## 4 Multi-Frequency Method  
In the previous section we showed that the number of data soon becomes huge for an electrically large investigation domain. Apart from the data storage requirements and the high computational burden, this would require an unfeasible sensing configuration or very long measurement time. To mitigate these problems, one is forced to work with a reduced (with respect to the NDF) number of data but aliasing then occurs. Since most electromagnetic sensors are able to acquire data at different frequencies at a relatively low cost, it appears useful to investigate whether it is possible to 'trade' space data for frequency data. In other words, we suggest to remedy the reduction of spatial measurements with frequency data [36]. According to our scattering model, since source information is unknown to the receivers, multiple frequency data cannot be processed simultaneously. Actually, each frequency needs to be processed separately. By single-frequency data, under-sampling gives rise to artifacts due to aliasing in the reconstructions. Therefore, each single-frequency under-sampled reconstruction will exhibit, besides the actual target reconstruction, also a number of replicas whose spatial positions, however, change with the frequency. This feature can be exploited in order to reduce the impact of the replicas by introducing a suitable 'fusion' strategy which combines the different single-frequency reconstructions. At this juncture we point out that the frequency fusion procedure cannot aim at correctly imaging the scattering objects, since it cannot compensate for the lack of information due to under-sampling. Rather, it is aimed at improving the detection and/or the localization of the scattering targets by reducing ambiguities due to aliasing replicas. In order to detail the multi-frequency fusion procedure, it is convenient to start from the one-dimensional version of Equation (4) and omit the unessential factors outside the integral,  
\[E(u)=\int_{-\chi_{in}}^{\chi_{in}}I_{eq}(\chi)\exp{(j2\pi u\chi)d\chi} \tag{10}\]  
where \(u=\sin\theta\), \(\chi_{in}=x_{in}/\lambda\), and \(\lambda=c/f\), \(f\) being the operating frequency and \(c\) the speed of light. According to the [PERSON] criterion, to reconstruct a current supported over \(\chi\in[-\chi_{in},\chi_{in}]\) the field must be sampled with the step \(\Delta u=1/(2\chi_{in})\). Assume that \(J_{eq}(\chi)=\delta(\chi)\exp{j\phi(\chi)}\) and  
Figure 4: Singular values of the operator \(\mathcal{A}\): \(21\times 21\) sampling points are taken over a square of side \(2u_{max}=\)\(\sqrt{2}\) in the \(u-v\) plane or \(2X_{max}=2h\) in the \(x-y\) plane; \(\chi_{in}=5\). The solid line refers to the uniform sampling in \(u-v\), the dashed line to the uniform sampling in \(x-y\). that the field is known within the interval \(u\in[-u_{max},u_{max}]\); this basically entails studying the point-spread-function (PSF) of the reconstruction procedure. In addition, assume that \(N\) samples are available at the step \(\Delta u=2u_{max}/N\). Accordingly, in the Fourier domain, the model data can be written as  
\[E(u)=\text{rect}[u/(2u_{max}))]\exp j\phi(0)\sum_{i}\delta(u-i\Delta u) \tag{11}\]  
and the corresponding image writes as  
\[|J_{eq}(\chi)|=|2u_{max}\sum_{i}\text{sinc}[2\pi u_{max}(\chi-i/\Delta u)]|\simeq\]  
\[\simeq 2u_{max}\sum_{i}|\text{sinc}[2\pi u_{max}(\chi-i/\Delta u)]| \tag{12}\]  
where \(\text{sinc}(x)=\sin(x)/x\) and in the last term we exploited the fact that the main beams of the different replicas do not overlap. On making explicit the frequency dependence, one obtains  
\[|J_{eq}(x,f)|\simeq 2u_{max}\sum_{i}|\text{sinc}[\pi(2u_{max}xf/c-iN)]| \tag{13}\]  
As can be seen, the reconstruction consists of a train of sinc pulses centered at \(\Delta x=\lambda N/(2u_{max})\) with the main lobe \(2l=\lambda/u_{max}\) large (see Figure 5). Consider now the reconstructions obtained at two different frequencies, say \(f_{1}\) and \(f_{2}>f_{1}\).