Principles of Modern Radar - Volume 2 1891121537

2.4 Nonlinear Frequency Modulated Waveforms 53whereD = 1 π cosh−1 [ 10 −PSR/20] (2.92)andS =¯n 2D 2 + (¯n − 0.5) 2 (2.93)1G =(2.94)1 + 2 ¯n−1 ∑F mThe exact shape of the Taylor weighting is a function of the desired peak sidelobe ratio(PSR) and a parameter ¯n. The Taylor weighting in equation (2.90) is centered at basebandand extends over the frequency range −πβ ≤ ≤ πβ. The square of the spectrummagnitude in equation (2.88) is set equal to equation (2.90) orm=1|X ()| 2 = W Taylor () (2.95)The requirement for a constant time-domain envelope is satisfied by setting the envelopeequal to 1 ora 2 (t) = 1 (2.96)Integrating equation (2.88) with respect to yieldsθ ′ () = k [12π G ∑¯n−1( )β F m m + 2msin + k 2β− πβ ≤ ≤ πβ (2.97)m=1The waveform’s group delay ist gd =−θ ′ () =− k [12π G∑¯n−1 + 2m=1β F mm( ) ] msin − k 2 − πβ ≤ ≤ πββ(2.98)The constants k 1 and k 2 are obtained by evaluating the group delay at the boundaryconditions. Evaluating equation (2.98) at t gd =− τ 2 when =−πβ, and at t gd = τ 2 when = πβ, yieldsandk 2 = 0 (2.99)k 1 =− τGβ(2.100)Making the substitutions for k 1 and k 2 into equation (2.98),t gd = 1[τ ∑¯n−1( )β F ] m m + 22π βmsin − πβ ≤ ≤ πβ (2.101)βm=1