Enter the nth sample of the discrete-time signal and the complex number into the calculator to determine the Z-Transform of the signal.

Bilateral transform X(z) = ฮฃ x[n]zโปโฟ. A rational expression needs a region of convergence (ROC) to identify its sequence. Numerical roots are approximate.

Up to 13 real samples; every entry is required.

Use a real number or a+bi, for example 0.8+0.3i. A value outside the ROC is only a rational extension.

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Z-Transform Formula

The following formula is used to calculate the Z-Transform of a discrete-time signal.

X(z) = ฮฃn = - โˆžโˆž x[n] zโปโฟ

Variables:

  • X(z) is the Z-transform of the signal
  • x[n] is the nth sample of the discrete-time signal
  • z is a complex number
  • n is the index of the discrete-time signal

To calculate the Z-Transform, for each sample of the discrete-time signal, multiply the sample by the complex number z raised to the power of the negative index of the sample. Sum all these results to get the Z-Transform of the signal.

What is a Z-Transform?

The Z-Transform is a mathematical technique used in signal processing and control theory for analyzing and representing discrete-time signals. It is a complex-valued function that transforms a sequence of numbers into a function of a complex variable, providing a complex-domain representation of the original sequence. Evaluation on the unit circle gives the ordinary frequency response only when the unit circle lies in the region of convergence. The Z-Transform is particularly useful for the analysis of linear, time-invariant systems and is a fundamental tool in digital signal processing.

How to Calculate Z-Transform?

The following steps outline how to calculate the Z-Transform using the given formula:

  1. First, determine the value of x[n] for each sample of the discrete-time signal.
  2. Next, raise the complex number z to the power of -n.
  3. Multiply each x[n] value with the corresponding (z^-n) value.
  4. Sum up all the products obtained in the previous step.
  5. The result is X(z) within its region of convergence. The finite-sequence mode treats samples outside the entered list as zero.

Example Problem:

Use the following variables as an example problem to test your knowledge:

x[n] = [1, 2, 3, 4]

z = 0.5

n = [0, 1, 2, 3]