**3. Range, Velocity Estimation by O**ff**set of LFM Waveform**

A single-chirped LFM waveform is defined as [12]

$$\exp(t) = \text{Rect}(\frac{t}{\tau\_0}) e^{j2\pi(f\_0 + \frac{\mu}{2}t^2)}\tag{5}$$

where the pulse duration is τ0, the center frequency is *f* 0, and the frequency chirping slope is μ. The sign of the slope μ indicates descending or ascending of the frequency increment. The steeper the slope, the more frequency difference in a fixed time period.

Figure 3 illustrates the phenomena of the range offset caused by the convolution of a reference simulated LFM waveform with a Doppler-shifted returning signal. The range offset is proportional to the amount of the Doppler shift (Δfd).

**Figure 3.** The linearity between target velocity and convolution range offset of a single chirped LFM waveform. The orange-squared line is the steeper chirping slope signal with pulse width (PW) = 2τ while the shows the flatter chirping slope signal with PW = τ in blue-circle line.

The range offset of convolution is linearly proportional to the moving object Δfd or Δν, which can be converted by Equation (3). The range offset changing rate is higher with the steeper chirping slope (pulse width (PW) = τ, orange-squared line) than the flatter chirping slope, PW = 2τ, LFM waveform (blue circle) under the same Δfd. Due to the linearity of a single chirp, the target velocity and detection range offset can be resolved from one another by the leaner ratio between fd/BW versus range offset Roft/μ. The equation is set up as follow [13]

$$\frac{\text{(PW} \times \text{C} / 2)}{\text{BW}} = \frac{\text{R}\_{\text{ofst}}}{\text{fd}} \tag{6}$$

where the pulse width is PW, speed of light is C, bandwidth is BW, the detection range offset is Rofst due to the Doppler shift fd.

Derived from Equation (6)

$$R\_{\text{ofst}} = \frac{\text{fd} \times (\text{PW} \times \text{C}/2)}{\text{BW}} \tag{7}$$

Rofst is proportional to fd in a single chirped LFM waveform. With a stationary target, the detecting range offset is zero, which is shown as the blue dot in the origin in Figure 3.

However, in the target searching stage, without further target information, it is difficult to resolve the true moving target location by a single chirped LFM waveform. Therefore, a two chirped LFM scheme is introduced for resolving the true location of a non-stationary moving target.

Extended from Equation (5), a two chirped LFM waveform with two opposite slopes is derived as follows [14]

$$\begin{array}{l} \text{sr}\_{2\text{chirp}}(t) \\ l = \text{Rect}(\frac{t - 0.25\tau\_0}{0.5\tau\_0})e^{j2\pi\left[f\_0(t - 0.25\tau\_0) + \frac{\mu}{2}\left(t - 0.25\tau\_0\right)^2\right]} \\ + \text{Rect}(\frac{t + 0.25\tau\_0}{0.5\tau\_0})e^{j2\pi\left[f\_0(t + 0.25\tau\_0) + \frac{\mu}{2}\left(t + 0.25\tau\_0\right)^2\right]} \end{array} \tag{8}$$

where the waveform is composed of half of positive slope μ and a half of negative slope −μ LFM within a PW = τ0.

In Figure 4, the example illustrates the phenomena of a detection pair of two equal range offset along with the true target position in the opposite direction after the matched filter detectors of a moving target with Δfd. When an up-down chirp referenced LFM signal (blue line) is shifted up by Δfd (red line), the matched filter detector has two correlation points at ±ΔR locations offset. The up-chirp signal matches the reference signal at the yellow dotted line position on the left while the down-chirp one has a matched point at the purple dotted line on the right. With the detection pair of two chirps, the true location of the target, which is zero, can be resolved unambiguously by the mean of two locations of the detection pair [14].

**Figure 4.** An up-down LFM waveform resolves two detection peaks from a non-stationary target by matched filter detectors.

Nevertheless, applying a two-chirped LFM waveform in multiple non-stationary targets scenarios, how to find the right pairs of the targets can be obscure without correct pairing information. There are three cases of ambiguous detection pair scenarios of two non-stationary moving targets.

Case 1 in Figure 5 demonstrates the detection pairs of two targets with the same velocity are right next to each other and the detection pairs are without crossover. Target one is at position zero, while target two is at position 3.3.

Case 2 in Figure 6 shows the detections of each target have one crossover with each other. Target one is at position zero, while target two is at position 0.86.

In Case 3 in Figure 7, the target one pair is enclosed by the target two detection pairs. The two targets are overlapped at position zero.

**Figure 5.** Case 1, an up-down chirp LFM waveform resolves four detection peaks by two moving targets. The detection pair of each target has no crossover.

**Figure 6.** Case 2, an up-down chirp LFM waveform resolves four detection peaks by two moving targets. The detection pair of each target has one side crossover.

**Figure 7.** Case 3, an up-down chirp LFM waveform resolves four detection peaks by two moving targets. The detection pair of target 1 is enclosed by the pair of target 2.

Observing from the examples in Figures 5–7, how to determine the right detection pair resolve the true location of each target is vague without the illustrative target-location marks. Thus, Doppler shift compensation (DSC) is applied to distinguish moving targets pair.

The DSC operation is computed as follows

$$\text{S}\_{\text{offset}}(\mathbf{t}) = \text{S}\mathbf{o}(\mathbf{t})^\ast \exp(\mathbf{j2}\pi \times \text{fd}\_{\text{offset}} \times \mathbf{t})\tag{9}$$

where the complex signal before DSC is S0(t), the DSC frequency step is fd\_offset I, and the signal after DSC is Soffset(t).

Figures 8 and 9 demonstrate the behavior of a detection pair after a series of DSC operations. The detection of up-chirp LFM moves toward a positive direction by ΔR in each DSC, while the up-chirp LFM moves toward left each time with the same amount of ΔR. Even if the pair position has crossover in the Figure 9 scenario, the ΔR movement rule for each detection in this DSC operation is still valid.

**Figure 8.** The range offset of a two-chirp detection pair of a moving target is processed by one fd\_offset DSC. Two detections both have ΔR offset, but in the opposite direction.

**Figure 9.** The range offset of a two-chirp detection pair of a moving target is processed by two fd\_offset DSC. Two detection lines crossover each other from one fd\_offset DSC (blue-dotted line) to two fd\_offset DSC (red line) with the same ΔR offset in opposite direction.
