## 2 Principles
### Receiving  
Joint phase-frequency manipulation also enables down-conversion reception and the beamforming of multi-band signals. As depicted in Figure 1b, a received multi-band signal, \(S_{r}(t)\), is modulated onto optical pulses with a repetition frequency of \(f_{s2}\). Afterphotodetection and low-pass filtering (with a bandwidth of DC-\(f_{s2}/2\)), the output signal, ignoring amplitude coefficients, can be expressed as:  
\[\begin{array}{ll}S_{2}(t)&=\{T_{M}[S_{r}(t)]\times u_{2}(t+\Delta t_{2})\}*h_{ oe}(t)*h_{LPF}(t)\\ &=\left\{\frac{1}{2}[1+\alpha\sum\limits_{k=1}^{k_{0}}\cos(2\pi f_{k}t+\pi \frac{B_{k}}{T_{k}}t^{2})]\sum\limits_{n=0}^{+\infty}\cos(2\pi nf_{s2}(t+ \Delta t_{2}))\right\}*h_{oe}(t)*h_{LPF}(t)\\ &=A\sum\limits_{k=1}^{k_{0}}\cos(i_{k}(2\pi(n_{k}f_{s2}-f_{k})t-\pi\frac{B_{k}} {T_{k}}t^{2}+2\pi n_{k}f_{s2}\Delta t_{2}))\\ &\quad\quad\text{where $i_{k}$ denotes a symbol, whose value is given by $i_{k}=\text{sgn}(n_{k}f_{s2}-f_{k})$. $h_{LPF}$ is the impulse response of the LPF. $k$ is the $k$-th band in the received multi-band signal. $n_{k}$ refers to the frequency component that satisfies $\mid\!n_{k}f_{s2}-f_{k}\!\mid<\!f_{s2}/2$. $S_{2}(t)$ is the multi-band signal down converted to DC-\(f_{s2}/2\). After being phase-shifted by an electronic phase shifter, the signal, \(S_{2}(t)\), can be expressed as:  
\[S_{2}(t)=A\sum\limits_{k=1}^{k_{0}}\cos(i_{k}(2\pi(n_{k}f_{s2}-f_{k})t-\pi \frac{B_{k}}{T_{k}}t^{2}+2\pi n_{k}f_{s2}\Delta t_{2})+\varphi_{2}), \tag{6}\]  
The equation demonstrates that optical pulses can simultaneously down-convert multi-band signals to baseband while imparting distinct phases to each band, enabling integrated multi-band beamforming and reception. The phase to be compensated for the beamforming of the signal, \(S_{2}(t)\), is still determined by its original frequency, as shown in Figure 1c. Therefore, for any aperture fill time, \(t_{d2}\), multi-band beamforming needs to meet the requirements of \(i_{k}2\pi n_{k}f_{s2}\Delta t_{2}+\varphi_{2}=2\pi f_{k}t_{d2}+2\pi\tau\) (where \(z\) is an integer). The above equation has solutions for \(\Delta t_{2}\) and \(\varphi_{2}\) only if \(f_{k}=(k-1)\Delta f+f_{1}\) (multi-band signals have equal frequency spacing like \(n_{k}f_{s2}\)) and \(i_{k}=i\) (all bands are on the same side of the reference frequency component of the optical pulses). Substituting these into the original equation, we obtain \(i2\pi n_{k}f_{s2}\Delta t_{2}+\varphi_{2}=2\pi((k-1)\Delta f+f_{1})t_{d2}+2 \pi\tau\). Consequently, \(\Delta t_{2}\) and \(\varphi_{2}\) need to satisfy:  
\[\Delta t_{2}=i\frac{\Delta f}{f_{s2}}t_{d2}-\frac{[i\Delta ft_{d2}]}{f_{s2}}; \;\varphi_{2}=2\pi(f_{1}-\Delta f)t_{d2}-2\pi[(f_{1}-\Delta f)t_{d2}]. \tag{7}\]  
The max adjustment range of \(\Delta t_{2}\) and \(\varphi_{2}\) is \(1/f_{s2}\) and \(2\pi\), respectively.  
The frequency, \(f_{k}\), in the multi-band signal and the repetition frequency, \(f_{s2}\), of the optical pulses are constrained by \(f_{k}=(k-1)\Delta f+f_{1}\), \(i_{k}=i\) and \(\mid\!n_{k}f_{s2}-f_{k}\!\mid<\!f_{s2}/2\). This means multi-band beamforming can only be realized when equally spaced frequencies, \(f_{k}\), and optical pulses frequency components, \(nf_{s2}\), are interleaved. Thus, the multi-band signals generated at the transmitter need to have equal frequency intervals, \((nf_{s1}+f_{0})-(nf_{s1}-f_{0})=((n+1)f_{s1}-f_{0})-(nf_{s1}+f_{0})\), which require that \(f_{0}=\!f_{s1}/4\) and \(\Delta f=\!f_{s1}/2\). To avoid spectrum aliasing of multi-band signals after down-conversion, the number of bands and the bandwidth of multi-band signals are limited by instantaneous bandwidth, \(f_{s2}/2\), which means \(\max(B_{k})\leq\mid(n_{k+1}f_{s2}-f_{k+1})-(n_{k}f_{s2}-f_{k})\!\mid\) and \(\Sigma B_{k}\leq\!f_{s2}/2\), which is \(B_{0}\leq\mid\!f_{s2}-f_{s1}/2\!\mid\) and \(k_{0}B_{0}\leq f_{s2}/2\).