Impulse Matching: Finding the Unit Impulse Response

[Corneo]

Impulse Matching

In regards to finding the unit impulse response of a system. We assume that $x(t) = \delta (t)$ and that the intials conditions at $t=0^_$ are all zero. The impulse response $h(t)$ therefore must consists of the systems's modes for when $t \geq 0^+$. But why is it that $h(t) = A \delta(t) + \text{modes}$ for $t \geq 0$?

 

[phantom_photon]

I guess the simplest explanation would be that if the domain of the response is extended to include t = 0 (or 0-) it will have an additional component: the system's response to the impulse while the impulse is being applied, which is of course a scalar multiple of the Dirac delta function.
Since the delta function is non-zero for t = 0- and 0 but not for 0+, the response for t >= 0+ will not involve the delta function at all, just it's residual effect (the system's characteristic oscillations). You will generally only be concerned with the response for t > 0+.
That's what I think it is, but I might be wrong. I appologise for the lack of mathematical rigour in this post.

 

[ravenprp]

The reason for this is because the unit impulse response represents the output of the system when a unit impulse is applied at the input. The impulse response captures the behavior of the system at a particular time, and since the input is a unit impulse, it only affects the system at that specific time. Therefore, the impulse response will only have a non-zero value at that time and will be zero for all other times.

The term A represents the amplitude of the unit impulse, which is typically set to 1. This means that the impulse response will have a value of A at the time of the impulse and will be zero for all other times. The remaining terms, referred to as the system's modes, represent the behavior of the system for t ≥ 0 when there is no input. These modes can be determined through various mathematical techniques such as differential equations or Laplace transforms.

In summary, the unit impulse response for a system can be represented as a combination of a scaled unit impulse and the system's modes. This allows us to fully understand the behavior of the system for any input, as the impulse response captures the system's response to a unit impulse, which can then be used to predict the response to any other input signal.