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

The z z -transform of a sequence x(n) x(n) is\newlineX(z)=1−4z−1+2z−21−3z−1+0.5z−2 X(z)=\frac{1-4 z^{-1}+2 z^{-2}}{1-3 z^{-1}+0.5 z^{-2}} \newlineIf the region of convergence includes the unit circle, find the DTFT of x(n) x(n) at ω=π/2 \omega=\pi / 2 .

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Q. The z z -transform of a sequence x(n) x(n) is\newlineX(z)=1−4z−1+2z−21−3z−1+0.5z−2 X(z)=\frac{1-4 z^{-1}+2 z^{-2}}{1-3 z^{-1}+0.5 z^{-2}} \newlineIf the region of convergence includes the unit circle, find the DTFT of x(n) x(n) at ω=π/2 \omega=\pi / 2 .
  1. Evaluate z-transform on unit circle: To find the Discrete-Time Fourier Transform (DTFT) of x(n)x(n) at a specific frequency, we need to evaluate the z-transform X(z)X(z) on the unit circle, where z=ejωz = e^{j\omega}. Here, ω\omega is the frequency variable, and jj is the imaginary unit.
  2. Substitute zz with ejωe^{j\omega}: Substitute zz with ejωe^{j\omega} in the given z-transform X(z)X(z) to find the DTFT. Since we are interested in the frequency ω=π2\omega = \frac{\pi}{2}, we will substitute zz with ejπ/2e^{j\pi/2}.
  3. Simplify expression with substitutions: The substitution gives us X(ejπ/2)=(1−4e−jπ/2+2e−jπ)(1−3e−jπ/2+0.5e−jπ)X(e^{j\pi/2}) = \frac{(1 - 4e^{-j\pi/2} + 2e^{-j\pi})}{(1 - 3e^{-j\pi/2} + 0.5e^{-j\pi})}.
  4. Evaluate complex exponentials: Simplify the expression by evaluating the complex exponentials. We know that ejπ/2=je^{j\pi/2} = j and ejπ=−1e^{j\pi} = -1. Therefore, e−jπ/2=−je^{-j\pi/2} = -j and e−jπ=−1e^{-j\pi} = -1.
  5. Combine like terms: Substitute these values into the expression to get X(j)=(1−4(−j)+2(−1))(1−3(−j)+0.5(−1))X(j) = \frac{(1 - 4(-j) + 2(-1))}{(1 - 3(-j) + 0.5(-1))}.
  6. Find magnitude and phase: Simplify the numerator and denominator separately. The numerator becomes 1+4j−21 + 4j - 2 and the denominator becomes 1+3j−0.51 + 3j - 0.5.
  7. Calculate DTFT at ω=π2\omega = \frac{\pi}{2}: Combine like terms in the numerator and denominator. The numerator simplifies to −1+4j-1 + 4j and the denominator simplifies to 0.5+3j0.5 + 3j.
  8. 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 ω=π2\omega = \frac{\pi}{2}.
  9. 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 ω=π2\omega = \frac{\pi}{2}.The magnitude of the numerator is ∣−1+4j∣=(−1)2+(4)2=1+16=17| -1 + 4j | = \sqrt{(-1)^2 + (4)^2} = \sqrt{1 + 16} = \sqrt{17}. The magnitude of the denominator is ∣0.5+3j∣=(0.5)2+(3)2=0.25+9=9.25| 0.5 + 3j | = \sqrt{(0.5)^2 + (3)^2} = \sqrt{0.25 + 9} = \sqrt{9.25}.
  10. 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 ω=π2\omega = \frac{\pi}{2}.The magnitude of the numerator is ∣−1+4j∣=(−1)2+(4)2=1+16=17| -1 + 4j | = \sqrt{(-1)^2 + (4)^2} = \sqrt{1 + 16} = \sqrt{17}. The magnitude of the denominator is ∣0.5+3j∣=(0.5)2+(3)2=0.25+9=9.25| 0.5 + 3j | = \sqrt{(0.5)^2 + (3)^2} = \sqrt{0.25 + 9} = \sqrt{9.25}.The phase of the numerator is atan2(4,−1)\text{atan2}(4, -1) and the phase of the denominator is atan2(3,0.5)\text{atan2}(3, 0.5). However, since we are only asked for the DTFT at ω=π2\omega = \frac{\pi}{2}, we do not need to calculate the phase explicitly.
  11. 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 ω=π2\omega = \frac{\pi}{2}.The magnitude of the numerator is ∣−1+4j∣=(−1)2+(4)2=1+16=17| -1 + 4j | = \sqrt{(-1)^2 + (4)^2} = \sqrt{1 + 16} = \sqrt{17}. The magnitude of the denominator is ∣0.5+3j∣=(0.5)2+(3)2=0.25+9=9.25| 0.5 + 3j | = \sqrt{(0.5)^2 + (3)^2} = \sqrt{0.25 + 9} = \sqrt{9.25}.The phase of the numerator is atan2(4,−1)\text{atan2}(4, -1) and the phase of the denominator is atan2(3,0.5)\text{atan2}(3, 0.5). However, since we are only asked for the DTFT at ω=π2\omega = \frac{\pi}{2}, we do not need to calculate the phase explicitly.The DTFT at ω=π2\omega = \frac{\pi}{2} is the ratio of the magnitude of the numerator to the magnitude of the denominator, which is 179.25\frac{\sqrt{17}}{\sqrt{9.25}}.
  12. 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 ω=π2\omega = \frac{\pi}{2}.The magnitude of the numerator is ∣−1+4j∣=(−1)2+(4)2=1+16=17| -1 + 4j | = \sqrt{(-1)^2 + (4)^2} = \sqrt{1 + 16} = \sqrt{17}. The magnitude of the denominator is ∣0.5+3j∣=(0.5)2+(3)2=0.25+9=9.25| 0.5 + 3j | = \sqrt{(0.5)^2 + (3)^2} = \sqrt{0.25 + 9} = \sqrt{9.25}.The phase of the numerator is atan2(4,−1)\text{atan2}(4, -1) and the phase of the denominator is atan2(3,0.5)\text{atan2}(3, 0.5). However, since we are only asked for the DTFT at ω=π2\omega = \frac{\pi}{2}, we do not need to calculate the phase explicitly.The DTFT at ω=π2\omega = \frac{\pi}{2} is the ratio of the magnitude of the numerator to the magnitude of the denominator, which is 179.25\frac{\sqrt{17}}{\sqrt{9.25}}.Calculate the final value of the DTFT at ω=π2\omega = \frac{\pi}{2}. The final answer is 179.25≈173.04\frac{\sqrt{17}}{\sqrt{9.25}} \approx \frac{\sqrt{17}}{3.04}.

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