Digital Filters: FIR vs IIR Comparison

Why Digital Filters?

Digital filters are preferred over analog for almost all modern applications.

Advantage	Description
Precision	Coefficients stored digitally—no component tolerance drift
Flexibility	Reconfigure via software, no soldering
Stability	FIR always stable; IIR poles controllable
Complexity	Arbitrary response shapes (linear phase, adaptive)
Integration	Single chip implements hundreds of filters
No aging	Performance doesn’t degrade over time

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FIR vs IIR: The Fundamental Choice

Aspect	FIR (Finite Impulse Response)	IIR (Infinite Impulse Response)
Impulse response	Finite length $M$	Infinite duration
Difference equation	$y[n] = \sum_{k=0}^{M-1} b_k x[n-k]$	$y[n] = \sum b_k x[n-k] - \sum a_k y[n-k]$
Transfer function	$H(z) = \sum b_k z^{-k}$ (all zeros)	$H(z) = \frac{\sum b_k z^{-k}}{\sum a_k z^{-k}}$ (poles + zeros)
Feedback	None (non-recursive)	Yes (recursive)
Stability	Always stable	Conditionally stable (poles inside unit circle)
Phase response	Exact linear phase (symmetric coeffs)	Nonlinear phase (usually)
Efficiency	More taps for sharp cutoff	Fewer coefficients for same spec
Roundoff noise	Lower	Higher (feedback accumulates error)
Design methods	Window, Parks-McClellan, frequency sampling	Bilinear transform, impulse invariance

When to Choose Which?

• FIR: Linear phase critical (audio, comms), adaptive filters, simple implementation
• IIR: Computational efficiency paramount, minimum phase acceptable, sharp cutoffs needed

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