Principles of Modern Radar - Volume 2 1891121537

7.4 Range-Doppler Algorithms 287where ∗ t denotes convolution. Note one feature of convolution in (7.36): a change in signon the dummy variable of integration between the two terms in the integral.Alternatively, filtering may be implemented in the frequency domain. Using theFourier transform to find the frequency response of the data, D(ω),and filter, H(ω),D(ω) =H(ω) =∫ ∞−∞∫ ∞−∞filtering is simply a multiplicative process with output, Y (ω).d(t)e − jωt dt (7.37)h(t)e − jωt dt (7.38)Y (ω) = D(ω)H(ω) (7.39)The matched filter is designed to maximize the signal-to-noise ratio (SNR) on aknown signal. The matched filter, h 0 (t), for a signal, s(t), in additive white Gaussiannoise (AWGN) ish 0 (t) = s ∗ (−t) (7.40)The matched filter is simply the conjugate of the time-reversed signal to which the filteris matched. Applying the matched filter in (7.40) to the convolution in (7.36) yieldsy(t) =∫ ∞−∞d (τ) s ∗ (τ − t)dτ (7.41)In contrast to the convolution equation, the dummy variable of integration has thesame sign in both terms, and one term is conjugated with respect to the other. These arethe hallmarks of not convolution but correlation. The matched filter output, then, is thecorrelation of the data with the conjugate of the signal of interest.The Fourier transform of the matched filter, H 0 (ω), can be found, through a reorderingof the conjugation operation on the Fourier integral, to beH 0 (ω) = S ∗ (ω) (7.42)Applying (7.42) to (7.39) yields the frequency-domain output of the matched filterY (ω) = D(ω)S ∗ (ω) (7.43)In anticipation of a SAR application we extend the matched filter development to twodimensions. The signal, s(u,t), constitutes the known system impulse response; in SAR,s(u,t) corresponds to the PSR. The matched filter in AWGN is the conjugate of the signalreversed in both dimensionsh 0 (u,t) = s ∗ (−u, − t) (7.44)