# Static and Moving Target Imaging Using Harmonic Radar
## 2 Harmonic Radar  
### Harmonic Radar Range Equation  
A nonlinear scatterer can be characterized by a scattering radar cross section (RCS) that depends upon (and generally increases with) the incident power density \(S_{i}\), as opposed to a linear scatterer for which the RCS may be assumed constant except at very high power levels. The nonlinear RCS, \(\sigma_{n}(S_{i})\) in m\({}^{2}\), is defined as that area intercepting the amount of power at frequency \(f_{t}\) which, when scattered isotropically at frequency \(f_{r}\), produces an echo equal to that observed from the scatterer [40]. For nonlinear scattering to occur, \(f_{r}\
eq f_{t}\); however, for harmonic scattering, \(f_{r}\) is a multiple of \(f_{t}\). Over the past few decades, several researchers have characterized the nonlinear scattering properties of a variety of targets, both analytically and experimentally, and have also derived different formulations for the nonlinear radar range equation [41; 42; 43; 44; 45; 46; 47; 48; 49].  
The radar range equation for a harmonic radar (\(f_{r}=nf_{t}\), where \(n\) is an integer) has been derived and it shows that, whereas the received signal is inversely proportional to the fourth power of the range for a conventional radar (i.e., \(R^{-4}\) dependence), the range law becomes increasingly unfavorable for successive higher harmonics, showing a dependence of \(R^{-(2n+2)}\)[50]. For a practical harmonic radar, the second or third harmonics are therefore generally preferred, yielding inverse sixth and inverse eighth power range laws, respectively. Even so, the higher rate of power reduction with range restricts the use of harmonic radar to close range applications (less than about 20 m), although they have the advantage of higher visibility when illuminated at higher power densities.  
Consider the block diagram of a radar signal path shown in Figure 1.  
In Figure 1, point A represents the transmitted waveform, point B represents the waveform impinging on the nonlinear target after traveling a distance \(R\), point C represents the response from the nonlinear target, and point D represents the signal returned to the radar. The radar range equation can be derived in the standard manner.  
We model the input-output relationship for the nonlinear device as a power series, as  
\[P_{\text{out}}=\sum_{n=1}^{\infty}d_{n}P_{\text{in}}^{n} \tag{1}\]  
where \(P_{\text{in}}\) is the input power incident on the target, \(P_{\text{out}}\) is the isotropically radiated output power by the target, and \(d_{n}\) is a scaling coefficient for the \(n\)th harmonic of the nonlinear device. The coefficient \(d_{n}\) has units of W\({}^{1-n}\). The term \(n=1\) represents the linear response, and the scaling coefficients generally drop off as \(n\) increases, i.e., \(d_{n}>d_{n+1}\).  
The radar range equation for the \(n\)th harmonic term has been derived as [51]  
\[P_{r}(nf_{t})=\frac{G_{r}(nf_{t})\lambda_{n}^{2}(P_{t}G_{t}(f_{t}))^{2}\sigma_ {n}}{(4\pi)^{n+2}R^{2n+2}} \tag{2}\]  
Figure 1: Block diagram showing a radar signal path.  
where \(f_{t}\) is the fundamental transmitted frequency, \(f_{r}=nf_{t}\) is the received frequency at the \(n\)th harmonic, \(P_{t}\) is the transmitted power, \(G_{t}(f_{t})\) is the gain of the transmit antenna at frequency \(f_{t}\), \(G_{r}(nf_{t})\) is the gain of the receive antenna at frequency \(f_{r}=nf_{t}\), \(\lambda_{n}\) is the wavelength corresponding to the \(n\)th harmonic at frequency \(nf_{t}\), \(R\) is the range to the target, and \(\sigma_{n}\) is the radar cross section of the target at the \(n\)th harmonic.  
The radar cross section, \(\sigma_{n}\), is given by [51]  
\[\sigma_{n}=d_{n}G_{t}(nf_{t})\left(A_{e}(f_{t})\right)^{n} \tag{3}\]  
where \(A_{e}(f_{t})\) is the effective aperture area of the target at the fundamental transmit frequency and \(G_{t}(nf_{t})\) is the target aperture's reradiation gain at the \(n\)th harmonic frequency \(nf_{t}\).