 Rectangular window

A rectangular window keeps all samples in a selected finite record with equal weight and sets the signal outside it to zero. For $N$ discrete samples, $w_n=1$ for $0\le n<N$ and zero otherwise. Calling this no taper does not remove the effect of having a finite record. SciPy's primary implementation documentation identifies the boxcar and rectangular window as this same all-ones choice.^[ref]

Its discrete-time Fourier transform at frequency $\nu$ in cycles per sample is a finite geometric sum:At integer $\nu$, the limiting value is $N$. The first zeros surrounding the central maximum are at $\nu =\pm 1/N$, so the central zero-to-zero width is $2/N$. The phase factor depends on the record's time origin; the magnitude response does not.

For a continuous centered record of duration $T$, $w(t)=1$ on $[-T/2,T/2]$ has transformwhere the normalized sinc function is $\mathrm{sinc}(x)=\sin (\pi x)/(\pi x)$. Multiplication by the finite window spreads a sharp spectral component according to this response, producing spectral leakage.

An original finite-record example quantifies an off-bin amplitude error. A complex tone of amplitude $A$ lying half a DFT bin from the nearest sampled frequency has normalized responseAt $N=64$ the ratio is about $0.6367$, or $-3.92\mathrm{dB}$ relative to the coherent peak. The coherent gain of a window is still one; it does not correct this scalloping loss. Zero-padding samples the same frequency response more densely rather than lengthening the measured record.

For independent equal-variance sample noise, rectangular weighting has equivalent noise bandwidth of one DFT bin under the usual coherent-amplitude normalization. A Hann window suppresses distant sidelobes but changes the central width and noise scaling. The primary analysis manual reports these separate window properties rather than one universal best window.^[ref]

A proposed test records a known sinusoidal input with an integer and then a non-integer number of cycles in the same record length, holding the sampling rate and amplitude fixed. Compare the raw DFT bins and the continuous window response. This separates finite-record leakage and bin placement from aliasing, which involves a different sampling ambiguity.