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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?

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  • $\begingroup$ it depend of course which function be taken as impulse response in my mind,if it is unit impulse response,then it's amplitude should be 1 $\endgroup$ – dato datuashvili May 20 '14 at 9:33
  • $\begingroup$ Interesting formula... Aren't you simply looking for convolution? $\endgroup$ – jojek♦ May 20 '14 at 9:45
  • $\begingroup$ Okay i figured it out. If i were to have a system in which i kick a brick (force of kick as the input) and then the brick moves forward (displacement of the brick would be output) then we can say that the response of a kick (if it were considered an impulse) would be $$h_k[n]$$ but if every kick were a multiple of the unit kick then in an LTI system, then the output would have to be a scalar multiple of that same unit response. thus mathematically $$x[n]h_k[n]$$ then we can give our system to someone and they can find the position of the brick at any point, n using the formula above. $\endgroup$ – Sach May 20 '14 at 10:06
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    $\begingroup$ So answering the question, the amplitude of the impulse response can be anything. Its just a response to a unit kick or impulse. then we can describe all the other responses as a scaled version of it. $\endgroup$ – Sach May 20 '14 at 10:09
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    $\begingroup$ @datodatuashvili if it is unit impulse response,then it's amplitude should be 1 Oh please. The adjective "unit" qualifies impulse, not impulse response. It is the response to a unit impulse that we are interested in, not the response to some impulse of arbitrary magnitude where the said response has been scaled to have an amplitude of $1$. $\endgroup$ – Dilip Sarwate May 20 '14 at 16:16
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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).

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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.

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