To calculate the Z-Transform (Z):
\[ Z = \sum x[n] \cdot (z^{-n}) \]
Where:
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.
Let's assume the following values:
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.
Let's assume the following values:
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.
