Using x_r(t) = m(t) cos(2Ï€f_ct) and e₀(t) = 2 cos (2Ï€f_ct + θ) for the assumed Costas PLL input and VCO output, respectively, verify that all signals shown at the various points in Figure 4.26 are correct. Assuming that the VCO frequency deviation is defined by dθ / dt = -K_ve_v(t), where e_v(t) is the VCO input and K_vis a positive constant, derive the phase plane. Using the phase plane, verify that the loop locks.

Figure 4.26

Demodulated m(t) cos 0 output Lowpass filter 2 cos (@t +0) m(1) sin 20 K sin 20 Lowpass filter VCO x,(1) = m(t) cos @d 90 phase shift 2 sin (@t +0) m(t) sin 0 Lowpass filter )