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[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
9. Multiplication in time property
x1[n].x2[n]=1/j2pi {X1[z]*X2[z]}
10. Accumulation property