$\mathcal Z$-transform if the output is given

1. How do you find the Z-transform?
2. What does the Z-transform tell us?
3. When to use Z-transform?
4. What does z mean in Z-transform?

How do you find the Z-transform?

To find the Z Transform of this shifted function, start with the definition of the transform: Since the first three elements (k=0, 1, 2) of the transform are zero, we can start the summation at k=3. In general, a time delay of n samples, results in multiplication by z-n in the 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.

When to use Z-transform?

z transforms are particularly useful to analyze the signal discretized in time. Hence, we are given a sequence of numbers in the time domain. z transform takes these sequences to the frequency domain (or the z domain), where we can check for their stability, frequency response, etc.

What does z mean in Z-transform?

Where, z is a complex variable and it is given by, z=rejω Where, r is the radius of a circle. Also, the unilateral or one-sided z-transform is defined as − Z[x(n)]=X(z)=∞∑n=0x(n)z−n.