Please do Exercise 3 part E-H and please show step by step and explainIn problems 3f,g,h, the series of samples for w1 and w2 should be equal TO EACH OTHER, but this doesn't mean that both series of samples have to be constant series. One example would be the case where the series of samples for w1 is 1,i,-1,-i,1,i,-1,-i, ... , and the series of samples for w2 is exactly the same. Digital signalsMost modern communications systems are digital. "Digital" means that in some sense the signal is associated 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 ael (here a is a complex number) sampled withsampling period & beginning att - 0 gives the sequence of values: a, a, a,... 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 act which, when sampled with sampling period6 = 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, 2,..takes 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 & such that the sampled signal takes values equal to the nth roots ofunity (your formula will be in terms of " and w).(d) Give the smallest possible sampling period & such that the complex wave exift/N sampled with period& gives the N'th roots of unity. (Your formula for & will be in terms of f.)(e) Given that the sampling period is equal to & and given that the sample when t=0 has value 1, find allpossible values for the angular frequency w such that the N+1th sample (when t = Nó) is also equalto 1 (in other words, the signal repeats itself after N sample intervals).(f) Given the sampling period & give an example of two different angular frequencies ₁,2 that willproduce exactly the same samples at the sample times t=0,8,25,... (the two frequencies w and ware called aliases of each other).(g) Building on the previous exercise, give necessary and sufficient conditions on the angular frequencieswy, w such that the signals ae and aes are equal at the sample times 0,5, 25,....(h) Given that is the sampling period and given an angular frequency w such that w<0, Give a formulafor the smallest possible positive angular frequency w, that will give the same samples as w.0The above exercise establishes the following facts:• For certain combinations of sample period and angular frequency, the signal repeats;• Given any integer N and sampling period 6, there are certain regularly-spaced angular frequenciesww2....such that signals with these angular frequencies will repeat every N samples;• Signals with different angular frequencies can result in the same samples. However, in such casesthere is always a smallest positive angular frequency that produces the given samples. OExercise 3XeiwtGive an example of a complex signal of the formwhich, when sampled with sampling period8=1 gives the values: 1, 1,-1,-1,₁ 1,₁ 1,-1,-1.00 (repeat)-As the complex signal xe , & € C sampled with samplingint2period & beginning at t=0 gives the sequence of valuesaixe ins3iwxe• for 6=1; the sequence of valves is x, xe'w xe 2iw02iwoO, xeMand it is equal to 1, 1,-1, -i , 1,... On comparing each termwe have α = 1 ⇒ x = 1xeiwi eiw = 1 ⇒ cosw+isinw=1=w = 17/1/₂Xe2iw=-1= eiπ = -1Biwix = 1₁xeand so on all the values holds✓ W = 17/1/₂w=complex signal form is e21(1/2)31(π/2)also eand exetin3iwi(7/2₂) t086(b) Notice that in (a), the four values assumed by the signalare the four roots of unity. Given that the signal frequencyis w, give a formula for a sample period & such that the sampledSignal 1, eiws, e2/w84... takes values equal to the 4th rootsof unity2iKT(b) fourth root of unity is given by e ²14T where K=0₁1₁2₁3for k=0 e º = 12ir/4 in/₂2for k=1 eFe4/4 imfor k=2, e=+3ir/2for k= 3, e•• Comparing these values with the sequence in (a), we canSay that the sampled signal, 1, eiws Liwsvalues equal to the 4th root of unityO=cos (1/2) +isin (7/₂) = i= COS (1T) + isin(TT) = -1Cos (3/2) +isin (3/2) = - i(C)Now generalize your answer to b for the case of arbitraryInth roots of unity, Given that the signal frequency is w,givea formula for 8 such that the sampled signal takesvalues equal to the nth roots of unity (your formula willbe in terms of n and up)2TTws⇒1/8 = 2πt w121411/06/1As we have seen that the sequence 1, eiws, eziwsland 1, eeiblinand they repeat are equalOn comparing 2nd term we get that w8 = 2π/²"PT2πi (n-1)000, ete27/0(d) Give the smallest possible sampling period of such thatsampled with period & gives thethe wave eNth roots of unity. (Your formula for & will bein terms of f)1By part C, smallest possible period & is 8 = 1/ /f(*.*QHIPS 2ni2kmiNN Nth root of unity e ² + ² ⇒ f S = 1 ⇒ 8 = 1/f)

Answered: (f) Given the sampling period & give an…

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...