Please do Exercise 4 and please do part A and B and please show step by step and explain Analyzing digital signals using complex roots of unityFor the remainder of this investigation, we will take our sampling period equal to 1, and consider signalsthat are linear combinations of complex waves with angular frequencies 27k/N, where k = 0,2,... N-1(i.e. f = k/N, k = 0,2,... N-1). Notice that these are all of the waves which have positive angularfrequencies less than 27 which also repeat after N samples. Any signal s(t) that is a linear combination ofthese waves can be written as follows:N-1s(t) = Σ' ake2nikt/Nk=0(1)The signal's information is contained in the complex amplitudes {a}, k = 0,... N-1. Hence we want torecover the amplitudes from the sampled signal.It turns out that properties of roots of unity will be crucial in obtaining a method to recover the ampli-tudes. To get at these properties, we will have to bring modular arithmetic also into the mix!Let = e27i/N ( is the Greek letter 'zeta') and recall that 5k are the Nth roots of unity for k = 0,... N - 1.Since these are roots of unity, that means they are all solutions to the equation zN - 1 = 0. This means thatthe linear factors z-k all divide zN - 1. Since we have N distinct linear factors, it follows that:ZN - 1 = (z-5°)... (z - GN-¹).(2)Exercise 4.(a) By computing the coefficient of zN−1 on the right hand side of Equation (2), show that the sum of Nthroots of unity is equal to 0. (Note that this coefficient must be equal to the coefficient of zN-1 on theleft-hand side, which is 0).(b) (This one is harder) Compute the coefficient of zN-2 on the right-hand side of Equation (2). What prop-erty of the N'th roots of unity can you conclude from this?0From Proposition 3.5.20 in the book we have: "Given a modular equation kx = m (mod N), wherek, m, N are integers. Then the equation has an integer solution for x if and only if m is an integer multipleof the greatest common divisor of k and N."

Question

Please do Exercise 4 and please do part A and B and please show step by step and explain Analyzing digital signals using complex roots of unityFor the remainder of this investigation, we will take our sampling period equal to 1, and consider signalsthat are linear combinations of complex waves with angular frequencies 27k/N, where k = 0,2,... N-1(i.e. f = k/N, k = 0,2,... N-1). Notice that these are all of the waves which have positive angularfrequencies less than 27 which also repeat after N samples. Any signal s(t) that is a linear combination ofthese waves can be written as follows:N-1s(t) = Σ' ake2nikt/Nk=0(1)The signal's information is contained in the complex amplitudes {a}, k = 0,... N-1. Hence we want torecover the amplitudes from the sampled signal.It turns out that properties of roots of unity will be crucial in obtaining a method to recover the ampli-tudes. To get at these properties, we will have to bring modular arithmetic also into the mix!Let = e27i/N ( is the Greek letter 'zeta') and recall that 5k are the Nth roots of unity for k = 0,... N - 1.Since these are roots of unity, that means they are all solutions to the equation zN - 1 = 0. This means thatthe linear factors z-k all divide zN - 1. Since we have N distinct linear factors, it follows that:ZN - 1 = (z-5°)... (z - GN-¹).(2)Exercise 4.(a) By computing the coefficient of zN−1 on the right hand side of Equation (2), show that the sum of Nthroots of unity is equal to 0. (Note that this coefficient must be equal to the coefficient of zN-1 on theleft-hand side, which is 0).(b) (This one is harder) Compute the coefficient of zN-2 on the right-hand side of Equation (2). What prop-erty of the N'th roots of unity can you conclude from this?0From Proposition 3.5.20 in the book we have: &#34;Given a modular equation kx = m (mod N), wherek, m, N are integers. Then the equation has an integer solution for x if and only if m is an integer multipleof the greatest common divisor of k and N.&#34;

Accepted Answer

As per the question we are given the following equation :
zN - 1 = (z - ζ0) ... (z - ζN-1)
Where ζ0…