Final answer:
To show that Z{Re(x[n])} = 1/2 {X(z) + X∗(z∗)} and Z{Im(x[n])} = 1/2 {X(z) - X∗(z∗)}, we can express Re(x[n]) and Im(x[n]) in terms of their complex forms. Then, we can apply the linearity and time-shifting properties of the Z-transform to derive the desired results.
Step-by-step explanation:
To show that Z{Re(x[n])} = 1/2 {X(z) + X∗(z∗)}, we can start by expressing Re(x[n]) in terms of its complex form. Re(x[n]) can be written as the sum of x[n] and its complex conjugate, divided by 2: Re(x[n]) = (x[n] + x∗[n]) / 2. Now, we can take the Z-transform of Re(x[n]): Z{Re(x[n])} = Z{(x[n] + x∗[n]) / 2}. Applying linearity and time-shifting properties of the Z-transform, we can write Z{Re(x[n])} as 1/2 {X(z) + X∗(z∗)}, where X(z) represents the Z-transform of x[n].
Similarly, to show that Z{Im(x[n])} = 1/2 {X(z) - X∗(z∗)}, we can express Im(x[n]) in terms of its complex form. Im(x[n]) can be written as the difference of x[n] and its complex conjugate, divided by 2: Im(x[n]) = (x[n] - x∗[n]) / 2. Taking the Z-transform of Im(x[n]), we have Z{Im(x[n])} = Z{(x[n] - x∗[n]) / 2}. Using linearity and time-shifting properties of the Z-transform, we can rewrite Z{Im(x[n])} as 1/2 {X(z) - X∗(z∗)}, where X(z) represents the Z-transform of x[n].