Properties of the z-Transform I

Certain properties provide a solid foundation for analyzing discrete-time systems using the Z-transform.

Considering two discrete-time signals, the property of linearity states that the Z-transform of a linear combination of signals is equal to the linear combination of their individual Z-transforms.

The time-shifting property means that if a signal is shifted in time, its Z-transform is multiplied by a factor that depends on the amount of the shift.

This property helps in analyzing the impact of time-domain delays or advances on the signal in the frequency domain.

Multiplying a signal in the time domain by an exponential factor corresponds to a scaling in the z-domain.

This property, known as frequency scaling, helps analyze how a signal's frequency characteristics are altered.

For time reversal, reversing the time axis of the signal corresponds to taking the reciprocal of the Z-transform variable in the z-domain.

Modulation of a signal by a cosine or sine function results in the signal's Z-transform being evaluated at shifted positions, showing how the frequency component affects its Z-transform.