F. Harris, On the Use of Windows…

This is a brief summary of a classic reference called “On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform” by Fredric J. Harris, PROCEEDINGS OF THE IEEE, VOL. 66, NO. 1, JANUARY 1978. Table 1 offers systems engineers the ability to quickly assess the effects of windowing on frequency, spatial or temporal transforms. In essence, windowing is a form of aperture filtering or shading that broadens a transform cell’s equivalent noise bandwidth, lowers the sidelobe level and, at 50% overlap, makes successive transforms almost independent (for “good” windows).

The paper describes many windows with desirable properties (and some with not so desirable ones). Several are easy to code for modeling purposes. Hanning is especially easy since, in the Fourier domain, its effects on a cell output are  the result of a phase weighted sum of that cell with the two adjacent Fourier cells. Dolph-Chebyshev has good properties: the narrowest mainlobe response for a given (uniform) sidelobe level. This window also has not so good properties, especially when used for spatial aperture shading of flexible arrays. The window exhibits a sidelobe suppression instability due to the alternating signs of its window coefficients.

The results in Harris’ paper are useful for both systems analysis and design. The following figure from Wikipedia defines the main lobe (or beam), sidelobe level and sidelobes of a (continuous) transform response.