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
A tiny hearing aid with a clear tube and earpiece resting on an open palm.
A hearing aid has almost no power or memory to spare. IIR filters match FIR sharpness with far fewer coefficients, which is why they fit.

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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