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signal (or the spectrum) of “prince.”

      1.3.3 Signal in the Joint Time‐Frequency (JTF) Plane

Although FT is very effective for demonstrating the frequency content of a signal, it does not give the knowledge of frequency variation over time. However, most of the real‐world signals have time‐varying frequency content such as speech and music signals. In these cases, the single‐frequency sinusoidal bases are not considered to be suitable for the detailed analysis of those signals. Therefore, JTF analysis methods were developed to represent these signals both in time and frequency to observe the variation of frequency content as the time progresses.

      There are many tools to map a time domain or frequency‐domain signal onto the JFT plane. Some of the most well‐known JFT tools are short‐time Fourier transform (STFT) (Allen 1977), Wigner–Ville distribution (Nuttall 1988), Choi–Willams distribution (Du and Su 2003), Cohen's class (Cohen 1989), and time‐frequency distribution series (TFDS) (Qian and Chen 1996). Among these, the most appreciated and commonly used one is the STFT or the spectrogram. STFT can easily display the variations in the sinusoidal frequency and phase content of local moments of a signal over time with sufficient resolution in most cases.

      The spectrogram transforms the signal onto two‐dimensional (2D) time‐frequency plane via the following famous equation:

      (1.17)

      This transformation formula is nothing but the short‐time (or short‐term) version of the famous FT operation defined in Eq. 1.1. The main signal, g(t) is multiplied with a shorter duration window signal, w(t). By sliding this window signal over g(t) and taking the FT of the product, only the frequency content for the windowed version of the original signal is acquired. Therefore, after completing the sliding process over the whole duration of the time‐domain signal g(t) and putting corresponding FTs side by side, the 2D STFT of g(t) is obtained.

      It is obvious that STFT will produce different output signals for different duration windows. The duration of the window affects the resolutions in both domains. While a very short‐duration time window provides a good resolution in the time direction, the resolution in the frequency direction becomes poor. This is because of the fact that the time duration and the frequency bandwidth of a signal are inversely proportional to each other. Similarly, a long duration time signal will give a good resolution in frequency domain while the resolution in the time domain will be bad. Therefore, a reasonable selection has to be bargained about the duration of the window in time to be able to view both domains with fairly good enough resolutions.

The shape of the window function has an effect on the resolutions as well. If a window with sharp ends is chosen, there will be strong side lobes in the other domain. Therefore, smooth waveform type windows are usually utilized to obtain well‐resolved images with less side lobes with the price of increased main beamwidth, i.e. less resolution. Commonly used window types are Hanning, Hamming, Kaiser, Blackman, and Gaussian.

Figure 1.3 The time‐frequency representation of the word “prince.”

An example of the use of spectrogram is demonstrated in Figure 1.3. The spectrogram of the sound signal in Figure 1.1 is obtained by applying the STFT operation with a Hanning window. This JFT representation obviously demonstrates the frequency content of different syllables when the word “prince” is spoken. Figure 1.3 illustrates that while the frequency content of the part “prin…” takes place at low frequencies, that of the part “..ce” occurs at much higher frequencies.

JTF transformation tools have been found to be very useful in interpreting the physical mechanisms such as scattering and resonance for radar applications (Trintinalia and Ling 1995; Filindras et al. 1996; Özdemir and Ling 1997; Chen and Ling 2002). In particular, when JTF transforms are used to form the 2D image of electromagnetic scattering from various structures, many useful physical features can be displayed. Distinct time events (such as scattering from point targets or specular points) show up as vertical line in the JTF plane as depicted in Figure 1.4a. Therefore, these scattering centers appear at only one time instant but for all frequencies. A resonance behavior such as scattering from an open cavity structure shows up as horizontal line on the JTF plane. Such mechanisms occur only at discrete frequencies but over all time instants (see Figure 1.4b). Dispersive mechanisms, on the other hand, are represented on the JTF plane as slanted curves. If the dispersion is due to the material, then the slope of the image is positive as shown in Figure 1.4c,d. The dielectric coated structures are the good examples of this type of dispersion. The reason for having a slanted line is because of the modes excited inside such materials. As frequency increases, the wave velocity changes for different modes inside these materials. Consequently, these modes show up as slanted curves in the JTF plane. Finally, if the dispersion is due to the geometry of the structure, this type of mechanism appears as a slanted line with a negative slope. This style of behavior occurs for such structures such as waveguides where there exist different modes with different wave velocities as the frequency changes as seen in Figure 1.4e,f.

Figure 1.4 Images of scattering mechanisms in the joint time–frequency plane. (a) Scattering center, (b) resonance, (c and d) dispersion due to material, (e and f) dispersion due to geometry of the structure.

An example of the use of JTF processing in radar application is shown in Figure 1.5 where spectrogram of the simulated backscattered data from a dielectric‐coated wire antenna is shown (Özdemir and Ling 1997). The backscattered field is collected from the Teflon‐coated wire (ε_r = 2.1) such that the tip of the electric field makes an angle of 60° with the wire axis as illustrated in Figure 1.5. After the incident field hits the wire, infinitely successive scattering mechanisms occur. The first four of them are illustrated on top of Figure 1.5. The first return comes from the near tip of the wire. This event occurs at a discrete time that exists at all frequencies. Therefore, this return demonstrates a scattering center‐type mechanism. On the other hand, all other returns experience at least one trip along the dielectric‐coated wire. Therefore, they confront a dispersive behavior. As the wave travels along the dielectric‐coated wire, it is influenced by the dominant dispersive surface mode called Goubau (Richmond and Newman 1976). Therefore, the wave velocity decreases as the frequency increases such that the dispersive returns are tilted to later times on the JTF plane. The dominant dispersive scattering mechanisms numbered 2, 3, and 4 are illustrated in Figure 1.5 where the spectrogram of the backscattered field is presented. The other dispersive returns with decreasing energy

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