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This module explains what is and how to use the Impulse Response of LTI systems.


  • The output of a discrete time LTI system is completely determined by the input and the system's response to a unit impulse.

    System output

    A discrete time system H takes the input x[n] and produces the output y[n].
    We can determine the system's output, y[n], if we know the system's impulse response, h[n], and the input, x[n].
  • The output for a unit impulse input is called the impulse response.
    An impulse input delta[n] going through a discrete time system H, producing the system's impulse response, h[n].
    delta[n] 'shocks' the system suddenly. h[n] is the response to the shock.

Example impulses

Since we are considering discrete time signals and systems, an ideal impulse is easy to simulate on a computer or some other digital device. It is simply a signal that is 1 at the point n = 0, and 0 everywhere else.

Lti systems and impulse responses

Finding system outputs

By the sifting property of impulses, any signal can be decomposed in terms of an infinite sum of shifted, scaled impulses.

x[n] k x[k] δ k [n] k x[k] δ [n-k]

The function δ k [n] δ [n-k] peaks up where n k .

The function δ[n-k]. It is simply 1 at point n and 0 everywhere else. Point n is marked on the graph. The function x[k]. It has a strange shape. Point n is marked on the graph.

Since we know the response of the system to an impulse and any signal can be decomposed into impulses, all we need to do to find the response of the system to any signal is to decompose the signal into impulses, calculate the system's output for every impulse and add the outputs back together. This is the process known as Convolution . Since we are in Discrete Time , this is the Discrete Time Convolution Sum .

Finding impulse responses

  • Theory:
    1. Solve the system's Difference Equation for y[n] with f[n]= δ[n]
    2. Use the Z-Transform
  • Practice:
    1. Apply an impulse input signal to the system and record the output
    2. Use Fourier methods
  • We will assume that h[n] is given for now.
    • The goal is now to comput the output y[n] given the impulse response h[n]and the input x[n].
      A system with impulse response h takes the input f and produces the output y.

Impulse response summary

When a system is "shocked" by a delta function, it produces an output known as its impulse response. For an LTI system, the impulse response completely determines the output of the system given any arbitrary input. The output can be found using discrete time convolution.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
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Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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