## Appendix B The Time Frequency Distribution of the SC-FDMA Signal
### The Equivalence of RF and Baseband Analysis  
Is it equivalent to analyze the time frequency distribution of the RF signal and the separated baseband signal? This section will firstly discuss the equivalence between the power spectrum of a quadrature modulated signal and the power spectrum of its corresponding baseband complex signal. The baseband complex signal is denoted as \(s_{b}(t)=I(t)+jQ(t)\), and the baseband spectrum is \(S_{b}(f)=\mathfrak{F}\{s_{b}(t)\}\). Then modulate \(s_{b}(t)\) to carrier with frequency \(f_{c}\), and the modulated RF signal is expressed as  
\[s(t)=s_{b}(t)e^{j2\pi f_{c}t} \tag{20}\]  
Apply Fourier transform to signal \(s(t)\), the magnitude spectrum of the RF signal is obtained as  
\[ \begin{split} S(f)=&\mathfrak{F}\left\{s_{b}(t)e^{ j2\pi f_{c}t}\right\}=\int_{-\infty}^{+\infty}s_{b}(t)e^{j2\pi f_{c}t}e^{-j2\pi f _{c}t}dt\\ =&\int_{-\infty}^{+\infty}s_{b}(t)e^{-j2\pi(f-f_{c })t}dt=S_{b}(f-f_{c})\end{split}  \tag{21}\]  
The baseband power spectrum is \(P_{b}(f)=|S_{b}(f)|^{2}\), and the power spectrum of the RF signal is \(P(f)=|S(f)|^{2}=|S_{b}(f-f_{c})|^{2}=P_{b}(f-f_{c})\). It is shown that power spectrum of the modulated signal can be obtained by moving the baseband power spectrum from center frequency 0 to \(f_{c}\). If the modulated signal is not complex but real, its power spectrum will appear at frequency \(\pm f_{c}\) simultaneously. Thanks to the power spectrum shifting characteristics between the baseband signal and its modulated signal, implementing a bandpass filter at radio front end is equivalent to implementing a low-pass filter at the baseband. However, the two dimensional time-frequency analyses between the baseband signal and the modulated signal are not equivalent. In section II, it is mentioned that there are two kinds of methods to analyze the signal in joint time-frequency domain. The first is linear analysis, such as STFT and Gabor transform. in which the signal is truncated by the analysis window before analyzing. While the second is bilinear analysis, and the main analysis method is Wigner distribution. Since the Wigner distribution has severe cross-term problems, researchers provide some improved methods to suppress the cross terms, such as Pseudo smoothing Wigner distribution and Choi-Williams distribution. In both of the two distributions, the signal is also truncated by the analysis window before analyzing. However, when applying Fourier transform to the truncated signal, the truncated length (or window length) affects the equivalence between the baseband signal and the modulated RF signal. For the modulated RF signal, if the window length is not integer multiples of the carrier cycle, the transition point will be introduced at the edge of extension period when implementing FFT calculation. The transition point will lead to carrier leakage, which will cause distortion to the time frequency distribution of the RF signal. In this case, the time frequency distribution of the RF signal disaccords with the time frequency distribution of the baseband complex signal. In Figure 7, the number of samples in each carrier cycle is \(N_{sample}=30\). Figure 7a is the time frequency distribution of the baseband complex signal with the window length \(L_{h}=40\), and Figure 7b is the time frequency distribution of the RF signal when the window length \(L_{h}=40\). It is shown that the energy center of Figure 7b has changed compared to Figure 7a. Since the truncated length is not integer multiples of the carrier cycle, it is \(1.1\) times in this example, the time frequency distribution of the RF signal has been distorted. Figure 7c is the time frequency distribution of the baseband complex signal with the window length \(L_{h}=50\), and Figure 7d is the time frequency distribution of the RF signal when the window length \(L_{h}=50\). It also shows the non-equivalence between the time frequency distribution of the baseband signal and that of the RF signal when the window length is not integer multiples of the carrier cycle.