# A Channel Compensation Technique Based on Frequency-Hopping Binary Offset Carrier Modulated Signal
## 3 Performance Analysis
### Acquisition Scheme  
The acquisition for FH-BOC signal is a search process containing a three-dimensional search on Doppler shift, spreading code, and a unique frequency-hopping pattern, which requires an additional searching dimension, compared to ordinary spreading signal. Generally, there are serial code search acquisition [27], parallel frequency search [28], and parallel code-phase search [29] serving as spreading signal acquisition scheme. Additionally, in order to accelerate the acquisition procedure, a parallel code-phase search is adopted, and the block diagram is shown in Figure 4. The input signal \(u^{\prime}(t)\) is the output of the preceding low-pass filter, and to perform a circular correlation with a local replica of the sub-carrier and PRN code in the time domain, which is presented as multiplication after FFT in the frequency domain. Then, after IFFT, the convolution results under different code-phase situations can be examined at the same time, which corresponds to a single preset Doppler shift frequency. As for the FH-BOC signal, the additional frequency-hopping search dimension has a great effect on the acquisition time and complexity to those who intercept and decipher signal unauthorizedly [30]. By defining the period of \(A_{r}\) as the same as a period of the pseudo-random noise code sequence \(T\), aligning these two parts uniformly can be achieved, in order to reduce the unnecessary time cost of calculation and complexity of demodulation, since the pattern can be derived from the spreading code phase in the meantime. Furthermore, coarse estimations of code phase \(\hat{\tau}_{acq}\) and Doppler shift \(\hat{f}_{d}\) will be regarded as initialized parameters, transferred to the following tracking loop and renewed for every period \(T\). Figure 3: Block diagram of designed receiver. The validity of our proposed system depends on the stability of the FH-BOC signal, which basically refers to the auto-correlation function. It can be expressed as  
\[ \begin{split} R(t,t+\tau)&=E(u(t),u^{*}(t+\tau))\\ &=E(\sum_{k=0}^{n-1}c_{k}p(t-kT_{c})\sum_{v=0}^{n-1}{c_{v}}^{*}p^{ *}(t+\tau-vT_{c}))\\ &=\sum_{k=0}^{n-1}\sum_{v=0}^{n-1}{E(c_{k}c_{k+v}}^{*})E(p(t-kT_{ c})p^{*}(t+\tau-(k+v)T_{c})),\end{split}  \tag{8}\]  
where \(E(c_{k}c_{k+v}^{*})\) denotes the ACF of spreading code. Supposing an ideal PRN sequence that possesses zero cross-correlation function (CCF) serves as the spreading code, then it has  
\[E(c_{k}c_{k+v}^{*})=\left\{ \begin{array}{cc}1,&v=0\\ 0,&v\
eq 0\end{array} \right. \tag{9}\]  
Hence, the ACF of FH-BOC signal can be derived as  
\[R(t,t+\tau)=\sum_{k=0}^{n-1}{E(p(t-kT_{c})p^{*}(t+\tau-kT_{c}))}, \tag{10}\]  
which is the ACF of frequency hopping pattern \(A_{r}\). Since adopting a cyclic repeating subcarrier pattern with the same period as spreading code, it can be concluded as  
\[R(t,t+\tau)\propto A_{r}A_{r+\sigma}^{*},\sigma=\tau/T. \tag{11}\]  
In the meantime, it can be inferred that  
\[A_{r}A_{r+\sigma}^{*}\leq|A_{r}|^{2}, \tag{12}\]  
where \(\sigma=0\) satisfies the equation demand, referring to \(\tau=0\). The simulation result on ACF of the FH-BOC signals, compared to the BOC signals, within a single centered chip duration is demonstrated in Figure 5.  
The main peak-to-maximum side-peak ratio (MSR) is an important factor in dealing with signal acquisition. In order to avoid Type I and Type II errors, and meanwhile, to improve system performance in occasions with considerable noise, MSR is a critical indicator for mitigating the ACF peak ambiguity problem and evaluating the capability of anti-interference. A higher main-peak for a maximum side-peak ratio indicates a cleaner, more stable signal with less unfavorable noise or distortion. Majorly, compared to the \(BOC(1,1)\) signal, as shown above, apparently the \(FH-BOC(1,1,4)\) signal produces a smaller MSR to gain a better performance on the signal acquisition. Additionally, the \(BOC(2,2)\) signal shows one more side-peak than \(BOC(1,1)\) signal; meanwhile, it leads to a larger fluctuation near the main peak.