Z-transform
Z-transform
It is a common practice when performing analyses that the entity under consideration is converted from one form to another that makes it easier to perform the study or operation. As demonstrated in the discussions on convolution and correlation, performing relatively simple DSP operations can be laborious. Therefore, it is desired to be able to simplify computation of such operations. A method to make this simplification is to analyse the signals in some domain other then the time domain. A common domain used for this purpose for discrete systems is called the z-domain. The operation of converting a signal in to the z-domain from the time or n-domain is called the z-transform. For a signal x[n], the z-transform is defined below
Where X(z) is the z-transform of the signal x[n], Z is a complex variable. It should be noted that the z-transform of a signal is always continuous. In the above equation, the z-transform definition extends for signals samples from negative infinity to positive infinity and thus this is called the Bilateral or Two-sided z-transform. There is also a Unilateral or One-sided z-transform which considers signal samples from zero to positive infinity i-e
As can be seen from both equations, the z-transform allows us to represent a signal as the sum of scaled terms of sequential powers of the variable z , this converting a signal from discrete samples in to a form of a polynomial. For a signal x[n] given as
The z-transform can be written as,
The Region Of Convergence (ROC) of the z-transform of a signal is defined as the values of z for which the geometric series defining the z-transform (equations above) converge and the z-transform is said to ‘exist’ within that range only.
The equation inverse z-transform can then be written as,
where the integration is performed over the region of convergence of X(Z). The inverse z-transform is usually computed using partial fractions, long division and seldom by using direct inversion as given by the above equation.
In order to demonstrate the usefulness of the z-transform in performing analyses, we will present some properties which will be used in proceeding sections.
Delay operator
The variable z is used for indicating the time delay in a signal i-e
and for time advance as
The intuition for this representation comes from the z-transform representation of a signal in which subsequent samples are written as scaled version of sequential powers of z. The variable z is called the Delay operator in such representations.
A DSP system can be expressed by the equation
The system can be represented using the z-transform as
Convolution
The convolution operation can be simplified by using the z-transform as
Thus the convolution operation shortens to a multiplication operation in the z-domain.
DSP System response
A DSP system can be described by its impulse response, however, the time domain representation of a DSP system does not allow the formulation of the system response (impulse response) directly (without letting the input equal to a train of impulses). However, the z-transform representation does allow for such a formulation. For a system with output y[n] and an input signal x[n], the system response is given as
Where H(Z) is the z-transform of the impulse response of the system, Y(Z) and X(Z) are the z-transforms of the output signal and input signal of the system respectively.
