Unilateral z-transform and regions of convergence



The unilateral z-transform of a discrete-time signal x[n] is given by the equation:

X(z) = Σ (x[n] * z^(-n))

where n ranges from 0 to infinity, and z is a complex variable.

To find the unilateral z-transform of a given signal, we need to substitute the signal into the above equation and perform the summation.



The given signal x1[n] is defined as:

x1[n] = (1/4)^n * u[n - 5]

where u[n] is the unit step function.

To find the unilateral z-transform of x1[n], we substitute the signal into the unilateral z-transform equation:

X1(z) = Σ ((1/4)^n * u[n - 5] * z^(-n))



The regions of convergence (ROC) are the set of values for which the z-transform converges. It is represented in terms of the complex variable z.

To determine the ROC, we need to analyze the given signal and identify the range of values for which the z-transform converges.

In the case of x1[n], the signal is a right-sided sequence since it is multiplied by the unit step function u[n - 5]. This means that the signal is nonzero only for n greater than or equal to 5.

Therefore, the ROC for x1[n] will be of the form |z| > R, where R is a positive real number. This is because for large enough values of n, the term (1/4)^n approaches zero, and the z-transform converges for all values of z except at z = 0.

In conclusion, the unilateral z-transform of x1[n] is X1(z) = Σ ((1/4)^n * u[n - 5] * z^(-n)), and the corresponding region of convergence (ROC) is |z| > R, where R is a positive real number.