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I'm trying to understand the expressiveness of the impulse response of an IIR. For example, I know that second order causal IIRs are capable of having impulse responses that are growing or decaying sinusoids. What else can they represent?

For a given order (causal or non causal) IIR, is there a description for the class (shape) of impulse responses it can have?

Karima Ma
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I'm assuming you're restricting your definition of an infinite impulse response filter to one that can be expressed as $$ y_n = \sum_{k=0}^N b_k x_{n - k} - \sum_{k=1}^N a_k y_{n - k} $$ where $y_n$ is the current output of the filter and $x_n$ is the current input to the filter.

In this case, there are $N$ possible modes which are a mix of $n^{p_m}d_m^n$ and $\delta(n - p_m)$. $\delta(n - p_m)$ is just a delayed Kroneker delta, and takes into account the fact that you can express a FIR filter or mixed IIR and FIR filter in the above form. Also, $d_m$ can be any complex number, with the constraint that if the coefficients $a_k$ are real, then any $d_m$ with a non-zero imaginary part must be accompanied by its complex conjugate.

This is basically just direct solutions to linear shift-invariant difference equations which are analogous to linear time-invariant differential equations in continuous time.

TimWescott
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  • thanks! I found these online notes useful too: https://tutorial.math.lamar.edu/classes/de/hohomogeneousde.aspx I didn't know that we can use similar tricks for solving differential equations to also solve recurrence relations. Looking into the theory of operators was also helpful for understanding the connection. In the case of IIRs we use the shift operator, in the case of differential equations we use the derivative operator. Exponentials are eigenfunctions of both of these operators and are thus solutions to the characteristic polynomials of differential equations / recurrence relations. – Karima Ma Apr 28 '22 at 22:31
  • I'm probably not as educated as I should be on difference equations -- I mostly understand them by analogy to differential equations -- but I've browsed tables of contents enough to know that there's books out there that cover the subject. If I only did signal processing, instead of a bit of everything, I'd have one on my shelf, and I'd have studied it. – TimWescott Apr 28 '22 at 23:26