9.2 DEFINITION OF z-TRANSFORM
The one-sided z-transform of a discrete time signal x(n) is given by the relation [78, 79]
(9.2)
where z is a complex number. We can write the z-transform in polynomial form:
(9.3)
We say that the signal x(n) in the time domain has an equivalent representation, X (z), in the z-domain.
The z-transform X (z) of the sequence x(n) is a polynomial of the different powers of z−1, such that x(i) is the coefficient of the ith power of z−1.
An important property of the z-transform is that the quantity z−1 in the z-domain corresponds to a time shift of 1 in the time domain. To prove this, we multiply X(z) by z−1 to obtain a new signal, Y(z):
(9.4)
The time domain representation y(n) is found by using the coefficients of the above polynomial. At time i, we find that
(9.5)
In effect, the term z−1 delayed each sample by one time step. We can write the relation between x(n) and y(n) as follows:
Multiplication by z−1 has the effect of delaying the signal ...
Get Algorithms and Parallel Computing now with the O’Reilly learning platform.
O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.