# Properties of z-Transforms

Prerequisite: What is Z-transform?

A z-Transform is important for analyzing discrete signals and systems. In this article, we will see the properties of z-Transforms. These properties are helpful in computing transforms of complex time-domain discrete signals.

**1. Linearity: **If we have two sequences x_{1}[n] and x_{2}[n], and their individual z-transforms as X_{1}(z) and X_{2}(z), then the linearity property permits us to write:

This is easily proved. First consider

Then, from the definition, we see:

**2. Time Shifting: **If we have a time-shifted sequence such as x[n-k], then its z-transform is given by Z{ x[n-k]} = z^{-k}X(z).

Let’s take n – k = m, i.e., n = k + m and y[n] = x[n-k]. Now here we are assuming that x[n] starts from n=0, hence x[n-k] starts from n=k, or n-k=0, or from m=0.

**3. Time reversal: **Time reversal property states that

We are going to formally prove this statement by taking y[n]=x[-n].

Now let’s take -n=m. Then

**4. Scaling in z domain: **When we multiply the signal sequence x[n] in the time domain with an exponential factor a^{n}, the equivalent z-transform of the new signal is scaled by a factor of a.

Basically, .

Proof is elementary and is shown below.

**5. Differentiation in z domain: **We know:

Differentiating with respect to z, we get

Hence, we can deduce that for k differentiations, we get

**6. Convolution: **Convolution of two sequences x[n] and h[n] is defined as

Now z-transforms of x[n] and h[n] are X(z) and H(z) respectively. Using this notation, we have

Hence, convolution in time domain is multiplication in z domain.

**7. Initial value theorem: **Initial value theorem gives us a tool to compute the initial value of the sequence x[n], that is, x[0] in the z domain by taking a limit of the value of X(z). It states that the following equivalence is feasible.

The proof, as before, relies on the definition of X(z).

Clearly, if we want to get x[0], we can make z approach to infinity so that all the other terms die out. What is left behind is precisely the statement of the theorem presented before.

**8. Final value theorem: **The final value theorem lets us know the final value of x[n], or the value at infinity of x[n], using appropriate limits of X(z).

It states that

If we take the z transform of x[n]-x[n-1], then we get

Now taking the limit z⇢1, we see that we get in the right hand side, which simplifies to x[\infty] basically. Hence the theorem is proved.