Differentiation in z-Domain Property of Z-Transform

The Z-transform is a mathematical tool which is used to convert the difference equations in discrete time domain into the algebraic equations in z-domain.

Mathematically, if $\mathit\mathrm<\left(\mathit\right)>$ is a discrete time function, then its Z-transform is defined as,

Differentiation in z-Domain Property of Z-Transform

Statement - The differentiation in z-domain property of Z-transform states that the multiplication by n in time domain corresponds to the differentiation in zdomain. This property is also called the multiplication by n property of Ztransform. Therefore, if

Then, according to the differentiation in z-domain property,

Proof

From the definition of Z-transform, we have,

Differentiating the above equation on both sides with respect to z, we get,

Also, it can be represented as,

Similarly, if the signal is multiplied by $\mathit^<\mathit>$ in time domain, then

Numerical Example

Find the Z-transform of the signal $\mathrm<\mathit\mathrm<\left(\mathit\right)>\:\mathrm\:\mathit^\mathrm\mathit\mathrm<\left(\mathit\right)>>$.

Solution

The given signal is,

Since Z-transform of the unit step sequence is given by,

Aging, using the multiplication by n property of Z-transform, we get,