Principles of Modern Radar - Volume 2 1891121537

2.4 Nonlinear Frequency Modulated Waveforms 55instantaneous frequency (normalized by β)0.50.40.30.20.10−0.1−0.2−0.3−0.4FIGURE 2-18 Theinstantaneousfrequency afterremoving the linearcomponent isviewed as oneperiod of a periodicfunction and ismodeled usingFourier seriescoefficients.−0.5−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5time (normalized by τ)where 0 = 2π/τ. For odd signals∞∑x(t) = 2 d k sin(m 0 t) (2.105)m=1The Fourier series coefficients d k are obtained by evaluating the integral∫d k = x(t) sin (m 0 t) dt (2.106)τwhere τ is one period of the waveform or, in this case, the pulse width. The coefficientsare obtained numerically by replacing the integral in (2.106) with a summation. Table 2-2TABLE 2-2Coefficients Used to Generate an NLFM WaveformIndex Coefficient Value Index Coefficient Value1 −0.0570894319 16 0.00033865012 0.0198005915 17 −0.00028743513 −0.0102418371 18 0.00024519374 0.0062655130 19 −0.00021009985 −0.0042068327 20 0.00018075516 0.0029960800 21 −0.00015607557 −0.0022222723 22 0.00013521108 0.0016980153 23 −0.00011748879 −0.0013271886 24 0.000102371010 0.0010560642 25 −0.000089424611 −0.0008525769 26 0.000078298112 0.0006965787 27 −0.000068704313 −0.0005748721 28 0.000060406914 0.0004785116 29 −0.000053210615 −0.0004012574 30 0.0000469533