Summary: The z-transform is a generalization of the DTFT as the Laplace transform is a generalization of the Fourier tranform. It allow the theory of complex variables to be used.
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The z-transform is an extension of the DTFT in a way that is analogous to the
Laplace transform for continuous-time signals being an extension of the
Fourier transform. It allows the use of complex variable theory and is
particularly useful in analyzing and describing systems. The question of
convergence becomes still more complicated and depends on values of
The z-transform (ZT) is defined as a polynomial in the complex variable
A unilateral z-transform is sometimes needed where the definition (2.23) uses a lower limit on the transform summation of zero. This allow the transformation to converge for some functions where the regular bilateral transform does not, it provides a straightforward way to solve initial condition difference equation problems, and it simplifies the question of finding the ROC. The bilateral z-transform is used more for signal analysis and the unilateral transform is used more for system description and analysis. Unless stated otherwise, we will be using the bilateral z-transform.
A few examples together with the above properties will enable one to solve and understand a wide variety of problems. These use the unit step function to remove the negative time part of the signal. This function is defined as
Notice that these are similar to but not the same as a term of a partial fraction expansion.
The z-transform can be inverted in three ways. The first two have similar procedures with Laplace transformations and the third has no counter part.
For example
We must understand the role of the ROC in the convergence and inversion of the z-transform. We must also see the difference between the one-sided and two-sided transform.
The FS coefficients are weights on the delta functions in a FT of the
periodically extended signal. The FT is the LT evaluated on the imaginary
axis:
The DFT values are samples of the DTFT of a finite length signal. The DTFT is the z-transform evaluated on the unit circle in the z plane.
It is important to be able to relate the time-domain signal