Consider the following problems related to the properties of the DTFT.

(a) For the signal x[n] = βⁿ u[n], β > 0, for what values of β you are able to find the DTFT of x[n] ,X(e^jω), from X(z)?

(b) Given the DTFT X(e^jω) = δ(ω), i.e., a continuous-frequency delta function, what is

i. The inverse DTFT x[n]?

ii. The inverse DTFT of X₁(e^jω) = X(e^{j(ωˆ’Ï€)}) = δ(ω ˆ’ Ï€)?

iii. The inverse of X₂ (e^jω ) = δ(ω) + δ( ˆ’ ω)?

(c) Is it true that

for any integer N > 1? what happens when N = 1?

Transcribed Image Text:

1+e-jo + . 1-e-joN +...+e-jw(N–1) %3D 1-e-jw