Name:_____________________________________ Roll Number:________________________________
Quiz-0----08-01-2024 || Communication—II (Digital Communication) || Credit: 0
1. (From solutions to problems…) The first class had me worried since many of you weren’t clear about what
problem the modulation scheme (a solution) was solving. So, let me put you on the spot. Consider a random
variable 𝑋 ∈ {1,2,3, … 𝑚} where corresponding probabilities 𝑝1 ≥ 𝑝2 ≥ 𝑝3 … . 𝑝𝑚 are known to you. 𝑋
“happened” and you have to “guess” what value it took.
a. What is the “best guess” you will make? For the answer you give, write clearly and mathematically the
optimization problem you posed for which your guess was the “best”.
b. If someone informed you additionally that an “even-number” occurred when 𝑋 happened, then what is
your “best guess” now? Relatedly, what is the (formal) optimization problem you solved now?
c. For the part (a), can you pose an optimization problem ad a solution, such that the “best guess” is
different from the one you had presented there?
2. (Pairing the parities.) You all know single-parity check, correct? Basically, for an even parity system if I want to
send 𝑛 bits (say in a binary vector 𝑋), I will “pad” an extra bit to ensure that the XOR of all 𝑛 + 1 bits (say in
vector 𝑋’ of size 𝑛 + 1) is equal to a 0 (for odd parity system, we will alternately require that this XOR value is 1).
What’s the use of this simple scheme? Well, it can detect an error in transmission of 𝑋’, i.e., if a single-bit (or
odd number of bits!) in 𝑋’ flips, then the XOR value will change and will flap an error somewhere (as the agreed
upon “parity” convention is broken).
a. Can you devise a scheme which can detect occurrence of both, one error or two errors (either may occur
without prior intimation), when sending 𝑋? You will need to “construct” a new 𝑋’ from the 𝑛-bit 𝑋 now.
b. If you have a scheme that works, criticize your scheme (what could improve potentially)?
3. (“You can check out anytime you like but you can never leave”(sic)..LZ) Imagine a (not-so-unlikely) scenario
where you want to run out from my class. There are 3 doors to get out, say D1, D2, D3, but some are rigged (I
won’t make it easy you know!). I have magically designed the hall, the doors and hallways, where exiting from
D1 gets you back into the class after 10 minutes of walking, exiting from D2 gets you your “freedom” after a 15
minute walk, while exiting from D3 gets you back into the class after 30 minutes of walking. To spice up the
problem, let’s assume that when unsuccessful and in the time you walk back, you lose your memory of the door
choice (a la the movie Ghajini/Memento), but you remain committed to exit (my teaching can do that, I think!)
and try again (perhaps endlessly?). What is the ^^“average” time to freedom?
___________________________________________________________________________
^^What is left unsaid here is what are you even averaging over. You should ponder upon that and use it clearly.
***Do all the rough work in your books and write only the clean answers in the space provided. Define math. variables (instead of carrying
long phrases in your answers). In a “credited” quiz, you will be provided “some” space in the paper itself and no access to books.***
-------Bring this sheet to class tomorrow------