Principles of Modern Radar - Volume 2 1891121537

5.2 CS Theory 157The combined space-time steering vector for a target is then given as the Kronecker productof the temporal and spatial steering vectors, that is,a( f s , f d ) = a t ( f d ) ⊗ a s ( f s )For a given range bin, the vector a( f s , f d ) represents the data that the radar would collectover a CPI if only a single target having unit scattering amplitude were present at theangle-Doppler location encoded by f s and f d . Specifically, the steering vector a( f s , f d )corresponds to a single column of the A matrix for this MTI problem.Let us discretize the frequency variables into N s ≥ J spatial frequency bins andN d ≥ K Doppler frequency bins spaced uniformly across the allowed ranges for eachvariable to obtain N = N s N d unique steering vectors. We can organize the resultingsteering vectors into a matrix A ∈ C M×N . Neglecting range ambiguities, we can definethe scene reflectivity function at a given range as x ∈ C N , where the rows of x areindexed by angle-Doppler pairs 8 ( f s , f d ). We then obtain the linear relationship betweenthe collected data and the scene of interest asy = Ax + ewhere e in this case will include the thermal noise, clutter, and other interference signals.A more realistic formulation would include measured steering vectors in the matrix A.Thus, we see that the data for a multichannel pulsed radar problem can be placed easilyinto the framework (5.1).5.2.2.4 CommentsThe overall message is that most radar signal processing tasks can be expressed in terms ofthe linear model (5.1). Additional examples and detailed references can be found in [17].We should also mention that both of these examples have used the “standard” basisfor the signal of interest: voxels for SAR imaging and delay-Doppler cells for MTI. Inmany cases, the signal of interest might not be sparse in this domain. For example, aSAR image might be sparse in a wavelet transform or a basis of canonical scatterers. 9 Asanother example, in [18], the authors explore the use of curvelets for compressing formedSAR images. Suppose that the signal is actually sparse in a basis such that x = α,with α sparse. In this case, we can simply redefine the linear problem asy = Aα + e (5.10)to obtain a model in the same form with the forward operator A. Indeed, many of theearly CS papers were written with this framework in mind with A = , where isthe measurement operator, and is the sparse basis. In this framework, one attempts todefine a measurement operator that is “incoherent” from the basis . We will not dwellon this interpretation of the problem and refer the interested reader to, for example, [19].8 Note that x in this context is the same image reflectivity function used in common STAP applications [15].9 Selection of the appropriate sparse basis is problem dependent, and we refer the reader to the literaturefor more detailed explorations.