In digital signal processing, we often have to convert a signal from its various representations. Interconversion between various domains like Laplace, Fourier, and Z is an important skill for any student. In this post, we will discuss the relationship between the three most common transformation methods. We will see the interconversion process both algebraically as well as graphically.
Relationship between Laplace transform and Z-transform
We employ the Laplace transform in DSP in analyzing continuous-time systems. Conversely, the z-transform is used to analyze discrete-time systems.
The Laplace transform converts differential equations into algebraic equations. Whereas the Z-transform converts difference equations (discrete versions of differential equations) into algebraic equations.
The Laplace transform maps a continuous-time function f(t) to f(s) which is defined in the s-plane. In the s-plane, s is a complex variable defined as:
Similarly, the Z-transform maps a discrete time function f(n) to f(z) that is defined in the z-plane. Here z is a complex variable defined as:
Consider a periodic train of impulses p(t) with a period T.
Now consider a periodic continuous time signal x(nT).
Take a product of the above two signals as shown below.
Taking Laplace transform of the above signal and using the identity
Which can be written as:
Compare this equation with that of z-transform
Thus we finally get the relation:
(Derived from the Impulse Invariant method)
(Derived from Bilinear Transform method)
Mapping the s-plane into the z-plane
Relationship between Fourier transform and Z-transform
And the equation for z-transform is
Replacing the value of z in the above equation using
For r = 1
That is, if we evaluate the above equation on the unit circle of the z-plane, we get:
If you compare the above equation with the formula of the fourier transform, you can observe that the RHS of both the equations is the same. Thus we can say that the z-transform of a signal evaluated on a unit circle is equal to the fourier transform of that signal.
Mapping between phase and frequency on the unit circle
In the z-plane, , is a phasor with r being the magnitude and ω being the angle. This angle can be represented as:
For a normal z-plane, this indicates that ω is going to vary from 0 to 2π. However, in this case, we have r=1 and we are trying to establish a relationship between FT and ZT. So we can say that since the fourier transform is equal to the z-transform, ω of the z-plane is equivalent to the frequencies of the fourier transform which by extension now vary from 0 to 2π too.
Now that you are comfortable with the interconversion between different domains, we can proceed to understand the peculiarities of each domain. If there is any query in your mind pertaining to the explanation above, let us know in the comments. We are always around to help.