THE DFT — a dart

THE DFT a signal weighed against every frequency — time becomes spectrum

The discrete Fourier transform answers “what frequencies are in here?” by correlating the signal against a phasor at each candidate frequency: X[k] = Σ x[n]·e^(−i·2πkn/N). Where the signal contains bin k, that sum builds up; elsewhere it cancels. The result is the spectrum — a magnitude and a phase at every frequency, the complete alternative view of the signal. The FFT (dart’s Cooley-Tukey cousin) computes it in N log N; the DFT is the definition it accelerates. It is the workhorse that turns time into frequency across all of science.

THE TECHNIQUE X[k] = Σ x[n]·(cos − i·sin)(2πkn/N) — correlate vs each freq

The demo runs a 16-point DFT of a bin-3 tone and shows bin 3 lights up far above bin 5 — the frequency found: live demo

HISTORY & CREDIT Fourier 1822 · Cooley-Tukey 1965 (FFT)

“A signal is a thing in time.” — it is equally a sum of frequencies; the DFT is the exact dictionary between the two. cited

the transform · X[k] = Σₙ x[n]·e^(−i2πkn/N) — correlate against every frequency.
the spectrum · |X[k]| magnitude and arg X[k] phase — the full alternate view.
1822 / 1965 · Joseph Fourier (the series); Cooley & Tukey (the fast algorithm).

A signal projected onto every frequency at once — time and spectrum, two names for one thing. time↔frequency

RECOMMEND FOR I-13 the spectrum, on the compiler

On the canonical compiler, a 16-point DFT of a bin-3 tone puts almost all energy in bin 3, far above bin 5:

$ i13 run cr_dft.i13 # 16-point DFT via sine LUT RUN OK · 2253 step(s) · peak stack 21 · call depth 17 bin3 = 6.40e13 -- energy at the tone's frequency bin5 = 6.07e6 -- essentially empty peak_at_3 = 1

Recommend as a NULL — a change of basis, and a theorem. The DFT re-expresses the signal in the frequency basis; the transform is a fixed linear map (every correct implementation gives the same spectrum), and its magnitude/phase are forall-pinned by the input. That makes it a theorem/witnessed structure (B39) — and a basis change whose invertibility is Parseval’s, not an enacted axis. It carries a phase spectrum (relevant to the PHASE TEST), but as a pinned property of a fixed map, not a droppable channel. NULL — the frame the whole batch lives inside.