Understanding the Impulse Response: A Key Tool for Identifying Unknown Systems

In summary, The impulse response of a system can be found by substituting the Dirac Delta function for the input and solving for h(t), the function relating the output and input. This works because the Dirac Delta function is an impulse and the output of a system with an impulse as an input is the impulse response. Impulse responses are important because they can fully characterize a linear time-invariant system. However, it is important to choose the appropriate input in order to get a precise result when identifying an unknown system.
  • #1
perplexabot
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Hi all. A problem I was working on required that I find the impulse response where I am given the output function ( y(t) ) of a system in terms of the input ( x(t) ). I read somewhere that to find the impulse response, you need a function that relates the output and the input, then you substitute h(t) for every y(t) and δ(t) (the impulse function) for every x(t) then finally solve for h(t) and that would be the impulse response. Assuming what I just stated is correct, my question is why does it work like that? Specifically why does substituting δ(t) for every x(t) lead to the impulse response?
 
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  • #2
δ(t) is the Dirac Delta function called an impulse. So for a system with impulse as an input the corresponding output would give you the response namely the impulse response.
Impulse response are significant because any linear time invariant system can be characterized by stating just its impulse response.
Hope this answers you query.
 
  • #3
The output is the convolution of the impulse response by the input. So if having the input and output, you "could" de-convolve the output by the input to deduce the impulse response.

Though, this is an inverse problem! The equations may look good, but the precision of the result may be extremely bad if the input and output you've observed do not stress the system enough.

Imagine that you're interested in a corner frequency at 1kHz and your input is a sine at 50Hz: you'll observe nothing interesting and deduce a very imprecise corner frequency.

That's why, if identifying an unknown system, we chose an excitation that stresses it properly, for instance a step input or a nearly-impulse input. Or a frequency sweep, equally good.
 

1. What is impulse response?

Impulse response is a mathematical representation of how a system responds to a brief input signal, also known as an impulse. It is used to analyze the behavior of systems in various fields such as engineering, physics, and signal processing.

2. How is the impulse response determined?

The impulse response can be determined through experiments or simulations, where a known impulse signal is applied to the system and the resulting output is measured. It can also be calculated mathematically using the system's transfer function.

3. Why is impulse response important?

Impulse response is important because it provides valuable insight into the characteristics of a system. It can be used to determine the stability, frequency response, and other properties of a system, making it a crucial tool in understanding and designing systems.

4. What are some common methods for finding impulse response?

Some common methods for finding impulse response include using Laplace transforms, Fourier transforms, and convolution. These methods can be used to calculate the impulse response of linear and time-invariant systems.

5. Can the impulse response change over time?

Yes, the impulse response of a system can change over time if the system is not linear or time-invariant. In such cases, the impulse response may vary depending on the input signal or the state of the system, making it more complex to determine.

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