Question

Compute the Discrete-Time Fourier Transform analytically for the following signals and plot the absolute values and the phase of the DTFT from-2π to 2π x[n] αηυ[n] for α-0.7 and 0.3 x[n]-δ[n-r] for τ-2 and 3 xInrk], for r -2 and 3 a. b. C. Please show your work step by step and include the formula for finding the absolute value of DTFT and the phase of DTFT.

Answer 1

X (n) = alphorn u (n) for alpha = 0.7 and 0.3 D.T.F.T x (e^jw) = sigma_n = -infinity^alpha x ()n) e^- jwn = sigma_n = a^infinity alpha^n e^- jwn = sigma_n = 0^infinity (alpha e^- jwn) = 1 + alpha e^- jwn + (alpha e^- jwn) + = 1/1 - alpha e^- jw x (e^jw) = 1/1 - alpha cos w + j alpha sin wVertical-bar times (e^jw) Vertical-bar = 1/squareroot (1 - alpha cos w)^2 + (alpha sin w)^2 = 1/squareroot 1 - 2 alpha cos w + alpha^2 case (i) alpha = 0.7Vertical-bar times (e^jw) Vertical-bar = 1/squareroot 1 - 2 alpha times 0.7 cos w + (0.7)^2 = 1/squareroot 1.49 - 1.4 cos w angle x (e^jw) = -Tan^- 1 [0.7 sin w/1 - 0.7 cos w] case (ii) alpha = 0.3 1 times (e^jw) = 1/squareroot 1.09 - 0.6 cos w angle x (e^jw) = -Tan^- 1 (0.3 sin w/1 - 0.3 cos w) b) x (n) = delta (n - T)