## 3 Time-Frequency Distribution of Communication Signals  
### Time-Frequency Distribution of the Binary Modulated Signal  
In a binary modulated system, e.g., a BPSK modulation system, the baseband signal is denoted as \(s_{b}(t)\). Then the modulated signal is expressed as  
\[s(t)=s_{b}(t)e^{j2\pi f_{0}t} \tag{12}\]  
where \(f_{0}\) is the center frequency of the modulated signal. In a communication system, the transmitted signal is pulse shaped at the baseband, and the pulse shaping filter is usually a raised cosine roll-off filter or a Gaussian filter. The waveform of the Gaussian pulse is plotted in Figure 2 with different BT values, where BT is the product of the Gaussian pulse's 3 dB bandwidth and the symbol duration. It is mentioned that the Gaussian pulse has no zero-crossing points, and it does not satisfy the zero-ISI criterion. However, thanks to its best time frequency concentration, it is still widely applied in the pulse shaping system. Figure 3a illustrates the pulse shaped baseband signal when the transmitted data sequence is \([1,-1,1,-1,1]\). The pulse shaping filter is a Gaussian filter with BT = 0.4, and the center frequency of its corresponding modulated signal is 11.52 MHz. Similarly, Figure 3b depicts the pulse shaped baseband signal and its modulated signal for transmitted data sequence \([1,-1,-1,-1,1]\)It is shown that the magnitude of the baseband signal changes with time when the continuous two data are different, such as \([1,-1]\) or \([-1,1]\). The absolute value of the magnitude reaches its maximum at each half symbol period, and falls to zero at the end of each symbol period. Therefore, the energy of the signal follows the same variation. In contrast, if the continuous two data are the same, such as \([1,1]\), the magnitude of the baseband signal remains positive for the whole two symbol periods. However, its magnitude still reaches its maximum value at each half symbol period, and falls to a smaller value at the end of each symbol period. Apply Fourier transform to the above mentioned signal, and the nominal bandwidth is obtained, which is 3 dB bandwidth for the Gaussian pulse. In current communication systems, engineers design bandpass filters or low-pass filters according to the nominal bandwidth, and filter the signal at radio front end or baseband, respectively. By bandpass or low-pass filter, the desired signal in the pass-band is retained and the interference and noise outside the pass-band are suppressed. However, the interference and noise in the pass-band will pass through the filter.  
The time frequency distribution of the Gaussian pulse shaping signal demonstrates that the bandwidth of the binary modulated pulse shaping signal is not constant, but varies with time, which is shown in Figure 4a,b. Figure 4a is the time frequency distribution corresponding to transmitted  
Figure 3: The baseband waveform and RF waveform of the pulse shaping signal. (**a**) Data stream \(\{1,-1,1,-1,1\}\); (**b**) data stream \(\{1,-1,-1,-1,1\}\).  
Figure 2: The Gaussian pulse waveform with different BTs.  
data \([1,-1,1,-1,1]\). It shows that if the continuous two data are different, the bandwidth of the signal reaches its maximum at the center of each symbol period, and shrinks to its minimal at the edge of each symbol period. Figure 4b is the time frequency distribution for transmitted data \([1,-1,-1,-1,1]\), the bandwidth of the signal still reaches its maximum at each half symbol period, and shrinks narrower at the end of each symbol period. In conclusion, the binary modulated pulse shaping signal has a time-varying spectrum, and the variation shows certain regularity.