Does the impulse response of a system have any physical meaning?

Question

In other words, what does the impulse response tell us about the characteristics of the system? For simplicity, let's assume we deconvolve a discrete output and a discrete input to obtain an impulse response that has n points. Does the amplitude of the first (or second, or third...) point indicate something? Does the length n tell us anything?

Answer 1

See this question for a long description of the impulse and frequency responses of a system.

In short, the impulse response of a discrete-time LTI system tells you something very specific. If the system has impulse response $h[n]$ and you apply an input signal of the following form:

$$ x[n] = \delta[n] = \begin{cases} 1, && n = 0 \\ 0, && \text{otherwise} \end{cases} $$

then the output of the system will be exactly $h[n]$. This input signal $\delta[n]$ is a discrete-time impulse, known as a Kronecker delta function. To your question specifically, the value of the first point in the impulse response indicates what the system's output will be at time instant $n = 0$ when you input an impulse. Likewise, the second value indicates what the system output will be at time instant $n = 1$, and so on.

This becomes a powerful tool for analysis when you combine it with the properties of linearity and time invariance that LTI systems have. You can decompose any discrete-time signal into a sum of multiple discrete-time impulses, all scaled and shifted in time in different ways. Since the system is linear and time-invariant, scaling and shifting the input signal has the same effect on the output, so if you input a signal of the form:

$$ a\delta[n-k] = \begin{cases} a, && n = k \\ 0, && \text{otherwise} \end{cases} $$

then the output of the system will be $ah[n-k]$; the scaling and time shift on the input just cause likewise scaling and time shifting on the output. Since for any discrete-time input signal, you can break it down into a sum of scaled and time-shifted impulses, you can do the same to the system's output; that output will just be a sum of correspondingly scaled and time-shifted impulse responses. That is, the system's output $y[n]$ for input signal $x[n]$ of length $K$ samples can be written as:

$$ y[n] = \sum_{k=0}^{K-1} x[k]h[n-k] $$

For each point $k \in [0, 1, \ldots , K-1]$, we take the corresponding value of the input signal and use it to scale a copy of the impulse response shifted by $k$ samples. The resulting output is the sum of all of these impulse responses. This is the discrete-time convolution sum.

Answer 2

The impulse response of an LTI (linear time invariant) system has the same physical meaning of the differential equation of a general dynamical system has.

So every output can be computed from the input and the impulse response convolution, without having to solve the differential equation. The frequency charactheristics can also be deduced from the Fourier transform of the impulse response.

Several properties of the (LTI) systems can be seen in the impulse reponse (and Fourier transform of it) expressions, such as slow system, fast system, wide band narrow band systems, overshoot, oscillations etc...