Suppose we wish to design a continuous-time generator that is capable of producing

Chapter 7, Problem 7.44

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Suppose we wish to design a continuous-time generator that is capable of producing sinusoidal signals at any frequency satisfying where w 1 and w2 are given positive numbers. Our design is to take the following form: We have stored a discrete-time cosine wave of period N; that is, we have stored x[O], ... , x[N - 1], where x[k] =cos N . Every T seconds we output an impulse weighted by a value of x[k], where we proceed through the values of k = 0, l, ... , N - 1 in a cyclic fashion. That is, Yp(kT) = x(k modulo N), or equivalently, ( 21Tk) Yp(kT) = cos N , and +oo (2 k) Yp(t) = k~oo cos ~ o(t - kT). (a) Show that by adjusting T, we can adjust the frequency of the cosine signal being sampled. That is, show that +oo Yp(t) = (cos wot) ~ o(t - kT), k= -oo where w0 = 21TI NT. Determine a range of values for T such that y pCt) can represent samples of a cosine signal with a frequency that is variable over the full range (b) Sketch Y p(jw ). The overall system for generating a continuous-time sinusoid is depicted in Figure P7.44(a). H(jw) is an idea11owpass filter with unity gain in its passband; that is, H(jw) = { 1 ' 0, ~._I_H_(j_w_) ----~~----- ...... y(t) (a) Figure P7 .44 lwl The parameter We is to be determined so that y(t) is a continuous-time cosine signal in the desired frequency band. (c) Consider any value of T in the range determined in part (a). Determine the minimum value of Nand some value for we such that y(t) is a cosine signal in the range w 1 :5 w :5 w2. (d) The amplitude of y(t) will vary, depending upon the value of w chosen between w 1 and w 2 . Thus, we must design a system G(jw) that normalizes the signal as shown in Figure P7.44(b). Find such a G(jw