When I began studying DSP (Digital Signal Processing), I was confounded by all the transformation of signals. There was the Laplace transform, the Fourier transform, and the Discrete Fourier transform and the z transform. Then there were all these planes like the s-plane, the z-plane, which looked a lot like the normal x-y axes of the familiar cartesian plane. Clearly, I was way out of my depths and could not figure out why we needed all these transforms. Or what they meant.
I found myself searching forums and Youtube videos that took the math out for a second and explained in layman terms what all these transforms did to a signal and why did we need them.
I found two excellent resources. This smoothie analogy for the Fourier transforms by Kalid and this overview of transforms by Abdel Helim Zekry.
This is my interpretation of what these signal transformations mean in the context of DSP and why we need them. Note that the scope of these transforms is far more wide-reaching than just DSP. This is just a crude attempt at creating a skyscraper look into the transforms.
So here we go. Let’s begin with an analogy and a story.
In some recipes of making a cheesecake, we start with solid cream cheese. Melt it into the cream batter while making the cake. And then cool it to a more solid form before we eat it. The accuracy of this recipe is not essential. I can’t bake a cake. But the important takeaway is that we start with one form of the cheese: cold, semi-solid. Melt it to make sure the ingredients mix up well. And then bring it back to the cold semi-solid form before eating it.
The important takeaway is that the cheese needs to be converted into a separate form for us to enhance its taste.
Similarly, the signal transforms that we use in DSP have the same purpose. The s-planes, the z-planes are used to contain an alternate form of the signals.
Signals, as we know, exist as a function of time. And to analyze and modify these signals in the physical world, we need to have some mathematical way of representing these equations. Let’s say we can do it using differential equations.
However, in the time domain, we cannot perform certain operations. For example, if you are listening to music, and you want to increase the bass or some audio component that lies in a specific frequency range. It will be much easier if we have the signal in the form of its frequencies. Then we can select the range of frequency we want to boost and do some operation on just that part.
When the signal is in the time domain, we don’t have direct access to the frequencies. In the frequency domain, however, we have direct access to the same signal’s frequencies.
This should now be giving you an idea of why we need to transform signals into different forms.
Similar to the frequency domain, the Laplace transform defines a new domain (or plane). The s-plane. Here, the complex variable s is defined as s = σ+jω, where ω is the frequency component of the signal. The importance of the s-plane is that it allows us to convert the differential equation of the time domain signal into an algebraic equation in the s-domain. Algebra is easier than calculus for us, as well as for machines. Thus, transforming the signal using the Laplace transform makes it easier to perform certain operations on it.
The smoothie analogy by Kalid defines the Fourier transform in a similar way. His approach defines the Fourier transform as something that filters out all the components of a smoothie. Now once we have the raw recipe, we can make whatever changes or analyses we want on the smoothie. To get back the original smoothie, we just blend the components back in.
In the next part, we will understand the specific advantages of each of the transforms. I hope this part allowed you to understand the need for these transformations.
Since signals operate in the time domain and are represented using differential equations, we can do something with them. But to be honest, working with these differential equations is a pain in the butt. So along comes this guy named Laplace, and he said that he will circumvent the differential equations and still process the signals. So he invented this new domain known as the s-domain. He gave us an equation to convert the continuous-time signal F(t) to this s-domain signal F(s). Amazingly, this equation also converted the differential equation of the time domain signal function to an algebraic function in the s-domain.
This was a landmark moment in the processing of signals.
Now, if we put the value of s=jω in the Laplace transfer function, we transform the signal to its frequency domain. This transformation is known as the Fourier transform. And jω is the complex part of the s-domain, and ω is the frequency. So now we have the original time-domain signal in its frequency domain.
The above transforms were for continuous-time signals — Analog, in other words.
For discrete-time sequences, we have the Z-transform and the Discrete Fourier Transform (DFT).
The Z-transform is the discrete-time version of the Laplace transform and exists in the z-domain. Here, z is a complex variable that relates to the s-complex variable of the Laplace transform as:
Here is a detailed relationship analysis between the Z-transform and the Laplace transform.
The Discrete Fourier Transform (DFT) is the discrete-time version of the Fourier transform. The relation between the Z-transform and the Fourier transform is given in detail over here.
Beyond this, we take the plunge into the mathematical part of the transforms, which you can glimpse by clicking the posts linked above.
In summary, the Laplace transform gives a way to represent a continuous-time domain signal in the s-domain. Additionally, it eases up calculations. A special case of the Laplace transform (s=jw) converts the signal into the frequency domain. This transformation is known as the Fourier transform.
For discrete-time sequences, the Z-transform is the Laplace’s equivalent. Transforming the discrete-time signal to the z-domain. The DFT is the discrete version of the Fourier transform. All of these transforms are interlinked and can be reversed to get the original signal.