For denoting the application of an FIR filter with impulse response $\mathbf{k}$ to a signal $\mathbf{x}$ in time domain, we can use the convolution operation, i.e., $\mathbf{x}*\mathbf{k}$. What is the best concise way to denote the application of an IIR filter to a signal in time domain? An IIR filter has, by definition, an impulse response that is infinitely long. Using convolution to denote its application seems odd to me. I am not asking about what an IIR filter is or how it is defined but how to denote it mathematically in a visually appealing way.


1 Answer 1


Standard notation for the time domain is the linear difference equation. See for example: https://ccrma.stanford.edu/~jos/filters/Difference_Equation_I.html

If you write it in the form

$$ \sum_i a_i y[n-1] = \sum_i b_i x[n-i] $$

it has a one to one correspondence with the z-transform.

$$H(z) = \frac{\sum_i b_i z^{-i}}{\sum_i a_i z^{-i}} $$

That's however NOT the best way to actually implement a time domain IIR filter. That would be a topic for a different question.

  • $\begingroup$ then it's like adding two convolutions. this is the Direct Form. $\endgroup$ Aug 12, 2021 at 21:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.