Output of an LTI system when the input is multiplied with 'n'

[jaus tail]

Homework Statement

Homework Equations

Find out the function of the system.
Apply that to input.

The Attempt at a Solution

From given input output, system is a differentiator.
So when input is n u[n]
output is differentiating it,
we get:
n (impulse response) + u[n]
Which is not in any of the option.
Book answer is B.
I'm struggling to get that.

 

[collinsmark]

Homework Statement

View attachment 219792

Homework Equations

Find out the function of the system.
Apply that to input.

The Attempt at a Solution

From given input output, system is a differentiator.
So when input is n u[n]
output is differentiating it,
we get:
n (impulse response) + u[n]
Which is not in any of the option.
Book answer is B.
I'm struggling to get that.

The problem statement seems to be mixing its use of square brackets and curved brackets. Just so we're on the same page, I like to use square brackets when working with discrete time, so I'm going to use them here.

Also, I'm going to assume that $\delta[n]$ is the Krockner delta function (which has a magnitude of 1 at n=0 and a magnitude of 0 elsewhere), which should not be confused with the Dirac delta function (which has infinite magnitude at n=0).

n      -3   -2   -1   0   1   2   3   4   5
u[n]    0    0    0   1   1   1   1   1   1
δ[n]    0    0    0   1   0   0   0   0   0

Now that we have that squared away, I think you are correct about the system being a differentiator.

So the first thing is to write out the sequence $n u[n]$. This is easy to calculate since you already know what $n$ is and what $u[n]$ is.

Differentiate your $n u[n]$ sequence, and what do you get?

[Edit: Also, for what it's worth, "differentiation" in continuous time becomes "difference" in discrete time. So can you find the difference equation for this system?]

 

[jaus tail]

 

[collinsmark]

How to find the difference equation?
From quora I found:
difference equation of f(n) = f(n+1) - f(n)

Close, but not quite perfect.

The equation does involve a difference between a couple of terms. Just make sure to get the right terms.

But here input is n u[n]
so how to find it's difference equation:
(n + 1) u [n + 1] - n u[n] ?

Start by writing out the sequence of $n u[n]$. Start at around n = -3 and work you way to around n = 5 or so.

 

[jaus tail]

Yeah I think i figured it out. I was editing my post when you replied. Is it right now?

 

[collinsmark]

Yeah I think i figured it out. I was editing my post when you replied. Is it right now?

Yes, that looks correct now to me.

(Edit: you skipped the number '4' in your n, but you seem to have worked it out correctly otherwise.)

 

[jaus tail]

Also gives me right answer of B. Thanks.