As shown in Eq. (4.25), a signal x[n] can be expressed as a sum of its even and odd components as x[n] = xe[n] + xo[n]. (a) If x[n] ⇐⇒ X(Ω), show for real x[n] that xe[n] ⇐⇒ Re {X(Ω)} and xo[n] ⇐⇒ j...

x[n] = xe[n] + xo[n].

(a) If x[n] ⇐⇒ X(Ω), show for real x[n]

that

xe[n] ⇐⇒ Re {X(Ω)}

and

xo[n] ⇐⇒ j Im {X(Ω)} .

(b) Verify the results of part (a) by finding the DTFT of the even and odd components of the signal (0.8)ⁿu[n].