**2. Ambiguity Function of LFM Waveform**

Linear frequency modulation (LFM) waveforms have wide modulation bandwidth compared with the relative narrow Doppler frequency (fd) variation of moving targets. It withstands Doppler interference by only sacrificing minimal energy loss in MF operation and it is also called pulse compression (PC) in the study.

Figure 1 shows the ambiguity plot at the zero-time delay of an LFM waveform with bandwidth=1MHz, pulse width = 1 s. In this chart, the frequency domain is normalized by the signal bandwidth for studying the relationship between the Doppler shift and the modulation bandwidth. The energy loss by the Doppler shift of the LFM wave is computed as follows

$$\text{Loss} = |\text{\textquotedblleft}/\text{BW}.\tag{1}$$

where ν is the Doppler frequency shift from the moving target. The sign is positive when the target is moving toward the observation point and the sign is negative otherwise.

The PC amplitude vs. fd /BW is defined as follows [12]

$$\text{Amp} = 1 - |\boldsymbol{\nu}| / \text{BW}.\tag{2}$$

The Doppler frequency fd caused by a moving target from a stationary observation point is calculated by

$$\mathbf{f\_d} = 2 \times f \mathbf{c} \times \mathbf{V} \text{el} / \text{speed of light.} \tag{3}$$

where the carrier frequency is *fc* and the object velocity is in the shorthand of *Vel*.

For covering object velocity up to 20 Mach, from Equations (1) and (2), the Doppler frequency is calculated as fd = 136 kHz with the carrier frequency *fc*= 3GHz and the fd/BW = 0.136. The energy under this velocity is

$$\text{Amp} = 10 \times \log\_{10}(0.868) = -0.6148(\text{dB}). \tag{4}$$

Unlike single tone pulse train waveform, the amplitude coverage of LFM has a linear decay without a vicious variation.

**Figure 1.** Normalized zero-time-delay ambiguity plot of linear frequency modulation (LFM) pulse. The vertical line marks the Doppler to bandwidth ratio of a moving target at fd/BW = 0.13.

Despite the robustly Doppler-shift endurance of the LFM waveform, however, the time delay response also contains a linear offset along with the Doppler-frequency shift. The time delay offset can be observed in Figure 2, the contour plot of the LFM ambiguity function. The layout of the time delay diagonally corresponds to the Doppler shift. This offset results in the range error to Pulse-Doppler radars processing non-stationary moving target detection by matched filter detectors.

**Figure 2.** The contour plot of the LFM ambiguity function.
