Special IIR Filters, Structures, and Finite Word-Length

Special IIR Filter Types (Proakis 5.4)

Digital Resonator

Narrowband bandpass with pole near unit circle at $z = re^{\pm j\omega_0}$: \(H(z) = \frac{1 - r}{1 - 2r\cos\omega_0 z^{-1} + r^2 z^{-2}}\)

• $r \approx 1$: Very sharp resonance
• Used for: Tone generation, frequency detection, equalizers

Notch Filter

Zero on unit circle at $z = e^{\pm j\omega_0}$ to reject specific frequency: \(H(z) = \frac{1 - 2\cos\omega_0 z^{-1} + z^{-2}}{1 - 2r\cos\omega_0 z^{-1} + r^2 z^{-2}}\)

• $r < 1$: Controls notch bandwidth
• Used for: 60 Hz hum removal, interference rejection

Comb Filter

Periodic notches/peaks at harmonics: \(H(z) = \frac{1 - z^{-N}}{1 - r z^{-N}} \quad \text{(comb with peaks)}\)

• Periodic in frequency with spacing $2\pi/N$
• Used for: Harmonic rejection, pitch detection, reverberation

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Group Delay & Phase Considerations

Filter Type	Phase Response	Group Delay $\tau_g(\omega) = -\frac{d\phi}{d\omega}$
FIR (symmetric)	Linear: $\phi(\omega) = -\alpha\omega$	Constant = $(M-1)/2$
IIR	Nonlinear	Frequency-dependent

Why group delay matters:

• Constant group delay = all frequencies delayed equally = no waveform distortion
• IIR filters cause phase distortion (different frequencies delayed differently)
• For audio/comms: Use FIR or zero-phase filtering (filtfilt)

Zero-Phase Filtering

% MATLAB: filtfilt applies filter forward then backward
y = filtfilt(b, a, x);  % Zero phase, magnitude squared

• Magnitude response: $

  H(\omega)

  ^2$

• Phase response: 0
• Non-causal (offline only)

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Filter Realization Structures

Direct Form I (from difference equation)

x[n] ──►[b0]──►[+]──►[b1]──►[+]──►[b2]──►[+]──► y[n]
         │     ▲      │     ▲      │     ▲
         │    z⁻¹     │    z⁻¹     │    z⁻¹
         ▼     │      ▼     │      ▼     │
        [-a1]──┘     [-a2]──┘     [-a3]──┘

\[y[n] = b_0 x[n] + b_1 x[n-1] + \dots - a_1 y[n-1] - a_2 y[n-2] - \dots\]

Pros: Direct from coefficients. Cons: High sensitivity for high order; $2N$ delays.

Direct Form II (Canonical)

x[n] ──►[+]──►[z⁻¹]──►[+]──►[z⁻¹]──►[+]──► ...
         ▲     │      ▲     │      ▲
         │    [b0]     │    [b1]   │
         │      │      │      │      │
        [-a1]  [b1]   [-a2]  [b2]   ...

Pros: Minimum delays ($N$). Cons: Large internal signal swings (overflow risk).

Cascade Form (Second-Order Sections - SOS)

H(z) = H_1(z) · H_2(z) · ... · H_K(z)

Each $H_k(z)$ is a biquad (2nd order): \(H_k(z) = \frac{b_{0k} + b_{1k}z^{-1} + b_{2k}z^{-2}}{1 + a_{1k}z^{-1} + a_{2k}z^{-2}}\)

Pros: Best numerical stability; standard for IIR implementation. Always use SOS for IIR!

Parallel Form

H(z) = H_1(z) + H_2(z) + ... + C

Partial fraction expansion. Less common.

FIR Structures

Structure	Description
Direct Form (Transversal)	$M$ taps, $M-1$ delays, $M$ multipliers
Cascade	Factor polynomial into 2nd-order sections
Linear Phase (Symmetric)	Fold symmetric taps → $\lceil M/2 \rceil$ multipliers

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Finite Word-Length Effects (Proakis Ch. 9)

Effect	Description	Mitigation
Coefficient quantization	Poles/zeros shift from ideal locations	Use SOS, increase precision
Roundoff noise	Accumulation of quantization errors	Scaling, noise shaping, higher precision
Limit cycles	Periodic oscillations in IIR due to rounding	Dithering, saturation arithmetic
Overflow	Sum exceeds representable range	Scaling, saturation, guard bits

Coefficient Quantization Sensitivity

• Direct Form I/II: High sensitivity for high-order filters (poles move significantly)
• SOS (Cascade): Low sensitivity — each biquad has low-order denominator

• Pole locations: Sensitivity $\propto 1/

  z - p_k

  $ near poles

Scaling Rules (for fixed-point)

1. L2-norm scaling: $\sum

   h[n]

   ^2 \le 1$ (prevents overflow for any input)

2. L1-norm scaling: $\sum

   h[n]

   \le 1$ (strict, for peak inputs)

3. L∞-norm scaling: $\max

   H(\omega)

   \le 1$ (frequency domain)

Always use SOS with proper scaling for fixed-point IIR implementation.

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Frequency Sampling Method (FIR)

Alternative to window method:

1. Sample desired $H_d(\omega)$ at $N$ equally spaced frequencies
2. Compute $h[n]$ via IDFT of samples
3. Adjust transition band samples for better control

Advantage: Exact match at sampled frequencies Disadvantage: Large ripples between samples; rarely used vs. Parks-McClellan

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Filter Design Specifications Template

Passband edge:      ωp = 0.2π rad/sample
Stopband edge:      ωs = 0.3π rad/sample
Passband ripple:    Ap = 1 dB
Stopband attenuation: As = 50 dB
Sampling rate:      Fs = 8000 Hz

FIR Design Checklist

• Choose window (Kaiser for specs-driven)
• Compute $M$ from transition width
• Ensure linear phase (odd $M$ for Type I)
• Verify frequency response meets specs

IIR Design Checklist

• Select prototype (Butterworth/Chebyshev/Elliptic)
• Pre-warp critical frequencies
• Compute order $N$
• Design analog $H(s)$
• Apply bilinear transform
• Convert to SOS for implementation
• Verify stability (all poles inside unit circle)
• Check frequency response

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MATLAB / Python Design Commands

% FIR via Window
N = 50; fc = 0.25;
b = fir1(N, fc, hamming(N+1));

% FIR via Parks-McClellan (optimal equiripple)
N = 50; f = [0 0.2 0.3 1]; a = [1 1 0 0]; w = [1 10];
b = firpm(N, f, a, w);

% IIR Butterworth
[N, Wn] = buttord(0.2, 0.3, 1, 50);
[b, a] = butter(N, Wn);
sos = tf2sos(b, a);  % Convert to SOS!

% IIR Chebyshev I
[N, Wn] = cheb1ord(0.2, 0.3, 1, 50);
[b, a] = cheby1(N, 1, Wn);
sos = tf2sos(b, a);

% IIR Elliptic
[N, Wn] = ellipord(0.2, 0.3, 1, 50);
[b, a] = ellip(N, 1, 50, Wn);
sos = tf2sos(b, a);

# Python (SciPy)
from scipy.signal import firwin, firpm, butter, cheby1, ellip, tf2sos

# FIR Window
b = firwin(51, 0.25, window='hamming')

# FIR Parks-McClellan
b = firpm(50, [0, 0.2, 0.3, 1], [1, 1, 0, 0], weight=[1, 10])

# IIR Butterworth
b, a = butter(4, 0.25, output='ba')
sos = tf2sos(b, a)

# IIR Chebyshev I
b, a = cheby1(4, 1, 0.25, output='ba')
sos = tf2sos(b, a)

# IIR Elliptic
b, a = ellip(4, 1, 50, 0.25, output='ba')
sos = tf2sos(b, a)

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