carriers, which results in an improved spectral efficiency. Therefore, the demand for higher data throughput over longer distances or under conditions of interference, signal fading, and multipath has driven the development of the MIMO-OFDM communication system.

Alamouti has discovered a simple transmission scheme known as the space–time coding that uses two transmit antennas [4]. Tarokh et al. has developed a generalization of the Alamouti transmission scheme based on the notion of orthogonal designs, which use more than two transmit antennas, leading to the concept of space–time block coding (STBC) [5]. More specifically, in an MIMO system, the STBC scheme allowed multiple copies of the same data stream to be transmitted using a number of transmitter antennas, which resulted in an improved data transfer reliability [6,7]. Therefore, the spatial and temporal variety gained from using STBC in MIMO-OFDM systems will lead to high-rate packet transmission without losing bandwidth, appropriate for high-throughput application [8]. An alternative to STBC in MIMO-OFDM systems is Space–Frequency Block Coding (SFBC), which provides diversity in both space and frequency by using multiple antennas and different sub-carriers. SFBC is especially beneficial in frequency-selective fading environments, offering robust performance with many antennas [9]. However, STBC is chosen in this work because it provides a simpler implementation with lower computational complexity compared to SFBC.

In a fast-paced communication scenario, the channel condition fluctuates rapidly, resulting in an apparent Doppler shift [10]. Adaptive modulation and coding (AMC) is a technology that adjusts the modulation and coding parameters to the dynamic wireless channel condition, hence increasing spectral efficiency [11]. The authors of [12,13,14] investigated adaptive modulation techniques for MIMO-OFDM systems and achieved improved spectral efficiency. An evaluation of the signal-to-noise ratio (SNR) at the receiver can be used to assess the channel state, and the value can be transmitted back to the transmitter via the feedback channel [15].

Generally, SNR estimators can be categorized into data-aided SNR (DA-SNR) estimators and non-data-aided SNR (NDA-SNR) estimators. The DA-SNR estimator uses transmitted pilot data to estimate the SNR at the receiver. This method increases estimation accuracy at the cost of decreased system throughput [16,17,18,19,20]. However, the NDA-SNR estimator blindly estimates the SNR without prior transmitted data. Hence, it does not incur throughput penalty [14,16,19,21,22,23,24]. Despite data overhead, the DA-SNR system outperformed NDA-SNR in terms of estimation accuracy [25,26].

However, some studies have offered methods for mitigating incurred throughput penalties in DA-SNR, such as utilizing OFDM synchronization preambles [17,18,20,27,28,29], also known as preamble-based SNR estimators. Numerous preamble structures have been proposed for time synchronization in OFDM systems [30,31,32]. The preamble-based SNR estimators that were proposed in [17,18,29] also made use of these preambles. The authors in [29] developed an SNR estimator for cooperative SISO-OFDM, which exploited the preamble structure proposed in [30]. However, in [18], an SNR estimator was developed for MIMO-OFDM systems and utilized the synchronization preamble structure proposed by Morelli et al. in [31].

The three key elements to consider when using SNR estimators are estimation accuracy, computational cost, and data overhead. Most proposed SNR estimators for MIMO-OFDM systems in the literature thus far focus on frequency domain SNR estimation, which lowers system performance due to inter-carrier interference [17,18,33]. Frequency domain SNR estimators execute SNR estimates after the OFDM signals have been demodulated. As a result, the computation of the Fast Fourier Transform (FFT) would add computational complexity to SNR estimation. In the literature, SNR estimation algorithms for MIMO-OFDM systems have been developed utilizing two methods: maximum likelihood (ML) [19] and second-order moment (M2) criteria [16,17,18,20]. The SNR algorithm based on these techniques was derived using probabilistic techniques, which are very complex and involve addition and multiplication operations. The ML method is more complex than the M2 approach because it requires optimization to maximize the likelihood function, which demands higher computational power. While ML can be more accurate, the M2 method is simpler and demands less computational power, providing a good balance of accuracy and complexity.

In [34], the suggested SNR estimator took the benefit of the OFDM synchronization preamble structure presented in [32], which employed Constant Amplitude Zero Autocorrelation (CAZAC) sequences. The SNR estimation algorithm of the SNR estimator in [34] was derived using the autocorrelation function, and the SNR estimation was performed in the time domain. However, the proposed preamble-based SNR estimator was employed in an adaptive modulation of a single-input single-output (SISO)-OFDM system. Against this background, this study extends the application of the preamble-based SNR estimator in [34] to an adaptive modulation STBC-assisted MIMO-OFDM system. This adaptation leverages the preamble-based SNR estimator to work effectively with STBC-decoded signals, enhancing signal reliability and diversity in MIMO-OFDM systems, which was not explored in the previous work. While the algorithm in [34] utilized a time domain approach, this study develops a new algorithm to estimate SNR in the frequency domain using the modified CAZAC preamble structure and utilizing M2 criteria. Let us denote the time domain and frequency domain preamble-based SNR estimators as CAZAC-TD and CAZAC-FD SNR estimators, respectively. The performance of both SNR estimators is evaluated when invoked in a non-adaptive STBC-assisted MIMO-OFDM system and is contrasted to the frequency domain preamble-based SNR estimator for the MIMO-OFDM system in [18], which employs M2 criteria in the SNR expression. Let us denote this frequency domain SNR estimator as the Milan-FD SNR estimator. Both CAZAC-TD and CAZAC-FD SNR estimators’ performances are also compared to the normalized Cramer–Rao bound (NCRB) to determine how well both SNR estimators may approach the theoretically best feasible performance, ensuring system efficiency is not compromised. The SNR estimator that exhibits superior performance is then invoked in an adaptive modulation STBC-assisted MIMO-OFDM system. Thus, the adaptive MIMO-OFDM system described in this paper combines the advantages of the preamble-based SNR estimator with the STBC technique.

The contributions of this work are summarized as follows:

• The adaptation of the CAZAC-TD SNR estimator to the STBC-decoded signal: This study extends the application of the CAZAC-TD SNR estimation algorithm to an adaptive modulation STBC-assisted MIMO-OFDM system. This adaptation leverages the preamble-based CAZAC-TD SNR estimator to work effectively with STBC-decoded signals, enhancing signal reliability and diversity in MIMO-OFDM systems.
• The development of the CAZAC-FD SNR estimator: A new CAZAC-FD SNR estimation algorithm based on M2 criteria, similar to the Milan-FD SNR estimator, is developed. The frequency domain version of the CAZAC-TD SNR estimator leverages the modified CAZAC preamble structure for synchronization, resulting in no throughput penalty.
• Comparative performance evaluation: The comparative evaluation of the newly developed CAZAC-FD SNR estimator against the existing Milan-FD SNR estimator, with both estimators derived using similar approaches, ensures a meaningful performance comparison and establishes a benchmark for future SNR estimation methods.
• The performance of preamble-based CAZAC-TD and CAZAC-FD MIMO SNR estimators is evaluated in non-adaptive $2\times 2$ and $4\times 4$ STBC-assisted MIMO-OFDM systems, with the normalized Cramer–Rao bound (NCRB) used as a benchmark for the best achievable performance. By comparing the normalized mean square error (NMSE) of the estimators with the NCRB, valuable insights are gained into how closely the performance of the estimators approaches the theoretical optimum. The modified CAZAC preamble structure, utilized in both time and frequency domain SNR estimators, proves effective in estimating SNR, thereby highlighting the dual-domain functionality of the modified structure.

The paper is organized as follows. Section 2 highlights the related research; Section 3 explains the overall system. Section 4 discusses the development of the preamble-based CAZAC-TD and CAZAC-FD SNR estimators, as well as the SNR threshold for modulation level selection in an adaptive modulation scheme. Section 5 presents the performance of the contrived SNR estimators and the performance of the adaptive STBC-assisted MIMO-OFDM system. Finally, Section 6 presents the conclusion.

2. Related Work

Table 1 lists relevant works on adaptive modulation methods in MIMO systems, covering target BER-based SNR thresholds [35,36], convolutional neural network (CNN)-based spectrogram classification [37,38], artificial neural network (ANN)-driven power spectral density (PSD) learning [39], channel state information (CSI)-driven adaptive algorithms [40], supervised learning [41], and signal-to-interference noise ratio (SINR)-based modulation coding scheme (MCS) selection. These MIMO schemes primarily focus on frequency domain estimation. These strategies have resulted in higher system throughput, spectrum efficiency, and BER performance. However, their reliance on computationally demanding preprocessing operations such as Fourier transforms and windowing causes delays, increases complexity, and makes them sensitive to dynamic channel conditions.

In frequency domain SNR estimation, approaches such as CNN-based PSD analysis [38] and ANN-based AMC [39] require significant computer resources to produce and interpret PSD images or extract spectral characteristics. Similarly, approaches like CSI estimation [40,41] suffer challenges in high-mobility or non-line-of-sight (NLOS) situations, where acquiring accurate CSI becomes increasingly challenging. Furthermore, supervised learning models [37,41] rely on huge, high-quality datasets, making them less adaptable to practical situations with varying fading scenarios and mobility.

Transitioning SNR estimation to the time domain offers a compelling alternative, especially in MIMO-OFDM systems where time synchronization is crucial. For example, employing CAZAC sequences initially intended for synchronization to perform SNR estimation provides two benefits. These time synchronization preambles have strong autocorrelation properties, allowing for precise and consistent SNR estimation while reducing overall system complexity [15,27,34,42]. Using CAZAC sequences for SNR estimation eliminates the need for additional preambles, simplifying the system’s design and increasing efficiency.

Time domain SNR estimation with CAZAC sequences also solves important issues in adaptive modulation. Their strong and consistent estimates make dynamic and optimal threshold selection for AMC possible, even in demanding situations with multipath fading, Doppler effects, and noise fluctuations. High-accuracy SNR estimation is especially useful for modern applications such as 5G [43], massive MIMO systems [36], unmanned aerial vehicle (UAV) communications [37], Internet of Things (IoT), and telemedicine systems [40], which require real-time adaptability and low latency.

To summarize, while previous SNR estimating techniques have greatly improved AMC in wireless systems, time domain estimation employing CAZAC sequences is a viable and efficient alternative. These approaches use the synchronization preamble for SNR estimation, which promises decreased complexity, robust performance, and seamless integration with adaptive modulation, making them a significant enabler for future wireless technologies across multiple domains.

Table 1. Summary of key studies on adaptive modulation methods in MIMO systems.

Year [Ref.]	Estimation Domain	Algorithm and Adaptive Criteria	Contribution	Challenges
2019 [35]	Post-FFT	SNR estimation. Target BER-based SNR switching thresholds.	The selection of the corresponding MIMO mode and its modulation size is based on the received SNR and target bit error rate for unipolar MIMO-OFDM visible light communication (VLC) systems. Improved spectral efficiency.	Developing AM techniques that minimize PAPR while maintaining efficient spectral utilization is critical. High complexity.
2019 [39]	Post-FFT	SNR estimation using ANN exploiting PSD values. Target BER-based SNR switching thresholds.	AMC scheme enabled by ANN-aided SNR estimation in the MIMO system. The PSD values are trained for SNR classification, and it is mapped to respective MCS sets. Improved accuracy of SNR estimation and throughput performance of the system.	Ensuring accurate SNR predictions across diverse channel conditions, such as multipath fading, Doppler shifts, and noise variations, is complex. A mismatch between training data and real-world conditions can lead to poor estimation performance.
2021 [37]	Post-FFT	CNN-trained data for SNR and Doppler estimation. SNR and Doppler shift-based adaptive switching.	It proposes a novel CNN-based joint classification method to characterize the SNR and AMC design using spectrogram images in the MIMO system. Improved accuracy of SNR estimation and throughput performance of the system.	Required optimized models and hardware accelerators to avoid processing delays due to windowing, Fourier transforms, and noise filtering. High complexity. Supervised learning requires a sufficient set of data.
2023 [40]	Post-FFT	CSI estimation. CSI-SNR switching thresholds.	Adaptive algorithm for use in telemedicine communication based on MIMO-OFDM WiMAX standard. Adaptive algorithms can improve the efficiency of the transmitted medical image in 3D MIMO-OFDM system.	The NLOS propagation and high mobility in 3D environments make accurate SNR, CSI, and Doppler estimation more challenging. High complexity to compute CSI table from received instantaneous SNR values.
2020 [36]	Pre-FFT	SINR estimation. Target BER-based SINR switching thresholds.	A third Link adaptation algorithm for an MIMO 5G system was formulated by varying both the modulation index and code rate, to yield an optimal algorithm that achieved the target BER with the highest data rate at any SNR. Improved system throughput.	Higher code rates delayed the achievement of the target BER while yielding higher data rates at high SNRs. High complexity due to simultaneous optimization over modulation schemes, coding rates, MIMO configurations, and scheduling strategies.
2018 [41]	Post-FFT	CSI estimation; singular value decomposition (SVD)-based SNR estimation. Target BER-based MCS thresholds.	A framework based on the supervised learning approach the k-nearest neighbor (k-NN) algorithm for AMC in MIMO-OFDM wireless systems is proposed, with the SVD of the channel matrix and SNR on each spatial stream extracted as a feature set. A classification scheme is then proposed to match channel implementations to different MCSs. Improved system throughput.	Collecting high-quality, labeled datasets for training supervised models is a critical challenge. The need for extensive datasets under diverse channel conditions (e.g., SNR, fading environments, mobility scenarios) increases the complexity of data acquisition.
2022 [43]	Post-FFT	SINR estimation with neural network-based MCS selection.	The paper describes an online deep learning (DL) algorithm for the adaptive modulation and coding in 5G Massive MIMO. The algorithm is based on a fully connected neural network, which is initially trained on the output of the traditional algorithm and then is incrementally retrained by the service feedback of its own output. Improved throughput.	Online DL models must process high-dimensional data in real time, which is computationally demanding. The presence of noise, errors, and missing values in real-time CSI and performance metrics can degrade model performance. Memory overhead.
2018 [44]	Post-FFT	Channel quality indicator (CQI) estimation. SINR estimates used to adapt to distinct modulation schemes are found through a CQI table lookup.	Performance of adaptive modulation scheme with CQI feedback in LTE MIMO system is presented. To compute the modulation scheme and the coding rate outputs, a table lookup operation with the CQI index is used with the measured SINR. Improved system efficiency.	CQI feedback is often delayed due to system latencies, leading to mismatches between the actual channel conditions and the reported CQI. Interference and noise levels affect the reliability of CQI feedback and the resulting modulation decisions with high complexity.
2020 [38]	Post-FFT	SNR estimation using CNN. Target BER-based SNR switching thresholds.	This paper proposes a highly accurate SNR estimation method for AMC by learning PSD images with a CNN in an MIMO OFDM system. Accurate SNR estimation and improved system throughput and BER.	Generating PSD images involves transforming time domain signals into the frequency domain, which is computationally expensive, so CNNs’ processing of it increases complexity. High-quality PSD images that accurately reflect channel conditions require precise signal processing that is challenging in low-SNR environments.

Table 1. Summary of key studies on adaptive modulation methods in MIMO systems.

Year [Ref.]	Estimation Domain	Algorithm and Adaptive Criteria	Contribution	Challenges
2019 [35]	Post-FFT	SNR estimation. Target BER-based SNR switching thresholds.	The selection of the corresponding MIMO mode and its modulation size is based on the received SNR and target bit error rate for unipolar MIMO-OFDM visible light communication (VLC) systems. Improved spectral efficiency.	Developing AM techniques that minimize PAPR while maintaining efficient spectral utilization is critical. High complexity.
2019 [39]	Post-FFT	SNR estimation using ANN exploiting PSD values. Target BER-based SNR switching thresholds.	AMC scheme enabled by ANN-aided SNR estimation in the MIMO system. The PSD values are trained for SNR classification, and it is mapped to respective MCS sets. Improved accuracy of SNR estimation and throughput performance of the system.	Ensuring accurate SNR predictions across diverse channel conditions, such as multipath fading, Doppler shifts, and noise variations, is complex. A mismatch between training data and real-world conditions can lead to poor estimation performance.
2021 [37]	Post-FFT	CNN-trained data for SNR and Doppler estimation. SNR and Doppler shift-based adaptive switching.	It proposes a novel CNN-based joint classification method to characterize the SNR and AMC design using spectrogram images in the MIMO system. Improved accuracy of SNR estimation and throughput performance of the system.	Required optimized models and hardware accelerators to avoid processing delays due to windowing, Fourier transforms, and noise filtering. High complexity. Supervised learning requires a sufficient set of data.
2023 [40]	Post-FFT	CSI estimation. CSI-SNR switching thresholds.	Adaptive algorithm for use in telemedicine communication based on MIMO-OFDM WiMAX standard. Adaptive algorithms can improve the efficiency of the transmitted medical image in 3D MIMO-OFDM system.	The NLOS propagation and high mobility in 3D environments make accurate SNR, CSI, and Doppler estimation more challenging. High complexity to compute CSI table from received instantaneous SNR values.
2020 [36]	Pre-FFT	SINR estimation. Target BER-based SINR switching thresholds.	A third Link adaptation algorithm for an MIMO 5G system was formulated by varying both the modulation index and code rate, to yield an optimal algorithm that achieved the target BER with the highest data rate at any SNR. Improved system throughput.	Higher code rates delayed the achievement of the target BER while yielding higher data rates at high SNRs. High complexity due to simultaneous optimization over modulation schemes, coding rates, MIMO configurations, and scheduling strategies.
2018 [41]	Post-FFT	CSI estimation; singular value decomposition (SVD)-based SNR estimation. Target BER-based MCS thresholds.	A framework based on the supervised learning approach the k-nearest neighbor (k-NN) algorithm for AMC in MIMO-OFDM wireless systems is proposed, with the SVD of the channel matrix and SNR on each spatial stream extracted as a feature set. A classification scheme is then proposed to match channel implementations to different MCSs. Improved system throughput.	Collecting high-quality, labeled datasets for training supervised models is a critical challenge. The need for extensive datasets under diverse channel conditions (e.g., SNR, fading environments, mobility scenarios) increases the complexity of data acquisition.
2022 [43]	Post-FFT	SINR estimation with neural network-based MCS selection.	The paper describes an online deep learning (DL) algorithm for the adaptive modulation and coding in 5G Massive MIMO. The algorithm is based on a fully connected neural network, which is initially trained on the output of the traditional algorithm and then is incrementally retrained by the service feedback of its own output. Improved throughput.	Online DL models must process high-dimensional data in real time, which is computationally demanding. The presence of noise, errors, and missing values in real-time CSI and performance metrics can degrade model performance. Memory overhead.
2018 [44]	Post-FFT	Channel quality indicator (CQI) estimation. SINR estimates used to adapt to distinct modulation schemes are found through a CQI table lookup.	Performance of adaptive modulation scheme with CQI feedback in LTE MIMO system is presented. To compute the modulation scheme and the coding rate outputs, a table lookup operation with the CQI index is used with the measured SINR. Improved system efficiency.	CQI feedback is often delayed due to system latencies, leading to mismatches between the actual channel conditions and the reported CQI. Interference and noise levels affect the reliability of CQI feedback and the resulting modulation decisions with high complexity.
2020 [38]	Post-FFT	SNR estimation using CNN. Target BER-based SNR switching thresholds.	This paper proposes a highly accurate SNR estimation method for AMC by learning PSD images with a CNN in an MIMO OFDM system. Accurate SNR estimation and improved system throughput and BER.	Generating PSD images involves transforming time domain signals into the frequency domain, which is computationally expensive, so CNNs’ processing of it increases complexity. High-quality PSD images that accurately reflect channel conditions require precise signal processing that is challenging in low-SNR environments.

3. System Description

Figure 1 shows the block diagram of the proposed adaptive modulation with STBC for a $2\times 2$ MIMO-OFDM system that invokes the CAZAC-TD and CAZAC-FD SNR estimators. The input data are initially modulated using the M-PSK scheme. The modulated signal is then converted to the time domain via the Inverse-Fast Fourier Transformation (IFFT). Then, a cyclic prefix $C_p$ is appended to each OFDM symbol, resulting in the OFDM signal vector, $S=(S_1,S_2,S_3,\dots ,S_i)$, where i represents the indexes of each OFDM symbol. A block of two scalar OFDM-modulated symbols $S_1$ and $S_2$ is STBC-encoded in two consecutive time slots according to the following code matrix:

$$\mathbf{G}=\left[\begin{array}{cc}{S}_1& S_2\\ {-S_2}^{*}& {S_1}^{*}\end{array}\right]$$

(1)

This matrix G consists of two rows that represent two time slots and two columns that represent two antennas. In the first time slot, the first transmit antenna $T_{x1}$ transmits symbol $S_1$ while symbol $S_2$ is transmitted from the second transmit antenna, $T_{x2}$. In the second time slot, symbol $-{S_2}^{*}$ is transmitted from the first transmit antenna $T_{x1}$, and the second transmit antenna $T_{x2}$ transmits symbol ${S_1}^{*}$. Here, symbols ${S_1}^{*}$ and ${S_2}^{*}$ denote the complex conjugates of $S_1$ and $S_2$, respectively. In STBC, the complex conjugates of symbols are utilized to form the G matrix, also known as the encoding matrix, which ensures orthogonality between the signals transmitted from different antennas. This orthogonality is essential for reducing interference between antennas and enhancing the receiver’s ability to recover the transmitted signals. As a result, the structure of the matrix enables accurate symbol decoding and improves the resistance of the system to fading and interference.

For transmission over a Rayleigh fading channel, the received signals are impacted by the channel matrix H, and the received signal Y matrix at each receiver antenna can be represented as:

$$\mathbf{Y}=\mathbf{HG}+\mathbf{Nc}$$

(2)

where $\mathbf{H}=\left[\begin{array}{c}{H}_{11}{H}_{12}\\ H_{21}{H}_{22}\end{array}\right]$ is the channel matrix that models the fading effects of the channel and $\mathbf{Nc}=\left[\begin{array}{c}{N}_{11}{N}_{12}\\ N_{21}{N}_{22}\end{array}\right]$ is the noise matrix, which represents the noise introduced at each receiver antenna. The received signal matrix Y, with scalar elements $Y_{11}$, $Y_{21}$, $Y_{12}$, and $Y_{22}$, represents the received signals at the respective antennas.

More specifically, at the receiver, during the first time slot, signal $Y_{11}$ is received at the first receiver antenna, $R_{x1}$, and $Y_{21}$ is received at the second receiver antenna, $R_{x2}$, as described below:

$$Y_{11}=H_{11}{S}_1+H_{12}{S}_2+N_{11}$$

(3)

$$Y_{21}=H_{21}{S}_1+H_{22}{S}_2+N_{21}$$

(4)

where $H_{11}$ and $H_{21}$ are the coefficients of channel matrix H for transmission from $T_{x1}$ to $R_{x1}$ and $R_{x2}$, respectively, while $H_{12}$ and $H_{22}$ are the coefficients of channel matrix H corresponding to the transmissions from $T_{x2}$ to $R_{x1}$ and $R_{x2}$, respectively. The noise signals at $R_{x1}$ and $R_{x2}$ during the first time slot are denoted as $N_{11}$ and $N_{21}$, respectively.

Subsequently, during the second time slot, the received signals $Y_{12}$ and $Y_{22}$ at $R_{x1}$ and $R_{x2}$, respectively, can be written as:

$$Y_{12}=-H_{11}{S}_2^{*}+H_{12}{S}_1^{*}+N_{12}$$

(5)

$$Y_{22}=-H_{21}{S}_2^{*}+H_{22}{S}_1^{*}+N_{22}$$

(6)

where $N_{12}$ and $N_{22}$ denote the noise signals at $R_{x1}$ and $R_{x2}$ during the second time slot. Thus, the transmitted OFDM symbols can be estimated using a Maximum Ratio Combiner (MRC) as follows:

$$\widehat{S_1}=H_{11}^{*}{Y}_{11}+H_{12}{Y}_{12}^{*}+H_{21}^{*}{Y}_{21}+H_{22}{Y}_{22}^{*}$$

(7)

$$\widehat{S_2}=H_{12}^{*}{Y}_{11}-H_{11}{Y}_{12}^{*}+H_{22}^{*}{Y}_{21}-H_{21}{Y}_{22}^{*}$$

(8)

where $Y_{12}^{*}$ and $Y_{22}^{*}$ are the complex conjugates of $Y_{12}$ and $Y_{22}$, respectively, while $H_{11}^{*}$, $H_{21}^{*}$, $H_{12}^{*}$, $H_{22}^{*}$ are the complex conjugates of $H_{11}$, $H_{21}$, $H_{12}$, $H_{22}$, respectively.

Finally, the STBC decoder decodes the received complex-valued symbols and is denoted as $Y_{dstbc}$. The instantaneous SNR is estimated using the OFDM symbols $Y_{dstbc}$. This value is fed to the transmitter via a feedback channel, as illustrated in Figure 1, and used for selecting the modulation level of M-PSK. The SNR thresholds’ criteria for the modulation level selection are presented in Section 4.3.

4. SNR Estimation

This section presents the development of the preamble-based CAZAC-TD and CAZAC-FD SNR estimation algorithms, along with the SNR threshold for modulation level selection in an adaptive modulation scheme. Both algorithms are derived under the assumption of signal transmission over an Additive White Gaussian Noise (AWGN) channel. To assess their robustness, simulations are conducted considering signal transmission over an Stanford University Interim-5 (SUI-5) channel in an NLOS environment. The results show consistent performance across both channel models, suggesting that the algorithms may generalize well to more complex, real-world scenarios. However, further validation in dynamic and practical conditions is needed to fully assess their applicability.

This work proposes employing CAZAC-TD and CAZAC-FD SNR estimators for $2\times 2$ and $4\times 4$ STBC-assisted MIMO-OFDM systems. Several preamble-based SNR estimators that utilized synchronization preambles for OFDM systems have been proposed in the literature to mitigate incurred throughput penalties in DA-SNR, as discussed in Section 1.

The innovative preamble structure for timing synchronization in OFDM systems was proposed by Suparna et al. [32]. For the purpose of time synchronization improvement, the proposed preamble was composed of one OFDM symbol with a short preamble structure having four equal parts, each of $N_S/4$ length, as depicted in Figure 2, where $N_S$ is the IFFT size, which represents the total sub-carriers per OFDM symbol. $C_Z$ denotes the CAZAC sequence, $P_N$ is the pseudo-noise (PN) sequence, and W denotes the weighted CAZAC sequence, which is obtained by bit-wise multiplication of $C_Z$ with the same length of $P_N$, while $S\left(W\right)$ represents the multiplication of the weighted CAZAC sequence, with W a scrambling function, which aims to mitigate the narrow-band interference effect. Let us denote this preamble structure as the Suparna preamble structure.

As a further development, the preamble structure is modified as illustrated in Figure 3, where the CAZAC sequence is used because of its good autocorrelation properties [42]. Figure 4 shows the proposed preamble structure with the added cyclic prefix $C_P$ of $N_S/4$ length that is utilized in the proposed CAZAC-TD and CAZAC-FD SNR estimators. Using IEEE802.16d OFDM standard, the CAZAC preamble is loaded on even sub-carriers, and the odd sub-carriers are nullified. The modified preamble structure is composed of one OFDM symbol having four equal parts, each part with $N_S/4=64$ bits length, where $N_S=256$ bits is the length of the signal without the cyclic prefix, while the cyclic prefix $C_P$ is added to mitigate inter-symbol interference and inter-carrier interference between the received OFDM symbols.

For both the CAZAC-TD and CAZAC-FD estimators, the modified CAZAC preamble is loaded in even sub-carriers, and the odd sub-carriers are not used (nulled). The proposed CAZAC-TD and CAZAC-FD SNR estimators are compared with the Milan-FD SNR estimator, presented in [18]. The Milan-FD SNR estimator utilized the preamble structure proposed in [31], which was composed of one OFDM symbol having Q equal parts, each of $N_S/Q$ length, where $Q>2$ and $N_S$ is the IFFT size. For the sake of benchmarking, the proposed SNR estimators are evaluated against the Milan-FD SNR estimator, which utilized the preamble structure in Figure 5, with Q = 4 and $N_S$ = 256 bits.

4.1. Time Domain SNR Estimation Using Autocorrelation

Figure 6 presents the flowchart of the CAZAC-TD SNR estimation algorithm. This algorithm utilizes the autocorrelation function of the STBC decoder output signal, $Y_{dstbc}$, to estimate the signal and noise power. The autocorrelation of the STBC decoder output signal, $Y_{dstbc}$, is given as:

$$R_{Y{Y}_{dstbc}}\left(k\right)=R_{XX}\left(k\right)+R_{N{N}_c}\left(k\right)$$

(9)

for $0\le k\le N_S-1$ denotes the autocorrelation of the transmitted signal, while $R_{N{N}_c}\left(k\right)$ denotes the autocorrelation of the noise signal for the AWGN channel with the noise variance, ${\sigma }^2$, given as:

$$R_{N{N}_c}\left(k\right)={\sigma }^2\delta \left(k\right)$$

(10)

where $\delta \left(k\right)$ represents the Dirac delta function. Similarly, the autocorrelation of the OFDM transmitted signal can be expressed as:

$$R_{XX}\left(k\right)=P_{SS}\delta \left(k\right)$$

(11)

where $\delta \left(k\right)$ represents the delta function and $P_{SS}$ is the signal power. Hence, at zero lag, the STBC-decoded signal’s autocorrelation consists of both the signal and the noise power. On the other hand, the autocorrelation of the transmitted OFDM signal consists of signal power only. Thus, the difference between the STBC-decoded signal’s autocorrelation value at zero lag and the estimated signal power can be used to estimate noise power. Figure 7 compares the autocorrelation of the transmitted OFDM signal and the corresponding received STBC-decoded signal over an SUI-5 channel at an SNR of 12 dB, utilizing the modified CAZAC preamble structure. Specifically, Figure 7a shows the autocorrelation plot of the transmitted OFDM signal S at SNR = 12 dB, and the autocorrelation plot of the corresponding STBC-decoded signal is shown in Figure 7b.

In Figure 7, the X-axis represents the lag between the signal and its shifted version, and the Y-axis represents the autocorrelation values at each lag. It can be seen from Figure 7 that there is one main peak at $L_S$, and there are two side peaks on its right and left sides. The two side peaks on the left side appeared at the specific lags of $L_S-N_S$ and $L_S-N_S/2$.

As a result, the estimation of the signal power can be written as:

$$P_{SS.est}=2R_{Y{Y}_{dstbc}}(L_S-N_S/2)-R_{Y{Y}_{dstbc}}(L_S-N_S)$$

(12)

where $L_S=N_S+N_{C_P}$ denotes the total length of the OFDM symbol of length $N_S$ with added $C_P$ and $N_{C_P}=N_S/4$ is the $C_P$ length. More specifically, the signal power is estimated by using the side peaks that result from the autocorrelation of the received STBC-decoded signal. The side peak at $(L_S-N_S)$ rises when the cyclic prefix $C_P$ overlaps with itself, while the side peak at $(L_S-N_S/2)$ rises when part of the preamble, along with the $C_P$, overlaps with itself. The subtraction between these two side peaks helps to eliminate the impact of the $C_P$, ensuring that the signal power estimation is not distorted by the cyclic prefix. This allows for a more accurate estimate of the true signal power.

As seen from Figure 7b, the main peak of the STBC-decoded signal’s autocorrelation includes the noise power. Hence, the noise power can be estimated by computing the difference between the STBC-decoded signal’s autocorrelation and the estimated signal power, $P_{SS.est}$. The estimation of noise power is given as:

$${\sigma }_{N.est}^2=R_{Y{Y}_{dstbc}}\left(L_S\right)-\left(P_{SS.est}\right)$$

(13)

where $R_{Y{Y}_{dstbc}}\left(L_S\right)$ is the maximum peak indicating the STBC-decoded signal’s autocorrelation value at zero lag. Therefore, the estimation of the SNR is given as:

$$SN{R}_{MIMO-est}={\displaystyle \frac{P_{SS.est}}{{\sigma }_{N.est}^2}}$$

(14)

Subsequently, the NMSE performance of the SNR estimator can be quantified using the following equation:

$$NMSE={\displaystyle \frac{1}{M}}\sum _{m=0}^{M-1}{\left({\displaystyle \frac{SN{R}_{MIMO-est}-SN{R}_{act}}{SN{R}_{act}}}\right)}^2$$

(15)

where the number of estimated instantaneous SNR at which the NMSE is quantified is denoted as M and the actual SNR is denoted as $SN{R}_{act}$.

4.2. Proposed Frequency Domain CAZAC-FD SNR Estimation

The CAZAC-FD SNR estimation algorithm is developed based on the second-order moment criteria where the SNR estimation is performed in the frequency domain after performing FFT processing on the STBC decoder output signal, $Y_{dstbc}$. The CAZAC-FD SNR estimator utilizes the modified CAZAC preambles, shown in Figure 4, having Q equal parts, each of $N_S/Q$ length, where Q > 2 and $N_S$ is the FFT size. For the CAZAC-FD estimator algorithm, where the modified CAZAC preamble, $X_P\left(m\right)$ for $m=0,\dots (N_S-1)$, is loaded on even sub-carriers, and the odd sub-carriers are not used (null). Therefore, the transmitted CAZAC preamble signal on the kth sub-carrier can be expressed as:

$$X\left(k\right)=X(2m+q)=\left\{\begin{array}{cc}{X}_P\left(m\right),\hfill & q=0\hfill \\ 0,\hfill & q=1\hfill \end{array}\right.$$

(16)

Thus, the received signal on the loaded sub-carrier can be expressed as [18]:

$$Y_{dstbc}\left(k\right)=Y_{dstbc}\left(2m\right)=\sqrt{S}{X}_p\left(m\right)H_p\left(m\right)+\sqrt{W}\sigma \left(m\right)$$

(17)

where S is the total transmit power and $H_p\left(m\right)$ is the channel response on the loaded sub-carriers. W is the noise power on each sub-carrier and $\sigma \left(m\right)$ is the corresponding sampled zero-mean AWGN with unit variance.

The received signal on the nulled sub-carriers consists of the only noise signal and is given as:

$$Y_{dstbc}\left(k\right)=Y_{dstbc}(2m+1)=\sqrt{W}\sigma (2m+1)$$

(18)

The CAZAC-FD SNR estimation algorithm was developed based on the second-order moment of the OFDM demodulated signal to estimate the SNR at the receiver. Thus, the STBC-decoded signal is OFDM-demodulated and denoted as $Y_{(d-ofdm)}$. Therefore, the second-order moment on $Y_{(d-ofdm)}$ by using equations in [18]:

$$P_{RS}={\displaystyle \frac{1}{N_s}}\sum _{k=1}^{N_s-1}{\left|Y_{(d-ofdm)}\left(2m\right)\right|}^2$$

(19)

Similarly, the received noise power from the nulled sub-carriers is given as:

$$P_{RN}={\displaystyle \frac{1}{N_s}}\sum _{m=1}^{N_s-1}{\left|Y_{(d-ofdm)}(2m+1)\right|}^2$$

(20)

Thus, the SNR estimation can be determined using the following:

$$SN{R}_{MIMO-est}={\displaystyle \frac{P_{RS}-P_{RN}}{P_{RN}}}$$

(21)

The NMSE performance of the CAZAC-FD SNR estimator can be quantified using Equation (15).

4.3. SNR Thresholds for Adaptive Modulation Switching

The SNR switching thresholds are determined from the bit error ratio (BER) curve for M-PSK that is invoked in the non-adaptive STBC assisted for a $2\times 2$ MIMO-OFDM system, when targeting a BER of about ${10}^{-3}$ [39,45].

Table 2. M-PSK modulation switching threshold for $2\times 2$ MIMO-OFDM system.

Modulation	SNR Threshold
QPSK	SNR ≤ 9 dB
8-PSK	9 dB < SNR ≤ 12 dB
16-PSK	12 dB < SNR ≤ 16 dB
32-PSK	16 dB < SNR ≤ 19 dB
64-PSK	SNR > 19 dB

outlines the switching thresholds that are derived from the BER curve for M-PSK for the transmission SUI-5 channel shown in Figure 8.

5. Results and Discussion

The performances of the CAZAC-TD and CAZAC-FD SNR estimators and the corresponding adaptive modulation scheme with STBC for the $2\times 2$ and $4\times 4$ MIMO-OFDM systems in Figure 1 are presented in this section.

Section 5.1 compares the performance of the SISO-OFDM system that uses the modified CAZAC preamble structure of Figure 3 to that of the Suparna preamble structure of Figure 2 using the autocorrelation plot and the NMSE performance, while the performance of the CAZAC-TD and CAZAC-FD SNR estimators that are invoked in the non-adaptive STBC-assisted $2\times 2$ and $4\times 4$ MIMO-OFDM systems are discussed in Section 5.2.

The performance of the proposed adaptive STBC-assisted MIMO-OFDM system that invokes the CAZAC-TD SNR estimator for transmission over the SUI-5 channel is presented in Section 5.3. Let us refer to this adaptive scheme as the AM-CAZAC-TD-MIMO system. The proposed AM-CAZAZ-TD-MIMO system employing M-PSK dynamically selects the modulation scheme to match the channel condition.

Table 3. Simulation parameters for OFDM system [47].

IFFT size, $N_S$ (bits)	256
Sampling frequency, $f_s$ (Hz)	$20\times {10}^6$
Sub-carrier spacing, $\Delta f_s$ (Hz)	$1\times {10}^5$
Symbol duration, $t_{symb}=1/\Delta f_s$ (s)	$1\times {10}^{-5}$
Guard interval duration, $t_{gt}=G_t\times t_{symb}$ (s), where $G_t=1/4$	$2.5\times {10}^{-6}$
OFDM symbol duration, $t_{ofdm}=t_{symb}+t_{gt}$ (s)	$1.25\times {10}^{-5}$

summarizes the simulation parameters used in this study, which are selected based on the IEEE802.16d standard [46,47]. Moreover, the modified CAZAC preamble of Figure 3, with a frame length of $L_S=320$ bits, is transmitted on even sub-carriers, while the odd sub-carriers are set to zero values. In this work, the MATLAB $rand(N,M)$ function is used to generate pseudorandom numbers of length N, producing an $N-by-M$ matrix. The MATLAB Communications Toolbox is used for STBC encoding and decoding, OFDM modulation and demodulation, and channel definition. Fourier transforms are applied using the FFT and IFFT functions to switch between the frequency and time domains. Autocorrelation for SNR estimation is performed with the $xcorr\left(data\right)$ function. Simulations are run on an HP ProBook 6560b laptop with an Intel® Core™ i5-2410M processor, 4 GB RAM, 256 GB HDD, AMD Radeon HD 6470M graphics, and Windows 7 Professional.

Meanwhile, the parameters used for the SUI-5 channel simulation are summarized in

Table 4. Simulation parameters for SUI-5 channel [48].

	Tap-1	Tap-2	Tap-3
Delay spread (μs)	0	4	10
Power 30° directional antenna (dB)	0	−11	−22
Rician K-factor	2	0	0

. The SUI-5 channel model is a multipath Rician channel that is adopted by the IEEE802.16d standard, comprising three outdoor-terrain categories’ real-time data, and the SUI-5 channel is the channel model for hilly terrain with high tree density [48,49].

5.1. Modified CAZAC Preamble Structure Performance

Figure 9 compares the autocorrelation of the OFDM received signal transmitted over an AWGN channel at an SNR of 12 dB, utilizing both the Suparna preamble structure and the modified CAZAC preamble structure. Specifically, Figure 9a illustrates the autocorrelation plot of the received OFDM signal, S, when the Suparna preamble structure is used in the SISO-OFDM system. In contrast, Figure 9b shows the autocorrelation plot of the received OFDM signal, S, when the modified CAZAC preamble structure is employed.

The OFDM system is very sensitive to frequency and timing offset because it can cause inter-carrier interference, resulting in performance degradation. It can be seen from Figure 9a that the autocorrelation plot for the Suparna preamble structure has a sharp peak to determine the start of the incoming signal at the receiver. Hence, it provides timing synchronization beneficial for OFDM system performance. However, the sub-peaks on both sides of the main peak can cause timing ambiguity when considering transmission over frequency-selective channels. On the other hand, there are no sub-peaks on both sides of the main peak in the autocorrelation plot for the modified preamble structure, as seen in Figure 9b. This leads to more precise synchronization and better SNR estimation.

The performance of an SISO-OFDM system utilizing the modified CAZAC preamble structure and the Suparna preamble structure is also compared in terms of NMSE, as shown in Figure 10. The SISO-OFDM system, which uses the CAZAC-modified preamble structure of Figure 3, outperforms the system with the Suparna preamble structure. It is evident that the modified CAZAC preamble has improved the NMSE by 2 dB at ${10}^{-2}$. These improvements highlight how the modified CAZAC sequences may improve OFDM system performance.

5.2. CAZAC-TD and CAZAC-FD SNR Estimators’ Performance

This section presents three sets of results to evaluate the performance of the CAZAC-TD and CAZAC-FD SNR estimators used in non-adaptive modulation schemes with STBC for the $2\times 2$ MIMO-OFDM system compared to that of the Milan-FD SNR estimator in terms of estimated SNR, NMSE, and BER. Furthermore, the performance of the CAZAC-TD and CAZAC-FD SNR estimators is examined when employed in the non-adaptive $4\times 4$ STBC-assisted MIMO-OFDM system. Each set of results considers transmission over both AWGN and SUI-5 channels to study the performance of the preamble-based SNR estimator under varying conditions. In this simulation, the systems use QPSK as the modulation technique, and the estimated SNR is calculated by average over 2000 iterations.

The first set of results evaluates the performance of the preamble-based SNR estimator in terms of the estimated SNR. Figure 11 compares the estimated SNR performance to the actual SNR for transmission over the AWGN channel. A zoomed-in portion of the plot is shown in the inset, providing a closer view of the details. This zoomed-in view illustrates that the estimated SNR using the CAZAC-TD SNR estimator matches the actual SNR for non-adaptive $2\times 2$ and $4\times 4$ STBC-assisted MIMO-OFDM systems transmitting over the AWGN channel. The CAZAC-FD SNR estimator for the non-adaptive $4\times 4$ STBC-assisted MIMO-OFDM system exhibits similar SNR estimation performance. However, in the $2\times 2$ STBC-assisted MIMO-OFDM system, the differences between the actual SNR and the estimated SNR using CAZAC-FD and Milan-FD SNR estimators are 0.12 dB and 0.16 dB, respectively.

However, as shown in the inset of Figure 12, SNR estimator performance deteriorated for transmission over the SUI-5 channel. In particular, the Milan-FD SNR estimator shows estimated SNR differences of about 0.26 dB, but the CAZAC-TD and CAZAC-FD SNR estimators used in the $2\times 2$ and $4\times 4$ STBC-MIMO-OFDM systems show estimated SNR differences below 0.2 dB. A good SNR estimator is the one with the smallest estimated SNR difference.

The performance of the CAZAC-TD and CAZAC-FD SNR estimators are also evaluated against the NCRB for a frequency-selective channel to assess how effectively the SNR estimators’ performance approaches the theoretical optimum. The Cramer–Rao bound (CRB) was derived in [50] as follows:

$$CR{B}_{SN{R}_{MIMO-Est}}={\left[{\displaystyle \frac{1+Q\left(SN{R}_{act}\right)}{\sqrt{N_s(M-1)}}}\right]}^2$$

(22)

where $N=256$ bits and $Q=4$ are the preamble parts, as discussed in Section 4. The variance of CRB can be found by taking the inverse of the Fisher information matrix (FIM) [50]. Hence, the NCRB can be obtained by dividing Equation (22) by $(SN{R}_{act}{)}^2$ and can be written as:

$$NCR{B}_{\left(SN{R}_{Est}\right)}={\displaystyle \frac{CR{B}_{SN{R}_{MIMO-Est}}}{{(SN{R}_{act})}^2}}$$

(23)

where $SN{R}_{MIMO-Est}$ for the CAZAC-TD and CAZAC-FD SNR estimators can be calculated using Equation (14) and Equation (21), respectively.

In the second set of results, the performance of the CAZAC-TD and CAZAC-FD SNR estimators is evaluated in terms of NMSE for transmission over AWGN and SUI-5 channels. The corresponding results are illustrated in Figure 13 and Figure 14, respectively. NMSE serves as a key metric for assessing the accuracy of the SNR estimations, with lower values indicating more precise estimations. Comparisons of the NMSE values for the two estimators against the Milan-FD estimator provide insight into the relative effectiveness of each method under different channel conditions. It is evident that the non-adaptive $2\times 2$ and $4\times 4$ STBC-assisted MIMO-OFDM system performs better and approaches closer to the NCRB when using the CAZAC-TD SNR estimator. The NMSE performance of the non-adaptive $2\times 2$ STBC-assisted MIMO-OFDM system that invoked CAZAC-TD and CAZAC-FD was further enhanced when invoked in the $4\times 4$ STBC-assisted MIMO-OFDM system for both the AWGN and SUI-5 channels.

In the third set of results, the CAZAC-TD and CAZAC-FD SNR estimators are also evaluated in terms of BER for AWGN and SUI-5 channels, as shown in Figure 15 and Figure 16, respectively. As seen from Figure 15, the $2\times 2$ non-adaptive STBC-assisted MIMO-OFDM system that employed the CAZAC-TD SNR estimator achieved a significant BER performance improvement compared to that of the corresponding CAZAC-FD and Milan-FD SNR estimators. Specifically, at BER = ${10}^{-4}$, the MIMO-OFDM system with the CAZAC-TD SNR estimator was capable of enhancing the achievable SNR performance by about 2 dB and 0.3 dB over the Milan-FD and CAZAC-FD SNR estimators, respectively, when communicating over the AWGN channel. The BER performance of the CAZAC-TD SNR estimator was further enhanced, exhibiting an improvement in SNR gain by about 0.2 dB when it was invoked in the $4\times 4$ STBC-assisted MIMO-OFDM system.

For transmission over the SUI-5 channel, at BER =${10}^{-4}$, the non-adaptive $2\times 2$ and $4\times 4$ STBC-assisted MIMO-OFDM system employing the CAZAC-TD SNR estimator exhibited SNR gain by about 0.75 dB and 0.18 dB, when comparing with Milan-FD and CAZAC-FD SNR estimators, respectively, as seen in Figure 16. For both the AWGN and SUI-5 channels, the CAZAC-TD SNR estimator outperformed the Milan-FD and CAZAC-FD SNR estimators, as seen in Figure 15 and Figure 16, respectively.

The floating point operations per second (FLOPs) complexity metric can be used to assess the complexity of the CAZAC-TD SNR estimation algorithm. Generally, FLOPs refer to the number of computations needed for a single SNR estimate. As stated in Section 4.1, CAZAC-TD SNR estimation is based on the autocorrelation of the STBC-decoded signal, which is performed in the time domain. The signal power and noise power are estimated using Equation (12) and Equation (13), respectively. Generally, the autocorrelation function of the STBC-decoded signal is the product of the STBC-decoded signal and its lagged version at each time step, and then these products are summed for all time steps within the overlapping range. Therefore, the computational complexity is only based on the multiplication of N bits and $(N-1)$ additions. Hence, the CAZAC-TD SNR estimator required $(N+N-1=2N-1)$ FLOPs for one SNR estimation, while the second-order moment criteria are applied to the OFDM-demodulated signal in both the Milan-FD and CAZAC-FD SNR estimators, which involves $(4N+2)$ FLOPs to compute one SNR estimate [18].

5.3. Adaptive Modulation Scheme Performance

This section presents two sets of performance results to evaluate the proposed adaptive scheme. The first set compares the BER and channel capacity performance of the adaptive AM-CAZAC-TD-MIMO system against a non-adaptive MIMO-OFDM scheme, highlighting the advantages of the adaptive approach. The second set examines the BER and throughput performance of the adaptive MIMO-OFDM scheme, comparing it with the adaptive SISO-OFDM scheme.

Figure 17 and Figure 18 show the performance of adaptive modulation in a $2\times 2$ STBC-assisted MIMO-OFDM system using the CAZAC-TD SNR estimator in terms of BER and channel capacity. Let us refer to this adaptive scheme as the AM-CAZAC-TD-MIMO system. The AM-CAZAC-TD-MIMO system uses M-PSK, with a switching SNR threshold, as shown in Table 2. More specifically, the adaptive MIMO-OFDM system adjusts its modulation level based on the estimated channel SNR, as outlined in Table 2. Its performance is compared to that of the non-adaptive $2\times 2$ STBC-assisted MIMO-OFDM system. Figure 17 shows that, at BER = ${10}^{-4}$, the AM-CAZAC-TD-MIMO system with M-PSK outperformed the non-adaptive STBC-assisted MIMO-OFDM system using 8-PSK by about 2 dB. This improvement is clearly reflected in the performance curves, where the adaptive AM-CAZAC-TD-MIMO system requires a lower SNR to achieve the same target BER. Moreover, when targeting a specific BER, the adaptive MIMO-OFDM system shows greater SNR reductions compared to the non-adaptive MIMO-OFDM system with a fixed modulation level, as the modulation order M increases. This suggests that the adaptive MIMO-OFDM scheme requires less SNR to maintain reliable performance across various modulation schemes.

The channel capacity is quantified using the following equation [51]:

$$C_{stbc-MIMO}=R·{log}_2(1+SN{R}_{MIMO-Est})$$

(24)

where $SN{R}_{MIMO-Est}$ for the CAZAC-TD SNR estimator can be calculated using Equations (14) and (21) and R is the STBC code rate with R = 1 for Alamouti’s code of a $2\times 2$ MIMO system [4,5].

Figure 18 shows that the AM-CAZAC-TD-MIMO system performed well in terms of channel capacity, reflecting variations in the level of modulation of M-PSK based on the SNR threshold in Table 2. For channel SNRs below 9 dB, the AM-CAZAC-TD-MIMO system performed similarly to the non-adaptive STBC-assisted MIMO-OFDM system using QPSK modulation in terms of channel capacity. The channel capacity performance improved significantly for channel SNRs ranging from 9 dB to 15 dB. Eventually, it achieved channel performance comparable to the non-adaptive STBC-assisted MIMO-OFDM system using 64-PSK modulation for channel SNRs greater than 15 dB. Thus, the enhanced robustness, coupled with the increased spectral efficiency, showcases the system’s ability to deliver consistent and reliable performance, even under varying channel conditions.

The second set of results focuses on the adaptive MIMO-OFDM system’s BER and channel capacity performance, comparing it against an adaptive SISO-OFDM scheme to demonstrate the benefits of employing multiple antennas. The adaptive SISO-OFDM system, which invokes the CAZAC-TD SNR estimator, is denoted as the AM-CAZAC-TD-SISO system. Both of these adaptive modulation systems use the M-PSK modulation technique. In contrast to the AM-CAZAC-TD-SISO system, the AM-CAZAC-TD-MIMO system demonstrated an SNR gain of about 4 dB, as shown in Figure 19.

Furthermore, Figure 20 illustrates that the AM-CAZAC-TD-MIMO system and the AM-CAZAC-TD-SISO system exhibit a notable improvement in channel capacity. Therefore, it was shown that the spatial diversity of the MIMO-OFDM system is key for the AM-CAZAC-TD-MIMO system’s improved performance.

6. Conclusions

The time domain and frequency domain preamble-based SNR estimators, known as CAZAC-TD and CAZAC-FD SNR estimators, respectively, that make use of the modified CAZAC preamble structure are presented in this study. These SNR estimators are compared to the Milan-FD SNR estimator proposed in [18] when used in a non-adaptive $2\times 2$ STBC-assisted MIMO-OFDM system. The performance of the CAZAC-TD and CAZAC-FD SNR estimators is further examined when employed in the non-adaptive $4\times 4$ STBC-assisted MIMO-OFDM system. The estimated SNR using the CAZAC-TD SNR estimator matches the actual SNR for non-adaptive $2\times 2$ and $4\times 4$ STBC-assisted MIMO-OFDM systems transmitting over the AWGN channel. However, in the $2\times 2$ STBC-assisted MIMO-OFDM system, the differences between the actual SNR and the estimated SNR using CAZAC-FD and Milan-FD SNR estimators are 0.12 dB and 0.16 dB, respectively. However, the SNR estimator performance deteriorated for transmission over the SUI-5 channel where the Milan-FD SNR estimator shows estimated SNR differences of about 0.26 dB, but the CAZAC-TD and CAZAC-FD SNR estimators used in the $2\times 2$ and $4\times 4$ STBC-MIMO-OFDM systems show estimated SNR differences below 0.2 dB. It was also demonstrated that the NMSE performance of the CAZAC-TD SNR estimator approached the theoretical limit set by NCRB for the non-adaptive $2\times 2$ and $4\times 4$ STBC-assisted MIMO-OFDM systems. Subsequently, the CAZAC-TD SNR estimator is invoked in an adaptive modulation scheme for a $2\times 2$ STBC-assisted MIMO-OFDM system employing M-PSK, denoted as the AM-CAZAC-TD-MIMO system. The AM-CAZAC-TD-MIMO system outperformed the non-adaptive STBC-assisted MIMO-OFDM system using 8-PSK by about 2 dB at BER =${10}^{-4}$. The adaptive SISO-OFDM system with M-PSK, also known as the AM-CAZAC-TD-SISO system, which invokes the CAZAC-TD SNR estimator, was compared to the proposed AM-CAZAC-TD-MIMO system in terms of channel capacity and BER. In contrast to the AM-CAZAC-TD-SISO system, the AM-CAZAC-TD-MIMO system demonstrated an SNR gain of about 4 dB. It was shown that the spatial diversity of the MIMO-OFDM system is key for the AM-CAZAC-TD-MIMO system’s improved performance.

7. Future Work

Future work will focus on three main areas: first, developing a preamble-based SNR estimator to address mobile environments, where factors like Doppler shifts and time-varying channel conditions impact SNR estimation. One potential application is in UAV communication systems, where knowledge of exact UAV communication channels is required [52]. In these systems, the Doppler effect arises due to flight speed, making it challenging for traditional methods. Employing orthogonal time frequency space (OTFS) modulation with a preamble could significantly improve SNR estimation in high-mobility scenarios like UAVs by enhancing Doppler resilience, synchronization accuracy, and channel estimation. Second, the study could be extended to massive MIMO systems, where two potential approaches can be considered for adapting the proposed SNR estimation method. First, decentralized estimation can be utilized, where clusters or individual antennas independently estimate the SNR. This approach reduces computational demands and enables the system to scale efficiently as the number of antennas increases. Second, exploiting spatial correlation between antennas can further enhance the estimation process. By sharing similar SNR estimates across neighboring antennas, this method improves the accuracy of SNR estimations while also reducing computational complexity, leveraging the natural similarity in signals received by spatially proximate antennas [53]. Lastly, an extension of this work could involve conducting a comparison study between the simulation results and real-time experimental measurements to provide a more comprehensive evaluation of the proposed methods, offering valuable insights into their performance in practical, real-world settings.