Homework #5 (due Monday, October 22, 2001)Communication TheoryECEN 4242Homework #5 (due Monday, October 22, 2001)Computer Experiments on Phase-Locked Loop A phase-locked loop (PLL) circuit, as shown by Figure 2.51, is a closed-loop system andconsists of three main components: (1) a phase comparator, (2) a loop filter, and (3) avoltage-controlled oscillator (VCO). A PLL circuit is usually used in coherent detectionand in frequency demodulation (frequency tracking). The phase comparator is simply a multiplier, which takes as inputs, two sinusoids andproduces a signal with two components (one is high frequency and the other one is lowfrequency) as follows )2/)()(cos()2/)()(4cos(2)2/)(2cos()).(2cos()(212121tttttfAAttfAttfAtecvccvcc (1)where tfdmkt01)(2)( (2)and tvdvkt02)(2)( (3) The loop filter is a low-pass filter that is used to cancel out the high frequency compo-nent of e(t). Thus the loop filter output will be ))(sin())()(sin()(21tKttKtve , (4)where K is a proportionality constant. The VCO is an oscillator whose output frequency is proportional to its input, which isin fact the signal v(t). This means that if there is any non-zero phase error e(t) it will ap-pear at the loop filter output and thus it will change the phase of the VCO output 2(t) sothat it becomes much closer to 1(t). If the PLL is operating near the optimum, then we can make the following approxima-tion:)())(sin( ttee. In this case the PLL behaves as a linear system and can be mod-eled as shown in Figure 2.52. In the case of Figure 2.52, three transfer functions are important to study:1. The open-loop transfer function G(s)=KL(s)/s, where L (s) is the transfer function ofthe loop filter, given in general by a first-order form, K is a constant, and s is theLaplace operator given by s=j2f.2. The closed-loop transfer function given by )(1)()()()(12sGsGsssH(5)13. The phase error transfer function )(11)(1)()()()()()(1211sGsHssssssFe (6) The transfer function L(s) of the loop filter has a first-order form sssL2111)( (7)where 12. Another commonly used form of L(s) is given by sasL 1)( (8) The simulation of a PLL, in general, consists of studying the behavior of the phase er-ror e(t) and the behavior of the VCO output phase 2(t) for a given input phase 1(t).This can be done using the transfer function F(s) and H(s), respectively. When the PLL isproperly functioning, the phase error e(t) should converge to zero and the VCO outputphase 2(t) should converge to the input phase 1(t) after a transient period of time.Questions1) Show that when L(s) is given by (7), the closed-loop transfer function H(s) can bewritten in the following standard form 22222)/2()(nnnnnsssKsH, where the pa-rameters n=2fn and , called the natural angular frequency and the damping factor,respectively, are to be determined.2) Show that when L(s) is given by (7), the phase error transfer function F(s) is given by2212)1()1()(ssKKsssF.3) Find similar expressions for the transfer functions H(s) and F(s) when L(s) takes theform given by (8).4) Show (for both cases of L(s)) that for an input phase 1(t) given by a unit step func-tion, we have lim e(t)=0 when t. (Hint: use Laplace transform property given by:0s )(lim)(limtsstee ).5) Modify and use the provided MATLAB script to compute and plot the phase errore(t) and the VCO output phase 2(t) for a unit step input phase 1(t), using L(s) givenby (7) with K=1, n=1, and =0.3, 0.707, 1.6) Comment your results. What value of the damping factor  do you think is a goodcompromise between fast response time and an under-damped oscillatory behavior?27) Repeat Question 5) for an input phase 1(t) given by one period of a square pulse withamplitude +1 and –1.NoteThe MATLAB script provided with this homework uses a state-space representation ofthe transfer function under analysis. For either transfer functions F(s) or H(s), the statespace representation is given by)()()()()()(tDutCxtytButAxtxdtd,where x(t) is the state vector, u(t) is the input, and y(t) is the output. The matrices A, B, C,and D can be obtained using the MATLAB function tf2ss.m. The implementation of this state space representation can be approximated by)()()()()()()(tDutCxtyttButtAxtxttxor by)()()()()()()1(iDuiCxiytiButiAxixixIn the MATLAB script, the time step size is taken as