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z-Transform Tutorial: z-Transform and Inverse z-Transform Examples & Functions

In analog or continuous systems the Fourier transform played a key role in analyzing signals and systems. The inverse z transform represents a time-domain sequence of a z transform function. We note that the z transform for digital signals or discrete-time signals is the digital counterpart of the Laplace transform for continuous-time signals.

Both the z Transform and the Laplace Transform have similar relationships with the Fourier Transform. We also note that the Fourier transform doesn’t converge for all sequences and it’s a useful generalization of the Fourier Transform. In other words, the z transform covers a broader class of signals than the Fourier Transform while being more convenient than the Fourier transform from an analytical perspective with respect to notation.

In future videos, we will consider some informal procedures in finding the inverse z transform. Specifically, they are the inspection method, partial fraction expansion, and the power series expansion.

Briefly the inspection method, consists simply of recognizing certain transform pairs in a table. These complex sequences appear quite frequently and so it would be useful to apply and use them directly.

As a result of using the inspection method, tables of z transforms are invaluable. When certain transforms can’t be found in the tables, however, then alternate expressions are sought and often broken down into simpler results.

If you’re familiar doing the partial fraction expansion in the Laplace domain, then the inverse z transform follows a similar process. Since the z- and Laplace transform are rational functions, then we can obtain a partial fraction expansion to easily identify sequences corresponding to the individual terms.

When using the partial fraction expansion, the X(z) function associated with the sequence x[n], can be expressed as a ratio of polynomials of 1/z. Such transforms occur frequently in the study of linear time-invariant systems. When the function X(z) is broken or decompose into simpler terms, poles and zeros of X(z) are easily recognized so that we can use the appropriate pairs found in the tables. Future articles will include examples on how to apply this method.

The final method is defining the expression X(z) as a power series expansion or Laurent series where the sequence values x[n] are coefficients of (1/z)^n. In fact, the above video used this method in determining any value of the sequence by finding the coefficient of the appropriate power of 1/z.

Examples will include the finite-length sequence, finding the inverse transform using long division, and finding the inverse transform of a left-sided sequence.

For more videos on these and other topics please contact Professor Santiago at john@e-liteworks.com or visit the Freedom University home page to sign up for additional and affordable access to other important videos.

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