Properties in Z - Transform Problem 04 | Z - Transform | Signals and Systems

TL;DR

Learn how to find the Z-transform and draw the ROC for a specific function using the properties of Z-transform and time scaling.

Transcript

click the bell icon to get latest videos from equator hello friends and today we are going to study a numerical number 4 and based on a properties of z-transform so first of all we'll see what was the question and then we want to its solution so the question is find the Z terms of X of n and draw its ROC now this one is important the question is po... Read More

Key Insights

• 🤪 The problem involves finding the Z-transform and ROC for a specific function.
• 🇦🇪 The function's ROC is determined to be outside the unit circle based on the presence of the unit step function.
• 🤪 The time scaling property is used to simplify the Z-transform equation.
• 🤪 The final Z-transform expression is derived to be 1.414Z / (Z^2 - 2.828Z + 1).
• 🤪 The video emphasizes the importance of understanding Z-transform formulas and properties for solving numerical problems.
• ⌛ Applying time scaling property saves time in the calculations but is optional for getting full marks.
• ⭕ The function's ROC being outside the unit circle signifies stability in the system.

Questions & Answers

Q: What is the specific question being solved in the video?

The video aims to find the Z-transform and draw the ROC for the function 0.5^n * sin(π/4 * n) * u(n).

Q: How is the Z-transform of sine Omega n calculated?

The Z-transform of sine Omega n is given by Z * sin(Omega)/ (Z^2 - 2Z * cos(Omega) + 1). The specific value of Omega can be substituted to get the actual transform.

Q: How is the time scaling property used in solving the problem?

The time scaling property is applied to the Z-transform equation by replacing Z with Z/0.5, as the scaling factor is 0.5 in this case. This allows simplification of the equation.

Q: What is the final Z-transform expression for the given function?

The Z-transform of 0.5^n * sin(π/4 * n) * u(n) is 1.414Z / (Z^2 - 2.828Z + 1).

Summary & Key Takeaways

• The video discusses solving a numerical problem involving finding the Z-transform and drawing the region of convergence (ROC) for a given function.

• The function is 0.5 raised to the power of n multiplied by sin(π/4 * n) and the unit step function u(n).

• The function's ROC is determined to be outside the unit circle.