Consider a continuous signal, \(x(t)\), which is sampled with sampling period \(T\). When sampled we can write the continuous signal, \(x(t)\) as follows,

\(x(t) = \sum_{n=0}^{\infty} x(t) \delta(t-nT)\)

or, \(x(t) = \sum_{n=0}^{\infty} x(nT) \delta(t-nT)\)

Taking Laplace transform,

\( X(s) = \sum_{n=0}^{\infty} x(nT) e^{-snT}\)

Let \(z= e^{-sT}\), where z is a complex variable,then we get,

\( X(z) = \sum_{n=0}^{\infty} x(nT) z^{-n}\)

Since T is a constant we have,

\( X(z) = \sum_{n=0}^{\infty} x(n) z^{-n}\)

The expression given above represents the Z-transform of the discrete-time signal x(n). The Z-transform is a mathematical representation of a discrete-time signal in the complex domain. It is similar to the Laplace transform used for continuous-time signals.

The first line of the expression represents the discrete-time signal x(n) in terms of the impulse train Î´(t-nT). This means that the signal is a sum of weighted impulses, where each impulse is delayed by a multiple of the sampling interval T. The second line of the expression is obtained by replacing t with nT in the first line.

The Laplace transform is then taken on both sides of the expression to convert it into the frequency domain. The resulting expression is a function of the complex variable s, which is related to the frequency of the signal.

In order to simplify the expression, a new variable z is introduced, which is related to the sampling interval T. The variable z is defined as \(z= e^{-sT}\), which represents a complex number with magnitude 1 and an angle of -sT in the complex plane.

The expression is then re-written in terms of z, which results in a summation of the signal x(n) multiplied by powers of z, where each power represents a delay of n samples. This is the Z-transform of the signal x(n), represented by X(z).

The Z-transform is a powerful tool in digital signal processing, as it can be used to analyze and manipulate discrete-time signals in the frequency domain. Application includes FIR and IIR filter design with Z-transform. The Z-transform of a filter, for example, can be used to determine its frequency response and stability characteristics.