The $z$ -transform of a sequence $x(n)$ is $\newline$ $X(z)=\frac{1-4 z^{-1}+2 z^{-2}}{1-3 z^{-1}+0.5 z^{-2}}$ $\newline$ If the region of convergence includes the unit circle, find the DTFT of $x(n)$ at $\omega=\pi / 2$ .

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Q. The

z

-transform of a sequence

x(n)

\newline

X(z)=\frac{1-4 z^{-1}+2 z^{-2}}{1-3 z^{-1}+0.5 z^{-2}}

\newline

If the region of convergence includes the unit circle, find the DTFT of

x(n)

\omega=\pi / 2

Evaluate z-transform on unit circle: To find the Discrete-Time Fourier Transform (DTFT) of $x(n)$ at a specific frequency, we need to evaluate the z-transform $X(z)$ on the unit circle, where $z = e^{j\omega}$ . Here, $\omega$ is the frequency variable, and $j$ is the imaginary unit.
Substitute $z$ with $e^{j\omega}$ : Substitute $z$ with $e^{j\omega}$ in the given z-transform $X(z)$ to find the DTFT. Since we are interested in the frequency $\omega = \frac{\pi}{2}$ , we will substitute $z$ with $e^{j\pi/2}$ .
Simplify expression with substitutions: The substitution gives us $X(e^{j\pi/2}) = \frac{(1 - 4e^{-j\pi/2} + 2e^{-j\pi})}{(1 - 3e^{-j\pi/2} + 0.5e^{-j\pi})}$ .
Evaluate complex exponentials: Simplify the expression by evaluating the complex exponentials. We know that $e^{j\pi/2} = j$ and $e^{j\pi} = -1$ . Therefore, $e^{-j\pi/2} = -j$ and $e^{-j\pi} = -1$ .
Combine like terms: Substitute these values into the expression to get $X(j) = \frac{(1 - 4(-j) + 2(-1))}{(1 - 3(-j) + 0.5(-1))}$ .
Find magnitude and phase: Simplify the numerator and denominator separately. The numerator becomes $1 + 4j - 2$ and the denominator becomes $1 + 3j - 0.5$ .
Calculate DTFT at $\omega = \frac{\pi}{2}$ : Combine like terms in the numerator and denominator. The numerator simplifies to $-1 + 4j$ and the denominator simplifies to $0.5 + 3j$ .
Final DTFT value: Now, we need to find the magnitude and phase of the complex number in the numerator and denominator to get the DTFT at $\omega = \frac{\pi}{2}$ .
Final DTFT value: Now, we need to find the magnitude and phase of the complex number in the numerator and denominator to get the DTFT at $\omega = \frac{\pi}{2}$ .The magnitude of the numerator is $| -1 + 4j | = \sqrt{(-1)^2 + (4)^2} = \sqrt{1 + 16} = \sqrt{17}$ . The magnitude of the denominator is $| 0.5 + 3j | = \sqrt{(0.5)^2 + (3)^2} = \sqrt{0.25 + 9} = \sqrt{9.25}$ .
Final DTFT value: Now, we need to find the magnitude and phase of the complex number in the numerator and denominator to get the DTFT at $\omega = \frac{\pi}{2}$ .The magnitude of the numerator is $| -1 + 4j | = \sqrt{(-1)^2 + (4)^2} = \sqrt{1 + 16} = \sqrt{17}$ . The magnitude of the denominator is $| 0.5 + 3j | = \sqrt{(0.5)^2 + (3)^2} = \sqrt{0.25 + 9} = \sqrt{9.25}$ .The phase of the numerator is $\text{atan2}(4, -1)$ and the phase of the denominator is $\text{atan2}(3, 0.5)$ . However, since we are only asked for the DTFT at $\omega = \frac{\pi}{2}$ , we do not need to calculate the phase explicitly.
Final DTFT value: Now, we need to find the magnitude and phase of the complex number in the numerator and denominator to get the DTFT at $\omega = \frac{\pi}{2}$ .The magnitude of the numerator is $| -1 + 4j | = \sqrt{(-1)^2 + (4)^2} = \sqrt{1 + 16} = \sqrt{17}$ . The magnitude of the denominator is $| 0.5 + 3j | = \sqrt{(0.5)^2 + (3)^2} = \sqrt{0.25 + 9} = \sqrt{9.25}$ .The phase of the numerator is $\text{atan2}(4, -1)$ and the phase of the denominator is $\text{atan2}(3, 0.5)$ . However, since we are only asked for the DTFT at $\omega = \frac{\pi}{2}$ , we do not need to calculate the phase explicitly.The DTFT at $\omega = \frac{\pi}{2}$ is the ratio of the magnitude of the numerator to the magnitude of the denominator, which is $\frac{\sqrt{17}}{\sqrt{9.25}}$ .
Final DTFT value: Now, we need to find the magnitude and phase of the complex number in the numerator and denominator to get the DTFT at $\omega = \frac{\pi}{2}$ .The magnitude of the numerator is $| -1 + 4j | = \sqrt{(-1)^2 + (4)^2} = \sqrt{1 + 16} = \sqrt{17}$ . The magnitude of the denominator is $| 0.5 + 3j | = \sqrt{(0.5)^2 + (3)^2} = \sqrt{0.25 + 9} = \sqrt{9.25}$ .The phase of the numerator is $\text{atan2}(4, -1)$ and the phase of the denominator is $\text{atan2}(3, 0.5)$ . However, since we are only asked for the DTFT at $\omega = \frac{\pi}{2}$ , we do not need to calculate the phase explicitly.The DTFT at $\omega = \frac{\pi}{2}$ is the ratio of the magnitude of the numerator to the magnitude of the denominator, which is $\frac{\sqrt{17}}{\sqrt{9.25}}$ .Calculate the final value of the DTFT at $\omega = \frac{\pi}{2}$ . The final answer is $\frac{\sqrt{17}}{\sqrt{9.25}} \approx \frac{\sqrt{17}}{3.04}$ .