Principles of Modern Radar - Volume 2 1891121537

10.2 Space-Time Signal Representation 465where n is the pulse index, and the factor of two accounts for two-way range; as in thespatial case, τ n is simply distance divided by the velocity of propagation.As a result of the time-varying change in range, a pulse-to-pulse phase rotation occursleading to a Doppler frequency offset. The phase for the n-th pulse is[ ]ro + nrφ n = τ n ω c = 4πλ o(10.29)This expression neglects bandwidth-related dispersion. Since radian frequency is the timederivative of phase, the corresponding Doppler frequency isf d = 1 dφ n2π dt= 2rλ o t = 2v rλ o. (10.30)v r denotes the radial velocity component, sometimes also called line-of-sight velocity orrange rate. Equation (10.30) indicates that motion along a radial line between radar andtarget leads to the Doppler frequency offset.The resulting temporal snapshot (temporal signal vector) corresponding to a pointscatterer with normalized Doppler ˜f d = f d T is x t = a t s t ( ˜f d ), where a t is the signalcomplex envelope (taken as perfectly correlated over the coherent dwell) ands t ( ˜f d ) = [ 1 e j2π ˜f de j2π(2 ˜f d )··· e j2π(N−1) ˜f d] T(10.31)s t ( ˜f d ) is known as the temporal steering vector. Comparing (10.31) and (10.19), we finda mathematical similarity between spatial and temporal responses after writing (10.19) ass s ( f sp ) = ā [ 1 e j2π f spe j2π(2 f sp)··· e j2π(M−2) f spe j2π(M−1) f sp ] T(10.32)where ā is a complex scalar. For this reason, the collection of N receive pulses is sometimescalled the temporal aperture.Based on this similarity between spatial and temporal responses, we find by analogyto our discussion of the spatial matched filter in Section 10.2.1 that the temporal matchedfilter requires the weight vector w t = s t ( ˜f d ). It is common to apply a real-valued weightingfunction to tailor the sidelobe response at the expense of broadened main lobe and loss ofintegration gain: w t = b t ⊙ s t ( ˜f d ), where b t is a real-valued function, such as a Hanningor Chebyshev weighting.Spatial or temporal matched filtering operations are coherent. The matched filteradjusts the phase of each sample and then sums the individual voltages. Referring to thecollection of N pulses as the coherent processing interval (CPI)—common in MTI radarparlance—emphasizes this notion.The temporal or spatial steering vectors in (10.31)–(10.32) assume a Vandermondestructure: each consecutive element of the vector is a multiple of the preceding term,increased by a single integer. This structure characterizes a linear phase ramp across thesampled aperture; the slope of the phase ramp corresponds to the particular frequency.Employing a fast Fourier transform (FFT) [9] enables the processor to efficiently testfor the various phase ramp slopes, thereby accomplishing matched filtering. A weightingfunction is directly applied to the data in this instance. Straddle loss—the loss in SNRdue to a mismatched filter response—occurs in practice since the precise target spatial orDoppler frequency is generally unknown. The difference between the precise linear phase