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<\/p>\n\n\n\nThe most important use of z-transform is its ability to completely characterize signals and linear systems in the most general ways possible. The z-transform is primarily used to convert the discrete time domain signal into discrete frequency domain signal. The z-transform for LTI system is defined as<\/p>\n\n\n\n
Due to its convolution property, the z-transform is a powerful tool to analyse LTI systems. From the convolution property Y(z)=H(z)X(z).<\/p>\n\n\n\n
Where:<\/p>\n\n\n\n
Y(z)= z-transform of system output<\/p>\n\n\n\n
H(z) = z-transform of impulse response<\/p>\n\n\n\n
X(z)= z-transform of system input<\/p>\n\n\n\n
A discrete time LTI system is causal if and only if the ROC of its system function is exterior of the circle including infinity. The ROC is the exterior of the outermost pole. H(z) is the ratio of polynomials in z, the order of the numerator cannot be greater than the order of the denominator.<\/p>\n\n\n\n
The LTI system is stable if and only if the ROC of the system function H(z) includes the unit circle |Z|=1. If it is a causal system it will be stable if and only if all the poles of H(z) lie inside the unit circle. <\/p>\n\n\n\n
The range of variation of z for which z-transform converges is called the region of convergence of z-transform.<\/p>\n\n\n\n
Properties of ROC of z-Transform<\/p>\n\n\n\n
R1<\/sub> <|z| < R2<\/sub><\/p>\n\n\n\n Rmin<\/sub> < |z| <Rmax<\/sub>. Where R1<\/sub>, R2<\/sub> are the magnitude of two poles of x(z)<\/p>\n\n\n\n