high-resolution spectral estimation =================================== The sensor imaging resolution is defined as **the minimum spatial separation of two targets revolvable by the radar which contains both range and azimuth resolutions**. For FMCW radars, the larger the sweeping bandwidth (B) is, the more range resolution is _. In fact, the minimum values in :ref:Table 1  are the resolutions. On the other hand, the angle of arrival resolution increases by increasing the number of transmitters and receivers which is equivalent to increasing the observation duration in space, hence it increases the resolution in the frequency domain or angle domain. However in literature, the resolution refers to the accuracy in estimating a target parameter rather than resolving two targets. Here, we use resolution as it is in literature Otherwise, in other sections, the resolution has the meaning as defined before. In order to obtain range information from :ref:eq11 , we take an FFT over fast-time. So, the signal along that axis will be transformed to the range domain. This is equivalent to collecting :math:\boldsymbol{s}(t_f) samples in the fast-time and taking FFT over the samples. Then, the signal model will be: .. eq16: .. math:: \tilde{\boldsymbol{x}} (f_{b_i}, t_s) = \Gamma A(\tilde{\boldsymbol{\theta}}) \tilde{\boldsymbol{y}} (f_{b_i},t_s) + \boldsymbol{e}(f_{b_i}, t_s ) \quad \qquad (16) with :math:f_{b_i} is the i’th target’s beat frequency, N and M are the number of samples in the fast-time and the number of receivers, respectively. The factor :math:\sqrt{N} is the result of FFT to maintain the same power in FFT and time domains. :math:\rho_i is related to the power received from the i’th target. In addition, the structure of :math:\tilde{\mathbf{y}} has changed and diagonal elements contain Doppler of targets at the range :math:f_{b_i}. Also note that :math:\tilde{\boldsymbol{\theta}} is different from the radar cube in time since now it contains the target angles at the particular range of :math:f_{b_i}. On the other hand, the dependency of :math:\tilde{\mathbf{y}} on :math:f_{b_i} is in the residual phase noise, thus for simplicity, it can be a noise term. Therefore, :math:\tilde{\mathbf{y}} is a noisy component with the dependency on :math:t_s only. Fig. 5_ shows the radar cube and the vector :math:\tilde{\mathbf{x}} location in the cube. .. _Fig. 5: .. image:: /figs/radar_cube_range_slow_time_antennas.png Fig. 5: Radar cube In order to detect the angle of targets, or more specifically vector :math:\boldsymbol{\theta}, one method is to use a second FFT along the receiver antennas; however, it requires to remove Doppler from the signal to properly estimate the angles since :math:\tilde{\mathbf{y}} contains Doppler which is an interference for the angle estimation. On the other hand, by using Capon filter _, we can estimate :math:\boldsymbol{\theta} such that the Doppler is eliminated in calculation of the covariance matrix of :math:\tilde{\mathbf{x}}: .. eq17: .. math:: R_{\tilde{\mathbf{x}}} (f_{b_i}) & = E(\tilde{\mathbf{x}}(f_{b_i} , t_s ) \tilde{\mathbf{x}}^H (f_{b_i} , t_s )) \nonumber \\ & = \Gamma A(\tilde{\boldsymbol{\theta}}) E\left[ \tilde{\mathbf{y}} (f_{b_i} , t_s ) \tilde{\mathbf{y}}^H ( f_{b_i} , t_s ) \right] A^H (\tilde{\boldsymbol{\theta}}) \Gamma^H \nonumber \\ &+ R_{\tilde{\mathbf{e}}}(f_{b_i}) \qquad \quad \quad\quad \quad\qquad \qquad \quad\qquad (17) If there is no phase noise, the middle term in the expectation is :math:\rho_i^2/M N \, I_K in which :math:I_K is an identitiy matrix of size K. Hence, .. eq18: .. math:: R_{\tilde{\mathbf{x}}} (f_{b_i} )= \frac{\rho_i^2}{M N} \Gamma A(\tilde{\boldsymbol{\theta}}) A^H (\tilde{\boldsymbol{\theta}}) \Gamma^H + R_{\tilde{\mathbf{e}}} (f_{b_i}) \quad\quad (18) This proves that the angle estimation with Capon filter reduces the effort to compute Dopplers before angle estimation. Fig. 6_ includes the integration direction to compute :math:R_{\tilde{\mathbf{x}}} (\mathbf{f}_{b} ) all ranges which lays out an output map of range-angles (range-azimuth or elevation). A sample map is shown on the range-angle face of the radar cube in Fig. 6_. .. _Fig. 6: .. image:: /figs/radar_cube_range_slow_time_capon_map.png Fig. 6: Capon map on one face of radar cube. References ---------- ..  M. Alizadeh, G. Shaker, J. C. M. D. Almeida, P. P. Morita, and S. SafaviNaeini, “Remote monitoring of human vital signs using mm-wave fmcw radar,” IEEE Access, vol. 7, pp. 54 958–54 968, 2019. ..  M. Alizadeh, H. Abedi, and G. Shaker, “Low-cost low-power in-vehicle occupant detection with mm-wave FMCW radar,” in 2019 IEEE SENSORS, pp. 1–4, ISSN: 1930-0395.