Please do Exercise 3 part A-D please show step by step and explain multiplying the sine and cosine components have different This produces to a signal having the mathemat-ical form a cos(wt) + b sin(wt). We can express this signal much more compactly if we make use of complexexponentials:Exercise 2.(a) Let z = a + bi, and express a cos(wt) + b sin(wt) in terms of z, z, eit, and e-it. (Hint: Express cos(wt)and sin(wt) in terms of eit and e-iwt.)(b) In (a), you should have obtained two terms. Show the two terms are complex conjugates of each other.(c) In the chapter on complex numbers, we showed that for any complex number w the real part of w(written as Re[w]) can be expressed as:Use this fact to show thatw+w2a cos(wt) + b sin(wt) = Re[z.eiwt].Re[w]=As a consequence of Exercise 2, we can treat real signals that are combinations of cosine and sine wavesas the real part of complex signals composed of complex exponentials. It turns out that this enables usto bring in the theory of complex numbers (in particular, complex roots of unity) to gain extremely usefulinsights into the nature of these signals. Digital signalsMost modern communications systems are digital. "Digital" means that in some sense the signal is associ-ated with a discrete set of discrete values rather than a continuous range of values. So when the signal isreceived it's not measured continuously, but rather it is sampled by taking signal values at fixed regular in-tervals. The fixed interval between samples is called the sampling period, which we will denote by 8 (Greekletter 'delta'). As specific example, the complex signal aeiwt (here a is a complex number) sampled withsampling period & beginning at t = 0 gives the sequence of values: a, aeis, ae2iw.... In theory this is aninfinite sequence, but it can repeat so that it only assumes a finite number of different values.Exercise 3.(a) Give an example of a complex signal of the form at which, when sampled with sampling period8 = 1, gives the values: 1, i,-1,-i,1, i,-1,-i... (the sequence repeats).(b) Notice that in (a), the four values assumed by the signal are the four 4th roots of unity. Given that thesignal frequency is w, give a formula for a sample period & such that the sampled signal 1, eis, eZiwstakes values equal to the 4th roots of unity.(c) Now generalize your answer to (b) for the case of arbitrary nth roots of unity. Given that the signalfrequency is w, give a formula for å such that the sampled signal takes values equal to the nth roots ofunity (your formula will be in terms of n and w).(d) Give the smallest possible sampling period & such that the complex wave e2rift/N sampled with period& gives the N'th roots of unity. (Your formula for 8 will be in terms of f.)

Question

Please do Exercise 3 part A-D please show step by step and explain multiplying the sine and cosine components have different This produces to a signal having the mathemat-ical form a cos(wt) + b sin(wt). We can express this signal much more compactly if we make use of complexexponentials:Exercise 2.(a) Let z = a + bi, and express a cos(wt) + b sin(wt) in terms of z, z, eit, and e-it. (Hint: Express cos(wt)and sin(wt) in terms of eit and e-iwt.)(b) In (a), you should have obtained two terms. Show the two terms are complex conjugates of each other.(c) In the chapter on complex numbers, we showed that for any complex number w the real part of w(written as Re[w]) can be expressed as:Use this fact to show thatw+w2a cos(wt) + b sin(wt) = Re[z.eiwt].Re[w]=As a consequence of Exercise 2, we can treat real signals that are combinations of cosine and sine wavesas the real part of complex signals composed of complex exponentials. It turns out that this enables usto bring in the theory of complex numbers (in particular, complex roots of unity) to gain extremely usefulinsights into the nature of these signals. Digital signalsMost modern communications systems are digital. &#34;Digital&#34; means that in some sense the signal is associ-ated with a discrete set of discrete values rather than a continuous range of values. So when the signal isreceived it's not measured continuously, but rather it is sampled by taking signal values at fixed regular in-tervals. The fixed interval between samples is called the sampling period, which we will denote by 8 (Greekletter 'delta'). As specific example, the complex signal aeiwt (here a is a complex number) sampled withsampling period & beginning at t = 0 gives the sequence of values: a, aeis, ae2iw.... In theory this is aninfinite sequence, but it can repeat so that it only assumes a finite number of different values.Exercise 3.(a) Give an example of a complex signal of the form at which, when sampled with sampling period8 = 1, gives the values: 1, i,-1,-i,1, i,-1,-i... (the sequence repeats).(b) Notice that in (a), the four values assumed by the signal are the four 4th roots of unity. Given that thesignal frequency is w, give a formula for a sample period & such that the sampled signal 1, eis, eZiwstakes values equal to the 4th roots of unity.(c) Now generalize your answer to (b) for the case of arbitrary nth roots of unity. Given that the signalfrequency is w, give a formula for å such that the sampled signal takes values equal to the nth roots ofunity (your formula will be in terms of n and w).(d) Give the smallest possible sampling period & such that the complex wave e2rift/N sampled with period& gives the N'th roots of unity. (Your formula for 8 will be in terms of f.)

Accepted Answer

Answered: Image /qna-images/answer/037991bd-d364-4fb8-a1f9-5eafda96396f.jpg