FIR Length, Latency, And Ringing
Canonical Definition
FIR length is the number of coefficients or taps in a finite impulse response filter. It sets the available time support and contributes to processing resources, while transition width, ripple, stop-band attenuation, phase distribution, group delay, and implementation latency depend on the design method and processing architecture.
Do Not Collapse These Into One Number
- length: number of coefficients
- order: length minus one
- time support: length divided by sample rate
- DFT bin spacing: sample rate divided by an analysis transform length
- transition width: designed change between passband and stopband
- group delay: phase slope, constant only for appropriate linear-phase designs
- system latency: filter, buffering, transport, converters, and other stages combined
Linear-Phase Delay
For a symmetric linear-phase FIR of order N:
group delay = N / 2 samples
seconds = N / (2 × sample rate)An arbitrary minimum-phase, maximum-phase, or mixed-phase FIR does not have that constant-delay response.
Why fs / N Is Not A Universal Correction Resolution
fs / N can describe a transform-bin spacing or a rough time-frequency scale in a specific context. It does not guarantee that a filter of length N can correct any feature of that width. Achievable response depends on transition bands, ripple, stop-band attenuation, phase target, window or optimization method, and the measured system.
Low-frequency correction often needs longer time support, but a deep spatial cancellation remains a poor inversion target regardless of tap count.
Pre- And Post-Main-Event Energy
A linear-phase design commonly distributes impulse energy on both sides of its aligned main event. A minimum-phase transform moves energy earlier and removes the constant linear-phase delay, but concentrates more energy after the initial event and changes phase. A mixed-phase design chooses a controlled compromise.
Call it pre-ringing only after the time reference and main event are defined. Inspect the actual impulse at realistic level rather than judging a filter label.
Practical Design Record
| Item | Record |
|---|---|
| Magnitude target | passbands, transitions, ripple, stop-band attenuation, correction range |
| Phase target | linear, minimum, mixed, or custom |
| Resources | sample rate, taps, MIPS, memory, block or partition size |
| Timing | filter delay, processing latency, transport, converter, total path |
| Verification | impulse, group delay, peak gain, output clipping, channel alignment, nearby positions |