Z-Transform Calculator

Formula

To calculate the Z-Transform (Z):

\[ Z = \sum x[n] \cdot (z^{-n}) \]

Where:

\(Z\) is the Z-Transform of the signal
\(x[n]\) is the \(n\)th sample of the discrete-time signal
\(z\) is a complex number
\(n\) is the index of the discrete-time 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 frequency domain representation of the original sequence. The Z-Transform is particularly useful for the analysis of linear, time-invariant systems and is a fundamental tool in digital signal processing.

Example Calculation 1

Let's assume the following values:

Discrete-Time Signal Samples (\(x[n]\)) = [1, 2, 3, 4]
Complex Number (\(z\)) = 2

Using the formula:

\[ Z = 1 \cdot (2^{-0}) + 2 \cdot (2^{-1}) + 3 \cdot (2^{-2}) + 4 \cdot (2^{-3}) = 1 + 1 + 0.75 + 0.5 = 3.25 \]

The Z-Transform is 3.25.

Example Calculation 2

Let's assume the following values:

Discrete-Time Signal Samples (\(x[n]\)) = [2, 4, 6, 8]
Complex Number (\(z\)) = 3

Using the formula:

\[ Z = 2 \cdot (3^{-0}) + 4 \cdot (3^{-1}) + 6 \cdot (3^{-2}) + 8 \cdot (3^{-3}) = 2 + 1.33 + 0.67 + 0.3 = 4.3 \]

The Z-Transform is 4.3.