# A Channel Compensation Technique Based on Frequency-Hopping Binary Offset Carrier Modulated Signal
## 2 Overview of the Designed System
### Signal Model  
In order to utilize the spectrum efficiently, I/Q modulation is implemented to allocate and separate the OFDM signal and FH-BOC signal. As the quadrature branch, an analytical expression for the FH-BOC signal can be formulated as  
\[u(t)=\sum_{k=0}^{n-1}c_{k}\cdot p(t-kT_{c}),0\leq t<T, \tag{1}\]  
within its period (\(T=1\) ms), and  
\[c_{k}\in\{\pm 1\},k=0,1,2,...,n-1 \tag{2}\]  
denotes the PRN code chip. Each chip duration is \(T_{c}\), and the length of the spreading code sequence is \(n\). In our system, the coarse/acquisition (C/A) code is adopted as the spreading code, which has been used in GPS signals [26]. Therefore, the code consists of \(n=1023\) chips and is sent at \(f_{0}=1.023\) MHz, and the code sequence repeats every millisecond as the single period \(T\). The expression \(p(. )\) represents the generation of the frequency-hopping square wave subcarrier with the same repeating period as the spreading code, which corresponds to the contemporary code chip and is formulated in sine-phased form as  
\[p(t-kT_{c})=sign(\sin(2\pi f_{sc}^{k}t)),kT_{c}\leq t<(k+1)T_{c}, \tag{3}\]  
where \(f_{sc}^{k}\) is a specific subcarrier frequency. Additionally, a row vector  
\[A_{r}=[a_{0},a_{1},...,a_{k},...,a_{n-1}],a_{k}\in\{\alpha:d:\beta\} \tag{4}\]  
defines the frequency-hopping pattern, where \(a_{k}\) denotes the kth subcarrier frequency through equation \(f_{sc}^{k}=a_{k}\cdot f_{0}\) belonging to hopset \(\{\alpha:d:\beta\}\). The hopset includes every possible single subcarrier frequency and can be presented by \(\{\alpha,\alpha+d,\alpha+2d,...,\beta\}\) as a monotonical increasing sequence, where \(d\) is the minimum frequency-hopping interval. According to [22], each subcarrier frequency \(f_{sc}\) is bound to be an integer multiple times that of \(f_{0}\). In other words, the hopset only contains an integer number, and the frequency-hopping pattern \(A_{r}\) is a decimal sequence controlling the frequency hopping, which is initialized at the very beginning. As for the in-phase component, a single OFDM symbol can be expressed in its discrete form as  
\[v(n)=\frac{1}{N}\sum_{k=0}^{N-1}X_{k}e^{j2\pi kn/N},n=0,1,2,...,N-1, \tag{5}\]  
where denoting \(k\)th data symbol as \(X_{k}\) that has been normalized as \(|X_{k}|=1\) and N-point inverse fast Fourier transform (IFFT) is performed. The duration of the single symbol \(v_{n}\) is equal to \(T_{sym}\). In order to achieve overlapping spectrum peaks, the subcarrier spacing \(f_{spa}=1/T_{sym}\) must be identical to the minimum frequency-hopping interval \(d\cdot f_{0}\). Generally, when \(d=1\), it can be determined that \(f_{spa}=f_{0}\) and, consequently, \(T_{sym}=1/1023\) us. Then, the cyclic prefix (CP) insertion is required to meet the guard interval, which can be interpreted as  
\[v(n)=\frac{1}{N}\sum_{k=0}^{N-1}X_{k}e^{j2\pi kn/N},n=-\frac{N}{4},-\frac{N}{4 }+1,...,N-1, \tag{6}\]  
where the length of cyclic prefix is decided as \(N/4\). Discrete OFDM symbols require up-sampling, since the sampling rate must surpass twice that of the carrier frequency \(f_{c}\) to meet the Nyquist criteria. Additionally, \(f_{c}\) itself must surpasses all the subcarrier frequencies in the OFDM symbols, which reach their highest frequency at \(N/T_{sym}\). In comparison, the FH-BOC signal can realize up-sampling easily by repeating, because each sampling point is located in its corresponding spread code chip, leading to identical value within every chip duration \(T_{c}\) consecutively. The up-sampling method on OFDM symbols is generally zero-padding or zero-inserting, which extends the input sequence by appending zeros before the IFFT process. However, such a method has a limitation that enlarged size must be equal to a power of two, which causes a mismatch on most sampling rates if there are any up-sampling points, except a power of two in a single OFDM symbol.