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# Unilateral Z-Transform

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.

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 ,

Next: Solving LCCDEs by Unilateral Up: Z_Transform Previous: System Algebra and Block
Ruye Wang 2014-10-28