FOURIER AND Z-TRANSFORM

The frequency function of a pulse at time $t_n=n\Delta t$ is $e^{i\omega n \Delta t} = (e^{i\omega \Delta t})^ n$.The factor $e^{i\omega\Delta t}$ occurs so often in applied work that it has a name:  

$$Z \eq e^{i\omega \, \Delta t}$$ (20)

With this Z, the pulse at time t_n is compactly represented as Zⁿ. The variable Z makes Fourier transforms look like polynomials, the subject of a literature called ``Z-transforms.'' The Z-transform is a variant form of the Fourier transform that is particularly useful for time-discretized (sampled) functions.

From the definition (20), we have $Z^2=e^{i\omega 2\Delta t }$, $Z^3=e^{i\omega 3\Delta t }$,etc. Using these equivalencies, equation (19) becomes  

$$B(\omega) \eq B(\omega (Z)) \eq \sum_n \ b_n \ Z^n$$ (21)