# FIR filter delay and window length

How long does it take for FIR filter to produce first output sample for given window / kernel / impulse (not sure what is best name) length?

Does it differ between minimum phase and linear phase variants? Lets say we have two FIR low pass filters,one minimum phase,other linear phase,both have 100 sample long impulse response/window length,how long will it take to obtain first output sample?

My guess is the minimum phase needs 100 sample long delay while the linear phase 50 samples.

## 1 Answer

For causal FIR linear phase filters the delay you ask for can simply be computed from the group delay and is given by (for an odd length symmetric FIR filter of length $L=2M+1$) $$\tau = - \frac{d\phi(\omega)}{d\omega} = - \frac{d-M\omega}{d\omega} = M$$

So it will shift every input component by $M$ samples at the output.

NOTE: you can always produce your output sample $y$ as soon as the first input sample $x$ arrives. The delay is related with the information processing and not the sample processing;i.e., your initial $M-1$ output samples will be called as the filter transient response, before the steady output is computed.

• Thank you very much for your help but my math skill is too low to get anything out of these equations.Could you please translate your answer into just words? – Sweeper Aug 25 '18 at 22:57
• if your FIR filter is of L=101 samples long, then its group delay will be M=50 samples... – Fat32 Aug 25 '18 at 23:00
• linear phase causal FIR filters cannot be minimum phase, because it has to maintain symmetry of its impulse response which is in the middle sample, but if you allow nonlinear phase (non-symmetric) FIR filters, then they can be mimimum phase. So there's a mistake in my generalization of minimum-phase to IIR filters. FIR can also be minimum phase (but not linear phase at the same time) – Fat32 Aug 25 '18 at 23:27
• DSP review is the book understanding digital signal processing from Richard Lyons. Very readable DSP book. Please find a copy of it and read its contents. – Fat32 Aug 26 '18 at 0:59
• If an IIR is stable you can in many cases, can truncate the impulse response and think of that as a FIR filter but the poles of an IIR filter are subject to round off so it can become unstable. It’s also easy to have a simple IIR filter have a very long impulse response so a FIR approximation will not be efficient and hence impractical. Treat IIR as IIR and FIR as FIR and avoid problems – user28715 Aug 26 '18 at 7:34