Channel Capacity - Heuristic proof

Quick recap of the channel capacity heuristic proof. Reference here: https://www.eecs.berkeley.edu/~dtse/Chapters_PDF/Fundamentals_Wireless_Communication_chapter5.pdf Assume we are transmitting a codeword in N dimensions. The dimension could represent time or any other signal characteristic. Assume it is corrupted with a noise vector (also N dimensions). Now each of the codewords can be represented as a point in N dimensional space. Assume also that we have a power constraint P (total power transmitted must not exceed P)- so all the received codewords must lie in an N-dimensional sphere of radius sqrt(N(P+sigma^2)). For error free decoding, all the received vectors (transmitted point + noise vector) should lie inside non-overlapping spheres inside this large sphere of radius sqrt(N(P+sigma^2)). The radius of this non-overlapping noise spheres has to be sigma^2 (noise variance) for a large N. Thus, max number of codewords that can be transmitted in this scenario can be geometrically deduced as sqrt(N(P+sigma^2))^N/sqrt(sigma^2)^N or equivalently the max number of bits that can be conveyed is log2(max_num_codewords). This yields the famous shannon expression for channel capacity. I.e no more information can be transmitted with the given power constraint on an AWGN channel with arbitrarily small error rate.

Today, while going through all the documents that have collected dust over the years, I started organizing them in to separate folders-by category. Bank related, income tax related, Visas, passport, stock certificates, real-estate investments, rental agreements, insurance etc.. the usual stuff. In the process, I came across my marks sheet from my under-grad. I stared it for a while. Most people hold themselves to some simple expectations. The expectation is that, one has to do what one is good at... regardless of the area of the job. "Well" here means whether one can do or has done new and innovative work in that area. For me, it is a bit late in life. However, to repeat a cliche, one should keep exploring. I am trying to recollect what I really-really enjoyed doing. What was I really-really good at? Should we all ask ourselves this question? May be the answer could even change from time to time. If you ask yourself that question often enough and you get an answer- I feel you are probably living a more fulfilling life. Let me know what u think.

What a wonderful day!

We choreographed a dance for dad's birthday. It was almost childish. In our conservative family environs, dancing itself was a big deal - so getting it right was not that important. The bigger part of "self growth" was for the first time in my life, I coordinated with people on the organization of an event. It felt so wonderful. We worked without taking ourselves too seriously but just wanted to make it fun for a bunch of people around us. The outcome was beautiful- none of us were perfect- but it was fun, nevertheless. I think the "key" learning that day was to not worry about getting things right- but just doing something different. My daughter was one of the best dancers.