biquad v0.1.0

Q15 direct-form I biquad section.

Application scenarios

• Audio tone shaping and anti-alias filters.
• Sensor smoothing where a first-order filter is not enough.
• A fixed-point DSP chain on an MCU without floating point.

Contract

void biquad_init(biquad_t *filter, int32_t b0, int32_t b1, int32_t b2,
                 int32_t a1, int32_t a2, uint8_t shift);
int32_t biquad_step(biquad_t *filter, int32_t sample);
void biquad_reset(biquad_t *filter);

• Coefficients are Q15; shift divides the accumulator by 2^shift.
• Direct form I: y = (b0*x + b1*x1 + b2*x2 - a1*y1 - a2*y2) / 2^shift.
• a1/a2 are stored with their normal sign (the equation subtracts them).
• No allocation; state is about 32 bytes.

Parameters

• Design coefficients with a tool such as SciPy or CMSIS-DSP, then scale to Q15.
• shift keeps the accumulator in range; 15 is common when coefficients are Q15.
• Cascade sections for higher-order filters.

Memory and timing

• 36 bytes; O(1), four 64-bit products per sample.

Integration

biquad_t lowpass;
biquad_init(&lowpass, 10923, 21845, 10923, -10923, 0, 15);
int32_t filtered = biquad_step(&lowpass, sample);

When not to use

• A single-pole response is enough; use ema.
• Coefficient design at runtime; use a floating-point design step offline.

Verification

make test covers unity gain, a three-tap average, impulse response, and reset.

Behavior reference: CMSIS-DSP arm_biquad_cascade_df1_q15 (Apache-2.0); this implementation is original.

int32_t b0;
int32_t b1;
int32_t b2;
int32_t a1;
int32_t a2;
int32_t x1;
int32_t x2;
int32_t y1;
int32_t y2;
uint8_t shift;