Playground · research instrument
DSP · AudioSet the microphone spacing, the element count, the steer angle and the frequency. The polar pattern, the half-power beamwidth, the directivity index and the spatial-aliasing ceiling are solved exactly, from the same module the page's unit tests drive.
Independent research instrument — not claimed as MakerPortal shipped product code. Methods, equations, assumptions, and limitations are disclosed so you can inspect what the page does and does not establish.
Array geometry
Speed of sound fixed at 343 m/s (dry air, 20 °C). It scales every frequency on this page linearly — at 0 °C the aliasing ceiling falls about 3%.
Directivity pattern · full plane
No grating lobe
2 elements at 50 mm, steered 0°, at 3430 Hz.
Solved for this geometry
The number that decides the design
50 Hz → 16 kHz, log scale
The curve is the reason small arrays disappoint. Directivity index is set by the aperture measured in wavelengths, so it falls towards 0 dB at low frequency for any element count, and past the aliasing ceiling it collapses again as the grating lobe re-admits energy from the direction you were rejecting.
N omnidirectional elements on the x axis at , a far-field plane wave from angle measured from broadside. The only thing an element at sees of is the direction cosine , and every consequence below follows from that one fact.
01 · The beam
The page evaluates this sum literally, one phasor per microphone. Its closed form is the periodic sinc with ; the unit test asserts the two agree to across 6,000 geometries, so the implementation is pinned to an expression written independently of it.
02 · Directivity index
Array gain against a spherically isotropic noise field — the honest stand-in for a diffuse room. With uniform weights the numerator is 1 and the denominator is a real double sum. At broadside the coherence matrix is the identity and exactly, which is the anchor the defaults sit on.
03 · Half-power beamwidth
Found on the pattern, not quoted from . That approximation is asymptotic in N and ignores the broadening when the beam is steered — at 40° off broadside the real lobe is 30% fatter than the formula says. When no −3 dB edge exists inside the readout says so instead of extrapolating.
04 · Spatial aliasing
A grating lobe is a second direction with the same full array gain — the spatial twin of aliasing in a sampled signal. The worst case over every steer angle is endfire, which gives the familiar ceiling. Steer only broadside and you are allowed an octave more spacing; the calculator prints both so that octave is a decision rather than an accident.
A beamformer applies a delay it already knows. A direction finder measures the delay it does not: across an aperture , then inverts to . The readouts above give you the whole budget — the widest arrival-time difference the aperture can produce is printed as max TDOA, and on a 50 mm two-microphone baseline it is 145.8 µs.
That number is what makes sample-rate arithmetic concrete. One sample period at 48 kHz is 20.833 µs, which is 14% of the whole usable range. Differentiating gives , so a single sample of timing error is worth 8.2° at broadside, 16.4° at 60°, and 47.2° at 80° on that baseline. The angular cost of pointing near the array axis is not a subtlety; it is the dominant error term.
Sub-sample interpolation — GCC-PHAT with a parabolic peak fit is the usual choice — buys back most of that, and it is the reason a practical two-microphone direction finder works at all. It does not touch the term, and it cannot touch the front-back ambiguity.
Named failure mode · front-back ambiguity
, so a talker at 30° and a talker at 150° produce identical delays at every element. The polar plot draws both lobes for exactly this reason. More microphones on the same line narrow both and remove neither. This is the single most common surprise when a line array meets a real room.
Named failure mode · the near field
Every curve here assumes plane waves. A source closer than a few aperture lengths presents a curved wavefront, the delays stop being linear in , and the steering vector is wrong in a way that broadens the main lobe and fills the nulls. On a 50 mm baseline that boundary is a few hundred millimetres — close enough to matter for a handheld device, far enough not to for a room.
Measured, not modelled
For what a seven-capsule 44 mm ring array actually delivers on real speech — SRP-PHAT direction tracking, a superdirective beam, and the octave-band rejection numbers off the recordings — read the UMA-8 field note. This page is the prediction; that one is the instrument.
Array + capture stack · 38 picks
$135.12BookDerives array factor AF=∑I_n exp(jn(kd cosθ+β)) and microstrip patch cavity model with fringing fields this sandbox implements for beam tilt and radiation efficiency.
$159.00HeadphonesReference monitoring headphones used for akous's binaural audio testing.
$13.90KitSOL cal standards + SMA jumper for NanoVNA H/H4 — SI Lab and RF Bench SOLT demos.
$206.99DiagnosticPortable spectrum analyzer + attenuator kit — EMI and spur checks next to the RF Bench.
$229.00Audio interfaceAudio interface used building Biquadia — 8-preamp USB I/O for real-time DSP testing.
$154.99HeadphonesClosed-back diffuse-field studio reference headphones (5 Hz–35 kHz). Used in acoustic labs and voice studios for speech monitoring, artifact isolation in neural audio codecs, and subjective listening evaluation.
$180.00MicrophoneBinaural stereo mic capturing HRTF — the spatial cues this head-tracked panner approximates via ITD/ILD and quaternion yaw driving StereoPannerNode.
$54.99MicrophoneCompact USB condenser mic used for nymic testing.
$149.00ApparatusInline discrete JFET preamplifier providing up to +25 dB of ultra-clean passive gain. Keeps audio interface preamps in their lowest-noise operating range when capturing speech with low-sensitivity dynamic microphones like the SM7B.
$9.99AudioMEMS mic breakout used for real-time DSP experiments feeding into Biquadia.
$80.49BookChapter on biquad Direct Form II transposed and pole-zero placement — move poles in pole-zero explorer and hear Q/ω0 map to ring frequency as Proakis derives.
$224.99Audio interface2-in/2-out 192 kHz / 24-bit USB audio interface with 120 dB dynamic range converters, ultra-low noise preamps (69 dB gain range), auto-gain, and hardware loopback for acoustic codec testing.
$72.88BookDefines Hammerstad-Jensen Z0, skin depth delta=sqrt(2/ωμσ), and crosstalk K_b backward coupling — closed-form behind this lab's per-unit-length L,C extraction and NEXT/FEXT formulas.
$12.69DiagnosticBudget 8-ch USB logic analyzer (sigrok/PulseView). Useful for RTOS GPIO timing and SI digital demos — not a Saleae substitute in bandwidth.
Audio interfacePortable 2-channel USB-C audio interface used for mobile Biquadia field recording.
$135.00BookDerives HRTF, ITD = d/c sinθ, and head-tracking compensation — the exact panning law this playground interpolates as you drag yaw.
$50.62BookBreakup modes, cone edge termination, and waveguide directivity index — chapters 7-9 explain the cone-breakup visualization and waveguide polar math this lab implements.
$233.00BookDerives S-parameter conversion Z=Z0(I+S)(I-S)^-1, Smith Chart mapping Γ=(Z-Z0)/(Z+Z0), and stub matching l=λ/2π arctan(B/Y0) this bench solves continuously during drag.
$105Microphone7-element MEMS microphone array with onboard XMOS DSP for raw channel output and real-time spatial audio / beamforming research.
$149.95MicrophoneCalibrated USB mic with individual cal file — measure your room's RT60 and modal peaks to validate the room-mode eigenfrequencies this calculator predicts.
$38.32BookL-network matching Q = sqrt(Rp/Rs-1) and component X = ±Q*Rs formulas — this bench's auto-match button solves these analytically then animates Smith Chart journey through series L then shunt C.
$168.00MicrophoneUSB condenser mic used for nymic testing.
Sensor100kHz-1.75GHz low-cost SDR for passively listening to patch array sweeps — compare simulator's beam steering phase Δφ = kd sinθ with over-the-air measured phase progression across 2.4GHz.
$8.99AudioUSB audio interface used in early Biquadia MEMS-mic prototyping.
$85.49DiagnosticHandheld VNA with touchscreen + SOL kit — S11/S21 lab companion for RF Bench and Antenna Sandbox.
$399.95HeadphonesOpen-back audiophile reference headphones with neutral acoustic response and 300-ohm aluminum voice coils. The universal standard for headphone frequency response calibration and DSP filter tuning.
$329.55Headphones300-ohm audiophile reference open-back headphones used for binaural dummy head calibration and earphone transfer function measurements.
MicrophoneWireless mic system used for akous's ambient/binaural field recording.
$319.00MicrophoneUSB/XLR hybrid mic used building and testing nymic.
$237.00MicrophonePortable stereo condenser mic kit used for Biquadia field/video capture.
$439.00MicrophoneCardioid dynamic vocal microphone with flat, wide-range frequency response (50 Hz–20 kHz) and internal air suspension shock isolation. The reference recording transducer for clean speech synthesis corpora and TTS model training.
$247.44BookDefines the DTFT, DFT, and complex Fourier coefficients c_k this visualizer computes — chapters 3-5 derive the exact reconstruction formula tracing your doodle.
$98.00HeadphonesReference studio headphones used for akous's binaural audio testing.
$530ApparatusBinaural dummy head acoustic fixture with anatomical silicone pinnae and Primo EM272 electret capsules for HRTF and headphone measurement.
$23.80Microcontroller600 MHz Cortex-M7 — high-rate control loops and DSP on the RTOS / FOC benches. Carried by SparkFun (third-party PJRC; tracked referral).
$26.80Microcontroller600 MHz M7 — flash the DF2T biquad cascade from Biquadia’s DSP core (pair with the Rev D audio shield for codec I/O) and hear the same filter your design exports here.
BookIntuitive pole-zero explanation — why a pole near unit circle rings, zero near kills — the exact intuition this pole-zero explorer makes tangible with drag-to-listen.
$124.53BookChapter on DFT and windowing explains spectral leakage and Gibbs phenomenon visible as overshoot when you draw sharp corners in the epicycles tracer.
Prices shown were retrieved from the Amazon Product Advertising API on 8 October 2026 and are indicative only — the price and availability on Amazon at the time of purchase apply.
Prices shown were each checked against the Amazon product listing between 8 August 2026 and 17 September 2026 and are indicative only — the price and availability on Amazon at the time of purchase apply.
More gear across every build: the full Gear list →
This is what runs in your browser above, with the argument validation stripped out. No FFT, no library, no adaptation — a delay-and-sum beamformer really is one loop over microphone positions.
Delay-and-sum + directivity index
const C = 343; // m/s, dry air at 20 degrees C
// Element positions, centred on the origin.
function positions(n, d) {
return Array.from({ length: n }, (_, i) => (i - (n - 1) / 2) * d);
}
// Normalised delay-and-sum response at arrival angle theta (radians from
// broadside), steered to theta_s. This IS the beamformer: one phasor per
// microphone, steering delay applied, summed, divided by N.
function response(n, d, thetaS, freq, theta) {
const k = (2 * Math.PI * freq) / C;
const u = Math.sin(theta) - Math.sin(thetaS);
let re = 0, im = 0;
for (const x of positions(n, d)) {
re += Math.cos(k * x * u);
im += Math.sin(k * x * u);
}
return Math.hypot(re, im) / n;
}
// Directivity against a spherically isotropic noise field.
// Gamma_mn = sinc(k * |x_m - x_n|); uniform weights make the numerator 1.
function directivityIndexDb(n, d, thetaS, freq) {
const k = (2 * Math.PI * freq) / C;
const x = positions(n, d);
let s = 0;
for (const xm of x) for (const xn of x) {
const delta = xm - xn;
const sinc = Math.abs(k * delta) < 1e-9 ? 1 : Math.sin(k * delta) / (k * delta);
s += sinc * Math.cos(k * delta * Math.sin(thetaS));
}
return 10 * Math.log10((n * n) / s);
}
// The only spacing rule that holds for every steer angle.
const aliasFreeCeilingHz = (d) => C / (2 * d);It is the simplest spatial filter there is. A plane wave arriving off-axis hits the microphones at different times; delay each channel by exactly the amount that cancels that difference for one chosen direction, sum the channels, and signals from that direction add coherently while everything else partially cancels. No adaptation, no statistics, no training — the only inputs are the microphone positions, the speed of sound and the angle you want to listen to. Every result on this page is that operation evaluated exactly, one frequency at a time.
Far less than most people expect, and the amount is computable rather than debatable. With uniform delay-and-sum weights the directivity index of a two-element array is 0.00 dB at 125 Hz on a 50 mm baseline, 0.30 dB at 1 kHz, 1.17 dB at 2 kHz and reaches exactly 3.01 dB only at the frequency where the spacing equals half a wavelength — 3430 Hz for 50 mm. Directivity index is set by the aperture measured in wavelengths, so a small array is close to omnidirectional through the whole low end no matter how the processing is described.
Above a certain frequency the array develops a grating lobe: a second direction with full array gain, indistinguishable from the one you steered at. A grating lobe enters the visible region when (d/λ)·(1 + |sin θs|) reaches 1. The worst case over all steer angles is endfire, |sin θs| = 1, which gives d = λ/2 and therefore f = c/(2d) — 3430 Hz for a 50 mm spacing. If you only ever steer broadside the onset is an octave higher at c/d, which is why the calculator reports both numbers instead of collapsing them into one rule of thumb.
Because a straight line of microphones physically cannot tell front from back. The response depends only on the direction cosine sin θ, and sin(180° − θ) = sin θ, so every arrival angle has a mirror twin on the other side of the array axis. Adding elements narrows both lobes equally and never removes the ambiguity. Breaking it requires geometry that is not a straight line — a second row, a baffle between the capsules, or a directional element.
Because it genuinely is. When the array is small compared with the wavelength, the response never falls 3 dB below the look direction anywhere in the visible half-space, so there is no half-power edge to report. The calculator returns no number in that case rather than extrapolating one, and the same applies at endfire, where the main lobe is cut off by the ±90° horizon on one side. A beamwidth wider than the space the array can see is not a beamwidth.
It will match the main-lobe shape closely and the nulls poorly. This model assumes ideal omnidirectional elements at exactly the stated positions in free space. A real build adds microphone-to-microphone sensitivity and phase mismatch, which fills the deep nulls first; a finite baffle, which scatters sound around the capsules; near-field wavefront curvature for sources within a few aperture lengths; and room reflections, which arrive from directions the beamformer is rejecting and dominate past the critical distance. All four move a measurement in the same direction: less rejection than the theory promises, never more.
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