### Theoretical model of echo signals

The intended target in this paper is a precession rotational symmetric warhead model, which is a simplified model of the ballistic missile target during the midcourse. The simulation data of this paper come from a 3D electromagnetic simulation software named computer simulation technology (CST). The diagram of the warhead model observed by radar is shown in Fig. 1.

A reference frame O-XYZ that takes the mess center of the model as the origin is established. Considering the model precessing around O-Z axis, the spin angular velocity is *Ω*, the precession angular velocity is *ω*, the precession angle is *θ*, the azimuth and pitch angles of the light of sight (LOS) are *α* and *β*, the initial distance between the radar and the model is *R*
_{0}. According to the theoretical calculation and the experimental measurement, every scattering center corresponds to a discontinuity in the Stratton-Chu integral, that is the discontinuities of the curvature or surface from the view point of geometry. For the model shown in Fig. 1, the scattering centers include the cone-top P1, the discontinuities in the intersection line of the plane *Γ* (plane *Γ* is composed of the LOS and the symmetrical axis of the model Oz) and the model P2, P3, P4, and P5.

Because the model is rotational symmetric, spin makes no difference to the echo modulation. Thus, the radial distance *r*(*t*) between the radar and any scattering center located at (*x*, *y*, *z*) on the target may be written as:

$$ \begin{array}{c} r(t)={R}_0+ \sin \beta \left( y \sin \theta + z \cos \theta \right)\\ {}+\left( x \cos \alpha + y \sin \alpha \cos \theta - z \sin \alpha \sin \theta \right) \cos \beta \cdot \cos \omega t\\ {}+\left( x \sin \alpha - y \cos \alpha \cos \theta + z \cos \alpha \sin \theta \right) \cos \beta \cdot \sin \omega t\\ {}\kern1.25em ={R}_0+ \sin \beta \left( y \sin \theta + z \cos \theta \right)+ A \cos \left(\omega t+\varphi \right)\end{array} $$

(1)

In this equation, *A* is amplitude modulated coefficient, and *φ* is the initial phase. The micro-Doppler *f*
_{
d
}(*t*) of the scattering center with a carrier frequency *f*
_{0} is:

$$ \begin{array}{l}{f}_d(t)=-\frac{2{f}_0}{c}\frac{ d r(t)}{ d t}=\frac{2{f}_0}{c} A\omega \sin \left(\omega t+\varphi \right)\\ {}\kern2.5em ={A}_{\omega} \sin \left(\omega t+{\varphi}_{\omega}\right)\end{array} $$

(2)

where *A*
_{
ω
} is the maximal micro-Doppler frequency, and *φ*
_{
ω
} is its initial phase. The micro-Doppler of scattering centers on the model are sinusoidally modulated with a period 2*π*/*ω*. Therefore, the extraction of precession parameters is equivalent to the estimation of periods, amplitudes and initial phases of sinusoidal curves in time-frequency images. In general, amplitudes and initial phases usually reflect the relative positions of scattering centers, which can be used for space reconstruction and imaging of micro-motion targets.

### Characteristics of Doppler aliasing

There is no sufficient available real data of precession targets in terahertz band to illustrate the characteristics of Doppler aliasing limiting by the current devices and conditions of our laboratory, and consequently the CST data are adopted to simulate the reality. It is obvious that micro-Doppler *f*
_{
d
} is in proportion to the carrier frequency *f*
_{0} through Eq. (2). The higher the *f*
_{0}, the more evident is the micro-Doppler effect. Therefore, micro-Doppler values in terahertz band are far larger than that in microwave band under identical motion conditions. Suppose the PRF of the transmitted signal is 1 KHz, and the maximum micro-Doppler value of a micro-motion scattering center is about 40 Hz when the carrier frequency is 10 GHz (Fig. 2a), while at 0.33 THz it would reach 1320 Hz, which substantially exceeds the up limit of PRF/2, hence aliasing is present (Fig. 2b).

In micro-Doppler aliasing situations, the observed Doppler values are no longer the correct values, but are projections on the interval from –PRF/2 to PRF/2. Estimation of the real Doppler value for SAR or ISAR becomes necessary as we cannot obtain perfect SAR/ISAR images without the real Doppler value.

### Aliasing effect on parameter estimation methods

Traditional parameter estimation methods such as the Fourier transform and the Inverse Radon transform (IRT) are not applicable for the aliasing situation because aliasing changes the signal properties and destroys the completeness of time-frequency curves.

The spectrum results of echo signals at 10 GHz and 0.33 THz are shown in Fig. 3, respectively. We can easily find that the abscissa values of the peaks in the spectra correspond to the maximal micro-Doppler values in non-aliasing situation. In aliasing situation on the contrary, abscissa values of the peaks are no longer correspond to the maximal micro-Doppler values but the aliasing values, and suggests that the Fourier transform is not suitable for aliasing situations.

The IRT is an effective method for the detection of sinusoidal curves. It can map sinusoidal curves in digital images to peaks in parameter spaces, and then obtain the parameters according to positions of peaks in parameter spaces. However, it is not applicable in micro-Doppler aliasing situations for the completeness of sinusoidal curves is destroyed. The IRT results of echo signals at 10 GHz and 0.33 THz are shown in Fig. 4, and we cannot obtain any useful information from the IRT result of aliasing micro-Doppler in Fig. 4b although the IRT is a powerful tool (see [13]). As a result, we have to study new parameter estimation algorithms for micro-Doppler aliasing situations.