How to read these. Each page states which of its numbers are computed, which come from a published figure, and which are illustrative. Several were generated with AI assistance and say so on the page. They are teaching instruments, not measurements of real hardware.
Labs
Demos
Sampling & aliasing playground
From lesson Aliasing
Raise a tone past half the sampling rate and watch — and hear — the pitch come back down. The replica spectrum, the folding map, a real Butterworth anti-alias filter and the wagon-wheel view of the same arithmetic.
Convolution step-through
From lesson Convolution
Build your own x and h by dragging the stems, then walk the lag while h flips, shifts and multiplies. Every product is written out term by term, the algebraic properties are recomputed on your sequences, and a computed room response lets you hear what a convolution is for.
Pole-zero / z-plane explorer
From lesson Poles, Zeros & Frequency Response
Drag poles and zeros on the z-plane and watch four views move together: magnitude, phase, impulse response and the pole-zero map itself.
DFT / FFT explorer
From lesson DFT Definition and Interpretation
Where spectral leakage comes from: change the tone, the window and the zero-padding, and watch the DFT bins respond.
Spectrum analyzer
From lesson Why DFT is Slow
The window figures of merit, computed from the coefficients: equivalent noise bandwidth N·Σw²/(Σw)², coherent gain (Σw)/N, scalloping loss and peak sidelobe level, each shown beside the published Harris (1978) value. Put a tone (plus noise) through a windowed FFT, detect the peak, and read its frequency and its coherent-gain-corrected amplitude 2|X_peak|/Σw. Watch the analyzer's central trade: a narrow window resolves close tones but hides weak ones; a wide one sees the weak tone but merges the close pair.
FIR / IIR filter designer
From lesson FIR vs. IIR
Design the same specification as FIR and as IIR at once, and see the trade-off the lessons describe in words — order, phase linearity and stopband.
Spectrogram / STFT explorer
From lesson Spectrogram
The time-frequency resolution trade-off, made physical: change the window length and overlap and watch what the spectrogram can and cannot resolve.
Detect a signal in noise
From lesson Auto-correlation & Cross-correlation
Bury a known template in seeded Gaussian noise, run the matched filter (a cross-correlation with the template), and watch the peak stand out where the raw record shows nothing. The processing gain 10*log10(N), the output SNR, and the ROC -- P_d against P_fa -- are all computed, with an optional Monte-Carlo overlay that converges onto the analytic curve.
Resample audio
From lesson Decimation (Downsampling)
Convert a signal from one sample rate to another by a rational factor L/M -- upsample by L, lowpass, downsample by M. The reduced L/M (48000 → 44100 = 147/160), the new sample count N' = round(N·L/M), and the anti-alias cutoff min(f_s, f_s')/2 are all computed, and zero-order-hold, linear and windowed-sinc interpolation are ranked by their reconstruction error (sinc < linear < ZOH). Downsample a high tone with the anti-alias filter off and watch it fold.