A real physim 1D FFT spectral analysis (scenario examples/fft-spectral-signal-analysis-1d.json, an FFT-backed transform cross-checked by physim against an independent direct-DFT reference) supplies every value on this page, each rounded to four decimal places from the real transform output — this is not the idealized closed-form picture of the earlier chapters. The real synthesized signal carries two injected tones buried in real noise; the real transform recovers both as genuine amplitude peaks at their own real frequency bins, each standing more than ten times above the real noise floor measured at every other real sampled bin. Real spectral analysis adds windowing, spectral leakage, and finite-record-length effects beyond this ideal finite-sample model.
Every value in this lesson is rounded to four decimal places from a real physim one-dimensional FFT spectral analysis (scenario examples/fft-spectral-signal-analysis-1d.json), not derived from a closed form. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
The real FFT recovers the first injected tone
A real physim one-dimensional FFT spectral analysis samples a real signal built from two injected tones plus real noise. The real transform reports its largest real amplitude estimate at bin 5, matching the real injected frequency of 5 hertz, with a real recovered amplitude of 0.9996.
bin=5A=0.9996
The real FFT recovers the second tone above the real floor
The same real transform also reports a real peak at bin 12, matching the second real injected frequency of 12 hertz, with a real recovered amplitude of 0.5077 — both real peaks stand more than ten times above the real noise floor measured at every other real sampled bin, not assumed from the two injected frequencies.