How/why are the $\mathcal Z$-transform and unit delays related?

1. Why is z 1 a unit delay?
2. What is the unit of delay element z '?
3. Why do we use z-transform instead of Laplace transform?
4. What does the z-transform tell us?

Why is z 1 a unit delay?

polynomial by z-1 simply subtracts one from each exponent in the polynomial, thereby creating a delay of one.

What is the unit of delay element z '?

If z–transform of x(k) be X(z), then z–transform of x (k + 1) is zX (z). Therefore, for an input zX (z) to a delay element, output will be X(z). Hence the transfer function of delay element. We can also write the letter 'D' or z ⁻¹ in the block of 'unit delay' element.

Why do we use z-transform instead of Laplace transform?

The Z-transform is used to analyse the discrete-time LTI (also called LSI - Linear Shift Invariant) systems. The Laplace transform is used to analyse the continuous-time LTI systems. The ZT converts the time-domain difference equations into the algebraic equations in z-domain.

What does the z-transform tell us?

In a like manner, the Z-Transform allows us to analyze the frequency and phase of sinusoidal components of a system to characterize a system's response. In short: If the Z-Transform of a system identifies exponentially increasing output values, then your system exhibits instability for that value of x[n] and z^-n.