## 5 Results and Discussion
### 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  
Figure 10: The NMSE performance invoking Suparma preamble structure and the modified CAZAC preamble structure for the AWGN channel. 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:  
\[CRB_{SNR_{MIMO-Est}}=\left[\frac{1+Q(SNR_{act})}{\sqrt{N_{s}(M-1)}}\right]^{2} \tag{22}\]  
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 \((SNR_{act})^{2}\) and can be written as:  
\[NCRB_{(SNR_{Est})}=\frac{CRB_{SNR_{MIMO-Est}}}{\left(SNR_{act}\right)^{2}} \tag{23}\]  
where \(SNR_{MIMO-Est}\) for the CAZAC-TD and CAZAC-FD SNR estimators can be calculated using Equation (14) and Equation (21), respectively. Figure 11: The estimated SNR performance for the AWGN channel with a zoomed-in view in the inset. 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 Figures 13 and 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.