Folding and Aliasing

Here is an java applet which lets you see the effect of sampling frequency and reconstruction of the signal. The signal is sampled, then goes through a low pass. You will see here what happens when sampling frequency is not large enough and the replicated spectrum overlap.

Requires Java.

Posted in Digital Communications

New look

In coming to this new format, I had to leave behind the board which has over 2000 members. There were quite a lot of people on it and I am sorry that it will not be accessible any more. However, this format may work even better, as now the discussions will be grouped by topic. For general questions, help etc. from others, please use the main page.

I may add a forum, if there is interest
Thank you.

Posted in DFT

Discrete-time Fourier Series and Fourier Transform

I have posted part 3 of the FFT tutorial at
It starts with the development of discrete signals, their properties, periodicity of discrete signals,
finding a set of basis harmonics and then finally the Discrete-time Fourier Transform.

The tutorial also includes Matlab code.
Let me know if I succeeded in making this topic easy(er) to grasp.


Posted in Digital Communications

Fourier Made easy part 2 – new version uploaded

I have uploaded a new version of the Fourier Made Easy part 2 tutorial. In this version, I have included more examples as well as Matlab code to plot some of the key pictures.  I get requests to include code, so I am going through my old work and including it as I can. I use Coware SPW for most of my “real” work and from it I can only include the pictures. Its purely block orientated and has no code to share.

This tutorial can be used for classroom use as well for anyone wanting to brush up on transform theory.
I am now helped in my work by Victor Levin, my son. He did the Matlab codes in this tutorial. Victor is a graduate student in EE at Georgia Tech. He is currently a TA at the Metz, France campus of GTech.

I recently went to France and saw the GTech campus and facilities. Very nice!

Charan Langton
October, 2012
Posted in DFT, Digital Communications, Fourier Transform Tutorial

Video that explains the complex exponential

While writing the second part of the Fourier tutorial, I came across this video which does a great job of explaining the complex exponential, its relationship to sinusoids and hence to the reason why signal processing math is done with exponentials. Nicely done.

Posted in DFT, Digital Communications, Fourier Transform Tutorial

Fourier tutorial part 1 – updated

I am going through my old material and am updating it. I have just uploaded FFT Part 1 after changing it in many places. If you have comments on this tutorial (any including typos!), please post them here. Charan Langton
Posted in DFT, Digital Communications, Fourier Transform Tutorial

Thanks to The Mathworks

The Mathworks company has just provided me a complementary copy of Matlab out of their “Author” program. So I want to take this space to thank them. For may of the demonstrations in my tutorials on and where I feel it necessary to pass on the code, I use Matlab. I have also started using Simulink and am beginning to make progress. I will make some of my Matlab models available with my communications tutorials. Progress on the book is slow. But in the meantime, please feel free to send me comments and errors you find. Thank you Naomi Fernandes and The Mathworks. Charan Langton

Posted in DFT, Digital Communications, Fourier Transform Tutorial

MIMO at last

I have posted the final (nearly so) version of my MIMO tutorial.

It took a while. In the writing of this paper, I relied heavily on a few books and papers. The three books I often found myself thumbing through, in order are: Wireless Communications by Andrea Goldsmith, Fundamentals of Wireless Communications by D. Tse and P. Viswanath and Digital Communications by Proakis, 5th Edition. The examples used in this paper are inspired by the Andrea Goldsmith book, the one I consider the best. She has a lot of examples in her book which truly help with understanding.

The other three books that I also found helpful were: Digital Communications by Barry, Less and Messerschmitt, Introduction to Space-Time Wireless Communications by Paulraj, Nabar and Gore and MIMO-OFDM Wireless Communications with Matlab book by Yong Soo-Cho, and Won Young Yan. I used the Matlab code in this book to create some of the graphs.

With papers, the one I read several times was by Gesbert’s and the one by Foschini. I read many others but these two standout. Agilent also has an excellent tutorial online. And of course the David Tse video lectures are fantastic. (See link below) He is quite funny! IEEE also has an audio lecture that is excellent.

It took me a while to get all my ideas together and I read many papers and checked nearly all the books written on the topic. While writing about these, I often could not figure out who the original source was. This is not good as I do want to credit to whom it is is due. If you the reader feel that I have not properly credited you or someone else in this paper, please do let me know.

In writing, Bernard wrote the first draft and then I added and subtracted from it. We are to do a book for Prentice Hall but the schedule is all gone out the window now. Hopefully it happens to other authors too.

The MIMO paper is now nearly 50 pages long but it is still lacking in many areas. I was not able to cover STC decoding, nor the code performance issues. The section on multi-user is also short. But I hope that what is here will help illuminate the topic and get you started.

Charan Langton

Posted in MIMO tutorial

MIMO tutorial at last

I have post a completed version of the MIMO tutorial on

Posted in MIMO, MIMO tutorial, MIMO tutorial signal processing digital communications, Uncategorized

MIMO Tutorial – Part One

I have posted part one of the MIMO tutorial. The first part is mostly concentrated on the capacity issues. You may think why even worry about capacity; We don’t seem to worry about the capacity of the traditional SISO links. We have to worry about the capacity possible with MIMO because that is the only way to know if this new-fangled technique requiring lots of antennas and receivers is even worth doing. And the conclusion is, no surprise, that MIMO is very much worth doing.

For same power that we might use in a regular link, if we just replace the transmitter and the receiver with a MIMO transmitter and a MIMO receiver, we have got ourselves as much as 3 to 4 times higher capacity. This is what we see when we went from 802.11 N to G. The gains came from the application of MIMO. You are already using MIMO in your router with its two antennas.

But what is MIMO, is it Multiple Access, NO. Is it modulation, No. Is it a type of receive signal processing like a rake receiver, No, again. It is just increasing the number of transmitters and receiver from one to many. The signal processing actually gets simpler, believe it or not.

MIMO is the future, but not for all applications. Line of sight links do not benefit from MIMO. So MIMO’s application is primarily in wireless phone and home networking.

As a topic to study, it seems complex and it is. Because you have to back track and start at the beginning by first looking at what it can offer. Then it very quickly leap frogs to trellis coding, block coding and these are not even for the purposes of error-correction. I will cover these in the next part.

Please let me know if it is easy to understand. MIMO does have a whole lot of new ideas, or actually old ideas combined in a new way. So it is confusing. Two books that I referred to a lot and like in order of preference are by Andrea Goldsmith (Stanford) and David Tse(Berkeley). David Tse actually has some terrific video movies on MIMO. Find them and watch them.

There are several other books that I used while writing this tutorial and I shall provide a list of these in part two.

BTW, if you have younger siblings or kids, be sure to look at my new Verbal Math books for kids. We have them in Kindle format now.

Posted in MIMO tutorial signal processing digital communications
Skip to toolbar