This video introduces the general strategy that we'll use to invert Z-Domain signals back into the discrete-time domain.In this course we avoid directly using the inverse Z-Transform equation as it requires performing contour integration in the complex plane.Instead, our strategy is to write the Z-Transform we're working with (e.g. X(z)) as a linear combination of first-order terms that we can lookup in a table. This process of writing X(z), which in general is a ratio of polynomials in z, as a linear combination of first-order terms is called Partial Fraction Expansion.Once expanded into first-order terms, we go back into the time-domain by picking either a right-sided or left-sided signal as appropriate based on the overall region of convergence (ROC) of the X(z).A specific example of this process is worked in the subsequent video.

The previous video outlined the general strategy we use in this course to invert a z-domain quantity back into the discrete-time domain.This video works a specific example of this process. Starting with X(z), we decompose X(z) into a linear combination of first-order terms using partial fraction expansion (PFE).For each term we then analyze the pole location and a decide on a right-sided or left-sided time-domain signal based on the pole location relative to the region of convergence of X(z).

The partial fraction expansion (PFE) approach to finding the inverse Z-transform is great for finding a time-domain equation that is valid for all time. Sometimes, finding just a few values of the time-domain signal is needed. In this case, finding the inverse Z-transform of a signal via long division can be used to compute time-domain samples of the signal one value at a time.

​In this video, we show how to use the long division approach to compute several values of a right-sided time-domain signal. For right-sided signals, this division operation should result in “z” raised to negative powers (e.g. z^-1, z^-2, etc.) since negative powers of z correspond to time-domain samples at positive values of time.

This video also provides an example of how to use the long division approach to compute the inverse Z-transform of a discrete-time signal. In the previous video, the inverse Z-transform of a right-sided signal was computed. In this example, the inverse Z-transform of a left-sided signal is considered. For left-sided signals, this division operation should result in “z” raised to positive powers (e.g. z^1, z^2, etc.) since positive powers of z correspond to time-domain samples at negative values of time.

This video is the first of several that involve working with the Transfer Function of a discrete-time LTI system. The transfer function H(z) is Z-domain quantity equal to the ratio of Y(z)/X(z) (e.g. the ratio of the system output and system input in the Z-domain).
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In this video, we start with a time-domain difference equation for a DT LTI system. We then transform this difference equation into the Z-domain and then solve for the transfer function H(z).

In the previous video we started with a system difference equation, and then solved for the system transfer function. The example presented here goes the other direction. We start with the transfer function H(z) of a discrete-time LTI system, and then we find the corresponding difference equation of the system.