Читать книгу Inverse Synthetic Aperture Radar Imaging With MATLAB Algorithms - Caner Ozdemir - Страница 41

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_o sampled 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. 1.25 and 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 5 to 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. 1.27 and 1.28 calculate the resolution in time as 0.1 ms and the duration of the signal as 1 ms.