Group Delay Calculator
Find how long a filter delays a signal, either from its FIR tap count or from measured phase at two nearby frequencies.
⏱️ What is Group Delay?
Group delay is the time delay a filter imposes on the amplitude envelope of a signal, defined mathematically as the negative derivative of phase response with respect to angular frequency, tau_g = -dphi/domega. In plain terms, it answers the question "how long does this filter take to respond?" at any given frequency.
Audio engineers care about group delay when designing crossovers and room-correction filters, since uneven group delay across frequencies smears the timing of transients even when the frequency magnitude response looks perfectly flat. Communications engineers track group delay in channel equalizers, since group delay variation (delay distortion) degrades data integrity in high-speed digital links. Control system designers watch group delay in feedback filters, since added latency directly reduces achievable control-loop bandwidth.
A common misconception is that group delay is the same everywhere for every filter. It is not. Linear-phase FIR filters are special: their group delay is exactly constant at every frequency, which is precisely why they are chosen whenever preserving a signal's waveform shape matters. Most IIR filters (and any non-symmetric FIR filter) have group delay that varies with frequency, distorting the relative timing of different frequency components even when they pass through with equal amplitude.
This calculator finds group delay two ways: directly from an FIR filter's tap count (the closed-form, exact case), or from two phase measurements at nearby frequencies using a finite-difference approximation that works for any filter type.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Audio FIR Low-Pass Filter
Linear-phase FIR, N = 101 taps, sampled at 8,000 Hz
Example 2 — High-Order Audio Crossover Filter
Linear-phase FIR, N = 257 taps, sampled at 44,100 Hz
Example 3 — Group Delay From Measured Phase
Phase = 0° at 1,000 Hz, phase = -18° at 1,010 Hz, sampled at 48,000 Hz
❓ Frequently Asked Questions
🔗 Related Calculators
What is group delay in signal processing?
Group delay is the time delay experienced by the amplitude envelope of a signal as it passes through a filter, defined as the negative derivative of phase with respect to angular frequency: tau_g = -dphi/domega. It measures how long a filter takes to respond at each frequency.
How do you calculate group delay for an FIR filter?
For a linear-phase FIR filter with N taps, group delay is constant at every frequency and equals (N - 1) / 2 samples. Divide by the sampling rate to convert to seconds: tau_g = (N - 1) / (2 x fs). A 101-tap filter at 8,000 Hz has a group delay of 50 samples, or 6.25 ms.
Why is FIR group delay always constant?
A symmetric (linear-phase) FIR filter's phase response is exactly a straight line through the origin, phi(omega) = -omega x (N-1)/2. The derivative of a straight line is constant, so the group delay (the negative of that derivative) is the same number at every frequency, with no phase distortion.
How do you find group delay from measured phase data?
Measure or compute phase at two closely spaced frequencies f1 and f2, then use the finite-difference approximation: tau_g = -(phi2 - phi1) / (2*pi*(f2 - f1)), with phase in radians. Keeping f2 close to f1 keeps the approximation accurate, since it estimates the true derivative -dphi/domega.
Does group delay change with sampling rate?
Group delay measured in samples depends only on the filter's tap count, not the sampling rate. But group delay measured in seconds is inversely proportional to sampling rate: the same filter run at a higher sampling rate produces a shorter delay in real time, because each sample represents less time.
What is the difference between group delay and phase delay?
Phase delay (-phi/omega) is the delay experienced by a single sinusoidal frequency component. Group delay (-dphi/domega) is the delay of the envelope formed by a narrow band of frequencies. The two are equal only when phase is exactly linear in frequency, which is exactly the case for a symmetric FIR filter.
Why do IIR filters have non-constant group delay?
IIR filters use feedback (poles), which generally produces a curved, non-linear phase response. Because group delay is the derivative of phase, a curved phase response gives a group delay that varies with frequency, smearing different frequency components by different amounts in time.
What happens if group delay is too large for a real-time application?
Excessive group delay adds latency between input and output, which matters for live audio monitoring, control-loop feedback, and communication systems with tight round-trip-time budgets. Reducing FIR tap count lowers delay but also reduces filter selectivity, so real designs trade the two against each other.
Can group delay be negative?
In a physically causal filter, group delay is generally non-negative on average, but locally it can dip negative in narrow frequency bands for certain all-pass or IIR designs (this does not violate causality; it reflects energy already present in the filter's transient response). This calculator's FIR mode always returns a non-negative value by construction.
How many samples of latency does a typical audio FIR filter add?
A common audio FIR crossover or EQ filter with 256 to 1,024 taps at a 48,000 Hz sample rate adds roughly 2.7 to 10.7 ms of group delay (half the tap count divided by the sampling rate), which is usually inaudible as a standalone delay but can matter for tight multi-speaker phase alignment.