Audio Biquad EQ Calculator
Compute normalized biquad filter coefficients (b0, b1, b2, a1, a2) for peaking, low-shelf, and high-shelf EQ at any sample rate. Shows the frequency response curve.
How to Use
- Pick filter type: peaking (bell EQ), low-shelf, high-shelf, low-pass, high-pass.
- Enter sample rate, center/cutoff frequency, gain in dB, and Q.
- Copy the normalized coefficients (b0, b1, b2, a1, a2) into your DSP code.
- The response plot shows magnitude in dB vs. frequency.
Coefficients
Formulas (Robert Bristow-Johnson cookbook)
About Biquad Filters
The biquad is the workhorse of digital audio processing. Robert Bristow-Johnson's "Audio EQ Cookbook" (1999) provided the standard formulas that nearly every EQ implementation uses — including Web Audio API, iOS Core Audio, and most DAW plugins. Five coefficients, one structure, decades of proven reliability.
Biquads are stable when poles are inside the unit circle. For Q values above ~20 at low frequencies, numerical precision matters — use double-precision floats or the transposed direct-form-II structure to avoid instability in 32-bit single precision. For most audio (20Hz-20kHz at 48kHz), direct-form-I in single precision is fine.
History of Digital Biquad Filters
The biquad (biquadratic) filter structure emerged from 1960s-70s digital signal processing research, when researchers at Bell Labs and elsewhere discovered that higher-order IIR filters could be factored into cascades of 2nd-order sections for better numerical stability. A 16th-order elliptic filter as a single monolithic difference equation is essentially unusable in fixed-point arithmetic; cascaded as 8 biquads, the same filter is robust and easily tunable.
Robert Bristow-Johnson's "Audio EQ Cookbook" (1999, circulated as a text file before any formal publication) gave the audio industry its lingua franca for digital EQ. Every plug-in developer, every DAW, every WebAudio implementation traces its biquad formulas back to this document. Before the cookbook, each company had its own slightly-different coefficient formulas, leading to subtle sonic differences when porting effects between platforms.
Today biquads run in the trillions of instances daily — every smartphone headphone EQ, every car audio tone control, every live-sound parametric rack unit is computing the same difference equation first formalized 60 years ago. WebAudio's BiquadFilterNode is a direct implementation of the cookbook; ARM's CMSIS-DSP and TI's C6000 DSP libraries both include optimized biquad kernels with bit-identical output.
About This Calculator
Pick filter type (peaking, low/high shelf, low/high pass, band-pass, notch), enter sample rate, center/cutoff frequency, gain in dB (for peaking and shelf), and Q. The tool applies Bristow-Johnson's cookbook formulas to compute b0, b1, b2, a1, a2 normalized with a0 = 1 — the exact same coefficient format used by WebAudio, iOS Core Audio, and most DSP libraries.
Copy the coefficients into your DSP code and implement with the direct-form-I difference equation: y[n] = b0·x[n] + b1·x[n-1] + b2·x[n-2] − a1·y[n-1] − a2·y[n-2]. For Q > 20 at low frequencies on embedded fixed-point DSPs, use transposed direct-form-II for better numerical stability. Everything runs client-side; no values leave your browser.
Frequently Asked Questions
What is a biquad filter?
A second-order IIR (infinite impulse response) filter, implemented as: y[n] = b0×x[n] + b1×x[n−1] + b2×x[n−2] − a1×y[n−1] − a2×y[n−2]. Five coefficients define its behavior. Audio DSP uses biquads as the basic building block for all EQ, filter, and shelf operations.
What is Q?
Q controls the bandwidth of peaking filters. Higher Q = narrower bump, lower Q = broader. For shelves, Q controls the slope transition sharpness. Common values: 0.707 (−3dB slope for shelves), 1.4 (musical bell), 5+ (notch/surgical).
What sample rate should I use?
Match your audio pipeline. CD/most consumer: 44.1 kHz. Pro: 48 kHz. Hi-res: 96 kHz or 192 kHz. Coefficients scale with sample rate — change the rate and recompute. Wrong sample rate at runtime produces detuned EQ.
What's the difference between peaking and shelf?
Peaking EQ boosts or cuts a narrow band around a center frequency — like a bell curve. Shelving boosts or cuts everything above (high-shelf) or below (low-shelf) a cutoff. Bass controls are usually low-shelf; treble is high-shelf. Parametric EQ mixes peaks in the mids.
Why are the coefficients normalized by a0?
The biquad difference equation has 6 coefficients; normalizing by a0 (dividing everything by it) reduces to 5 and simplifies the runtime math. All major DSP libraries (WebAudio, DSP textbooks, most embedded DSPs) use normalized forms.
Common Use Cases
Parametric EQ
Each band: one peaking biquad. Full 10-band EQ = 10 biquads in series. Tune center frequencies, Qs, and gains for any EQ curve.
Crossover Network
Low-pass + high-pass biquads create 2-way speaker crossover at a chosen frequency (Linkwitz-Riley: two cascaded Butterworths).
Tone Controls
Bass = low-shelf at ~100Hz. Treble = high-shelf at ~10kHz. Both with Q ≈ 0.707 for smooth response.
Notch / Anti-Hum
Very high Q peaking (Q > 10) with negative gain to remove 60Hz line hum or narrow-band noise.
Anti-Alias / DC Block
Low-pass biquad at fs/2 × 0.4 to prevent aliasing before downsampling; high-pass at ~20Hz to remove DC and sub-sonic content.
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