Finite Impulse Response filters — the difference equation, the impulse response, unconditional stability, and the celebrated linear-phase property that makes FIR indispensable.
An FIR filter outputs a weighted sum of the current and M past input samples. No feedback — the output never re-enters the calculation. This one structural choice delivers all of FIR's best properties.
For an FIR filter, the impulse response h[n] equals the coefficient sequence b_k — nothing more. The output is just the convolution of the input with this finite sequence.
No poles → no instability. BIBO stability requires all poles inside the unit circle. FIR filters have only zeros — their poles are trivially at the origin. Any coefficient values you choose produce a stable filter.
The frequency response is the DTFT of h[n] evaluated on the unit circle — a polynomial in e^{−jω}. Longer filters (more terms) → sharper transitions, flatter passbands.
Symmetric coefficients (h[n] = h[M−n]) → linear phase: ∠H(e^{jω}) = −(M/2)ω. Every frequency is delayed by the same M/2 samples — waveform shape is preserved exactly.
- Symmetric h[n] → zero phase distortion (Type I & II)
- Antisymmetric h[n] → 90° shift + linear phase (Type III & IV)
- Symmetry also halves unique multiplications needed
- IIR filters cannot achieve exact linear phase
- Audio EQ & crossovers — linear phase preserves music fidelity
- Pulse shaping (RRC) in 4G/5G modems & Wi-Fi
- Anti-aliasing & anti-imaging in sample-rate converters
- ECG/EEG processing — waveform shape must not be distorted
- Very long filters: overlap-add FFT method for efficiency
- FIR = weighted sum of M+1 input samples, zero feedback
- Impulse response h[n] = coefficient sequence b_k
- No poles → unconditionally BIBO stable for any coefficients
- Symmetric h[n] → linear phase → no waveform distortion
- Cost: M+1 MACs/sample; long filters use overlap-add FFT
- Used everywhere: audio, comms, multirate, biomedical