Principles of Modern Radar - Volume 2 1891121537

218 CHAPTER 6 Spotlight Synthetic Aperture Radarspecific main lobe and sidelobe behavior. Interesting features of numerous such windowingfunctions are described by Harris [12].When we let B grow to be arbitrarily large, the rect function becomes infinitelywide and the corresponding sinc funtion grows infinitesimally narrow. Without rigorousjustification of the assertion, we simply state the following transform pair:1 ⇔ δ(t) (6.11)where δ(t) is the well-known Dirac delta function that takes the value 1 at t = 0 and zeroeverywhere else [9]. For our purposes, the important feature of the delta function is thatequations (6.3) and (6.11) can be combined to find that its Fourier transform is a complexsinusoid:e − jωt 0⇔ δ(t − t 0 ) (6.12)This property holds in higher dimensions, and it is critical to understanding many aspectsof SAR imaging. Each point in a scene can be idealized as a shifted and weighted deltafunction where the weight corresponds to the reflectivity at the location implied by theshift.6.2.3 Spatial Frequency and Plane WavesThe familiar symbols t and ω are used to represent time and temporal frequency, respectively.Equally useful is the notion of spatial frequency, symbolized as k with units ofradians per meter. While temporal frequency ω represents the number of radians a wavecycles through per unit time, the spatial frequency represents the number of radians aharmonic (or single frequency) wave passes through per unit length. The wavenumber isoften written as k = 2π/λ, or equivalently k = ω/c since c = λf . The wavenumbervector k = [k x k y k z ] T is used to describe the spatial frequency in multiple dimensions.The Fourier transform of a shifted delta function, given in equation (6.12), also holdsin higher dimensions:δ(x − x 0 ) ⇔ exp{− jk · x 0 } (6.13)The two-dimensional case is obtained when x 0 has no z component. These two- and threedimensionalcomplex sinusoids are called plane waves because contours of constant phaseare parallel. Some examples are shown in the right-hand column of Figure 6-2. Given thata delta function in one domain transforms into a plane wave in the other, the frequency ofthis plane wave is proportional to the impulse’s distance from the origin. The orientationof the plane wave corresponds to the location of the delta function around the origin. Inthis fashion each point in the SAR image corresponds to a unique spatial frequency, ortwo-dimensional complex sinusoid in the Fourier domain.The SAR image is just like any other signal in that it is composed of (and can beexpressed as) the weighted sum of sinusoids. SAR practitioners rely heavily on thisFourier-oriented mindset. The relevance to SAR is that we can infer δ(x − x 0 ) by makingmeasurements in the frequency domain and then Fourier transforming the result toobtain the image. The amplitude and location of the delta function will be shown to correspondto the amplitude of the scene reflectivity at a particular location on the ground.Since the imaging process is linear, superposition holds and we are able to reconstruct the