The property of Accumulation is derived by expressing the accumulated sum and applying the time-shifting property to solve for the Z-transform.

It states that summing a discrete-time signal produces another signal whose Z-transform equals the Z-transform of the original signal multiplied by z over z minus 1.

The convolution property shows that convolving two signals in the time domain results in the product of their Z-transforms in the frequency domain.

This is valid for both causal and noncausal signals.

Applying the time-shifting property to the time-domain equation helps verify the convolution property.

The initial value theorem relates the initial value of a signal to its Z-transform. For a signal x[n], the initial value is the limit of X(z) as z approaches infinity.

Similarly, the final value theorem states that the final value is the limit of 1 minus the inverse of z multiplied by X(z) as z approaches one.

It applies only if x exists at infinity and all the poles are inside a unit circle except at z  equal to one.