Читать книгу Inverse Synthetic Aperture Radar Imaging With MATLAB Algorithms - Caner Ozdemir - Страница 74

In radar theory, the concept of Doppler frequency describes the shift in the center frequency of an incident EM wave due to movement of radar with respect to target. The basic concept of Doppler shift in frequency has been conceptually demonstrated through Figure 2.10 and is defined as

(2.66)

where v_r is the radial velocity along the radar line of sight (RLOS) direction. Now, we will demonstrate how the shift in the phase (also in the frequency) of the reflected signal from a moving target constitutes. Let us consider an object moving toward the radar with a speed of v_r. The radar produces and sends out pulses with the PRF value of f_PR. Every pulse has a time duration (or width) of τ. The illustration of Doppler frequency shift phenomenon is given in Figure 2.22. The leading edge of the first pulse hits the target (see Figure 2.22a). After a time advance of Δt. the trailing edge of the first pulse hits the target as shown in Figure 2.22b. During this time period, the target traveled a distance of

(2.67)

Figure 2.22 Illustration of Doppler shift phenomenon: (a) the leading edge of the first pulse in hitting the target at t = 0; (b) the trailing edge of the first pulse in hitting the target at t = Δt; (c) the trailing edge of the second pulse is hitting the target at t = dt. During this period, the target traveled a distance of D = v_r × dt.

Looking at the situation in Figure 2.22b, it is obvious that the pulse distance before the reflection is equal to the distance traveled by the leading (or trailing) edge of the pulse plus the distance traveled by the target as

(2.68)

Similarly, the pulse distance after the reflection is equal to the distance traveled by the leading (or trailing) edge of the pulse minus the distance traveled by the target as

(2.69)

Dividing these last two equations yields

(2.70)

On the left‐hand side of this equation, c terms are canceled, whereas Δt terms are canceled on the right‐hand side. Then, the pulse width after the reflection can be written in terms of the original pulse width as

(2.71)

The term (c − v_r)/(c + v_r) is known as the dilation factor in the radar community. Notice that when the target is stationary (v_r = 0), then the pulse duration remains unchanged (τ' = τ) as expected.

Now, consider the situation in Figure 2.22c. As trailing edge of the second pulse is hitting the target, the target has traveled a distance of

(2.72)

within the time frame of dt. During this period, the leading edge of the first pulse has traveled a distance of

(2.73)

On the other hand, the leading edge of the second pulse has to travel a distance of (c/f_PR − D) at the instant when it reaches the target. Therefore,

(2.74)

Solving for dt yields

(2.75)

Putting Eq. 2.75 into 2.72, one can get

(2.76)

The new PRF for the reflected pulse is

(2.77)

Substituting Eq. 2.75 to the above equation, one can get the relationship between the PRFs of incident and reflected waves as

(2.78)

If the center frequency of the incident and reflected waves are f₀ and , these two frequencies are related to each other with the same factor:

(2.79)

To find the Doppler shift in the frequency, f_D, we should subtract the center frequency of the incident wave from the center frequency of the reflected wave as

(2.80)

Since it is also obvious that the target velocity is very small compared to the speed of light (i.e. v_r ≪ c), Eq. 2.81 simplifies as

(2.81)

where λ₀ is the wavelength corresponding to the center frequency of f₀. For the target that is moving away from the radar, the Doppler frequency shift has a negative sign as

(2.82)

Figure 2.23 Doppler shift is caused by the target's radial velocity, v_r.

It is clear from these equations that Doppler frequency shift is directly proportional to the velocity of the target. If the velocity increases, the shift in the frequency increases as well. If the target is stationary with respect to radar (v_r = 0), then the Doppler frequency shift is zero. The velocity v_r in the above equation corresponds to the velocity along the RLOS direction. If the target is moving along another direction, v_r refers to the velocity projected toward the direction of radar. If the target's velocity is v as illustrated in Figure 2.23, then the Doppler frequency shift in terms of the target's original velocity will be equal to

(2.83)