Real-Time IIR Notch Filter

Overview

Designed and implemented dual second-order IIR notch filters to remove interference tones from audio signals in real-time, achieving 30 Hz bandwidth with -3dB attenuation at target frequencies.

Highlights

• Pole-zero placement method for precise notch design at 774.7 Hz and 2231.2 Hz
• Cascaded second-order IIR sections running at 48 kHz sample rate
• Floating-point implementation with Q15 conversion for audio I/O
• Verified performance using network analyzer and FFT spectrum analysis

Tech Stack

Transfer Function & Pole-Zero Placement Method

The notch filter design employed the pole-zero placement method with the general second-order transfer function: H(z) = [1 - 2cos(Ω₀)z⁻¹ + z⁻²] / [1 - 2r·cos(Ω₀)z⁻¹ + r²z⁻²] where Ω₀ is the digital notch frequency (radians/sample) and r is the pole radius. The digital frequencies were calculated from analog frequencies using Ω = 2πf₀/Fs. Zeros are placed exactly on the unit circle at angles ±Ω₀, creating complete attenuation at the notch frequency. Poles are positioned at the same angles but at radius r < 1, just inside the unit circle. The pole radius r controls the notch bandwidth - values closer to 1 produce narrower notches with sharper roll-off.

Notch Filter Design - 774.7 Hz

The first notch filter was designed using the pole-zero placement method to remove the 774.7 Hz interference tone. Digital Frequency: Ω₁ = 2πf₁/Fs = 2π(774.7)/48000 = 0.1014 rad/sample Pole Radius (30 Hz bandwidth): r = 1 - (BW·π)/Fs = 1 - (30π)/48000 = 0.99804 Filter Coefficients: b = [1.000000, -1.989725, 1.000000] a = [1.000000, -1.985825, 0.996077] The frequency response shows the notch centered precisely at 774.7 Hz with sharp attenuation exceeding -40 dB.

Figure 2: Frequency response of first notch filter at 774.7 Hz

Notch Filter Design - 2231.2 Hz

Figure 3: Frequency response of second notch filter at 2231.2 Hz

The second notch filter targets the 2231.2 Hz interference tone using identical design methodology. Digital Frequency: Ω₂ = 2πf₂/Fs = 2π(2231.2)/48000 = 0.2921 rad/sample Pole Radius (30 Hz bandwidth): r = 0.99804 (same as Filter 1) Filter Coefficients: b = [1.000000, -1.915304, 1.000000] a = [1.000000, -1.911554, 0.996084] The frequency response demonstrates sharp attenuation at 2231.2 Hz while maintaining unity gain in adjacent bands. Both filters use identical pole radius values, ensuring consistent 30 Hz bandwidth despite different center frequencies.

Pole-Zero Plot - 774.7 Hz Filter

The pole-zero plot for the first notch filter visualizes the strategic placement in the z-plane. Zeros are positioned exactly on the unit circle at angles ±0.1014 radians (±5.81°), corresponding to the 774.7 Hz notch frequency. Poles are placed at radius r = 0.99804 at the same angles, positioned just inside the unit circle. This geometric configuration creates complete attenuation at the notch frequency (where zeros are located) while the nearby poles ensure narrow bandwidth and minimal impact on adjacent frequencies.

Figure 5: Pole-zero plot showing precise placement for 774.7 Hz notch filter

Pole-Zero Plot - 2231.2 Hz Filter

Figure 6: Pole-zero plot showing precise placement for 2231.2 Hz notch filter

The second filter's pole-zero plot shows zeros on the unit circle at angles ±0.2921 radians (±16.73°), corresponding to the 2231.2 Hz notch frequency. Poles are positioned at the same angles but at radius r = 0.99804, just inside the unit circle. The larger angle compared to the first filter reflects the higher notch frequency. Both filters use identical pole radius values, ensuring consistent 30 Hz bandwidth despite different center frequencies. This demonstrates the scalability of the pole-zero placement method across the frequency spectrum.

Real-Time Hardware Implementation

The STM32F746-DISCO implementation used 48 kHz sampling rate with dual-channel processing: LEFT channel provided pass-through audio for comparison, RIGHT channel processed audio through both cascaded notch filters. The C code implemented the difference equations using single-precision floating-point arithmetic (float32_t) for filter calculations. Static variables stored filter states (x1, x2, y1, y2) between samples, maintaining causality. Input samples were converted from Q15 fixed-point format to floating-point, processed through both filters in series, then converted back to Q15 for audio output. The LINE IN connector received audio from a laptop or network analyzer.

Cascaded Filter Architecture

The two notch filters were implemented in series (cascaded) rather than as a single fourth-order filter. This approach improves numerical stability and simplifies coefficient calculation. The output of Filter 1 becomes the input to Filter 2, with independent state variables for each filter preventing cross-contamination. Cascading second-order sections is standard DSP practice for implementing higher-order filters, as it avoids the numerical precision issues associated with direct-form higher-order filters. Each filter maintained its own difference equation and state history.

Key Learnings & Results

• Pole-zero placement method provides intuitive control over notch frequency and bandwidth through geometric positioning in the z-plane
• Cascading second-order sections yields better numerical stability than designing higher-order filters directly
• High-precision filter coefficients (format long in MATLAB) are critical for matching theoretical and measured responses
• Real-time audio processing requires careful management of floating-point arithmetic and state variables in C
• Network analyzer provides accurate hardware verification by measuring relative channel responses
• Notch filters effectively remove tonal interference while preserving broadband audio content and maintaining subjective audio quality