Sound is pressure changing over time. Your ear and brain are extraordinarily good at noticing when that pressure repeats at a steady rate — we call that pitch. Machinery rarely produces a single pure tone; it produces stacks of periodic components plus hiss. The Fourier transform is the mathematical tool that answers a deceptively simple question: if I record a few milliseconds of microphone data, which frequencies are present and how loud is each one? Sonarish builds its spectrum analyzer and scrolling spectrogram on that question, running entirely on your phone without sending audio to a server.

The discrete Fourier transform, computed efficiently as the fast Fourier transform or FFT, takes N consecutive audio samples and produces N complex numbers representing sine and cosine contributions at equally spaced frequencies. We rarely display the full complex result. For noise and fault work we care about magnitude — how strong each frequency bin is — usually converted to decibels so quiet and loud components share one readable axis.

From one FFT to a moving picture

A single FFT snapshot tells you what happened in one short slice of time. Real machines and rooms change slowly but not infinitely slowly, so Sonarish uses a short-time Fourier transform: chop the continuous microphone stream into overlapping windows, FFT each window, and plot magnitude against time. Window size trades frequency resolution against time resolution. At forty-eight kilohertz sample rate, a window of two thousand forty-eight samples yields bin width of about eleven point seven hertz — fine enough to separate a hundred twenty hertz mains hum from its second harmonic at two hundred forty. A four thousand ninety-six sample window halves that bin width but reacts more slowly to sudden clicks.

We multiply each window by a Hann function before transforming. That tapering reduces spectral leakage — the smearing of energy into neighboring bins when a tone's frequency falls between bin centers. For tonal machinery with sharp harmonics, Hann is a sensible default. Hop size at fifty percent overlap keeps the scrolling waterfall visually smooth without doubling CPU cost unnecessarily.

Reading what the graph means

A peak at one hundred twenty hertz in a country with sixty hertz mains often marks a two-pole induction motor spinning at three thousand six hundred RPM, because electrical frequency times pole pairs maps to mechanical rotation. Bearing wear introduces sidebands that are not simple multiples of shaft speed — energy appearing at unexpected offsets is sometimes the first audible clue before vibration analysts confirm fault frequencies in a report. Broadband hiss rising across many bins suggests turbulence, brush arcing, or loose panels rattling without a single dominant tone.

Sonarish labels peaks automatically when signal-to-noise ratio exceeds about six decibels above a locally estimated noise floor, but the app does not pretend to diagnose your washing machine. It shows the spectrum honestly and lets experience — or our baseline comparison workflow — interpret change over time.

Performance and threading on a phone

FFT work never runs inside the audio capture callback. That path is reserved for copying samples into a lock-free ring buffer, as described in life on the audio thread. A background queue pulls blocks, applies window and FFT using platform libraries — Accelerate vDSP on iOS, a compact FFT implementation on Android — and hands magnitude arrays to the UI thread. Even when transforms complete faster, we cap spectrum redraws at fifteen to thirty hertz because humans cannot interpret a flickering bar chart at sixty hertz and battery matters in a factory walk-through.

Common pitfalls are worth naming because they confuse beginners and reviewers alike. Aliasing — folding high frequencies into low ones — is largely handled by the phone's audio hardware anti-alias filters, but you still should not point a mic at a whistle near the Nyquist limit and trust the graph blindly. Confusing decibels relative to digital full scale with decibels sound pressure level invalidates any comparison to environmental law; our A-weighting and decibels post explains that distinction. Shrinking window size to make peaks look sharper actually widens bins and merges adjacent harmonics into one fat bar — pretty UI, worse physics.

Phyzix includes a live microphone FFT view for classroom demos where students clap, whistle, and watch bins light up. Wheria does not need spectral analysis, but the same sampling discipline — respect Nyquist, never block real-time capture — runs through every uranashel app that touches a microphone or accelerometer at high rate.