Procedure

Region of Convergence (ROC)

This section requires entering of number of ROCs possible for the corresponding pole-zero plot provided beside. Click on "Check Number" to check if your answer is correct, which would be displayed in the Observations box at the bottom. If the answer is right, you will be required to enter all the ROCs, in an increasing annular order, from $0$ to $\infty$, and click on "Check" button to verify your answers. Steps to be done are as follows

1. Enter the number of ROCs possible for the corresponding pole-zero plot provided beside.
2. Click on "Check Number" button to verify your answer and move further.
3. In the new fields that appear, enter all the ROCs, in an increasing annular order, from $0$ to $\infty$.
4. Click on "Check" button to verify your answers.

Stability and Causality

This section requires entering the ROC number in the field as referenced in the ROC List on the left, corresponding to stable and causal ROC. The ROCs for the given pole-zero plot on the right, will already be provided in a box on the left. Click on "Check" to check if your answer is correct, which would be displayed in the Observations box at the bottom. Steps to be done are as follows

1. Enter the ROC number in the field as referenced in the ROC List on the left, corresponding to stable and causal.
2. Click on "Check" button to verify your answer.

Pole-Zero (Real)

This section requires entering of values of real poles and zeros of a polynomial function given by $$H(z) = \frac{(z-a)(z-b)(z-c)}{(z-d)(z-e)(z-f)}$$ and visualizing the pole-zero plot and the magnitude response when evaluated on the unit circle. Click on the "Plot" button to visualize the plots. Steps to be done are as follows

1. Enter the values in any of the six fields (fields can be left empty too), named "a", "b", "c", "d", "e" and "f".
2. Click on "Plot" button to visualize the plots in the figures.

The plots are obtained and they represent the pole-zero plot and the magnitude response evaluated on the unit circle, for the entered set of real pole(s) and zero(s).

Pole-Zero (Imaginery)

This section requires entering of values of real and imaginery poles and zeros of a polynomial function given by $$H(z) = \frac{(z-z_{1})(z-z_{1}^{})}{(z-z_{2})(z-z_{2}^{})}$$ and visualizing the pole-zero plot and the magnitude response when evaluated on the unit circle. Here, we consider conjuage pairs of poles and zeros following that it is a real system. The poles are zeros are specified using $$z_{1} = a+jb,\ z_{2} = c+jd$$ and $a,\ b,\ c,$ and $d$ are real numbers. Click on the "Plot" button to visualize the plots. Steps to be done are as follows

1. Enter the values in any of the four fields (fields can be left empty too), named "a", "b", "c" and "d".
2. Click on "Plot" button to visualize the plots in the figures.

The plots are obtained and they represent the pole-zero plot and the magnitude response evaluated on the unit circle, for the entered set of complex pole(s) and zero(s).

Filtering

This section requires entering of values of real and imaginery poles and zeros of a polynomial function given by $$H(z) = \frac{(z-z_{1})(z-z_{1}^{})}{(z-z_{2})(z-z_{2}^{})}$$ and designing a filter whose magnitude response when evaluated on the unit circle would follow the type of filter as shown in the specification box on the left. Here, we consider conjuage pairs of poles and zeros following that it is a real system. The poles are zeros are specified using $$z_{1} = a+jb,\ z_{2} = c+jd$$ and $a,\ b,\ c,$ and $d$ are real numbers. Click on the "Check" button to visualize the plot and verify the working of your filter. Steps to be done are as follows

1. Enter the values in any of the four fields (fields can be left empty too), named "a", "b", "c" and "d".
2. Click on "Check" button to visualize the plot and verify your answer.

The plot obtained represents the magnitude response evaluated on the unit circle, for the entered set of complex pole(s) and zero(s), with the x-axis as frequency ($w$) to understand the low-pass and high-pass components.