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.