FIR Filter Group Delay
Group delay and time lag of a linear-phase FIR filter from its tap count and sample rate.
Your recent runs (stored only in your browser)
No calculations yet — results land here so you can compare runs.
The engineering
A linear-phase FIR filter has a phase response that is a straight line versus frequency, which makes its group delay — the negative slope of that phase — a single constant across the whole band. For an N-tap symmetric (or antisymmetric) FIR the delay is exactly (N−1)/2 samples, so every frequency component is shifted in time by the same amount and no waveform shape is distorted. This is the property that makes FIR filters the default choice when phase fidelity matters.
The gotcha is latency: doubling the tap count to sharpen the transition band directly doubles the delay. When N is odd the order (N−1) is even and the delay lands on a whole sample; when N is even you get a half-sample delay, which is fine for filtering but matters if you're aligning channels sample-for-sample. If you need zero net delay offline, run the data forward and backward (filtfilt) to cancel the phase.
Where this math comes from
The constant-group-delay property of symmetric FIR filters fell out of the digital-filter theory developed through the 1960s, once the FFT made frequency-domain design and analysis practical. Engineers realized that if the impulse response is symmetric about its midpoint, the phase is forced to be linear — a purely structural guarantee, not something you tune for.
John Proakis and Dimitris Manolakis codified the four linear-phase FIR types and their (N−1)/2 group delay in Digital Signal Processing, the text that trained a generation of DSP engineers. The result underpins everything from audio crossovers to radar pulse-compression front ends.
- 1965Cooley & TukeyFFT algorithm makes frequency-domain FIR design and analysis practical.
- 1972Parks & McClellanEquiripple linear-phase FIR design formalizes the symmetric-tap approach.
- 1996Proakis & ManolakisDigital Signal Processing pins the four linear-phase types and the (N−1)/2 group delay.
See the full timeline of the math behind every calculator →
Runs entirely in your browser — nothing you enter leaves this page. Your recent runs are stored only on your device.