Inverse Synthetic Aperture Radar Imaging With MATLAB Algorithms. Caner Ozdemir

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Inverse Synthetic Aperture Radar Imaging With MATLAB Algorithms - Caner Ozdemir


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polarizedRLRight–LeftRLOSRadar line of sightRRRight–rightRxReceiver[S]Polarization scattering matrixSACShift and correlateSARSynthetic aperture radarSBRShooting and bouncing raySFCWStepped frequency continuous wavesincSinus cardinalisSNRSignal‐to‐noise ratioSPUSignal processing unitSTFTShort‐time Fourier transformTCRTrihedral corner reflectorTFDSTime‐frequency distribution seriesTWIRThrough‐the‐wall imaging radarTWRThrough‐the‐wall radarTxTransmitterVVerticalVHVertical–horizontalVNAVector network analyzerVVVertical–verticalω‐kAFrequency‐wavenumber algorithm

      Fourier transform (FT) is a common and useful mathematical tool that is utilized in innumerous applications in science and technology. FT is quite practical especially for characterizing nonlinear functions in nonlinear systems, analyzing random signals, and solving linear problems. FT is also a very important tool in radar imaging applications as we shall investigate in the forthcoming chapters of this book. Before starting to deal with the FT and inverse Fourier transform (IFT), a brief history of this useful linear operator, and its founders are presented.

      1.1.1 Brief History of FT

      Jean Baptiste Joseph Fourier, a great mathematician, was born in 1768, Auxerre, France. His special interest in heat conduction led him to describe a mathematical series of sine and cosine terms that could be used to analyze propagation and diffusion of heat in solid bodies. In 1807, he tried to share his innovative ideas with researchers by preparing an essay entitled as On the Propagation of Heat in Solid Bodies. The work was examined by Lagrange, Laplace, Monge, and Lacroix. Lagrange's oppositions caused the rejection of Fourier's paper. This unfortunate decision cost colleagues to wait for 15 more years to meet his remarkable contributions to mathematics, physics, and especially on signal analysis. Finally, his ideas were published thru the book The Analytic Theory of Heat in 1822 (Fourier 1955).

      1.1.2 Forward FT Operation

      The FT can be simply defined as a certain linear operator that maps functions or signals defined in one domain to other functions or signals in another domain. The common use of FT in electrical engineering is to transform signals from time domain to frequency domain or vice‐versa. More precisely, forward FT decomposes a signal into a continuous spectrum of its frequency components such that the time signal is transformed to a frequency domain signal. In radar applications, these two opposing domains are usually represented as “spatial‐frequency (or wave‐number)” and “range (distance).” Such use of FT will be often examined and applied throughout this book.

      The forward FT of a continuous signal g(t) where −∞ < t < ∞ is described as

      where

represents the forward FT operation that is defined from time domain to frequency domain.

is a complex phasor representation for a sinusoidal function with the single frequency of “fi.” This signal oscillates with the single frequency of “fi” and does not contain any other frequency component. Multiplying the signal in interest, g(t) with
provides the similarity between each signal, that is, how much of g(t) has the frequency content of “fi.” Integrating this multiplication over all time instants from −∞ to ∞ will sum the “fi” contents of g(t) over all time instants to give G(fi) that is the amplitude of the signal at the particular frequency of “fi.” Repeating this process for all the frequencies from −∞ to ∞ will provide the frequency spectrum of the signal represented as G(f). Therefore, the transformed signal represents the continuous spectrum of frequency components; i.e. representation of the signal in “frequency domain.”

      1.1.3 IFT

      This transformation is the inverse operation of the FT. IFT, therefore, synthesizes a frequency‐domain signal from its spectrum of frequency components to its time domain form. The IFT of a continuous signal G(f) where −∞ < f < ∞ is described as

      where the IFT operation from frequency domain to time domain is represented by

.

      (1.3)

      and the corresponding alternative pair is given by

      (1.4)

      Based on these notations, the properties of FT are listed briefly below.

      1.2.1 Linearity

      If G(f) and H(f) are the FTs of the time signals g(t) and h(t), respectively, the following equation is valid for the scalars a and b.

      (1.5)

      Therefore, the FT is a linear operator.

      1.2.2 Time Shifting

      If the signal is shifted in time with a value of to, then the corresponding frequency signal will have the form of

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