 Fourier frequency

For a record of $N$ samples separated by time $\Delta t$, the $j$th Fourier frequency has angular frequencyThe total record duration is $T=N\Delta t$; writing $2\pi j/T$ uses that duration rather than the sample count. Frequencies differing by $2\pi /\Delta t$ give the same discrete Fourier mode. For a real record, the distinct positive interior modes have $1\le j<N/2$; the zero mode and, for even $N$, the Nyquist frequency are separate real modes.

Their sine and cosine sample vectors are mutually orthogonal. If the samples are independent centered Gaussian random variables with a common variance, projections on these orthogonal vectors are independent Gaussian random variables. Negative-frequency coefficients are instead fixed by discrete Fourier conjugate symmetry; they are not additional independent measurements.