How to compute the inverse Z-transform

1. How do you find the inverse Z-transform by long division?
2. What is the inverse Z-transform of 1 z?

How do you find the inverse Z-transform by long division?

Long Division Method to Calculate Inverse Z-Transform

As the determination of the inverse Z-transform of X(z) is only the determination of sequence x(n), i.e., if x(n) is causal then x(0),x(1),x(2),... or if x(n) is anti-causal,thenx(0), x(−1),x(−2),...

What is the inverse Z-transform of 1 z?

The Z-transform of a sequence an is defined as A(z)=∑∞n=−∞anz−n. In your case, A(z)=1/z=z−1, so this must mean an=0 for all n≠1, and a1=1. We don't need any fancy computations in this example, we just read off the one nonzero coefficient directly from A.