The *unilateral* z-transform of an arbitrary signal is defined as

When the unilateral z-transform is applied to find the transfer function of an LTI system, it is always assumed to be causal, and the ROC is always the exterior of a circle. The unilateral z-transform of any signal is identical to its bilateral Laplace transform. However, if , the two z-transforms are different. Some of the properties of the unilateral z-transform different from the bilateral z-transform are listed below.

**Time Advance**

where we have assumed .**Time Delay**

where . Similarly, we have

where . In general, we have

**Convolution**

If both and are causal, i.e., for , the unilateral and bilateral z-transforms are identical.**Time Difference**

**Proof:**

**Time Accumulation**

**Initial Value Theorem**If , i.e., for , then

**Proof:**

All terms with become zero as as , except the first one which is always .**Final Value Theorem**If , i.e., for , then

**Proof:**

i.e.

Letting in the above, we get

where .

**Example:**

This signal is right sided starting at (i.e., ). By definition, the bilateral z-transform of is

It was assumed that . The unilateral z-transform of this signal is

If we assume zero initial condition ,