DSP 101
M8 · L1
Module 8 — IIR Filter Design
What is an IIR Filter?

Infinite Impulse Response filters — the recursive difference equation, poles and zeros, the stability condition, and the dramatic efficiency advantage that makes IIR indispensable.

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DSP 101
M8 · L1
The Core Idea
Inputs and Past Outputs

An IIR filter uses both past inputs and past outputs. This feedback loop creates resonance — a single impulse rings forever in theory, hence "infinite impulse response."

IIR Difference Equation
y[n] = \sum_{k=0}^{M} b_k\, x[n-k] - \sum_{k=1}^{N} a_k\, y[n-k]
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DSP 101
M8 · L1
Transfer Function
Poles & Zeros

The Z-transform gives a rational transfer function H(z) = B(z)/A(z). The numerator creates zeros; the denominator creates poles. Poles near the unit circle produce sharp resonances.

Zeros
from B(z) numerator
Poles
from A(z) denominator
∞
impulse response length
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DSP 101
M8 · L1
The Critical Constraint
Poles Inside the Unit Circle

BIBO stability requires all poles strictly inside the unit circle (|z_pole| < 1). A pole on or outside the circle → output grows without bound. This must be verified explicitly.

Unlike FIR
FIR filters have no poles and are always stable. IIR filters require careful design — and careful implementation in fixed-point arithmetic where quantization can nudge poles outward.
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DSP 101
M8 · L1
The Big Advantage
Dramatically Fewer Coefficients

An IIR filter achieves the same sharpness as FIR with far fewer coefficients. A 6th-order elliptic IIR can match a 1000-tap FIR — making IIR the only viable choice on power-constrained hardware.

  • Sharp bandpass: FIR needs ~255 taps; IIR needs ~10 coefficients
  • Elliptic notch: FIR ~1023 taps; IIR ~6 coefficients (170× fewer)
  • Lower memory, lower power, lower latency
  • Ideal for hearing aids, modems, real-time control loops
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DSP 101
M8 · L1
The Trade-off
Nonlinear Phase

IIR poles introduce frequency-dependent group delay — different frequencies are delayed by different amounts, distorting waveform shape. FIR can achieve exact linear phase; IIR cannot.

  • Matters for: ECG/EEG, radar pulses, digital data eye diagrams
  • Doesn't matter for: audio tone controls, channel filtering, noise rejection
  • Offline fix: zero-phase filtering (forward + backward pass)
  • Real-time fix: use FIR instead, or accept the distortion
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DSP 101
M8 · L1
Where IIR Filters Live
IIR in the Real World
  • Audio EQ & tone controls — biquad sections, adjustable in real time
  • Channel filtering in GSM & DECT mobile handsets
  • Digital PID controllers & phase-locked loop filters
  • Anti-aliasing & DC blocking for ADC front-ends
  • Noise reduction in hearing aids & voice interfaces
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DSP 101
Knowledge Check

Check whatstuck

Four questions on IIR filters — feedback, poles, stability and the efficiency trade.

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DSP 101
M8 · L1
Key Takeaways
What You Learned
  • IIR = weighted sum of past inputs AND past outputs (feedback)
  • Transfer function H(z) = B(z)/A(z) is rational with poles and zeros
  • Stability ⟺ all poles inside unit circle — must verify explicitly
  • Far fewer coefficients than FIR for same sharpness (up to 170× fewer)
  • Trade-off: nonlinear phase distorts waveform shape
  • Preferred for audio, channel filtering, control, low-power DSP
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