 Peak phasor versus positive-frequency Fourier coefficient

Peak phasor versus positive-frequency Fourier coefficient distinguishes a real sinusoid's reconstruction amplitude from one coefficient in its two-sided exponential expansion. For $v(t)=\mathrm{Re}(\hat{V}{e}^{i\omega t})$,Thus the positive-frequency Fourier coefficient is half the peak amplitude phasor. The negative-frequency coefficient is its conjugate. A real sinusoid needs both; a complex exponential input has a different one-sided reconstruction.

The distinction also controls a sampled record. For $N$ evenly spaced samples of a real sinusoid with exactly $k$ cycles in the record and $0<k<N/2$, use the unscaled forward discrete Fourier transformOrthogonality of the discrete exponentials gives $X_k=N\hat{V}/2$. ConsequentlyThe default SciPy forward FFT has exactly this unscaled convention; its forward and orthonormal modes have different factors.^[ref]

For an original example, sample $v_m=3\cos (2\pi 5m/64)-4\sin (2\pi 5m/64)$ volts. Then $\hat{V}=3+4i$ volts and $X_5=96+128i$ volts. Dividing by 64 gives the positive-frequency coefficient $1.5+2i$ volts; doubling it recovers the peak phasor. The RMS amplitude is $5/\sqrt{2}\approx 3.536$ volts.

These exact relations assume coherent sampling of the stated sinusoid without aliased extra components. DC and, for even $N$, the Nyquist bin do not have distinct conjugate partners and must not be doubled. With a noninteger number of cycles or a windowed record, finite-record phasor extraction with coherent sampling must be distinguished from a generic bin readout; spectral leakage and the window's amplitude normalization alter the estimate. The parent peak amplitude phasor specifies the continuous-time convention before any sampled transform is chosen.