Question: Let φ = 2ωcv/c. Show that aωc is approximately equal to ωc +φ. It follows then that, for ω > 0, F(ω), the Fourier transform of the echo f(t), is approximately AaeibωΨ(ω + φ). Because the Doppler shift affects positive and negative frequencies differently, it is convenient to construct a related signal having only positive frequency components. Let G(ω)=2F(ω) for ω > 0 and G(ω) = 0 otherwise. Let g(t) be the inverse Fourier transform of G(ω). Then, the complex-valued function g(t) is called the analytic signal associated with f(t). The function f(t) is the real part of g(t); the imaginary part of g(t) is the Hilbert transform of f(t). Then, the demodulated analytic signal associated with f(t) is h(t) with Fourier transform H(ω) = G(ω+ωc). Similarly, let γ(t) be the demodulated analytic signal associated with ψ(t).