Calculating Unit Impulse Response

[paul_harris77]

I am trying to teach myself DSP, owing to bad lecture notes. In particular at the moment I'm trying to calculate impulse responses for LTI systems, given the system equation. I would really appreciate it if someone could tell me if my working and assumptions below are correct for the following question:

Homework Statement

Find the unit impulse response of the following LTI system:

y[n+2] + y[n+1] + y[n] = x[n+1] - x[n]

Homework Equations

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The Attempt at a Solution

Rearrange for y[n+2]:

y[n+2] = x[n+1]-x[n]-y[n+1]-y[n]

Since input is an impulse, and output is impulse response, can replace x with delta (&) and y with h.

h[n+2] = &[n+1] - &[n] - h[n+1] - h[n]

By the definition of impulse response, it is a zero state response, i.e. h[n] = 0 for all n<=0 - IS THIS CORRECT?

Therefore h[0] = 0

Starting iteration at n = -1

h[1] = &[0] - &[-1] -h[0] - h[-1]
h[1] = 1-0-0-0 = +1

n=0
h[2] = &[1] - &[0] - h[1] - h[0]
h[2] = 0 -1 -1 -0 = -2

All delta (&) terms are zero from now on.

n=1
h[3] = -h[2] -h[1]
h[3] = +2 -1 = +1

n=2
h[4] = -h[3] - h[2]
h[4] = -1 +2 = +1

Continuing in this way shows that h[n] = {0, +1, -2, +1, +1, -2, +1, +1, -2, +1, +1, -2, ...)
i.e. oscillatory response.

Please could someone tell me if my working is all correct, and if this is the correct answer?

Many thanks

Paul Harris

 

[klondike]

I am trying to teach myself DSP, owing to bad lecture notes. In
Since input is an impulse, and output is impulse response, can replace x with delta (&) and y with h.

h[n+2] = &[n+1] - &[n] - h[n+1] - h[n]
By the definition of impulse response, it is a zero state response, i.e. h[n] = 0 for all n<=0 - IS THIS CORRECT?
Paul Harris

Hi Paul,

Your answer seems fine to me.

You don't need to guess what h[0] is. It can be calculated by letting n=-2 in your last formula. This is a second order difference equation, by "find impulse response", you can assume h[-2]=h[-1]=0.

I would rewrite it as h[n]=&[n-1]+&[n-2]-h[n-1]-h[n-2], with h[n] identically 0 for n<0 which means causal and system is completely at rest before getting tickled. Actually, you only need to ensure h[-2] and h[-1] = 0. This is equivalent to finding particular integral of a second order differential equation with initial conditions as y(0)=0 and dy/dt(0)=0.

Solving h[n] this way is said to use "brutal force". You can use z-transform and it gives you more insights.

I suggest reviewing Signals and Systems first before jumping into DSP.

Good luck!

 

[paul_harris77]

Hi Klondike

Thanks for the reply. Glad I'm on the right track :) We've been chucked in at the deep end with no prior experience in SP of any kind and just a load of notes with definitions and no explanation! There was a previous exam question like this yet no examples on how to do it...

Once again thanks!

Best wishes

Paul