Understanding IQ in Communication Systems: Complex Numbers (Originally posted on Wednesday, October 14, 2020)

So, as far as this blog goes this is not going to be one of those other blogs where profanity runs amok (though it can be, and it will be in some points here and there). The concept of signals and application of IQ data in communication system math in our area is rather a in-depth and detailed concept heavy area. Like IQ modulation has it's roots in complex numbers, and how actually these things play a role in defining digital modulation concepts and how constellation points are actually defined in IQ mode, and what does that mean as an equivalent in it's analog domain. But first things as they go, understanding what is an complex number is an essential part of this blog post an not-so brief introduction is given here. At this point I frankly don't know how I am split up these parts and publish as an blog post article, but I'll make a decision about it as I go through that bolg post. Now with the second part of this. The following article can be considered a the theroretical math part one needs to understand with SDR, thereby this can be considered a part 2 of out SDR 101 article And the last this this article contains a mix of hand written notes and ipython notebooks that I have written in jupyter in my undergrad days and modified a bit for the present day, which I will link along with the refernce that I used for this article at the end of this part/article. 
Complex Numbers What is a complex number? 
In traditional notion we represent complex number by 𝑗 or 𝑖 , but in our case we use it's former representation, so that I am free to use the i as a variable in this notebook. Developling a notion for this is something that atleast my school teaches missed while teaching the concept in my high school math classes. And it took me first year of engineering to understand it's sheer beauty and how this can make my analysis and imagination simple at most times. Complex numbers are often called as imaginary numbers. But to establish what is imaginary we first need to establish what is a real quantity. Consider next cell where we give a number line and represnt a real number for establishing the concept of real quantity. 

 Now from the above figure we have an simple numpy array that has values along the x-axis alone. Due to tradition and representation let us name the x-axis as real axis in communication systems as 𝐼 or the In-phase component and y-axis as imaginary in communication system as 𝑄 or the quadrature component. 

 In the next part let us generate some random numbers and convert them into complex numbers and plot them in an above diagram which is called as an "Argand" or "Complex" plane diagram. In out aricle and world we'll be calling it Complex Plane diagram. 


 From the above plot let us consider only one set of values from all of the points and explain it here.
 Now, as we see from the above paragraph the expression is then expanded as . 

But why is it okay to represent a complex number in such a way with nautral log as its base is the question that comes across ones mind first. The answer lies with this German Mathamatician named Euler. So, by his infinite series expansion we get something like below. 

 Now people with some exposure to trignometry can recognize the expression and can see that the alternating expressions in the third line are actually higher order powers of sine and cosine expression that is given in the infinite series, and hence that expression for this weird math representation. Now with the help another German Mathematician named Gauss, we can now represent the notation in complex plane. Now, the notation j can be sufficiently expressed in the complex plane even if the equation has a minus sign in between them, hence the operator j means for us to rotate along the real axis by a shift of 90 degrees in the counter-clockwise direction (CCW). Hence for us IQ signals purpose we can use the aforementioned expression as a complex number that is a function of time. With the above notions I am concluding this article and will upload the ipynb notebook for this in the git and provide a link below. For actual IQ representation and connecting this concept to out signals domain will be done in the later part of this series. 



git link
 https://github.com/Infiltra/Complex-Numbers

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