## 4 Procedure and Analysis of Range Differentiation in 3D MMW Imaging  
To evaluate the capability of the GDD integrated with FMCW radar in capturing depth information, we conducted an experimental procedure focusing on range differentiation while imaging objects of varying dimensional profiles. By obtaining range data through beat frequency analysis, we sought to establish the potential for creating three-dimensional MMW images using the GDD system. Initially, a flat \"H\" shape, as described in Section 3 and illustrated in Figure 4, was positioned as the object. Radiation was focused on the central portion of the \"H,\" marked by a red dotted square in Figure 7a, to emphasize the region of interest. The GDD's position was adjusted to receive reflected signals specifically from this central section. Raw data collected from the oscilloscope were then processed through FFT analysis to identify the beat frequency. For this configuration, the beat frequency was determined to be 67.03 kHz, as shown in Figure 7b. Following this, the experiment was repeated with another \"H\" shape featuring a 25 mm central projection, as illustrated in Figure 8a. Radiation was focused on central projection to evaluate the system's ability to detect depth variations. Using the same setup, the beat frequency for the 3D shape was found to be 66.03 kHz, as shown in Figure 8b. This difference in beat frequency value aligns closely with the 25 mm projection in the 3D model, confirming the system's sensitivity to subtle depth variations. This preliminary result demonstrates that the GDD, when integrated with FMCW radar, can effectively detect range differences. Figure 7: (a) Flat “H”-shaped object used for imaging, with the central portion marked by a red dotted square, and (b) the beat frequency determined through FFT analysis for the focused region (right). In interpreting the results of our imaging experiments, calculating the optical path difference (OPD) was crucial, as this reflects the effective range difference rather than the physical distance alone. For our setup, the beat frequency of 67.03 kHz, as shown in Figure 6, indicated a calculated range of 1.617 m, which can be denoted as R\({}_{1}\). The experiment was then repeated with an \"H\" shape featuring a 25 mm central projection, as illustrated in Figure 9, yielding a beat frequency of 66.03 kHz as seen in Figure 7 and a range of 1.593 m, which can be denoted as R\({}_{2}\). This range difference (R\({}_{1}\)-R\({}_{2}\)) of 24.1 mm aligns closely with the expected 25 mm projection, verifying the system's ability to detect fine depth variations accurately. Figure 8: (**a**) Three-dimensional “H”-shaped object with a 25 mm central projection. The red dotted square and arrows highlight the elevated central region, emphasizing the depth variation., and (**b**) the beat frequency determined through FFT analysis for the central projection (right). Figure 9: Schematic representation of the “H”-shaped object featuring a 25 mm central projection used for 3D depth evaluation. The experimental results were in close alignment with theoretical predictions based on Equation (1) [12], with only minor discrepancies observed. \[f_{b}=\frac{\Delta f}{T_{s}}.\ \frac{2R}{C} \tag{1}\]  
In the equation, \(f_{b}\) represents the beat frequency, \(\Delta f\) corresponds to the full frequency swing, T\({}_{s}\) stands for the chirp sweep time, 'C' denotes the velocity of light in free space, and 'R' corresponds to the target range. Minor discrepancies between the experimental and theoretical values can be attributed to limitations in range resolution (\(\delta R\)) and beat frequency resolution (\(\delta f_{b}\)), which were estimated at approximately 0.02 m and 1 kHz, respectively, using Equation (2) [13] and Equation (3) [14]:  
\[\delta f_{b}=\frac{1}{T_{s}} \tag{2}\]  
\[\delta R=\frac{C}{2\Delta f} \tag{3}\]  
The depth resolution in the proposed system is influenced by several factors, including the chirp parameters and the response time of the GDD.