u/Collez_boi

Hey folks. I'm back. Gave my DESE interview today. I appeared for my M.Tech (Res) interview today, applied through CFTI mode. Let's jump straight to the transcript.

There were four (or five?) professors, but it was led by two of them. Let's call them Prof. S and Prof. C. (Prof S is someone whose work I found pretty cool. Safe to say that they were the senior-most faculty and the one I'm most aligned with. Fortunately, he was the one to start the interview). I have also been given plenty of A4 plain sheets and a simple pen to work out my logic on.

S: "Introduce yourself."

I tell them about myself. I'm a 26' ECE UG at XYZ Uni and my broad area of interest is Signal Processing.

Follow up: "What do you know about Signal Processing?"

I tell them about my BTP (Final Year Project) Thesis that I've been pursuing at IITM for almost a year. I describe them the general problem statement, the difficulty and the more or less "empirical" solution that I found for it and how it worked. It has now been pushed into prod at the industry (a defense vertical) that my lab is working with. The most exposure I've had so far is in array processing.

S: "Okay, let's start with some questions?"

We begin the main interview.

S: "So tell me, do you know what DNA is?"

I was already amused. An interesting start. "If I remember my biology classes right, it's Deoxyribonucleic Acid."

S: "Correct. Now, you know about nucleotides?"

I was smiling. I did remember from my JEE days, but of course not now! "I... don't recall, sir."

S: 'Don't worry. So there's A, G, C and T in DNA. They can get paired in different ways to produce sequences. Right?"

I nod my head, following along.

S: "Let's generalize this now. I'll give you some constraints as well. Assume the nucleotides are x1, x2, x3 and x4. Draw them as nodes, in a square or a rhombus. And assume that x1 and x2 can't pair, and so can't x2 and x4."

I do as told.

S: "Now represent these in a convenient way so that it'll be true for any N length sequence."

I don't understand at first what he means and I ask him to elaborate. Then he tells me he wants it as a graph (or think of it like an FSM-like representation). Accounting for self loops as well. (x1 can be paired with x1 and so on)

I make the directed graph. Double directed edges for the pairs, self loops and keeping in mind the constraints.

S: "Now assume an N length sequence. Provided that it ends in x1, write the number of sequences that are possible."

I first try a combinatorics approach. He halts me, saying that's not the direction he wants me to go in. He then proceeded to pen down an appropriate ad-hoc notation that we can use for representation. Basically, an equation in "n" (say n is the number of "steps" you take to count each nucleotide), then what can be my possible x_k(n-1)'s?

For instance, x1 can form pairs with itself, x3 and x4, but not x2. So I'd write

x_1(n) = x_1(n-1) + x_3(n-1) + x_4(n-1)

and he told me to proceed in similar fashion and write an equation each for each node.

S: "Now have you heard of state-space representations?"

Me: *Vaguely remembering my Control Theory course: "Yes... vaguely."

S: "Good. Then write this in a matrix form."

I see what he meant, so I write it in a matrix vector equation.

x(n) = A times x(n-1)
Where the structure of A is such that each row corresponds to a row, and each column corresponds to the connections allowed. 0 for forbidden, 1 for allowed. (Graph theory enthusiasts, I hear you. Yes, akin to an incidence matrix.)

S: "Now generalize it for all n's, meaning tell me how do I write this for some arbitrary kth node in my N length sequence."

I derive it rather naively and fundamentally, taking n=0 first, then n=1, seeing a pattern and generalizing to

x(k) = A^k times x(k-1).

Cool.

S: "Now tell me what is the growth rate of A."

Me: O_o
internally: *tf is a growth rate?*
I ask for clarification. We exchange thoughts back and forth a bit and the first thing I deduce is he wants some numerical relation.

Me: "Can we consider the eigen value decomposition by diagonalizing the matrix A?" (He had given a hint prior by deliberately using a variable named lambda. That was very nice of him. There was a slight joke about that here where I acknowledged the cheeky naming did in fact push me to think that way)

I screw up here. I accidentally raise the powers of the eigen vector matrices to K as well. Huh. Something seems strange.

S: "You're wrong. Can you recheck it?"

I realize my fuckup. We go on a slight tangent to the first principles. I elaborate what eigen values and eigen vectors are in terms of a linear transformation, and correct myself providing the reasoning why I was wrong.

The prof then told me (I have NO memory of how) that the "growth rate" is then proportional to the logarithm (base 2) of this normalized dominant eigen value.

I nodded in confusion.

S: "So since we've made a recursive series, you know about Fibbonnaci series right?"

I write it down.

S: "Write the recursive equation for it and then write it as a function of n."

I blank out again. Heh? I stare at the paper aggressively for a while. He then pushes me towards Z-transform. Oh. Silly me.

I write the z transform, and working it out leaves us with x(z) (Z^2 - z + 1) = 0, now x(z) can't be zero, so we find the roots of the characteristic equation.

Finally, my x(n) = c_0 times lambda1 + c_1 times lambda2 where lambda1 and 2 are my roots. I can get c0 and c1 from initial conditions given prior.

S is done asking all they wanted to.

C: "Let's deal with some probability?"

Oh boy. Here we go again. (I f*cked up a probability question in my ECE interview the day prior)

so C gave me a uniform CRV x distributed over 0 to 1.2, told me to draw the pdf, asked me what does the value at a particular x, say x=0.3 MEAN. I fumbled up again and the classic question of probabiliity vs likelihood was asked.

I said they aren't the same. Likelihood is the value at of the pdf at that x and probability is an area.

C tried ricking me again. "So probability is an area. Supposed I do a coin toss and say probability is area. How are you going to draw the pdf for that? Where is the area?"

Me: "Sir, this is a discrete random variable. The 'area' would be the height of the Dirac delta function describing the pdf." (I have NO idea if what I said is correct or not. I just tried using my intuition.)

C: "Okay. And can pdf be negative?"

Drumroll please, the MOST EPIC fuck-up of all time. "yyyeah?"

They literally crash out. I realize what I said. _why tf did I say that?_ I correct myself.

C: "Okay now give me the CDF of this."

I fuck up again by integrating from x=0 to... 1/2 (bruh). Then they tell me to slow down. I'm going too fast and making mistakes. I revert to first principles, state that I need to find Pr(X<=x) and THEN integrate from x=0 to x. I give them the expression for the CDF.

C: "Draw it. What happens before 0? what happens after 1/2?"

I correctly (wow so difficult) draw it.

C: "okay. Now consider another random variable y=x^2. How will you draw the pdf and cdf for this?"

I talk about the Jacobian and the intuition of it. I screw up something there though in my derivation of the new pdf and I end up in a situation where my pdf can be negative. So tey tell me to drop the Jacobian idea (since I fucked it up lol) and derive it using first principles.

I finally find out the pdf and cdf with a few more fuck ups her and there and some encouraging pushes from their side that bring me to the solution.

C: (smiling) "I think we've troubled you enough."

S: "Before you leave, are you willing to consider a PhD?"

Me: "I believe that is up to the panel to decide if my I have the calibre."

S: "That's okay. But are YOU open to it?"

Me: "Yes."

Awkward pause. "May I leave? :3" They all started laughing. They assured me I need not take permission, and that my interview was over and I can take my leave.

Estimated approx duration: One hour. I filled four A4 sheets with my derivations/deductions, front and back side both.

I don't know how I did, but I had a BLAST. I loved it. That first question was very unique and fun.

A while later when I was stuffing my face with Dosa cuz I had missed lunch prof. S called me. "Can you meet me at my office? I would like to discuss our future prospects with you."

I wasn't able to go to his office because it was already late and I had just started eating, so we talked on the phone. The end. :)

P.S; Pardon me if I may have recollected something wrong, do let me know. It's a little hazy and a strange mixture of a clear and foggy memory, but I've got the jist of it right. Cya around. :D