Principles of Modern Radar - Volume 2 1891121537

288 CHAPTER 7 Stripmap SARand in the frequency domain it is the conjugate of the signalTime-domain matched filtering is a 2-D convolutionH 0 (k u ,ω) = S ∗ (k u ,ω) (7.45)y(u,t) = d(u,t) ∗ u,t s ∗ (−u, − t) (7.46)which reduces to a 2-D correlation with the known signaly(u,t) =∫ ∞ ∫ ∞−∞ −∞d (υ,τ) s ∗ (υ − u,τ − t)dυdτ (7.47)Note the use of two dummy variables of integration, υ and τ. The frequency-domainoutput is simply the product of the data and the conjugate of the signalY (k u ,ω) = D(k u ,ω)S ∗ (k u ,ω) (7.48)The mathematics become more complicated when the signal is spatially variant; that is,the impulse response is a function of the target location. In that case we define the knownsignal impulse response as s(u,t; u ′ , t ′ ), where u and t are the support for the signal in theraw data as before, and u ′ and t ′ denote the signal source (e.g., scatterer location). Theconvolution integral takes the formy(u,t) =∫ ∞ ∫ ∞−∞ −∞d (υ,τ) s ∗ (υ,τ; u,t)dυdτ (7.49)Figure 7-31 highlights the role of the variables in (7.49); output location can be thoughtof as an image pixel. Instead of generating just one matched filter and sliding it throughthe data in a correlation process, a different matched filter must be generated for everyoutput (u,t). This may be computationally intensive, depending upon how difficult it isto generate the impulse response. Worse yet, there is no frequency-domain expression forspatially variant filtering.7.4.2 Range Stacking AlgorithmWe can build an image formation algorithm by rewriting the spatially variant form ofthe matched filter in (7.49) using source and output locations given by down-range andcross-range (x,r) and data location given by (u,t)f (x,r) =∫ ∞ ∫ ∞−∞ −∞d(u,t)h ∗ (u,t; x,r)dudt (7.50)∞ ∞y (u, t) = ∫∫d(υ,t)s ∗ (u, t ; u, t)dudt−∞−∞Output Location Indices Into Data Source LocationFIGURE 7-31Two-dimensional matched filtering for a spatially variant signal.