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

Suppose we wish to design a continuous-time generator that is capable of producing sinusoidal signals at any frequency satisfying

where 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[0],. . . , x[N - 1], where

Every T second we output an impulse weighted by a value of x[k], where we proceed through the values of k= 0, 1, . . . , N – 1 in a cyclic fashion. That is,

or equivalently,

and

(a) Show that by adjusting T, we can adjust the frequency of the cosine signal being sampled. That is, show that

where . Determine a range of values for T such that can rep-resent samples of a cosine signal with a frequency that is variable over the full range

(b) Sketch

The overall system for generating a continuous-time sinusoid is depicted in Figure P7.44(a). H(jω) is an ideal lowpass filter with unity gain in its pass-band; that is,

The parameter 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 N and some value for , such that y(t) is a cosine signal the range

(d) The amplitude of y(t) will vary, depending upon the value of co chosen between . Thus, we must design a system G(jω) that normalizes shown in Figure P7.44(b). Find such a G(jω).