Principles of Modern Radar - Volume 2 1891121537

6.7 Phase Error Effects 247dB re: Ideal Peak0−10−20−30−400π/2π2πFIGURE 6-18Point scattererresponse as affectedby various amountsquadratic phaseerror correspondingto β = 0, 1,2,and4.−50−3 −2 −1 0 1 2 3Normalized Cross−Rangeexponential is another quadratic-phase complex exponential. In the image domain, thewidth of G(x) is found to be 4πβθ int , where θ int is the integration angle of the collection.Furthermore, we see that the width of the resulting PSR is determined by the accumulatedphase of g(θ). Thus, larger values of β and longer synthetic apertures serve to widen G(x).Approximating G(x) as a rect function, we find that the quadratic phase error smears thecross-range PSR causing the image to be defocused. This effect is shown in Figure 6-18where the PSR is corrupted by quadratic phase errors equal to 2πβθint 2 /4 for several valuesof β. The QPE is characterized by its maximum value, found at the edges of the syntheticaperture, where θ =±θ int /2 and θ int = 1.The quadratic phase error is important for analyzing SAR performance because it canbe used to model a number of significant effects. For example, it can be used to predictthe consequences of using the tomographic paradigm when the radiated wavefronts arein fact spherical. The QPE can also be used to model the appearance of moving objectsin spotlight SAR imagery, as the cross-range component of target velocity and line-ofsightacceleration both create additional quadratic phase on top of the ideal response. Ifthe QPE is small, the moving object will be mildly defocused. If it is large, the objectcan be smeared across many cross-range bins. Especially severe QPE can spread theenergy of the moving object so much that it falls below the reflectivity of the surroundingterrain. In such cases, the moving object is invisible in the image, but may be detectedby comparison with another image of the same scene using coherent change detectiontechniques.6.7.3 Sinusoidal Phase ErrorsThe sinusoidal phase error g(θ) = exp{ jφ 0 sin(2πγθ)} is the third type of phase error wediscuss, where φ 0 is the zero-to-peak amplitude of the phase error, and γ is its frequency.Our analysis is based largely on the work by Fornaro [32]. The function g(x) can bemanipulated into a more useful form by employing the generating function for the Besselfunction of the first kind:{ ( ξexp t − 1 )}=2 t∞∑t n J n (ξ) (6.33)n=−∞