Caner Ozdemir - Inverse Synthetic Aperture Radar Imaging With MATLAB Algorithms

The duration, the bandwidth, and the resolution are important parameters while transforming signals from time domain to frequency domain or vice versa. Considering a discrete time‐domain signal with a duration of T o= 1/ f osampled N times with a sampling interval of T s= T o/ N , the frequency resolution (or the sampling interval in frequency) after applying the DFT can be found as

(1.25)

The spectral extend (or the frequency bandwidth) of the discrete frequency signal is

(1.26)

For the example in Figure 1.11, the signal duration is 1 ms with N = 10 samples. Therefore, the sampling interval is 0.1 ms. After applying the expressions in Eqs. 1.25and 1.26, the frequency resolution is 100 Hz and the frequency bandwidth is 1000 Hz. After taking the DFT of the discrete time‐domain signal, the first entry of the discrete frequency signal corresponds to zero frequency and negative frequencies are located in the second half of the discrete frequency signal as seen in Figure 1.11b. After the DFT operation, therefore, the entries of the discrete frequency signal should be swapped from the middle to be able to form the frequency axis correctly as shown in Figure 1.11c. This property of DFT will be thoroughly explored in Chapter 5to demonstrate its use in ISAR imaging.

Figure 1.11 An example of DFT operation: (a) discrete time‐domain signal, (b) discrete frequency‐domain signal without FFT shifting, (c) discrete frequency‐domain signal with FFT shifting.

Similar arguments can be made for the case of IDFT. Considering a discrete frequency‐domain signal with a bandwidth of B sampled N times with a sampling interval of Δf , the time resolution (or the sampling interval in time) after applying IDFT can be found as

(1.27)

The time duration of the discrete time signal is

(1.28)

For the frequency‐domain signal in Figure 1.11b or c, the frequency bandwidth is 1000 Hz with N = 10 samples. Therefore, the sampling interval in frequency is 100 Hz. After applying IDFT to get the time‐domain signal as in Figure 1.11a, the formulas in Eqs. 1.27and 1.28calculate the resolution in time as 0.1 ms and the duration of the signal as 1 ms.

Aliasing is a type of signal distortion due to undersampling the signal of interest. According to Nyquist–Shannon sampling theorem (Shannon 1949), the sampling frequency f sshould be equal to or larger than twice the maximum frequency content; f maxof the signal to be able to perfectly reconstruct the original signal:

(1.29)

Since processing the analog radar signal requires sampling of the received data, the concept of aliasing should be taken into account when dealing with radar signals.

The imaging of a target using electromagnetic waves emitted from radars is mainly based on the phase information of the scattered waves from the target. This is because of the fact that the phase information is directly related to the range distance of the target. In the case of monostatic radar configuration as shown in Figure 1.12a, let the scattering center on the target be at R distance away from the radar. Then, the scattered field Es from this scattering center on the target has a complex scattering amplitude, A and a phase factor that contains the distance information of the target as follows:

(1.30)

As is obvious from Eq. 1.30, there exists a Fourier relationship between the wave number, k , and the distance, R . Provided that the scattered field is collected over a bandwidth of frequencies ( Figure 1.12b), it is possible to pinpoint the distance R by Fourier transforming the scattered field data as depicted in Figure 1.12c. The plot of scattered field versus range is called the range profile , which is an important phenomenon in radar imaging. Range profile is, in fact, nothing but the one‐dimensional range image of the target. An example is illustrated in Figure 1.13where the range profile of an airplane is shown. The concept of range profiling will be thoroughly investigated in Chapter 4, Section 4.3.

Figure 1.12 (a) Monostatic radar configuration, (b) scattered field versus frequency, (c) range profile of the target.

Figure 1.13 Simulated range profile of an airplane.

Another main usage of FT in radar imaging is the ISAR imaging. In fact, ISAR can be regarded as the 2D range and cross‐range profile image of a target. While the range resolution is achieved by utilizing the frequency diversity of the backscattered signal, the cross‐range resolution is gathered by collecting the backscattered signal over different look angles of the target. An example of ISAR imaging for the same airplane is demonstrated in Figure 1.14where both the CAD view and the constructed ISAR image for the airplane are shown. The concept of ISAR imaging will be examined in great detail in Chapter 4.

The FT operations are also extensively used in synthetic aperture radar (SAR) imaging as well. Since the SAR data are usually huge and processing this amount of data is an extensive and time‐consuming task, the FTs are usually utilized to speed up signal processing procedures such as range and azimuth compression.

An example of SAR imagery is given in Figure 1.15where the image was acquired by spaceborne imaging radar‐C/X‐band synthetic aperture radar (SIR‐C/X‐SAR 1997) onboard the space shuttle Endeavour in 1994 ( www.jpl.nasa.gov/radar/sircxsar/capecod2.html). This SAR image covers an area of Cape Cod, Massachusetts. The details of SAR imagery will be explored in Chapter 3.

Figure 1.14 Simulated 2D ISAR image of an airplane.