The FIR Filter Coefficient Calculator helps engineers, DSP students, and audio developers design finite impulse response (FIR) filters by automatically computing the filter coefficients (also called taps). These coefficients define how input signals are weighted and summed to produce the desired filtering effect. The tool saves time and minimizes errors compared to manual calculation.
This tool is part of the Digital Signal Processing (DSP) Calculator category.
Formula of Fir Filter Coefficient Calculator
1. FIR Filter Output
A basic FIR filter applies a weighted sum of past input samples:
y[n] = Σ (from k = 0 to N−1) [ h[k] × x[n−k] ]
Where:
- y[n] = output at time n
- x[n−k] = input sample at time n−k
- h[k] = filter coefficient at tap k
- N = number of taps (equal to filter order + 1)
2. Coefficient Calculation (Window Method)
For a simple low-pass FIR filter, the ideal impulse response is:
h[k] = ( sin(2πf_c (k − M)) ) / (π (k − M))
Where:
- f_c = normalized cutoff frequency (desired cutoff frequency divided by sampling frequency)
- k = index (0 to N−1)
- M = (N−1)/2 (the center tap)
Special case at k = M:
h[M] = 2 × f_c
3. Apply a Window Function
To reduce ripple and improve performance, the ideal coefficients are multiplied by a window:
h_windowed[k] = h[k] × w[k]
Where:
- w[k] = window function value at tap k
Common window functions include Hamming, Hanning, and Blackman.
Common Window Functions Table
| Window Name | Main Lobe Width | Side Lobe Attenuation | Typical Use |
|---|---|---|---|
| Rectangular | Narrowest | Poor attenuation | Basic use, high ripple |
| Hamming | Wider | Better attenuation | General DSP filters |
| Hanning | Similar to Hamming | Slightly lower attenuation | Smooth signal edges |
| Blackman | Widest | High attenuation | High precision filters |
Example of Fir Filter Coefficient Calculator
Design a low-pass FIR filter:
Specs:
- Cutoff frequency: 1 kHz
- Sampling frequency: 8 kHz
- Filter order: 20 (N = 21 taps)
- Window: Hamming
Steps:
- Compute normalized cutoff frequency:
f_c = 1 kHz / 8 kHz = 0.125 - Calculate M:
M = (21 − 1)/2 = 10 - For each k (0 to 20), compute:
h[k] = sin(2π × 0.125 × (k − 10)) / (π (k − 10)) If k = M (10):
h[10] = 2 × 0.125 = 0.25 - Multiply by Hamming window w[k].
The calculator automates all these steps instantly.
Most Common FAQs
Higher order (more taps) means better frequency separation but more processing time. Typical orders range from 20 to 100 for audio or general DSP.
Ideal filters have infinite response. Windows limit this but control unwanted ripples and leakage, improving practical performance.
Yes. FIR filters are inherently stable and predictable, making them suitable for real-time applications, although high orders may increase computation time.