What is the amplitude of the impulse resonse

Question

I'm learning how to construct a discrete signal as a sum of impulse responses of the system. The final formula is then:

$$ y[n]= \sum_{k= -\infty}^\infty x[n] h_{k}[n] $$

My question is, what is the amplitude of the impulse response, because i thought an impulse response was already a scaled response to the unit impulse inputted into the system? But here they are multiplying again with a scaling factor of x[n]? So is the unit impulse's amplitude just 1 then?

Answer 1

In case of discrete signals, the amplitude of unit impulse is 1.

It does not mean that your impulse response should have unit amplitude. The impulse response can have any real valued amplitude in it.

Note: The area under unit impulse in continuous time domain is 1 not the amplitude. The amplitude of a unit impulse approaches infinity as the area under remains constant (=1).

Answer 2

Impulse response means the output of an LTI system when the input is a unit impulse.

eg. $ y[n]=x[n]+2x[n-1]-x[n-2]$

is a system. where, y[n] is the output and x[n] is the input. Then the unit impulse response of the system is calculated by substituting $x[n]=\delta[n]$

then, the output $y[n]=\delta[n]+2\delta[n-1]-\delta[n-2]$

ie. impulse response, h[n]={1,2,-1}

Hence values in the impulse response may take any real number.

Impulse response is used to characterize an LTI system. That is if we know the impulse response of a system we can calculate the output for any input x[n] by finding convolution of impulse response (h[n]) with the input(x[n]).

Output of a system with impulse response h[n] and input x[n] is given by the equation (convolution), $ y[n]= \sum_{k= -\infty}^\infty x[k] h[n-k] $

Here the input x[n] does not have any connection with impulse response h[n].

Note : Unit Impulse is a standard signal with unit amplitude at 0 and 0 at all other instants. But a unit impulse response is the output of an LTI system when the input is a unit impulse. It is the characteristics of an LTI system and the values may take any real number and is different for different system.