# Properties of z-Transforms

• Last Updated : 11 May, 2022

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 x1[n] and x2[n], and their individual z-transforms as X1(z) and X2(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 an, 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 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, 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.

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