Principles of Modern Radar - Volume 2 1891121537

50 CHAPTER 2 Advanced Pulse Compression Waveform Modulationsor = φ ′ (t) (2.81)where a single prime denotes the first derivative with respect to the independent variable.The relationship in equation (2.81) suggests that the derivative of the phase function (i.e.,the instantaneous frequency) at a given value of time t determines the spectral responseat a given frequency and that and t are one-to-one provided φ ′ (t) is monotonic.Applying the inverse Fourier transform to the waveform spectrumx(t) = 1 ∫|X ()| exp ( jθ ()) exp ( jt) d (2.82)2πand again using the PSP, Key [19] derives a similar relationshipt =−θ ′ () (2.83)relating time and the derivative of the spectrum’s phase.Group delay is a measure of the relative time delay between a waveform’s frequencycomponents. Group delay, t gd , is formally defined as the negative of the first derivative ofthe spectrum’s phase function, or t gd =−θ ′ (). If the group delay is a constant, then allfrequencies are time coincident. If the group delay is linear, the frequency of the waveformvaries linearly with time and produces a rectangular-shaped spectrum as in the case of anLFM waveform. Higher-order group delays may be used to shape a waveform’s spectrum.2.4.1.2 Inverse FunctionsAn inverse relationship between instantaneous frequency and group delay exists. To showthis, solve equation (2.81) for tEquating equations (2.83) and (2.84)or equivalentlyt = φ ′−1 () (2.84)φ ′−1 () =−θ ′ () (2.85)φ ′ (t) =−θ ′−1 (t) (2.86)As illustrated in Figure 2-16, equations (2.85) and (2.86) define an inverse relationshipbetween group delay and instantaneous frequency that is commonly exploited in the designof NLFM waveforms. Cook [26] states that the inverse relationship may be visualized byrotating a plot of the group delay (versus frequency) clockwise by 90 ◦ and flipping theresult about the frequency axis to obtain a plot of instantaneous frequency versus time. Thegraphical technique may be applied in a computer to transform samples of a waveform’sgroup delay versus frequency into an instantaneous frequency versus time response.2.4.1.3 Parametric EquationsKey [19] uses equations (2.79), (2.81), (2.82), and (2.83), a Taylor series expansion, andseveral approximations to arrive at the following parametric relationships:|X ( t )| 2 ≈ 2π a2 (t)(2.87)|φ ′′ (t)|