We introduce the concept of the Z-Transform. The Z-Transform is a general transform used for discrete-time signals. It's continuous-time counterpart is the Laplace Transform.

A key component of the Z-Transform is the complex exponential signal z^k. We analyze this class of signal and show that it is an eigenfunction of discrete-time linear time-invariant (DT LTI) systems. The output of a DT LTI system due to input z^k is just H(z)z^k, i.e. the output is the same as the input except it has been scaled by the complex number H(z).

The previous two videos introduced and provided motivation for use of the Z-Transform. We now formally define both the Bilateral Z-Transform and Bilateral Inverse Z-Transform.

The Z-transform definition, which includes an infinite summation over time, doesn't necessarily converge. This video discusses conditions on x[k]z^-k for which it does converge. The values of z for which the Z-transform converges is called the Region of Convergence (ROC).conditions

The Z-Transform is a function of the complex number z. This video sketches/represents the complex quantity z in the complex plane in polar format, i.e. z = re^(j\Omega). We also show that for r = 1, the set of points z = e^(j\Omega) is the unit circle, and the Z-Transform evaluated on the unit circle is equivalent to the Discrete-Time Fourier Transform (DTFT).

Given the discrete-time signal x[k], we use the definition of the Z-Transform to compute its Z-Transform X(z) and region of convergence (ROC). The discrete-time signal x[k] in this problem is a finite length signal.

Given the discrete-time signal x[k], we use the definition of the Z-Transform to compute its Z-Transform X(z) and region of convergence (ROC). The discrete-time signal x[k] in this problem is a right-sided signal, i.e. it has an infinite number of non-zero samples for positive time.

Given the discrete-time signal x[k], we use the definition of the Z-Transform to compute its Z-Transform X(z) and region of convergence (ROC). The discrete-time signal x[k] in this problem is a left-sided signal, i.e. it has an infinite number of non-zero samples for negative time.

This video summarizes the Z-Transform Example #1, #2, and #3 worked in the previous videos. We see that one must specify both X(z) and the region of convergence to uniquely determine the discrete-time signal x[k]. Also, finite-length signals, right-sided signals, and left-sided signals will always have a ROC that consists of the entire complex plane (except possibly z = 0 or infinity), points outside of a circle, or points inside of a circle, respectively.