# Non Data-Aided SNR Estimation for UAV OFDM Systems
## 3 The Proposed NDA SNR Estimation Method  
Suppose the sample sequence of the received OFDM signal is \(r\left(k\right),k=1,\ldots,N_{r}\), where \(k\) and \(N_{r}\) are the \(k\)-th sampling point and the total number of sampling points, respectively. Provided that the parameters have been estimated accurately and \(N\) is the length of the valid signal data. Define \(y(k)\) as  
\[y(k)=r(k)-r(k-N). \tag{4}\]  
It is assumed that the frequency offset has been perfectly estimated and compensated without considering the influence of the dispersion channel, then, the received signal are given as  
\[y(k)=s(k)+n(k)-s(k-N)-n(k-N), \tag{5}\]  
where \(s(k)\) and \(n(k)\) represent the signal component and noise component of the \(k\)-th slot, respectively. According to the characteristics of the OFDM signal with CP, there are two different situations. When \(s(k)=s(k-N)\), \(s(k)\) and \(s(k-N)\) represent the copied OFDM symbol and CP symbol, respectively. When \(s(k)\
eq s(k-N)\), \(s(k)\) and \(s(k-N)\) represent two samples from different systems, respectively. In addition, the transmitted signal, which is composed of a large number of independent information flows, can be accurately modeled as a complex Gaussian process. Since each term of \(y(k)\) obeys Gaussian distribution, \(\left|y(k)\right|\) obeys Rayleigh distribution. Suppose that the power of the desired signal is \(\sigma_{s}^{2}\) and the noise power is \(\sigma_{n}^{2}\) in \(y(k)\) and the power of the complex Gaussian variable \(z\) is \(\sigma^{2}\), then, (6) can be established. \[E\left[\left|z\right|\right]=\sqrt{\frac{\pi}{2}}\sigma. \tag{6}\]  
When \(\vartheta_{m}(n)=v_{m}(n+N)-v_{m}(n)\), it can be easily proved that \(\vartheta_{m}(n)\) is a Gaussian noise with mean zero, variance \(\sigma_{\vartheta}^{2}=2\sigma_{v}^{2}\) and the proof process is  
\[ \begin{split} E\left[\left|\vartheta_{m}\left(n\right)\right|^{ 2}\right]&=E\left[\left|v_{m}\left(n+N\right)-v_{m}\left(n\right) \right|^{2}\right]\\ &=E\left[\left(v_{m}\left(n+N\right)-v_{m}\left(n\right)\right) \left(v_{m}\left(n+N\right)-v_{m}\left(n\right)\right)^{*}\right]\\ &=E\left[\left|v_{m}\left(n+N\right)\right|^{2}\right]+E\left[ \left|v_{m}\left(n\right)\right|^{2}\right]=2\sigma_{v}^{2}.\end{split}  \tag{7}\]  
Figure 2: OFDM model signals over UAV channel. Therefore, when \(s(k)=s(k-N)\), we can get  
\[y_{abs}(k)=E\left[\left|y\left(k\right)\right|\right]=\sqrt{\frac{\pi}{2}}\cdot \sqrt{2}\sigma_{n}=\sqrt{\pi}\sigma_{n}, \tag{8}\]  
otherwise, when \(s\left(k\right)\
eq s\left(k-N\right)\), we obtain  
\[y_{abs}(k)=E\left[\left|y\left(k\right)\right|\right]=\sqrt{\frac{\pi}{2}}\cdot \sqrt{2\left(\sigma_{s}^{2}+\sigma_{n}^{2}\right)}=\sqrt{\pi}\cdot\sqrt{ \sigma_{s}^{2}+\sigma_{n}^{2}}, \tag{9}\]  
provided that \(y_{abs}(k)\) is a periodic sequence and its period is the symbol total length. The total length is known before estimating SNR. Divide the sequence according to the symbol total length and stacking each part. This stacking method can get a better statistical result based on little intercepting data. \(y_{acc}\) is expressed as  
\[y_{acc}(i)=\frac{1}{M}\sum_{j=1}^{M}y_{abs}\left(i+j\cdot N_{T}\right), \tag{10}\]  
where \(M\) is the complete intercepted OFDM symbolic number, \(N_{T}\) is the OFDM symbol total length. Therefore, the sequence length of \(y_{acc}\) always equals the symbol total length. Owing to the influence caused by multipath channel, there is a transition part between the maximum and minimum platforms. This part is caused by the CP with ISI. Figure 3 illustrates the characteristic curve of \(y_{acc}\) after the fourth-order Haar wavelet denoising. From Figure 3, we observe that the maximum and minimum platforms are attained by utilizing the characteristic curve after the wavelet denoising. There is a minimum platform in the characteristic curve after wavelet denoising, which also exists in the original characteristic minimum platform part.