# A Channel Compensation Technique Based on Frequency-Hopping Binary Offset Carrier Modulated Signal
## 3 Performance Analysis
### Matched Filters  
The idea of matched filters is widely implemented in digital communication system, in order to achieve various purposes, such as SNR improving [31], pulse shaping [32], or out-of-band emission reducing in certain multiple-input multiple-output (MIMO) scenarios, such as Universal filtered multi-carrier [33] and filtered-OFDM [34]. The dynamic channel condition causes frequency-selective fading and Doppler frequency shift, leading to inter-symbol interference (ISI) and inter-carrier interference (ICI) [35], which impose distortion on the OFDM signal waveforms. Although the FH-BOC modulated signal is capable of avoiding such problems, the OFDM signal requires multipath interference suppression and waveform restoring. As for our proposed system, the square-root raised cosine (SRRC) filters, which are two parts divided from raised cosine filter, are adopted for pulse shaping on the OFDM signal, being capable of mediating negative affection. The normalized frequency response of raised cosine filter  
\[H(f)=\left\{ \begin{array}{cc}1,&|f|\leq\frac{1-\beta}{2T}\\ \frac{1}{2}[1+\cos(\frac{\pi T}{\beta}[|f|-\frac{1-\beta}{2T}])],&\frac{1- \beta}{2T}<|f|\leq\frac{1+\beta}{2T}\\ 0,&|f|>\frac{1+\beta}{2T}\end{array} \right. \tag{25}\]  
is a piecewise-defined function for \(0<\beta\leqslant 1\), where \(\beta\) is a roll-off factor and \(T_{s}\) is the reciprocal of symbol rate. The crucial is the roll-off zone between, which is determined by \(\beta\) controlling its smoothness and bandwidth. Correspondingly, the impulse response of this filter  
\[h(t)=\frac{1}{T}\text{sinc}(\frac{\pi t}{T})\frac{\cos(\frac{\pi\beta t}{T})}{ 1-\left(\frac{2\beta t}{T}\right)^{2}} \tag{26}\]  
Figure 10: Structure diagram of digital loop filter of DLL. is a sinc function shape, both waveforms have been plotted in Figure 11. Additionally, it can be recognized \(h(t)\) has zero-crossings at multiple times of \(T_{s}\), so that every previous sampling decision remains no interference on current sampling one, referring to its capability of eliminating ISI during symbol stream. Therefore, raised cosine filter itself is generally split into two square-root raised cosine filters and being applied in both transmitter and receiver. Separated filter parts are required to satisfy the equation \(H_{T}(f)\cdot H_{R}(f)=H(f)\), which leads to \(|H_{T}(f)|=|H_{R}(f)|=\sqrt{|H(f)|}\). However, there exists the Doppler shift that has been changing the signal waveform, making the matched filter no longer able to realize the ISI cancellation on the received signal. Therefore, ISI minimization is an alternative and feasible solution for confronting Doppler effect instead, which is accessed by adjusting value of the roll-off factor \(\beta\). In fact, the characteristics of the raised cosine filter vary with different roll-off factors and can be noticed in Figure 11. When \(\beta\) approaches 0, the roll-off zone gradually becomes narrow, as \(H(f)\) is almost into a rectangular shape in frequency domain; meanwhile, \(h(t)\) has been a complete sinc function shape. As \(\beta=1\), the frequency response \(H(f)\) is a pure raised cosine, and the impulse response seems much flatter at every other sampling time. Practically, the pulse shape \(h(t)\) only has finite support in time, for it has been truncated by spanning symbols. The result of truncation is the emergence of non-zero side lobes of frequency response, that the PSD fluctuates in waves beyond the roll-off zone. Additionally, the smaller the factor \(\beta\) is, the better the side lobes suppression that can be achieved, which is a contradiction to the desired flat impulse response. Additionally, even though ideal SRRC filters are not realizable in realistic [36], making use of them still earns benefits. In fact, an intuitive pulse shaping illustration of matched filter is suggested in Figure 12, with waveform comparisons for both the real component and imagined component within a short time.