I only need help with part b. The solutions from my professor are provided. Plea
ID: 3718690 • Letter: I
Question
I only need help with part b. The solutions from my professor are provided. Please explain how to get part b. Thank you!!!
3. A real-valued 8-periodic sequence (i)o has a discrete Fourier transform whose values at the following points are given 3(0) 0,3(1) = a, x(2) =-1,3(4) =-4,3(5) = b where, a and b are unknown real numbers, and is the complex number such that i2--1. a. Prove that x(8-k-3(k) for each k 0, 1, 2, . . . ,7. b. Find the missing values of (k), in terms of a and b. c. Assume that 2(0) = 0 and that llxll-Li=0 |X(k)12-27. Find all possible value(s) of a and b.Explanation / Answer
In relation to part b, it is given that the value of a x(8-k) = complement of x(k) i.e.
As the missing values are x(3),x(6),x(7) and x(8)
thus x(3) = x(8-5) = complement of x(5) = complement of (b) = b (as be is a real value)
similarily:
x(6) = x(8-2) = -1
x(7) = x(8-1) = a
x(8) = x(8-0) = 0
thus as x is a periodic sequence hence: sum (x(i)) = 0;
hence x(0) + x(1) + x(2) + x(3) + x(4) + x(5) + x(6) + x(7) + x(8) = 0
hence 0 + a - 1 + b - 4 + b - 1 + a + 0 = 0 => 2a + 2b = 6 => a+b=3
Rest is self explanatory with the given answer!