# How to calculate the total time delay of a first order linear filter on a signal?

I have a non-periodic signal. I apply a first order linear IIR filter on it. The filter will introduce phase shift to the signal that will appear as a total time delay. Is there a formula I can use to calculate the net time delay on the signal? such that the maximum correlation is achieved when the signal is shifted back using this time delay.

• Often for low-pass filters, we use the group delay and phase delay at 0 Hz, DC, to characterize the overall delay of the filter even though it is not correct for higher frequencies. Jul 18, 2021 at 4:26

## 1 Answer

The filter will introduce phase shift to the signal that will appear as a total time delay.

Not really. For an IIR filter the phase shift will be a function of frequency and it will (in general) not correspond to a single delay. It's a "frequency dependent delay"

such that the maximum correlation is achieved when the signal is shifted back using this time delay.

That's an interesting definition. Many "standard" IIR filters are minimum phase. In this case it's probably best to assume a net delay of zero to maximize correlation. However, this will depend a lot on the specific filter and the spectrum of the input signal. The spectrum of the input will "weigh" different frequencies differently so you get a different mix of the frequency dependent delay.

• Minimum phase causal filters will still have a finite delay so assuming a net delay of zero will likely not have the maximum correlation. The OP’s definition makes sense to me for all cases when the group delay variation is small. For all cases delay is the negative time derivative of phase so using correlation when there is a lot of variation in delay across frequency is problematic. Jul 16, 2021 at 13:46