Explain the duality between the discrete and continuous domains in signal processing. How does the Fourier transform illustrate this duality?

1-Find the Fourier-transform (FT), for the following signals: (a) x1(n) = -8(n + 1) + 8(n –- 1). As this is an odd function, show that the FT is a pure imaginary function of frequency. (b) x2(n) = -8(n)+8(n – 2). Calculate it in two ways, first directly, and second by relating it to the signal in the previous part, and then using the properties of the Fourier transform. (c) x3(n) = 8(n + 1) + 5 + 8(n – 1). As this is an even function, show that the FT is a real function of frequency. (d) vi(n) = (–0.9)"-lu(n – 1), (e) v2(n) = (-0.9)"u(n – 1), (f) h(n) = 2[u(n) – u(n – 80)].

FIND DTFT ( NEED A NEAT HAND WRITING SOLUTION ONLY OTHERWISE WILL LEAVE A DOWNVOTE)