Principles of Modern Radar - Volume 2 1891121537

5.2 CS Theory 153at position r m ∈ R 3 and a transmitter at t m ∈ R 3 operating at wavenumber k m = ω m /c =2π f m /c, where c is the speed of propagation. The RF transmission frequency is given inhertz as f m . Data in this format can of course be synthesized from a pulsed system bymatch filtering and transforming the processed signal to the frequency domain. One canthink of these data as phase history data for a multistatic SAR collection after dechirp orderamp processing [5,6].Consider an isotropic point scatterer located at an arbitrary position q ∈ R 3 . Assumingscalar wave propagation governed by the Helmholtz equation, the Green’s function (section7.3 of [7]) describing the spherical wave emanating from a point source t m and impingingon the point q is given byG(t m , q, k m ) = e jk m‖t m −q‖ 24π ‖t m − q‖ 2(5.2)To obtain the field received by a point source with an arbitrary transmit waveforma m (k) = ∫ ∞−∞ âm(t)e jkt dt, the frequency-domain Green’s function G(t m , q, k m ) is modulatedby the complex-valued frequency-domain waveform 3 a m (k m ); thus the field receivedat point q is a m (k m )G(t m , q, k m ). Similarly, if the scattering potential at q is given by x(q),then the measured field for scattering experiment m from the single point target would bey m = x(q)a m (k m )G(t m , q, k m )G(q, r m , k m ) (5.3)The antenna patterns for the transmit and receive antennas are omitted for simplicity. If welinearize the scalar data model using the Born approximation [9] and discretize the sceneinto N voxels with locations q n and reflectivities x = [x(q 1 )x(q 2 )...x(q N )] T , then wecan obtain the total measured field for experiment m asN∑y m = x(q n )a m (k m )G(t m , q n , k m )G(q n , r m , k m ) (5.4)n=1We can express the complete data collection in the form of (5.1) by defining A ∈ C M×Nwith A mn = a m (k m )G(t m , q n , k m )G(q n , r m , k m ) and denoting the additive measurementnoise as e.The inner products of the columns of A can be related to values of the familiarpoint spread function (PSF) from SAR imaging [10]. Under the far-field assumption‖t‖ 2 ≫‖q‖ 2 , the PSF reduces to the standard form encountered in parallel-ray computedtomography [6,11]. An example of this development in the time domain with arbitrarywaveforms can be found in [12].5.2.2.2 The Ambiguity FunctionWe next present two moving target indication (MTI) examples. The first example modelsthe data for a single pulse using a single radar channel to discriminate delay and Doppler.The two-dimensional function describing the coupling between delay and Doppler for aradar waveform is known as the ambiguity function.Consider the case of a radar with narrow fractional bandwidth and monostatic operation,that is, co-located transmitter and receiver antennas. Working in the time domain,3 This is because Green’s function is a fundamental solution [8] to the scalar wave equation (∇ 2 +k 2 )G =−δ, where δ is the Dirac delta function.