# Experimental Test of Continuous Wave Frequency Diverse Array Doppler Radar +
## 5 Data Collection IV: Micro-Doppler
### Time-Frequency Analysis of Simulated Reflections  
We use the short-time Fourier transform (STFT), a linear time-frequency analysis method, to analyze the received samples. Linear time-frequency analysis is selected for its ease of interpretation, as it lacks the cross-terms found in bilinear time-frequency analysis methods [55; 56].  
The STFT is defined as [55; 56]  
\[X(t,f)=\int_{-\infty}^{+\infty}x(\tau)h(\tau-t)e^{-j2\pi f\tau}\,d\tau, \tag{23}\]  
where \(x(\cdot)\) is the captured signal, \(h(\cdot)\) is the window function, and \(f\) is frequency of the Fourier kernel.  
Our Fourier transform (FT) window is a \(400\cdot 10^{3}\) sample Blackman window, and the sample overlap between successive FT windows is \(10\cdot 10^{3}\) samples. These parameters yield time and frequency resolutions of, respectively, \(t_{res}\approx 0.40\) s and \(f_{res}=2.5\) Hz.  
Time-frequency analysis of the simulated fast-time micro-Doppler return is shown in Figure 18 through Figure 19. The simulations use the operational parameters given above in Section 5.2. A 5 second zoom in is shown to provide a clearer view of key time-frequency features produced by the rotating blade(s):  
1. Periodic blade flashes.
2. Strong scattering from the blade tip during each periodic flash.
3. A sinusoidal trace produced by the tip of the rotating blade.  
The enumerated items above are annotated in Figure 18 to provide a link between the verbal description and the associated time-frequency behavior. Note the similarity between Figures 18, 20 and Figures 19, 21, with Figures 19, 21 being the summation of Figures 18, 20 and a \(180^{\circ}\) phase-shifted version of Figures 18, 20, respectively.  
Figure 17: Photograph of the different transmit and receive antenna setups for Data Collection IV.  
Figure 18: Time–frequency analysis of the simulated micro-Doppler response from a single blade propeller with operational parameters \(f_{0}=5.90\) GHz, \(L_{1b}=0.165\) m, and \(\Omega=5.24\) rad/s.  
Figure 19: Time–frequency analysis of the simulated micro-Doppler response from a two-blade propeller with operational parameters \(f_{0}=5.90\) GHz, \(\Delta f=20\) kHz, \(L_{2b}=0.145\) m, and \(\Omega=5.24\) rad/s.  
Figure 20: Time–frequency analysis of the simulated micro-Doppler response from a single-blade propeller with operational parameters \(f_{0}=5.90\) GHz, \(\Delta f=20\) kHz, \(L_{1b}=0.165\) m, and \(\Omega=5.24\) rad/s.  
Figure 23: Time–frequency analysis of the experimentally captured micro-Doppler response from a single-blade propeller with operational parameters \(f_{0}=5.90\) GHz, \(\Delta f=20\) kHz, \(L_{1b}=0.165\) m, and \(\Omega=5.24\) rad/s.  
Figure 22: Time–frequency analysis of the experimentally captured micro-Doppler response from a single-blade propeller with operational parameters \(f_{0}=5.90\) GHz, \(L_{1b}=0.165\) m, and \(\Omega=5.24\) rad/s.  
Figure 24: Time–frequency analysis of the experimentally captured micro-Doppler response from a two-blade propeller with operational parameters \(f_{0}=5.90\) GHz, \(L_{2b}=0.145\) m, and \(\Omega=5.24\) rad/s.  
Figure 25: Time–frequency analysis of the experimentally captured micro-Doppler response from a two-blade propeller with operational parameters \(f_{0}=5.90\) GHz, \(\Delta f=20\) kHz, \(L_{2b}=0.145\) m, and \(\Omega=5.24\) rad/s.