− 1)·Δf, is sent. Each subwave has a time duration of τ and is of T distance away from the adjacent subwave. The total frequency bandwidth, B, and the frequency increment (or resolution), Δf, can be readily calculated as below:
(2.51)
(2.52)
The SFCW signal can be used to estimate the range of a possible target in the following manner. Suppose that the target is at the range distance of Ro from the radar. With a single measurement of monostatic SFCW radar, the phase of the backscattered wave is proportional to the range as given in the following equation:
Figure 2.14 SFCW signal in time‐frequency plane.
(2.53)
Here, Es is the scattered electric field, A is the scattered field amplitude, and k is the wavenumber vector corresponding to the frequency vector of f = [fo f1 f2⋯fN−1]. The number 2 in the phase corresponds to the two‐way propagation between radar‐ to‐ target and target to radar. It is obvious that there is FT relationship between (2k) and (R). Therefore, it is possible to resolve the range, Ro, by taking the inverse Fourier transform (IFT) of the output of the SFCW radar. The resulted signal is nothing but the range profile of the target. The range resolution is determined by the Fourier theory as
where BWk and BWf ≜ B are the bandwidths in wavenumber and frequency domains, respectively. The maximum range is then determined by multiplying the range resolution by the number of SFCW pulses:
Figure 2.15 Range profile of a point target is obtained with the help of SFCW radar processing.
We will demonstrate the operation of SFCW radar with an example. Let us consider a point target which is 50 m away from the radar. Suppose that the SFCW radar's frequencies change from 2 to 22 GHz with the frequency increments of 2 MHz. Using Eqs. 2.54 and 2.55, one can easily find the range resolution and the maximum range as 0.75 cm and 75 m, respectively. Applying the Matlab routine “Figure 2.15.m” to the synthetic backscattered data, the range profile of this point target can be obtained as plotted in Figure 2.15. It is clearly seen from the figure that the point target at the range of 50 m is perfectly pinpointed.
2.6.4 Short Pulse
One of the simplest radar waveforms is the short pulse (or impulse) whose time duration is usually on the order of a few nanoseconds. As calculated in Eq. 2.54, the range resolution of a pulsed radar is equal to
(2.56)
where B is the frequency bandwidth of the pulse. According to the Fourier theory, the frequency bandwidth, B of a pulse is also inversely proportional to its pulse duration as
(2.57)
which means that the range resolution is proportional to its pulse duration as
Therefore, to have a good range resolution, the duration of a pulse has to be as small as possible. Common short pulse waveforms are rectangular pulse, single‐tone pulse, and single wavelet pulse of different forms. In Figure 2.16a, a rectangular pulse‐shape wave is shown, and the spectrum of this signal is plotted in Figure 2.16b. In the frequency domain, a sinc‐type pattern is obtained as expected.
Another common single‐pulse shape is a single sine signal as plotted in Figure 2.17. Since the time‐domain pulse is smoother when compared to the rectangular pulse (see Figure 2.17a), the spectrum widens, and sidelobe levels decrease as expected according to the Fourier theory as depicted in Figure 2.17b.
Another popular short‐duration waveform is called the wavelet signal. Wavelets are much smoother than the sine pulse; therefore, they provide less sidelobes in the frequency domain. In Figure 2.18a, a Mexican‐hat type wavelet whose mathematical function is given below is shown below:
(2.59)
Since this signal is much smoother than the previous short pulse waveforms that we have presented, the frequency extent of this wavelet is extremely broad. Therefore, it provides an ultrawide band (UWB) spectrum as most of the other short‐duration wavelets do as shown in Figure 2.18b.
While these short pulses are good for providing a wide spectrum, they are not practical in terms of providing sufficient energy. This is because of the fact that it is not possible to put great amount of power onto a very small pulse. To circumvent this problem, the pulse is modulated by altering the frequency as time continues to pass. The common practice is to use a chirp waveform to be able to put enough energy onto the pulse, as will be investigated next.
2.6.5 Chirp (LFM) Pulse
As explained in the previous paragraph, it will not be possible to use a sufficiently wide pulse and achieve a wide bandwidth. If a broadband spectrum is achieved with an unmodulated, or constant‐frequency pulse (as in
