# Is a LTI filter completely characterized by its frequency response

I know that we use the frequency response to determine the properties gain and phase delay of a filter. However, I was wondering if this is enough for a complet characterization of the the filter for e.g. in terms of noise attenuation.

For example: I have a low pass filter. Am I guaranteed to attenuate e.g. colored noise?

Cheers, rot8

• but i would say that initial conditions are not properties of the filter but are property of the initial states of the filter. in general, i would say that the frequency response (you need both gain and phase) completely characterize the input/output relationship of the filter, just as the impulse response does in the time domain. But if there is pole/zero cancellation there could be internal states that don't show up in the output. – robert bristow-johnson Jun 21 '19 at 19:41
• @StanleyPawlukiewicz we had a similar discussion with MattL about initial conditions and LCCDE and LTI ness, however once the term LTI system is triggered then it shall also imply initial rest and therefore its frequency response is uniquely specifying its impulse response and that latter completely describes the (observable part of the) LTI system... – Fat32 Jun 21 '19 at 21:18
• @StanleyPawlukiewicz yes overlap save (block based) processing requires initial conditions (or previous block outputs) but it's not about characterizing the system...anyway not a big deal... – Fat32 Jun 21 '19 at 21:51
• @Fat32 what about if the filter is causal, anti causal, or no causal – user28715 Jun 21 '19 at 23:47
• @StanleyPawlukiewicz unlike Z-transform which requires a region of convergence to distinguish between causal, anti-causal seqeunces, Fourier transform alone can distinguish between causal or anti causal sequences. That's why it's said to uniquely representing $h[n]$. In particular if $h[n]$ is causal with $H(w)$ then $h[-n]$ will be anticausal with $H(-w)$ and you will have two different Fourier trasforms... – Fat32 Jun 22 '19 at 9:12