## Appendix B The Time Frequency Distribution of the SC-FDMA Signal
### The Time Frequency Distribution of I/Q Separated Signals  
From the time frequency analysis of the single carrier pulse shaping quadrature modulated signal, it indicates that the energy of the signal does not occupy the whole passband all the time. Its energy concentrates in some time frequency areas, and leaves some low-energy areas, as shown in Figure 5a,c. This provides us a chance to suppress the interference and noise in the low-energy areas with the time-varying filter, which is designed according to the energy concentration of the pulse shaping quadrature modulated signal.  
However, the physical analysis and time-frequency distribution of the QAM modulated signal show that its energy concentration is determined by the in-phase and quadrature transmitted data. Since the transmitted data are random generated, the energy distribution of the QAM signal varies with the pulse shaped baseband signals \(I(t)\) and \(Q(t)\), where \(I(t)\) and \(Q(t)\) are associated with \(M\) continuous data. Here, \(M\) is decided by the truncated length of the impulse response of the Gaussian filter. Therefore, \(I(t)\) and \(Q(t)\) have \(2^{2M-2}\) kinds of combination, which makes the signal have \(2^{2M-2}\) kinds of time-varying spectrums. This makes it difficult to generate the pass region for the time-varying filter. One solution is to generate the pass region from the received signal. However, since the received signal has been contaminated by the interference and noise, the energy distribution of the received signal is quite different from the desired signal, hence it is hard to generate the pass region from the received signal directly. Another solution is to match the time-frequency distribution of the received signal with \(2^{2M-2}\) local pass region templates, and choose the most matching pass region template as the pass region of the time-varying filter. However, this solution also has its disadvantages. On one hand, the computational complexity is high; on the other hand, the time-frequency distribution of the received signal is distorted by the noise, thus wrong pass region templates may be chosen as the pass region of the time-varying filter. Therefore, the uncertainty of the time-frequency distribution of the quadrature modulated signal makes it hard to apply the time-varying filter.  
At the receiver, the quadrature modulated signal is demodulated by the demodulator and the signal is separated into an in-phase signal and a quadrature signal. If there is no noise or interference, the in-phase signal \(I_{r}(t)=I(t)\), and the quadrature signal \(Q_{r}(t)=Q(t)\). The time frequency distribution of the separated signal is just the time frequency distribution of the baseband pulse shaped signal, as shown in Figure 4a,b.  
It is shown that the time frequency distribution of the separated I/Q signal has the following characteristics: (1) the energy center of the separated signal is zero without shifting. (2) the time frequency distribution shows a certain regularity. If the continuous two data are different, the  
Figure 6: The time frequency distribution of the SC-FDMA signal.  
bandwidth of the signal reaches its maximum value at the center of each symbol period, and shrinks to its minimal value at the edge of each symbol period. If the continuous two or \(N\) data are the same, the bandwidth of the signal retains wide in the whole two or \(N\) symbol periods, but still reaches its maximum value at the center of each symbol period, and shrinks narrower at the edge of each symbol period. Therefore, if the signal is time-varying filtered with the proposed pass region, more interference and noise could be suppressed with less distortion to the desired signal. Meanwhile, since the filter processing is implemented at the baseband, the complexity is reduced.