Figure 1.5 JTF image of a backscattered measured data from a dielectric‐coated wire antenna using spectrogram.
1.4 Convolution and Multiplication Using FT
Convolution and multiplication of signals are often used in radar signal processing. As listed in Eqs. 1.12 and 1.13, convolution is the inverse operation of multiplication as the FT is concerned, and vice versa. This useful feature of the FT is widely used in signal and image processing applications. It is obvious that the multiplication operation is significantly faster and easier to deal with when compared to the convolution operation, especially for long signals. Instead of directly convolving two signals in the time domain, therefore, it is much easier and faster to take the IFT of the multiplication of the spectrums of those signals as shown below:
(1.18)
In a dual manner, convolution between the frequency‐domain signals can be calculated in a much faster and easier way by taking the FT of the product of their time‐domain versions as formulated below:
(1.19)
1.5 Filtering/Windowing
Filtering is the common procedure that is used to remove undesired parts of signals such as noise. It is also used to extract some useful features of the signals. The filtering function is usually in the form of a window in the frequency domain. Depending on the frequency inclusion of the window in the frequency axis, the filters are named low‐pass (LP), high‐pass (HP), or band‐pass (BP).
The frequency characteristics of an ideal LP filter are depicted as dashed line in Figure 1.6. Ideally, this filter should pass frequencies from DC to the cut‐off frequency; fc and should stop higher frequencies beyond. In real practice, however, ideal LP filter characteristics cannot be realized. According to the Fourier theory, a signal cannot be both time limited and band limited. That is to say, to be able to achieve an ideal band‐limited characteristic as in Figure 1.6, then the corresponding time‐domain signal should theoretically extent to infinity which is of course not possible for realistic applications. Since all practical human‐made signals are time limited, i.e. it should start and stop at specific time instants, the frequency contents of these signals extent to infinity. Therefore, an ideal filter characteristic as the one in Figure 1.6 cannot be realizable; but, only the approximate versions of it can be implemented in real applications. The best implementation of practical low‐pass filter characteristic was achieved by Butterworth (Daniels 1974) and Chebyshev (Williams and Taylors 1988). The solid line in Figure 1.6 demonstrates a real LP filter characteristic of Butterworth type.
Figure 1.6 An ideal and real LP filter characteristics.
Windowing procedure is usually applied to smoothen a time‐domain signal, therefore, filtering out higher frequency components. Some of the popular windows that are widely used in signal and image processing are Kaiser, Hanning, Hamming, Blackman, and Gaussian. A comparative plot of some of these windows is given in Figure 1.7.
The effect of windowing operation is illustrated in Figure 1.8. A time‐domain signal of a rectangular signal is shown in Figure 1.8a and its FT is provided in Figure 1.8b. This function is, in fact, a sinc (sinus cardinalis) function and has major side lobes. For the sinc function, the highest side lobe is approximately 13 dB lower than the apex of the main lobe. This much of contrast, of course, may not be sufficient in some imaging applications. As shown in Figure 1.8c, the original rectangular time‐domain signal is Hanning windowed. Its corresponding spectrum is depicted in Figure 1.8d where the side lobes are highly suppressed thanks to the windowing operation. For this example, the highest side lobe level is now 32 dB below the maximum value of the main lobe which provides better contrast when compared to the original, non‐windowed signal.
Figure 1.7 Some common window characteristics.
Figure 1.8 Effect of windowing. (a) Rectangular time signal, (b) its Fourier spectrum: a sinc signal, (c) Hanning windowed time signal, (d) corresponding frequency‐domain signal.
A main drawback of windowing is the resolution decline in the frequency signal. The FT of the windowed signal has worse resolution than the FT of the original time‐domain signal. This feature can also be noticed from the example in Figure 1.8. By comparing the main lobes of the figures on the right, the resolution after windowing is almost twice as bad when compared to the original frequency‐domain signal. A comprehensive examination of windowing procedure will be presented later on, in Chapter 5.
1.6 Data Sampling
Sampling can be regarded as the preprocess of transforming a continuous or analog signal to a discrete or digital signal. When the signal analysis has to be done using digital computers via numerical evaluations, continuous signals need to be converted to the digital versions. This is achieved by applying the common procedure of sampling. Analog‐to‐digital (A/D) converters are common electronic devices to accomplish this process. The implementation of a typical sampling process is shown in Figure 1.9. A time signal s(t) is sampled at every Ts seconds such that the discrete signal, s[n], is generated via the following equation: