ABSTRACT
Mathematically, the z -transform is a mapping between complex sequences and analytical functions on the complex plane. Given a discrete-time signal x [n ], the z -transform of x [n ] is formally defined as the complex function of a complex variable z ∈
X (z ) = x [n ]= ∞∑ n=−∞
x [n ]z−n (6.1)
Contrary to the Fourier transform (as well as to other well-known transforms such as the Laplace transform or the wavelet transform), the z -transform is not an analysis tool per se, in that it does not offer a new physical insight on the nature of signals and systems. The z -transform, however, derives its status as a fundamental tool in digital signal processing from two key features:
• Its mathematical formalism, which allows us to easily solve constantcoefficient difference equations as algebraic equations (and this was precisely the context in which the z -transform was originally invented).
