CU Boulder 10-28-2002ECE Dept.Homework #7(Due Wednesday, November 6, 2002)Computer Experiment on Phase-Locked Loop (PLL)CU Boulder 10-28-2002ECE Dept.Communication TheoryECEN 4242Homework #7 (Due Wednesday, November 6, 2002)Problem 1 5.4-2 (From Text book)Problem 2 5.4-3 (From Textbook)Problem 3Computer Experiment on Phase-Locked Loop (PLL)A phase-locked loop (PLL) circuit, as shown by Figure 1, is a closed-loop system and consists ofthree main components: (1) a phase comparator, (2) a loop filter, and (3) a voltage-controlled oscilla-tor (VCO). A PLL circuit is usually used in coherent detection and in frequency (FM) demodulation(or frequency tracking). The phase comparator is simply a multiplier, which takes as inputs, two sinusoids and produces asignal with two components (one is high frequency and the other one is low frequency) as follows )2/)()(cos()2/)()(2cos(2)2/)(cos()).(cos()(00000tttttAAttAttAteiicicici (1)where tfidmkt0)()((2)and tdekt0000)()((3) The loop filter is a low-pass filter that is used to suppress the high frequency component of e(t).Thus the loop filter output will be ))(sin(*)())()(sin()(00ttKhttKteei ,(4)where K is a proportionality factor, h(t) is the impulse response of the loop filter, and * denotes theconvolution operation.1The VCO is an oscillator whose output frequency is proportional to its input signal, which is infact the signal e0(t). This means that if there is any non-zero phase error e(t) its effect will appear atthe loop filter output and thus it will change the phase of the VCO output 0(t) so that the latter be-comes much closer to i(t). If the PLL is operating near the optimum, then we can make the following approximation:)())(sin( ttee. In this case the PLL behaves as a linear system and can be modeled as shown inFigure 2. In the case of Figure 2, three transfer functions are important to study:1. The open-loop transfer function L(s)=KH(s)/s, where H(s) is the transfer function of the loop fil-ter, given in general by a first-order form, K is a constant, and s is the Laplace operator given by s= j.2. The closed-loop transfer function given by )(1)()()()(0sLsLsssGi(5) where )(si and )(0s are, respectively, Laplace transforms of i(t) and 0(t).3. The phase error transfer function )(11)(1)()()()()()(0sLsHssssssFiiie (6) where )(se is the Laplace transform of e(t). The transfer function H(s) of the loop filter has a first-order form sssH2111)((7) where 12. Another commonly used form of H(s) is given by sasH 1)((8)The simulation of a PLL, in general, consists of studying the behavior of the phase error e(t) and thebehavior of the VCO output phase 0(t) for a given input phase i(t). This can be done using thetransfer function F(s) and G(s), respectively. When the PLL is properly functioning, the phase errore(t) should converge to zero and the VCO output phase 0(t) should converge to the input phase i(t)after a transient period of time.In general, the transfer function of the loop filter H(s) should be designed to ensure the conver-gence of e(t) to zero and the convergence of 0(t) to i(t) as t .2Questions1) Show that when H(s) is given by (7), the closed-loop transfer function G(s) can be written in thefollowing standard form 22222)/2()(nnnnnsssKsG, where the parameters n=2fn and ,called the natural angular frequency and the damping factor, respectively, are to be determined interms of K, 1, and 2.2) Show that when H(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 G(s) and F(s) when H(s) takes the form givenby (8).4) Show (for both cases of H(s)) that for an input phase i(t) given by a unit step function, we havelim e(t)=0 when t. (Hint: use Laplace transform property given by: 0s )(lim)(limtsstee ).5) Use and modify (if necessary) the provided MATLAB script to compute and plot the phase errore(t) and the VCO output phase 0(t) for a unit step input phase i(t), using H(s) given by (7) withK=1, n=1, and  = 0.3, 0.5, 0.707, 1. What values of 1 and 2 are used for each case?6) Comment your results. Which value of the damping factor  do you think is a good compromisebetween fast response time and an under-damped oscillatory behavior?7) Repeat Questions 5) and 6) for an input phase i(t) given by a periodic square wave with a period50 s and amplitude alternating between +2 and –2. For convenience, consider only two periods.NoteThe MATLAB script provided with this homework uses a state-space representation of the transferfunction under analysis. For either transfer functions F(s) or H(s), the state space representation isgiven by )()()()()()(tDutCxtytButAxtxdtd,(9)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 canbe obtained using the MATLAB function tf2ss.m, which converts a transfer function into a state-space representation.The implementation of this state space representation can be approximated by3)()()()()()()(tDutCxtyttButtAxtxttx(10) or by )()()()()()()1(kDukCxkytkButkAxkxkx(11)In the MATLAB script, the time